arXiv:math/0211159v1 [math.DG] 11 Nov 2002
The entropy formula for the Ricci flow and its geometric applications Grisha Perelman∗ February 1, 2008
Introduction 1. The Ricci flow equation, introduced by Richard Hamilton [H 1], is the evolution equation dtd gij (t) = −2Rij for a riemannian metric gij (t). In his seminal paper, Hamilton proved that this equation has a unique solution for a short time for an arbitrary (smooth) metric on a closed manifold. The evolution equation for the metric tensor implies the evolution equation for the curvature tensor of the form Rmt = △Rm + Q, where Q is a certain quadratic expression of the curvatures. In particular, the scalar curvature R satisfies Rt = △R + 2|Ric|2 , so by the maximum principle its minimum is non-decreasing along the flow. By developing a maximum principle for tensors, Hamilton [H 1,H 2] proved that Ricci flow preserves the positivity of the Ricci tensor in dimension three and of the curvature operator in all dimensions; moreover, the eigenvalues of the Ricci tensor in dimension three and of the curvature operator in dimension four are getting pinched pointwisely as the curvature is getting large. This observation allowed him to prove the convergence results: the evolving metrics (on a closed manifold) of positive Ricci curvature in dimension three, or positive curvature operator St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email:
[email protected] or
[email protected] ; I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992, to the SUNY at Stony Brook in the Spring of 1993, and to the UC at Berkeley as a Miller Fellow in 1993-95. I’d like to thank everyone who worked to make those opportunities available to me. ∗
1
in dimension four converge, modulo scaling, to metrics of constant positive curvature. Without assumptions on curvature the long time behavior of the metric evolving by Ricci flow may be more complicated. In particular, as t approaches some finite time T, the curvatures may become arbitrarily large in some region while staying bounded in its complement. In such a case, it is useful to look at the blow up of the solution for t close to T at a point where curvature is large (the time is scaled with the same factor as the metric tensor). Hamilton [H 9] proved a convergence theorem , which implies that a subsequence of such scalings smoothly converges (modulo diffeomorphisms) to a complete solution to the Ricci flow whenever the curvatures of the scaled metrics are uniformly bounded (on some time interval), and their injectivity radii at the origin are bounded away from zero; moreover, if the size of the scaled time interval goes to infinity, then the limit solution is ancient, that is defined on a time interval of the form (−∞, T ). In general it may be hard to analyze an arbitrary ancient solution. However, Ivey [I] and Hamilton [H 4] proved that in dimension three, at the points where scalar curvature is large, the negative part of the curvature tensor is small compared to the scalar curvature, and therefore the blow-up limits have necessarily nonnegative sectional curvature. On the other hand, Hamilton [H 3] discovered a remarkable property of solutions with nonnegative curvature operator in arbitrary dimension, called a differential Harnack inequality, which allows, in particular, to compare the curvatures of the solution at different points and different times. These results lead Hamilton to certain conjectures on the structure of the blow-up limits in dimension three, see [H 4,§26]; the present work confirms them. The most natural way of forming a singularity in finite time is by pinching an (almost) round cylindrical neck. In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries, and then to continue running the Ricci flow. The exact procedure was described by Hamilton [H 5] in the case of four-manifolds, satisfying certain curvature assumptions. He also expressed the hope that a similar procedure would work in the three dimensional case, without any a priory assumptions, and that after finite number of surgeries, the Ricci flow would exist for all time t → ∞, and be nonsingular, in the sense that the normalized curvatures ˜ Rm(x, t) = tRm(x, t) would stay bounded. The topology of such nonsingular solutions was described by Hamilton [H 6] to the extent sufficient to make sure that no counterexample to the Thurston geometrization conjecture can 2
occur among them. Thus, the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds. In this paper we carry out some details of Hamilton program. The more technically complicated arguments, related to the surgery, will be discussed elsewhere. We have not been able to confirm Hamilton’s hope that the solution that exists for all time t → ∞ necessarily has bounded normalized curvature; still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below; by our earlier (partly unpublished) work this is enough for topological conclusions. Our present work has also some applications to the Hamilton-Tian conjecture concerning K¨ahler-Ricci flow on K¨ahler manifolds with positive first Chern class; these will be discussed in a separate paper. 2. The Ricci flow has also been discussed in quantum field theory, as an approximation to the renormalization group (RG) flow for the two-dimensional nonlinear σ-model, see [Gaw,§3] and references therein. While my background in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RG flow. In this picture, t corresponds to the scale parameter; the larger is t, the larger is the distance scale and the smaller is the energy scale; to compute something on a lower energy scale one has to average the contributions of the degrees of freedom, corresponding to the higher energy scale. In other words, decreasing of t should correspond to looking at our Space through a microscope with higher resolution, where Space is now described not by some (riemannian or any other) metric, but by an hierarchy of riemannian metrics, connected by the Ricci flow equation. Note that we have a paradox here: the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale; moreover, if we allow Ricci flow through singularities, the regions that are in different connected components at larger distance scale may become neighboring when viewed through microscope. Anyway, this connection between the Ricci flow and the RG flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation. 3. The paper is organized as follows. In §1 we explain why Ricci flow can be regarded as a gradient flow. In §2, 3 we prove that Ricci flow, considered as a dynamical system on the space of riemannian metrics modulo diffeomorphisms and scaling, has no nontrivial periodic orbits. The easy (and known) 3
case of metrics with negative minimum of scalar curvature is treated in §2; the other case is dealt with in §3, using our main monotonicity formula (3.4) and the Gaussian logarithmic Sobolev inequality, due to L.Gross. In §4 we apply our monotonicity formula to prove that for a smooth solution on a finite time interval, the injectivity radius at each point is controlled by the curvatures at nearby points. This result removes the major stumbling block in Hamilton’s approach to geometrization. In §5 we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble. In §6 we try to interpret the formal expressions , arising in the study of the Ricci flow, as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension. The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricci flow. This formula is rigorously proved in §7; it may be more useful than the first one in local considerations. In §8 it is applied to obtain the injectivity radius control under somewhat different assumptions than in §4. In §9 we consider one more way to localize the original monotonicity formula, this time using the differential Harnack inequality for the solutions of the conjugate heat equation, in the spirit of Li-Yau and Hamilton. The technique of §9 and the logarithmic Sobolev inequality are then used in §10 to show that Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. The results of sections 1 through 10 require no dimensional or curvature restrictions, and are not immediately related to Hamilton program for geometrization of three manifolds. The work on details of this program starts in §11, where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits of finite time singularities ( they must satisfy a certain noncollapsing assumption, which, in the interpretation of §5, corresponds to having bounded entropy). Then in §12 we describe the regions of high curvature under the assumption of almost nonnegative curvature, which is guaranteed to hold by the Hamilton and Ivey result, mentioned above. We also prove, under the same assumption, some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball. Finally, in §13 we give a brief sketch of the proof of geometrization conjecture. The subsections marked by * contain historical remarks and references. See also [Cao-C] for a relatively recent survey on the Ricci flow.
4
1
Ricci flow as a gradient flow
R 1.1. Consider the functional F = M (R + |∇f |2)e−f dV for a riemannian metric gij and a function f on a closed manifold M. Its first variation can be expressed as follows: Z δF (vij , h) = e−f [−△v + ∇i ∇j vij − Rij vij M
=
Z
−vij ∇i f ∇j f + 2 < ∇f, ∇h > +(R + |∇f |2 )(v/2 − h)]
M
e−f [−vij (Rij + ∇i ∇j f ) + (v/2 − h)(2△f − |∇f |2 + R)],
where δgij = vij , δf = h, v = g ij vij . Notice that v/2 − h vanishes identically iff the measure dm = e−f dV is kept fixed. Therefore, theR symmetric tensor −(Rij +∇i ∇j f ) is the L2 gradient of the functional F m = M (R + |∇f |2 )dm, where now f denotes log(dV /dm). Thus given a measure m , we may consider the gradient flow (gij )t = −2(Rij + ∇i ∇j f ) for F m . For general m this flow may not exist even for short time; however, when it exists, it is just the Ricci flow, modified by a diffeomorphism. The remarkable fact here is that different choices of m lead to the same flow, up to a diffeomorphism; that is, the choice of m is analogous to the choice of gauge. 1.2 Proposition. Suppose that the gradient flow for F m exists for t ∈ [0, T ]. R n Then at t = 0 we have F m ≤ 2T M dm. R Proof. We may assume M dm = 1. The evolution equations for the gradient flow of F m are (gij )t = −2(Rij + ∇i ∇j f ), ft = −R − △f, and F m satisfies
Ftm
=2
Z
|Rij + ∇i ∇j f |2 dm
(1.1) (1.2)
Modifying by an appropriate diffeomorphism, we get evolution equations (gij )t = −2Rij , ft = −△f + |∇f |2 − R,
(1.3)
and retain (1.2) in the form Ft = 2
Z
|Rij + ∇i ∇j f |2 e−f dV 5
(1.4)
Now we compute Z Z 2 2 2 2 −f (R + △f ) e dV ≥ ( (R + △f )e−f dV )2 = F 2 , Ft ≥ n n n and the proposition follows. 1.3 Remark. The functional F m has a natural interpretation in terms of Bochner-Lichnerovicz formulas. The classical formulas of Bochner (for one-forms) and Lichnerovicz (for spinors) are ∇∗ ∇ui = (d∗ d + dd∗ )ui − Rij uj and ∇∗ ∇ψ = δ 2 ψ − 1/4Rψ. Here the operators ∇∗ , d∗ are defined using the riemannian volume form; this volume form is also implicitly used in the definition of the Dirac operator δ via the requirement δ ∗ = δ. A routine computation shows that if we substitute dm = e−f dV for dV , we get m modified Bochner-Lichnerovicz formulas ∇∗m ∇ui = (d∗m d + dd∗m )ui − Rij uj ∗m m 2 m m m and ∇ ∇ψ = (δ ) ψ − 1/4R ψ, where δ ψ = δψ − 1/2(∇f ) · ψ , Rij = m Rij +∇i ∇j f , Rm = 2△f −|∇f |2 +R. Note that g ij Rij = R+△f 6= Rm . How∗m m m ever, we do have the Bianchi identity ∇i Rij = ∇i Rij −Rij ∇i f = 1/2∇j Rm . R R m Now F m = M Rm dm = M g ij Rij dm. 1.4* The Ricci flow modified by a diffeomorphism was considered by DeTurck, who observed that by an appropriate choice of diffeomorphism one can turn the equation from weakly parabolic into strongly parabolic, thus considerably simplifying the proof of short time existence and uniqueness; a nice version of DeTurck trick can be found in [H 4,§6]. The functional F and its first variation formula can be found in the literature on the string theory, where it describes the low energy effective action; the function f is called dilaton field; see [D,§6] for instance. m The Ricci tensor Rij for a riemannian manifold with a smooth measure has been used by Bakry and Emery [B-Em]. See also a very recent paper [Lott].
2
No breathers theorem I
2.1. A metric gij (t) evolving by the Ricci flow is called a breather, if for some t1 < t2 and α > 0 the metrics αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism; the cases α = 1, α < 1, α > 1 correspond to steady, shrinking and expanding breathers, respectively. Trivial breathers, for which the metrics gij (t1 ) and gij (t2 ) differ only by diffeomorphism and scaling for each pair of
6
t1 and t2 , are called Ricci solitons. (Thus, if one considers Ricci flow as a dynamical system on the space of riemannian metrics modulo diffeomorphism and scaling, then breathers and solitons correspond to periodic orbits and fixed points respectively). At each time the Ricci soliton metric satisfies an equation of the form Rij + cgij + ∇i bj + ∇j bi = 0, where c is a number and bi is a one-form; in particular, when bi = 12 ∇i a for some function a on M, we get a gradient Ricci soliton. An important example of a gradient shrinking soliton is the Gaussian soliton, for which the metric gij is just the euclidean metric on Rn , c = 1 and a = −|x|2 /2. In this and the next section we use the gradient interpretation of the Ricci flow to rule out nontrivial breathers (on closed M). The argument in the steady case is pretty straightforward; the expanding case is a little bit more subtle, because our functional F is not scale invariant. The more difficult shrinking case is discussed in section 3. 2.2. DefineR λ(gij ) = inf F (gij , f ), where infimum is taken over all smooth f, satisfying M e−f dV = 1. Clearly, λ(gij ) is just the lowest eigenvalue of the operator −4△+R. Then formula (1.4) implies that λ(gij (t)) is nondecreasing in t, and moreover, if λ(t1 ) = λ(t2 ), then for t ∈ [t1 , t2 ] we have Rij +∇i ∇j f = 0 for f which minimizes F . Thus a steady breather is necessarily a steady soliton. 2.3. To deal with the expanding case consider a scale invariant version ¯ ij ) = λ(gij )V 2/n (gij ). The nontrivial expanding breathers will be ruled λ(g out once we prove the following ¯ is nondecreasing along the Ricci flow whenever it is nonpositive; Claim λ moreover, the monotonicity is strict unless we are on a gradient soliton. (Indeed, on an expanding breather we would necessarily have dV R /dt > 0 1 d for some t∈[t1 , t2 ]. On the other hand, for every t, − dt logV = V RdV ≥ ¯ can not be nonnegative everywhere on [t1 , t2 ], and the claim apλ(t), so λ plies.) Proof of the claim. ¯ dλ(t)/dt ≥ 2V 2/n
R
|Rij + ∇i ∇j f |2 e−f dV + n2 V (2−n)/n λ
R
R 2V 2/n [ |Rij + ∇i ∇j f − n1 (R + △f )gij |2 e−f dV + R
1 n(
−RdV ≥
R (R + △f )2 e−f dV − ( (R + △f )e−f dV )2 )] ≥ 0,
where f is the minimizer for F .
7
2.4. The arguments above also show that there are no nontrivial (that is with non-constant Ricci curvature) steady or expanding Ricci solitons (on closed M). Indeed, the equality case in the chain of inequalities above requires that R+△f be constant on M; on the other hand, the Euler-Lagrange equation for the minimizer f is 2△f − |∇f |2 + R = const. Thus, △f − |∇f |2 = const = 0, R because (△f − |∇f |2)e−f dV = 0. Therefore, f is constant by the maximum principle. 2.5*. A similar, but simpler proof of the results in this section, follows im2 mediately from [H 6,§2], where Hamilton checks that the minimum of RV n is nondecreasing whenever it is nonpositive, and monotonicity is strict unless the metric has constant Ricci curvature.
3
No breathers theorem II
3.1. In order to handle the shrinking case when λ > 0, we need to replace our functional F by its generalization, which contains explicit insertions of the scale parameter, to be denoted by τ. Thus consider the functional Z n W(gij , f, τ ) = [τ (|∇f |2 + R) + f − n](4πτ )− 2 e−f dV , (3.1) M
restricted to f satisfying Z
n
(4πτ )− 2 e−f dV = 1,
(3.2)
M
τ > 0. Clearly W is invariant under simultaneous scaling of τ and gij . The evolution equations, generalizing (1.3) are (gij )t = −2Rij , ft = −△f + |∇f |2 − R +
n , τt = −1 2τ
(3.3)
The evolution equation for f can also be written as follows: 2∗ u = 0, where n u = (4πτ )− 2 e−f , and 2∗ = −∂/∂t − △ + R is the conjugate heat operator. Now a routine computation gives Z n 1 dW/dt = 2τ |Rij + ∇i ∇j f − gij |2 (4πτ )− 2 e−f dV . (3.4) 2τ M Therefore, if we let µ(gij , τ ) = inf W(gij , f, τ ) over smooth f satisfying (3.2), and ν(gij ) = inf µ(gij , τ ) over all positive τ, then ν(gij (t)) is nondecreasing 8
along the Ricci flow. It is not hard to show that in the definition of µ there always exists a smooth minimizer f (on a closed M). It is also clear that limτ →∞ µ(gij , τ ) = +∞ whenever the first eigenvalue of −4△ + R is positive. Thus, our statement that there is no shrinking breathers other than gradient solitons, is implied by the following Claim For an arbitrary metric gij on a closed manifold M, the function µ(gij , τ ) is negative for small τ > 0 and tends to zero as τ tends to zero. Proof of the Claim. (sketch) Assume that τ¯ > 0 is so small that Ricci n flow starting from gij exists on [0, τ¯]. Let u = (4πτ )− 2 e−f be the solution of the conjugate heat equation, starting from a δ-function at t = τ¯, τ (t) = τ¯ − t. Then W(gij (t), f (t), τ (t)) tends to zero as t tends to τ¯, and therefore µ(gij , τ¯) ≤ W(gij (0), f (0), τ (0)) < 0 by (3.4). Now let τ → 0 and assume that f τ are the minimizers, such that 1 1 W( τ −1 gij , f τ , ) = W(gij , f τ , τ ) = µ(gij , τ ) ≤ c < 0. 2 2 The metrics 21 τ −1 gij ”converge” to the euclidean metric, and if we could extract a converging subsequence from f τ , we would get a function f on Rn , R n such that Rn (2π)− 2 e−f dx = 1 and Z n 1 [ |∇f |2 + f − n](2π)− 2 e−f dx < 0 Rn 2 The latter inequality contradicts the Gaussian logarithmic Sobolev inequality, due to L.Gross. (To pass to its standard form, take f = |x|2 /2 − 2 log φ and integrate by parts) This argument is not hard to make rigorous; the details are left to the reader. 3.2 Remark. Our monotonicity formula (3.4) can in fact be used to prove a version of the logarithmic Sobolev inequality (with description of the equality cases) on shrinking Ricci solitons. Indeed, assume that a metric gij satisfies Rij − gij − ∇i bj − ∇j bi = 0. Then under Ricci flow, gij (t) is isometric to (1 − 2t)gij (0), µ(gij (t), 12 − t) = µ(gij (0), 21 ), and therefore the monotonicity formula (3.4) implies that the minimizer f for µ(gij , 12 ) satisfies Rij + ∇i ∇j f − gij = 0. Of course, this argument requires the existence of minimizer, and justification of the integration by parts; this is easy if M is closed, but can also be done with more efforts on some complete M, for instance when M is the Gaussian soliton. 9
3.3* The no breathers theorem in dimension three was proved by Ivey [I]; in fact, he also ruled out nontrivial Ricci solitons; his proof uses the almost nonnegative curvature estimate, mentioned in the introduction. Logarithmic Sobolev inequalities is a vast area of research; see [G] for a survey and bibliography up to the year 1992; the influence of the curvature was discussed by Bakry-Emery [B-Em]. In the context of geometric evolution equations, the logarithmic Sobolev inequality occurs in Ecker [E 1].
4
No local collapsing theorem I
In this section we present an application of the monotonicity formula (3.4) to the analysis of singularities of the Ricci flow. 4.1. Let gij (t) be a smooth solution to the Ricci flow (gij )t = −2Rij on [0, T ). We say that gij (t) is locally collapsing at T, if there is a sequence of times tk → T and a sequence of metric balls Bk = B(pk , rk ) at times tk , such that rk2 /tk is bounded, |Rm|(gij (tk )) ≤ rk−2 in Bk and rk−n V ol(Bk ) → 0. Theorem. If M is closed and T < ∞, then gij (t) is not locally collapsing at T. Proof. Assume that there is a sequence of collapsing balls Bk = B(pk , rk ) at times tk → T. Then we claim that µ(gij (tk ), rk2 ) → −∞. Indeed one can take fk (x) = − log φ(disttk (x, pk )rk−1 ) + ck , where φ is a function of one variable, equal 1 on [0, 1/2], decreasing on [1/2, 1], and very close to 0 on [1, ∞), and ck is a constant; clearly ck → −∞ as rk−n V ol(Bk ) → 0. Therefore, applying the monotonicity formula (3.4), we get µ(gij (0), tk + rk2 ) → −∞. However this is impossible, since tk + rk2 is bounded. 4.2. Definition We say that a metric gij is κ-noncollapsed on the scale ρ, if every metric ball B of radius r < ρ, which satisfies |Rm|(x) ≤ r −2 for every x ∈ B, has volume at least κrn . It is clear that a limit of κ-noncollapsed metrics on the scale ρ is also κ-noncollapsed on the scale ρ; it is also clear that α2 gij is κ-noncollapsed on the scale αρ whenever gij is κ-noncollapsed on the scale ρ. The theorem above essentially says that given a metric gij on a closed manifold M and T < ∞, one can find κ = κ(gij , T ) > 0, such that the solution gij (t) to the Ricci flow starting at gij is κ-noncollapsed on the scale T 1/2 for all t ∈ [0, T ), provided it exists on this interval. Therefore, using the convergence theorem of Hamilton, we obtain the following 10
Corollary. Let gij (t), t ∈ [0, T ) be a solution to the Ricci flow on a closed manifold M, T < ∞. Assume that for some sequences tk → T, pk ∈ M and some constant C we have Qk = |Rm|(pk , tk ) → ∞ and |Rm|(x, t) ≤ CQk , whenever t < tk . Then (a subsequence of ) the scalings of gij (tk ) at pk with factors Qk converges to a complete ancient solution to the Ricci flow, which is κ-noncollapsed on all scales for some κ > 0.
5
A statistical analogy
In this section we show that the functional W, introduced in section 3, is in a sense analogous to minus entropy. 5.1 Recall that the partition R function for the canonical ensemble at temperature β −1 is given by Z = exp(−βE)dω(E), where ω(E) is a ”density of states” measure, which does not depend on β. Then one computes the ∂ average energy < E >= − ∂β log Z, the entropy S = β < E > + log Z, and 2
∂ the fluctuation σ =< (E− < E >)2 >= (∂β) 2 log Z. Now fix a closed manifold M with a probability measure m, and suppose that our system is described by a metric gij (τ ), which depends on the temperature τ according to equation (gij )τ = 2(Rij + ∇i ∇j f ), where Rdm = udV, u = n (4πτ )− 2 e−f , and the partition function is given by log Z = (−f + n2 )dm. (We do not discuss here what assumptions on gij guarantee that the corresponding ”density of states” measure can be found) Then we compute Z n 2 < E >= −τ (R + |∇f |2 − )dm, 2τ M Z S=− (τ (R + |∇f |2) + f − n)dm, M
σ = 2τ 4
Z
M
|Rij + ∇i ∇j f −
1 gij |2 dm 2τ
Alternatively, we could prescribe the evolution equations by replacing the t-derivatives by minus τ -derivatives in (3.3 ), and get the same formulas for Z, < E >, S, σ, with dm replaced by udV. Clearly, σ is nonnegative; it vanishes only on a gradient shrinking soliton. < E > is nonnegative as well, whenever the flow exists for all sufficiently small τ > 0 (by proposition 1.2). Furthermore, if (a) u tends to a δ-function as τ → 0, or (b) u is a limit of a sequence of functions ui , such that each ui 11
tends to a δ-function as τ → τi > 0, and τi → 0, then S is also nonnegative. In case (a) all the quantities < E >, S, σ tend to zero as τ → 0, while in case (b), which may be interesting if gij (τ ) goes singular at τ = 0, the entropy S may tend to a positive limit. If the flow is defined for all sufficiently large τ (that is, we have an ancient solution to the Ricci flow, in Hamilton’s terminology), we may be interested in the behavior of the entropy S as τ → ∞. A natural question is whether we have a gradient shrinking soliton whenever S stays bounded. 5.2 Remark. Heuristically, this statistical analogy is related to the description of the renormalization group flow, mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states, whereas in the former those states are suppressed by the exponential factor. 5.3* An entropy formula for the Ricci flow in dimension two was found by Chow [C]; there seems to be no relation between his formula and ours. The interplay of statistical physics and (pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics, developed by Hawking et al. Unfortunately, this subject is beyond my understanding at the moment.
6
Riemannian formalism in potentially infinite dimensions
When one is talking of the canonical ensemble, one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature (thermostat). In this section we attempt to describe such an embedding using the formalism of Rimannian geometry. ˜ = M × SN × R+ with the following metric: 6.1 Consider the manifold M g˜ij = gij , g˜αβ = τ gαβ , g˜00 =
N + R, g˜iα = g˜i0 = g˜α0 = 0, 2τ
where i, j denote coordinate indices on the M factor, α, β denote those on the SN factor, and the coordinate τ on R+ has index 0; gij evolves with τ by the backward Ricci flow (gij )τ = 2Rij , gαβ is the metric on SN of 1 constant curvature 2N . It turns out that the components of the curvature tensor of this metric coincide (modulo N −1 ) with the components of the matrix Harnack expression (and its traces), discovered by Hamilton [H 3]. One can also compute that all the components of the Ricci tensor are equal 12
to zero (mod N −1 ). The heat equation and the conjugate heat equation on ˜ for functions and volume M can be interpreted via Laplace equation on M forms respectively: u satisfies the heat equation on M iff u˜ (the extension of ˜ u = 0 mod N −1 ; similarly, u ˜ constant along the SN fibres) satisfies △˜ u to M N−1 ˜ u∗ = satisfies the conjugate heat equation on M iff u˜∗ = τ − 2 u˜ satisfies △˜ −1 ˜. 0 mod N on M ˜ isometric 6.2 Starting from g˜, we can also construct a metric g m on M, −1 to g˜ (mod N ), which corresponds to the backward m-preserving Ricci flow ( given by equations (1.1) with t-derivatives replaced by minus τ -derivatives, n dm = (4πτ )− 2 e−f dV ). To achieve this, first apply to g˜ a (small) diffeomorphism, mapping each point (xi , y α , τ ) into (xi , y α, τ (1 − 2f )); we would get a N m −1 metric g˜ , with components (mod N ) m = (1 − g˜ijm = g˜ij , g˜αβ
2f f m m m m )˜ gαβ , g˜00 = g˜00 − 2fτ − , g˜i0 = −∇i f, g˜iα = g˜α0 = 0; N τ
then apply a horizontal (that is, along the M factor) diffeomorphism to get g m satisfying (gijm )τ = 2(Rij + ∇i ∇j f ); the other components of g m become (mod N −1 ) m gαβ = (1 −
1 N 2f m m )˜ gαβ , g00 = g˜00 − |∇f |2 = ( − [τ (2△f − |∇f |2 + R) + f − n]), N τ 2 m m m gi0 = gα0 = giα =0
Note that the hypersurface τ =const in the metric g m has the volume form τ N/2 e−f times the canonical form on M and SN , and the scalar curvature of this hypersurface is τ1 ( N2 + τ (2△f − |∇f |2 + R) + f ) mod N −1 . Thus the entropy S multiplied by the inverse temperature β is essentially minus the total scalar curvature of this hypersurface. 6.3 Now we return to the metric g˜ and try to use its Ricci-flatness by interpreting the Bishop-Gromov relative volume comparison theorem. Con˜ g˜) centered at some point p where τ = 0. Then sider a metric ball in (M, clearly the shortest geodesic between p and an arbitrary point q is always orthogonal to the SN fibre. The length of such curve γ(τ ) can be computed as Z τ (q) r N + R + |γ˙ M (τ )|2 dτ 2τ 0 Z τ (q) p √ 3 1 = 2Nτ (q) + √ τ (R + |γ˙ M (τ )|2 )dτ + O(N − 2 ) 2N 0 13
R τ (q) √ Thus a shortest geodesic should minimize L(γ) = 0 τ (R + |γ˙ M (τ )|2 )dτ , an expression defined entirely in terms of M. Let L(qM ) denote the p corre˜ of radius 2Nτ (q) sponding infimum. It follows that a metric sphere in M centered at p is O(N −1 )-close τ = τ (q), and its volume pthe hypersurface R to 1 N can be computed as V (S ) M ( τ (q) − 2N L(x) + O(N −2 ))N dx, so the ratio p N +n n is just constant times N − 2 times of this volume to 2Nτ (q) Z n 1 L(x))dx + O(N −1 ) τ (q)− 2 exp(− p 2τ (q) M
The computation suggests that this integral, which we will call the reduced volume and denote by V˜ (τ (q)), should be increasing as τ decreases. A rigorous proof of this monotonicity is given in the next section. 6.4* The first geometric interpretation of Hamilton’s Harnack expressions was found by Chow and Chu [C-Chu 1,2]; they construct a potentially degenerate riemannian metric on M × R, which potentially satisfies the Ricci soliton equation; our construction is, in a certain sense, dual to theirs. Our formula for the reduced volume resembles the expression in Huisken monotonicity formula for the mean curvature flow [Hu]; however, in our case the monotonicity is in the opposite direction.
7
A comparison geometry approach to the Ricci flow
7.1 In this section we consider an evolving metric (gij )τ = 2Rij on a manifold M; we assume that either M is closed, or gij (τ ) are complete and have uniformly bounded curvatures. To each curve γ(τ ), 0 < τ1 ≤ τ ≤ τ2 , we associate its L-length Z τ2 √ τ (R(γ(τ )) + |γ(τ ˙ )|2 )dτ L(γ) = τ1
(of course, R(γ(τ )) and |γ(τ ˙ )|2 are computed using gij (τ )) Let X(τ ) = γ(τ ˙ ), and let Y (τ ) be any vector field along γ(τ ). Then the first variation formula can be derived as follows: δY (L) = 14
Z
τ2
τ1
√ Z
τ (< Y, ∇R > +2 < ∇Y X, X >)dτ = τ2
Z
τ2 τ1
√ τ (< Y, ∇R > +2 < ∇X Y, X >)dτ
√ d τ (< Y, ∇R > +2 < Y, X > −2 < Y, ∇X X > −4Ric(Y, X))dτ dτ τ1 Z τ2 τ2 √ √ 1 τ < Y, ∇R − 2∇X X − 4Ric(X, ·) − X > dτ = 2 τ < X, Y > τ1 + τ τ1 (7.1) Thus L-geodesics must satisfy =
1 1 ∇X X − ∇R + X + 2Ric(X, ·) = 0 2 2τ
(7.2)
Given two points p, q and τ2 > τ1 > 0, we can always find an L-shortest curve γ(τ ), τ ∈ [τ1 , τ2 ] between them, and every such L-shortest curve √ is Lgeodesic. It is easy to extend this to the case τ1 = 0; in this case τ X(τ ) has a limit as τ → 0. From now on we fix p and τ1 = 0 and denote by L(q, τ¯) the L-length of the L-shortest curve γ(τ ), 0 ≤ τ ≤ τ¯, connecting p and q. In the computations below we pretend that shortest L-geodesics between p and q are unique for all pairs (q, τ¯); if this is not the case, the inequalities that we obtain are still valid when understood in the barrier sense, or in the sense of distributions. √ The first variation formula (7.1) implies that ∇L(q, τ¯) = 2 τ¯X(¯ τ ), so 2 2 2 that |∇L| = 4¯ τ |X| = −4¯ τ R + 4¯ τ (R + |X| ). We can also compute √ √ √ Lτ¯ (q, τ¯) = τ¯(R + |X|2 )− < X, ∇L >= 2 τ¯R − τ¯(R + |X|2) To evaluate R + |X|2 we compute (using (7.2)) d (R(γ(τ )) + |X(τ )|2 ) = Rτ + < ∇R, X > +2 < ∇X X, X > +2Ric(X, X) dτ 1 1 = Rτ + R + 2 < ∇R, X > −2Ric(X, X) − (R + |X|2 ) τ τ 1 = −H(X) − (R + |X|2), (7.3) τ where H(X) is the Hamilton’s expression for the trace Harnack inequality (with t = −τ ). Hence, 3 1 τ¯ 2 (R + |X|2 )(¯ τ ) = −K + L(q, τ¯), 2
15
(7.4)
where K = K(γ, τ¯) denotes the integral a few times below. Thus we get
R τ¯ 0
3
τ 2 H(X)dτ, which we’ll encounter
√ 1 1 Lτ¯ = 2 τ¯R − L + K 2¯ τ τ¯
(7.5)
2 4 |∇L|2 = −4¯ τR + √ L − √ K (7.6) τ¯ τ¯ Finally we need to estimate the second variation of L. We compute Z τ¯ √ 2 δY (L) = τ (Y · Y · R + 2 < ∇Y ∇Y X, X > +2|∇Y X|2 )dτ 0
=
Z
τ¯ 0
√
τ (Y · Y · R + 2 < ∇X ∇Y Y, X > +2 < R(Y, X), Y, X > +2|∇X Y |2 )dτ
Now d < ∇Y Y, X >=< ∇X ∇Y Y, X > + < ∇Y Y, ∇X X > +2Y ·Ric(Y, X)−X·Ric(Y, Y ), dτ so, if Y (0) = 0 then δY2 (L) = 2 < ∇Y Y, X > Z
τ¯ 0
√
τ¯+
√ τ (∇Y ∇Y R + 2 < R(Y, X), Y, X > +2|∇X Y |2 + 2∇X Ric(Y, Y ) − 4∇Y Ric(Y, X))dτ, (7.7)
where we discarded the scalar product of −2∇Y Y with the left hand side of (7.2). Now fix the value of Y at τ = τ¯, assuming |Y (¯ τ )| = 1, and construct Y on [0, τ¯] by solving the ODE ∇X Y = −Ric(Y, ·) +
1 Y 2τ
We compute d 1 < Y, Y >= 2Ric(Y, Y ) + 2 < ∇X Y, Y >= < Y, Y >, dτ τ 16
(7.8)
so |Y (τ )|2 = ττ¯ , and in particular, Y (0) = 0. Making a substitution into (7.7), we get HessL (Y, Y ) ≤ Z τ¯ √ τ (∇Y ∇Y R + 2 < R(Y, X), Y, X > +2∇X Ric(Y, Y ) − 4∇Y Ric(Y, X) 0
1 2 )dτ +2|Ric(Y, ·)|2 − Ric(Y, Y ) + τ 2τ τ¯ To put this in a more convenient form, observe that
d Ric(Y (τ ), Y (τ )) = Ricτ (Y, Y ) + ∇X Ric(Y, Y ) + 2Ric(∇X Y, Y ) dτ
so
where
1 = Ricτ (Y, Y ) + ∇X Ric(Y, Y ) + Ric(Y, Y ) − 2|Ric(Y, ·)|2, τ Z τ¯ √ √ 1 τ H(X, Y )dτ , HessL (Y, Y ) ≤ √ − 2 τ¯Ric(Y, Y ) − τ¯ 0
(7.9)
H(X, Y ) = −∇Y ∇Y R−2 < R(Y, X)Y, X > −4(∇X Ric(Y, Y )−∇Y Ric(Y, X)) 1 −2Ricτ (Y, Y ) + 2|Ric(Y, ·)|2 − Ric(Y, Y ) τ is the Hamilton’s expression for the matrix Harnack inequality (with t = −τ ). Thus √ 1 n (7.10) △L ≤ −2 τ R + √ − K τ τ A field Y (τ ) along L-geodesic γ(τ ) is called L-Jacobi, if it is the derivative of a variation of γ among L-geodesics. For an L-Jacobi field Y with |Y (¯ τ )| = 1 we have d |Y |2 = 2Ric(Y, Y ) + 2 < ∇X Y, Y >= 2Ric(Y, Y ) + 2 < ∇Y X, Y > dτ Z τ¯ 1 1 1 1 = 2Ric(Y, Y ) + √ HessL (Y, Y ) ≤ − √ τ 2 H(X, Y˜ )dτ , (7.11) τ¯ τ¯ τ¯ 0 where Y˜ is obtained by solving ODE (7.8) with initial data Y˜ (¯ τ ) = Y (¯ τ ). ˜ Moreover, the equality in (7.11) holds only if Y is L-Jacobi and hence d |Y |2 = 2Ric(Y, Y ) + √1τ¯ HessL (Y, Y ) = τ1¯ . dτ 17
Now we can deduce an estimate for the jacobian J of the L-exponential map, given by LexpX (¯ τ ) = γ(¯ τ ), where γ(τ ) is the L-geodesic, starting at p √ ˙ ) as τ → 0. We obtain and having X as the limit of τ γ(τ n 1 3 d logJ(τ ) ≤ − τ¯− 2 K, dτ 2¯ τ 2
(7.12)
with equality only if 2Ric + √1τ¯ HessL = τ1¯ g. Let l(q, τ ) = 2√1 τ L(q, τ ) be the reduced distance. Then along an L-geodesic γ(τ ) we have (by (7.4)) d 1 1 1 3 l(τ ) = − l + (R + |X|2 ) = − τ¯− 2 K, dτ 2¯ τ 2 2 n
so (7.12) implies that τ − 2 exp(−l(τ ))J(τ ) is nonincreasing in τ along γ, and monotonicity is strict unless we are on a gradient shrinking soliton. Integrating M, we get monotonicity of the reduced volume function R over −n ˜ 2 V (τ ) = M τ exp(−l(q, τ ))dq. ( Alternatively, one could obtain the same monotonicity by integrating the differential inequality lτ¯ − △l + |∇l|2 − R +
n ≥ 0, 2¯ τ
(7.13)
which follows immediately from (7.5), (7.6) and (7.10). Note also a useful inequality l−n ≤ 0, (7.14) 2△l − |∇l|2 + R + τ¯ which follows from (7.6), (7.10).) ¯ τ ) = 2√τ L(q, τ ), then from (7.5), On the other hand, if we denote L(q, (7.10) we obtain ¯ τ¯ + △L ¯ ≤ 2n L (7.15) ¯ τ¯) − 2n¯ Therefore, the minimum of L(·, τ is nonincreasing, so in particular, the minimum of l(·, τ¯) does not exceed n2 for each τ¯ > 0. (The lower bound for l is much easier to obtain since the evolution equation Rτ = −△R − 2|Ric|2 implies R(·, τ ) ≥ − 2(τ0n−τ ) , whenever the flow exists for τ ∈ [0, τ0 ].) 7.2 If the metrics gij (τ ) have nonnegative curvature operator, then Hamilton’s differential Harnack inequalities hold, and one can say more about the behavior of l. Indeed, in this case, if the solution is defined for τ ∈ [0, τ0 ], then H(X, Y ) ≥ −Ric(Y, Y )( τ1 + τ01−τ ) ≥ −R( τ1 + τ01−τ )|Y |2 and 18
H(X) ≥ −R( τ1 + τ01−τ ). Therefore, whenever τ is bounded away from τ0 (say, τ ≤ (1 − c)τ0 , c > 0), we get (using (7.6), (7.11)) |∇l|2 + R ≤
Cl , τ
(7.16)
and for L-Jacobi fields Y 1 d log|Y |2 ≤ (Cl + 1) dτ τ
(7.17)
7.3 As the first application of the comparison inequalities above, let us give an alternative proof of a weakened version of the no local collapsing theorem 4.1. Namely, rather than assuming |Rm|(x, tk ) ≤ rk−2 for x ∈ Bk , we require |Rm|(x, t) ≤ rk−2 whenever x ∈ Bk , tk − rk2 ≤ t ≤ tk . Then the 1 proof can go as follows: let τk (t) = tk −t, p = pk , ǫk = rk−1 V ol(Bk ) n . We claim n that V˜k (ǫk rk2 ) < 3ǫk2 when k is large. Indeed, using the L-exponential map we can integrate over Tp M rather than M; the vectors in Tp M of length at most 1 1 −2 ǫ 2 k
give rise to L-geodesics, which can not escape from Bk in time ǫk rk2 , so n
their contribution to the reduced volume does not exceed 2ǫk2 ; on the other −1
hand, the contribution of the longer vectors does not exceed exp(− 12 ǫk 2 ) by the jacobian comparison theorem. However, V˜k (tk ) (that is, at t = 0) stays bounded away from zero. Indeed, since min lk (·, tk − 12 T ) ≤ n2 , we can pick a point qk , where it is attained, and obtain a universal upper bound on lk (·, tk ) by considering only curves γ with γ(tk − 12 T ) = qk , and using the fact that all geometric quantities in gij (t) are uniformly bounded when t ∈ [0, 21 T ]. Since the monotonicity of the reduced volume requires V˜k (tk ) ≤ V˜k (ǫk rk2 ), this is a contradiction. A similar argument shows that the statement of the corollary in 4.2 can be strengthened by adding another property of the ancient solution, obtained as a blow-up limit. Namely, we may claim that if, say, this solution is defined for t ∈ (−∞, 0), then for any point p and any t0 > 0, the reduced volume function V˜ (τ ), constructed using p and τ (t) = t0 − t, is bounded below by κ. 7.4* The computations in this section are just natural modifications of those in the classical variational theory of geodesics that can be found in any textbook on Riemannian geometry; an even closer reference is [L-Y], where they use ”length”, associated to a linear parabolic equation, which is pretty much the same as in our case. 19
8
No local collapsing theorem II
8.1 Let us first formalize the notion of local collapsing, that was used in 7.3. Definition. A solution to the Ricci flow (gij )t = −2Rij is said to be κ-collapsed at (x0 , t0 ) on the scale r > 0 if |Rm|(x, t) ≤ r −2 for all (x, t) satisfying distt0 (x, x0 ) < r and t0 − r 2 ≤ t ≤ t0 , and the volume of the metric ball B(x0 , r 2 ) at time t0 is less than κrn . 8.2 Theorem. For any A > 0 there exists κ = κ(A) > 0 with the following property. If gij (t) is a smooth solution to the Ricci flow (gij )t = −2Rij , 0 ≤ t ≤ r02 , which has |Rm|(x, t) ≤ r0−2 for all (x, t), satisfying dist0 (x, x0 ) < r0 , and the volume of the metric ball B(x0 , r0 ) at time zero is at least A−1 r0n , then gij (t) can not be κ-collapsed on the scales less than r0 at a point (x, r02 ) with distr02 (x, x0 ) ≤ Ar0 . Proof. By scaling we may assume r0 = 1; we may also assume dist1 (x, x0 ) = A. Let us apply the constructions of 7.1 choosing p = x, τ (t) = 1 − t. Arguing as in 7.3, we see that if our solution is collapsed at x on the scale r ≤ 1, then the reduced volume V˜ (r 2 ) must be very small; on the other hand, V˜ (1) can 1 is large. not be small unless min l(x, 12 ) over x satisfying dist 1 (x, x0 ) ≤ 10 2 ¯ Thus all we need is to estimate l, or equivalently L, in that ball. Recall that ¯ satisfies the differential inequality (7.15). In order to use it efficiently in L a maximum principle argument, we need first to check the following simple assertion. 8.3 Lemma. Suppose we have a solution to the Ricci flow (gij )t = −2Rij . (a) Suppose Ric(x, t0 ) ≤ (n − 1)K when distt0 (x, x0 ) < r0 . Then the distance function d(x, t) = distt (x, x0 ) satisfies at t = t0 outside B(x0 , r0 ) the differential inequality 2 dt − △d ≥ −(n − 1)( Kr0 + r0−1 ) 3 (the inequality must be understood in the barrier sense, when necessary) (b) (cf. [H 4,§17]) Suppose Ric(x, t0 ) ≤ (n − 1)K when distt0 (x, x0 ) < r0 , or distt0 (x, x1 ) < r0 . Then d 2 distt (x0 , x1 ) ≥ −2(n − 1)( Kr0 + r0−1 ) at t = t0 dt 3 R Proof of Lemma. (a) Clearly, dt (x) = γ −Ric(X, X), where γ is the shortest geodesic between x0 and X is its unit tangent vector, On the other Pn−1x and ′′ hand, △d ≤ k=1 sYk (γ), where Yk are vector fields along γ, vanishing at 20
x0 and forming an orthonormal basis at x when complemented by X, and s′′Yk (γ) denotes the second variation along Yk of the length of γ. Take Yk to be parallel between x and x1 , and linear between x1 and x0 , where d(x1 , t0 ) = r0 . Then △d ≤ =
Z
γ
n−1 X
s′′Yk (γ)
=
Z
d(x,t0 )
r0
k=1
−Ric(X, X)+
Z
0
−Ric(X, X)ds+
r0
(Ric(X, X)(1 −
Z
0
r0
(
s2 n−1 (−Ric(X, X)) + )ds 2 r0 r02
n−1 2 s2 )+ )ds ≤ dt +(n−1)( Kr0 +r0−1 ) 2 2 r0 r0 3
The proof of (b) is similar. Continuing the proof of theorem, apply the maximum principle to the ¯ 1 − t) + 2n + 1), where d(y, t) = function h(y, t) = φ(d(y, t) − A(2t − 1))(L(y, 1 ), and distt (x, x0 ), and φ is a function of one variable, equal 1 on (−∞, 20 1 1 rapidly increasing to infinity on ( 20 , 10 ), in such a way that 2(φ′ )2 /φ − φ′′ ≥ (2A + 100n)φ′ − C(A)φ,
(8.1)
¯ + 2n + 1 ≥ 1 for t ≥ 1 by the for some constant C(A) < ∞. Note that L 2 remark in the very end of 7.1. Clearly, min h(y, 1) ≤ h(x, 1) = 2n + 1. On 1 the other hand, min h(y, 12 ) is achieved for some y satisfying d(y, 12 ) ≤ 10 . Now we compute ′′ ¯ ¯ > +(L ¯ t −△L)φ ¯ 2h = (L+2n+1)(−φ +(dt −△d−2A)φ′ )−2 < ∇φ∇L (8.2)
¯ + 2n + 1)∇φ + φ∇L ¯ ∇h = (L
(8.3)
At a minimum point of h we have ∇h = 0, so (8.2) becomes ¯ + 2n + 1)(−φ′′ + (dt − △d − 2A)φ′ + 2(φ′ )2 /φ) + (L ¯ t − △L)φ ¯ 2h = (L (8.4) 1 1 whenever φ′ 6= 0, and since Ric ≤ n − 1 in B(x0 , 20 ), Now since d(y, t) ≥ 20 we can apply our lemma (a) to get dt − △d ≥ −100(n − 1) on the set where φ′ 6= 0. Thus, using (8.1) and (7.15), we get
¯ + 2n + 1)C(A)φ − 2nφ ≥ −(2n + C(A))h 2h ≥ −(L This implies that min h can not decrease too fast, and we get the required estimate. 21
9
Differential Harnack inequality for solutions of the conjugate heat equation
9.1 Proposition. Let gij (t) be a solution to the Ricci flow (gij )t = −2Rij , 0 ≤ n t ≤ T, and let u = (4π(T − t))− 2 e−f satisfy the conjugate heat equation 2∗ u = −ut − △u + Ru = 0. Then v = [(T − t)(2△f − |∇f |2 + R) + f − n]u satisfies 1 2∗ v = −2(T − t)|Rij + ∇i ∇j f − gij |2 (9.1) 2(T − t) Proof. Routine computation. Clearly, this proposition immediately implies the monotonicity formula (3.4); its advantage over (3.4) shows up when one has to work locally. 9.2 Corollary. Under the same assumptions, on a closed manifold M,or whenever the application of the maximum principle can be justified, min v/u is nondecreasing in t. 9.3 Corollary. Under the same assumptions, if u tends to a δ-function as t → T, then v ≤ 0 for all t < T. Proof. If h satisfies the ordinary heat Requation ht = △h R with respect to d d the evolving metric gij (t), then we have dt hu = 0 and dt hvR≥ 0. Thus we only need to check that for everywhere positive h the limit of hv as t → T is nonpositive. But it is easy to see, that this limit is in fact zero. 9.4 Corollary. Under assumptions of the previous corollary, for any smooth curve γ(t) in M holds −
1 1 d 2 f (γ(t), t) ≤ (R(γ(t), t) + |γ(t)| ˙ )− f (γ(t), t) dt 2 2(T − t)
(9.2)
Proof. From the evolution equation ft = −△f + |∇f |2 − R + 2(Tn−t) and v ≤ 0 we get ft + 12 R− 12 |∇f |2 − 2(Tf−t) ≥ 0. On the other hand,− dtd f (γ(t), t) = ˙ 2 . Summing these two inequalities, −ft − < ∇f, γ(t) ˙ >≤ −ft + 21 |∇f |2 + 12 |γ| we get (9.2). 9.5 Corollary. If under assumptions of the previous corollary, p is the point where the limit δ-function is concentrated, then f (q, t) ≤ l(q, T − t), where l is the reduced distance, defined in 7.1, using p and τ (t) = T − t. 22
Proof. Use (7.13) in the form 2∗ exp(−l) ≤ 0. 9.6 Remark. Ricci flow can be characterized among all other evolution equations by the infinitesimal behavior of the fundamental solutions of the conjugate heat equation. Namely, suppose we have a riemannian metric gij (t) evolving with time according to an equation (gij )t = Aij (t). Then we have ∂ ∂ − △ and its conjugate 2∗ = − ∂t − △ − 21 A, so that theR heat operator 2 = ∂t R d uv = ((2u)v − u(2∗ v)). (Here A = g ij Aij ) Consider the fundamental dt n solution u = (−4πt)− 2 e−f for 2∗ , starting as δ-function at some point (p, 0). R ¯ Then for general Aij the function (2f¯+ ft )(q, t), where f¯ = f − f u, is of the order O(1) for (q, t) near (p, 0). The Ricci flow Aij = −2Rij is characterized ¯ by the condition (2f¯ + ft )(q, t) = o(1); in fact, it is O(|pq|2 + |t|) in this case. 9.7* Inequalities of the type of (9.2) are known as differential Harnack inequalities; such inequality was proved by Li and Yau [L-Y] for the solutions of linear parabolic equations on riemannian manifolds. Hamilton [H 7,8] used differential Harnack inequalities for the solutions of backward heat equation on a manifold to prove monotonicity formulas for certain parabolic flows. A local monotonicity formula for mean curvature flow making use of solutions of backward heat equation was obtained by Ecker [E 2].
10
Pseudolocality theorem
10.1 Theorem. For every α > 0 there exist δ > 0, ǫ > 0 with the following property. Suppose we have a smooth solution to the Ricci flow (gij )t = −2Rij , 0 ≤ t ≤ (ǫr0 )2 , and assume that at t = 0 we have R(x) ≥ −r0−2 and V ol(∂Ω)n ≥ (1 − δ)cn V ol(Ω)n−1 for any x, Ω ⊂ B(x0 , r0 ), where cn is the euclidean isoperimetric constant. Then we have an estimate |Rm|(x, t) ≤ αt−1 + (ǫr0 )−2 whenever 0 < t ≤ (ǫr0 )2 , d(x, t) = distt (x, x0 ) < ǫr0 . Thus, under the Ricci flow, the almost singular regions (where curvature is large) can not instantly significantly influence the almost euclidean regions. Or , using the interpretation via renormalization group flow, if a region looks trivial (almost euclidean) on higher energy scale, then it can not suddenly become highly nontrivial on a slightly lower energy scale. Proof. It is an argument by contradiction. The idea is to pick a point ¯ (¯ x, t) not far from (x0 , 0) and consider the solution u to the conjugate heat equation, starting as δ-function at (¯ x, t¯), and the corresponding nonpositive function v as in 9.3. If the curvatures at (¯ x, t¯) are not small compared to 23
R t¯−1 and are larger than at nearby points, then one can show that v at time t is bounded away from zero for (small) time intervals t¯ − t of the order of R −1 |Rm| (¯ x, t¯). By monotonicity we conclude that v is bounded away from zero at t = 0. In fact, using (9.1) and an appropriate cut-off function, we can show that at t = 0 already the integral of v over B(x0 , r) is bounded away from zero, whereas the integral of u over this ball is close to 1, where r can be made as small as we like compared to r0 . Now using the control over the scalar curvature and isoperimetric constant in B(x0 r0 ), we can obtain a contradiction to the logarithmic Sobolev inequality. Now let us go into details. By scaling assume that r0 = 1. We may also 1 . From now on we fix α and denote by assume that α is small, say α < 100n Mα the set of pairs (x, t), such that |Rm|(x, t) ≥ αt−1 . Claim 1.For any A > 0, if gij (t) solves the Ricci flow equation on 0 ≤ 1 , and |Rm|(x, t) > αt−1 + ǫ−2 for some (x, t), satisfying 0 ≤ t ≤ ǫ2 , Aǫ < 100n 2 t ≤ ǫ , d(x, t) < ǫ, then one can find (¯ x, t¯) ∈ Mα , with 0 < t¯ ≤ ǫ2 , d(¯ x, t¯) < (2A + 1)ǫ, such that |Rm|(x, t) ≤ 4|Rm|(¯ x, t¯), (10.1) whenever
1
(x, t) ∈ Mα , 0 < t ≤ t¯, d(x, t) ≤ d(¯ x, t¯) + A|Rm|− 2 (¯ x, t¯)
(10.2)
Proof of Claim 1. We construct (¯ x, t¯) as a limit of a (finite) sequence (xk , tk ), defined in the following way. Let (x1 , t1 ) be an arbitrary point, satisfying 0 < t1 ≤ ǫ2 , d(x1 , t1 ) < ǫ, |Rm|(x1 , t1 ) ≥ αt−1 + ǫ−2 . Now if (xk , tk ) is already constructed, and if it can not be taken for (¯ x, t¯), because there is some (x, t) satisfying (10.2), but not (10.1), then take any such (x, t) for (xk+1 , tk+1 ). Clearly, the sequence, constructed in such a way, satisfies |Rm|(xk , tk ) ≥ 4k−1|Rm|(x1 , t1 ) ≥ 4k−1ǫ−2 , and therefore, d(xk , tk ) ≤ (2A + 1)ǫ. Since the solution is smooth, the sequence is finite, and its last element fits. Claim 2. For (¯ x, t¯), constructed above, (10.1) holds whenever 1 1 1 t¯ − αQ−1 ≤ t ≤ t¯, distt¯(x, x¯) ≤ AQ− 2 , 2 10
(10.3)
where Q = |Rm|(¯ x, t¯). Proof of Claim 2. We only need to show that if (x, t) satisfies (10.3), then it must satisfy (10.1) or (10.2). Since (¯ x, t¯) ∈ Mα , we have Q ≥ αt¯−1 , so 1 1 −1 t¯− 2 αQ ≥ 2 t¯. Hence, if (x, t) does not satisfy (10.1), it definitely belongs to 24
1 1 AQ− 2 . On the other Mα . Now by the triangle inequality, d(x, t¯) ≤ d(¯ x, t¯) + 10 hand, using lemma 8.3(b) we see that, as t decreases from t¯ to t¯ − 12 αQ−1 , 1 the point x can not escape from the ball of radius d(¯ x, t¯) + AQ− 2 centered at x0 . Continuing the proof of the theorem, and arguing by contradiction, take sequences ǫ → 0, δ → 0 and solutions gij (t), violating the statement; by reducing ǫ, we’ll assume that
|Rm|(x, t) ≤ αt−1 + 2ǫ−2 whenever 0 ≤ t ≤ ǫ2 and d(x, t) ≤ ǫ
(10.4)
1 → ∞, construct (¯ x, t¯), and consider solutions u = (4π(t¯ − Take A = 100nǫ n − 2 −f x, t¯), t)) e of the conjugate heat equation, starting from δ-functions at (¯ and corresponding nonpositive functions v. −1 ¯ ˜ ¯ 1 Claim R 3.As ǫ, δ → 0, one can find times t ∈ [t − 2 αQ , t], such that the integral B v stays bounded away from zero, where B is the ball at time t˜ of p radius t¯ − t˜ centered at x¯. Proof of Claim 3(sketch). The statement is invariant under scaling, so we can try to take a limit of scalings of gij (t) at points (¯ x, t¯) with factors ¯ Q. If the injectivity radii of the scaled metrics at (¯ x, t) are bounded away from zero, then a smooth limit exists, it is complete and has |Rm|(¯ x, t¯) = 1 1 and |Rm|(x, t) ≤ 4 when t¯ − 2 α ≤ t ≤ t¯. It is not hard to show that the fundamental solutions u of the conjugate heat equation converge to such a R solution on the limit manifold. But on the limit manifold, B v can not be zero for t˜ = t¯ − 21 α, since the evolution equation (9.1) would imply in this case that the limit is a gradient shrinking soliton, and this is incompatible with |Rm|(¯ x, t¯) = 1. If the injectivity radii of the scaled metrics tend to zero, then we can change the scaling factor, to make the scaled metrics converge to a flat manifold with finite injectivity radius; in this case it is not hard to choose t˜ in R such a way that B v → −∞. R The positive lower bound for − B v will be denoted by β. Our next goal is to construct an appropriate cut-off function. We choose √ ˜ ˜ t) = d(y, t) + 200n t, and φ is ), where d(y, it in the form h(y, t) = φ( d(y,t) 10Aǫ a smooth function of one variable, equal one on (−∞, 1] and decreasing to zero on [1, 2]. Clearly, h vanishes at t = 0 outside B(x0 , 20Aǫ); on the other hand, it is equal to one near (¯ x, t¯). 1 100n 100n 1 ′′ √ ≥ 0 Now 2h = 10Aǫ (dt − △d + √t )φ′ − (10Aǫ) 2 φ . Note that dt − △t + t ′ on the set where φ 6= 0 − this follows from the lemma 8.3(a) and our
25
assumption (10.4). WeR may alsoRchoose φ so that φ′′R≥ −10φ, (φ′R)2 ≤ 10φ. 1 Now we can compute ( M hu)t = M (2h)u ≤ (Aǫ) 2 , so M hu |t=0 ≥ M hu |t=t¯ R R R 1 t¯ −2 −hv, − (Aǫ)2 ≥ 1 − A . Also, by (9.1), ( M −hv)t ≤ M −(2h)v ≤ (Aǫ) 2 M R t¯ −2 so by Claim 3, − M hv |t=0 ≥ βexp(− (Aǫ)2 ) ≥ β(1 − A ). From now on we”ll work at t = 0 only. Let u˜ = hu and correspondingly ˜ f = f − logh. Then Z Z −2 β(1 − A ) ≤ − hv = [(−2△f + |∇f |2 − R)t¯ − f + n]hu M
=
Z
M
R
˜2
M
[−t¯|∇f| − f˜ + n]˜ u+ ≤
Z
M
Z
M
[t¯(|∇h|2 /h − Rh) − hlogh]u
[−t¯|∇f˜|2 − f˜ − n]˜ u + A−2 + 100ǫ2
( Note that M −uh log h does Rnot exceed theR integral of u over ¯ ≥ 1 − A−2 , B(x0 , 20Aǫ)\B(x0 , 10Aǫ), and B(x0 ,10Aǫ) u ≥ M hu ¯ = φ( d˜ )) where h 5Aǫ Now scaling the metric by the factor 21 t¯−1 and sending ǫ, δ to zero, we get a sequence of metric balls with radii going to infinity, and a sequence R n of compactly supported nonnegative functions u = (2π)− 2 e−f with u → 1 R and [− 21 |∇f |2 − f + n]u bounded away from zero by a positive constant. We also have isoperimetric inequalities with the constants tending to the euclidean one. This set up is in conflict with the Gaussian logarithmic Sobolev inequality, as can be seen by using spherical symmetrization. 10.2 Corollary(from the proof) Under the same √ assumptions, √ n we also 2 have at time t, 0 < t ≤ (ǫr0 ) , an estimate V olB(x, t) ≥ c t for x ∈ B(x0 , ǫr0 ), where c = c(n) is a universal constant. 10.3 Theorem. There exist ǫ, δ > 0 with the following property. Suppose gij (t) is a smooth solution to the Ricci flow on [0, (ǫr0 )2 ], and assume that at t = 0 we have |Rm|(x) ≤ r0−2 in B(x0 , r0 ), and V olB(x0 , r0 ) ≥ (1 − δ)ωn r0n , where ωn is the volume of the unit ball in Rn . Then the estimate |Rm|(x, t) ≤ (ǫr0 )−2 holds whenever 0 ≤ t ≤ (ǫr0 )2 , distt (x, x0 ) < ǫr0 . The proof is a slight modification of the proof of theorem 10.1, and is left to the reader. A natural question is whether the assumption on the volume of the ball is superfluous. 10.4 Corollary(from 8.2, 10.1, 10.2) There exist ǫ, δ > 0 and for any A > 0 there exists κ(A) > 0 with the following property. If gij (t) is a 26
smooth solution to the Ricci flow on [0, (ǫr0 )2 ], such that at t = 0 we have R(x) ≥ −r0−2 , V ol(∂Ω)n ≥ (1 − δ)cn V ol(Ω)n−1 for any x, Ω ⊂ B(x0 , r0 ), and (x, t) satisfies A−1 (ǫr0 )2 ≤ t ≤ (ǫr0 )2 , distt (x, x√0 ) ≤ Ar0 , then gij (t) can not be κ-collapsed at (x, t) on the scales less than t. 10.5 Remark. It is straightforward to get from 10.1 a version of the Cheeger diffeo finiteness theorem for manifolds, satisfying our assumptions on scalar curvature and isoperimetric constant on each ball of some fixed radius r0 > 0. In particular, these assumptions are satisfied (for some controllably smaller r0 ), if we assume a lower bound for Ric and an almost euclidean lower bound for the volume of the balls of radius r0 . (this follows from the LevyGromov isoperimetric inequality); thus we get one of the results of Cheeger and Colding [Ch-Co] under somewhat weaker assumptions. 10.6* Our pseudolocality theorem is similar in some respect to the results of Ecker-Huisken [E-Hu] on the mean curvature flow.
11
Ancient solutions with nonnegative curvature operator and bounded entropy
11.1. In this section we consider smooth solutions to the Ricci flow (gij )t = −2Rij , −∞ < t ≤ 0, such that for each t the metric gij (t) is a complete non-flat metric of bounded curvature and nonnegative curvature operator. Hamilton discovered a remarkable differential Harnack inequality for such solutions; we need only its trace version Rt + 2 < X, ∇R > +2Ric(X, X) ≥ 0
(11.1)
and its corollary, Rt ≥ 0. In particular, the scalar curvature at some time t0 ≤ 0 controls the curvatures for all t ≤ t0 . We impose one more requirement on the solutions; namely, we fix some κ > 0 and require that gij (t) be κ-noncollapsed on all scales (the definitions 4.2 and 8.1 are essentially equivalent in this case). It is not hard to show that this requirement is equivalent to a uniform bound on the entropy S, defined as in 5.1 using an arbitrary fundamental solution to the conjugate heat equation. 11.2. Pick an arbitrary point (p, t0 ) and define V˜ (τ ), l(q, τ ) as in 7.1, for τ (t) = t0 − t. Recall that for each τ > 0 we can find q = q(τ ), such that l(q, τ ) ≤ n2 . 27
Proposition.The scalings of gij (t0 − τ ) at q(τ ) with factors τ −1 converge along a subsequence of τ → ∞ to a non-flat gradient shrinking soliton. Proof (sketch). It is not hard to deduce from (7.16) that for any ǫ > 0 one can find δ > 0 such that both l(q, τ ) and τ R(q, t0 − τ ) do not exceed τ )) ≤ ǫ−1 τ¯ for some τ¯ > 0. δ −1 whenever 12 τ¯ ≤ τ ≤ τ¯ and dist2t0 −¯τ (q, q(¯ Therefore, taking into account the κ-noncollapsing assumption, we can take ¯ ij . We may gij )τ = 2R a blow-down limit, say g¯ij (τ ), defined for τ ∈ ( 21 , 1), (¯ assume also that functions l tend to a locally Lipschitz function ¯l, satisfying (7.13),(7.14) in the sense of distributions. Now, since V˜ (τ ) is nonincreasing and bounded away from zero (because the scaled metrics are not collapsed near q(τ )) the limit function V¯ (τ ) must be a positive constant; this constant n is strictly less than limτ →0 V˜ (τ ) = (4π) 2 , since gij (t) is not flat. Therefore, on the one hand, (7.14) must become an equality, hence ¯l is smooth, and on the other hand, by the description of the equality case in (7.12), g¯ij (τ ) must ¯ ij + ∇ ¯ i∇ ¯ j ¯l − 1 g¯ij = 0. If this soliton be a gradient shrinking soliton with R 2τ is flat, then ¯l is uniquely determined by the equality in (7.14), and it turns n out that the value of V¯ is exactly (4π) 2 , which was ruled out. 11.3 Corollary. There is only one oriented two-dimensional solution, satisfying the assumptions stated in 11.1, - the round sphere. Proof. Hamilton [H 10] proved that round sphere is the only non-flat oriented nonnegatively curved gradient shrinking soliton in dimension two. Thus, the scalings of our ancient solution must converge to a round sphere. However, Hamilton [H 10] has also shown that an almost round sphere is getting more round under Ricci flow, therefore our ancient solution must be round. 11.4. Recall that for any non-compact complete riemannian manifold M of nonnegative Ricci curvature and a point p ∈ M, the function V olB(p, r)r −n is nonincreasing in r > 0; therefore, one can define an asymptotic volume ratio V as the limit of this function as r → ∞. Proposition.Under assumptions of 11.1, V = 0 for each t. Proof. Induction on dimension. In dimension two the statement is vacuous, as we have just shown. Now let n ≥ 3, suppose that V > 0 for some t = t0 , and consider the asymptotic scalar curvature ratio R = lim supR(x, t0 )d2 (x) as d(x) → ∞. (d(x) denotes the distance, at time t0 , from x to some fixed point x0 ) If R = ∞, then we can find a sequence of points xk and radii rk > 0, such that rk /d(xk ) → 0, R(xk )rk2 → ∞, and 28
R(x) ≤ 2R(xk ) whenever x ∈ B(xk , rk ). Taking blow-up limit of gij (t) at (xk , t0 ) with factors R(xk ), we get a smooth non-flat ancient solution, satisfying the assumptions of 11.1, which splits off a line (this follows from a standard argument based on the Aleksandrov-Toponogov concavity). Thus, we can do dimension reduction in this case (cf. [H 4,§22]). If 0 < R < ∞, then a similar argument gives a blow-up limit in a ball of finite radius; this limit has the structure of a non-flat metric cone. This is ruled out by Hamilton’s strong maximum principle for nonnegative curvature operator. Finally, if R = 0, then (in dimensions three and up) it is easy to see that the metric is flat. 11.5 Corollary. For every ǫ > 0 there exists A < ∞ with the following property. Suppose we have a sequence of ( not necessarily complete) solutions (gk )ij (t) with nonnegative curvature operator, defined on Mk × [tk , 0], such that for each k the ball B(xk , rk ) at time t = 0 is compactly contained in Mk , 1 R(x, t) ≤ R(xk , 0)√= Qk for all√(x, t), tk Qk → −∞, rk2 Qk → ∞ as k → ∞. 2 Then V olB(xk , A/ Qk ) ≤ ǫ(A/ Qk )n at t = 0 if k is large enough. Proof. Assuming the contrary, we may take a blow-up limit (at (xk , 0) with factors Qk ) and get a non-flat ancient solution with positive asymptotic volume ratio at t = 0, satisfying the assumptions in 11.1, except, may be, the κ-noncollapsing assumption. But if that assumption is violated for each κ > 0, then V(t) is not bounded away from zero as t → −∞. However, this is impossible, because it is easy to see that V(t) is nonincreasing in t. (Indeed, Ricci √ flow decreases the volume and does not decrease the distances faster than C R per time unit, by lemma 8.3(b)) Thus, κ-noncollapsing holds for some κ > 0, and we can apply the previous proposition to obtain a contradiction. 11.6 Corollary. For every w > 0 there exist B = B(w) < ∞, C = C(w) < ∞, τ0 = τ0 (w) > 0, with the following properties. (a) Suppose we have a (not necessarily complete) solution gij (t) to the Ricci flow, defined on M × [t0 , 0], so that at time t = 0 the metric ball B(x0 , r0 ) is compactly contained in M. Suppose that at each time t, t0 ≤ t ≤ 0, the metric gij (t) has nonnegative curvature operator, and V olB(x0 , r0 ) ≥ wr0n . Then we have an estimate R(x, t) ≤ Cr0−2 + B(t − t0 )−1 whenever distt (x, x0 ) ≤ 14 r0 . 29
(b) If, rather than assuming a lower bound on volume for all t, we assume it only for t = 0, then the same conclusion holds with −τ0 r02 in place of t0 , provided that −t0 ≥ τ0 r02 .
Proof. By scaling assume r0 = 1. (a) Arguing by contradiction, consider a sequence of B, C → ∞, of solutions gij (t) and points (x, t), such that distt (x, x0 ) ≤ 41 and R(x, t) > C + B(t − t0 )−1 . Then, arguing as in the proof of claims 1,2 in 10.1, we can find a point (¯ x, t¯), satisfying distt¯(¯ x, x0 ) < 1 −1 ′ ′ ¯ ¯ , Q = R(¯ x, t) > C + B(t − t0 ) , and such that R(x , t ) ≤ 2Q whenever 3 1 −1 ¯ t − AQ ≤ t′ ≤ t¯, distt¯(x′ , x¯) < AQ− 2 , where A tends to infinity with B, C. Applying the previous corollary at (¯ x, t¯) and using the relative volume comparison, we get a contradiction with the assumption involving w. (b) Let B(w), C(w) be good for (a). We claim that B = B(5−n w), C = C(5−n w) are good for (b) , for an appropriate τ0 (w) > 0. Indeed, let gij (t) be a solution with nonnegative curvature operator, such that V olB(x0 , 1) ≥ w at t = 0, and let [−τ, 0] be the maximal time interval, where the assumption of (a) still holds, with 5−n w in place of w and with −τ in place of t0 . Then at time t = −τ we must have V olB(x0 , 1) ≤ 5−n w. On the other hand, from lemma 8.3 (b) we see that the ball B(x0 , 41 ) at time t = −τ contains the ball √ √ B(x0 , 14 − 10(n − 1)(τ C + 2 Bτ )) at time t = 0, and the volume of the former is at least as large as the volume of the latter. Thus, it is enough to choose τ0 = τ0 (w) in such a way that the radius of the latter ball is > 51 . Clearly, the proof also works if instead of assuming that curvature operator is nonnegative, we assumed that it is bounded below by −r0−2 in the (time-dependent) metric ball of radius r0 , centered at x0 . 11.7. From now on we restrict our attention to oriented manifolds of dimension three. Under the assumptions in 11.1, the solutions on closed manifolds must be quotients of the round S3 or S2 × R - this is proved in the same way as in two dimensions, since the gradient shrinking solitons are known from the work of Hamilton [H 1,10]. The noncompact solutions are described below. Theorem.The set of non-compact ancient solutions , satisfying the assumptions of 11.1, is compact modulo scaling. That is , from any sequence of such solutions and points (xk , 0) with R(xk , 0) = 1, we can extract a smoothly converging subsequence, and the limit satisfies the same conditions. Proof. To ensure a converging subsequence it is enough to show that whenever R(yk , 0) → ∞, the distances at t = 0 between xk and yk go to infinity as well. Assume the contrary. Define a sequence zk by the requirement 30
that zk be the closest point to xk (at t = 0), satisfying R(zk , 0)dist20 (xk , zk ) = 1. 1 We claim that R(z, 0)/R(zk , 0) is uniformly bounded for z ∈ B(zk , 2R(zk , 0)− 2 ). Indeed, otherwise we could show, using 11.5 and relative volume comparison 1 in nonnegative curvature, that the balls B(zk , R(zk , 0)− 2 ) are collapsing on the scale of their radii. Therefore, using the local derivative estimate, due to W.-X.Shi (see [H 4,§13]), we get a bound on Rt (zk , t) of the order of R2 (zk , 0). Then we can compare 1 = R(xk , 0) ≥ cR(zk , −cR−1 (zk , 0)) ≥ cR(zk , 0) for some small c > 0, where the first inequality comes from the Harnack inequality, obtained by integrating (11.1). Thus, R(zk , 0) are bounded. But now the existence of the sequence yk at bounded distance from xk implies, via 11.5 and relative volume comparison, that balls B(xk , c) are collapsing - a contradiction. It remains to show that the limit has bounded curvature at t = 0. If this was not the case, then we could find a sequence yi going to infinity, such 1 that R(yi , 0) → ∞ and R(y, 0) ≤ 2R(yi, 0) for y ∈ B(yi, Ai R(yi , 0)− 2 ), Ai → ∞. Then the limit of scalings at (yi, 0) with factors R(yi, 0) satisfies the assumptions in 11.1 and splits off a line. Thus by 11.3 it must be a round infinite cylinder. It follows that for large i each yi is contained in a round 1 cylindrical ”neck” of radius ( 12 R(yi , 0))− 2 → 0, - something that can not happen in an open manifold of nonnegative curvature. 11.8. Fix ǫ > 0. Let gij (t) be an ancient solution on a noncompact oriented three-manifold M, satisfying the assumptions in 11.1. We say that a point x0 ∈ M is the center of an ǫ-neck, if the solution gij (t) in the set {(x, t) : −(ǫQ)−1 < t ≤ 0, dist20 (x, x0 ) < (ǫQ)−1 }, where Q = R(x0 , 0), is, after scaling with factor Q, ǫ-close (in some fixed smooth topology) to the corresponding subset of the evolving round cylinder, having scalar curvature one at t = 0. Corollary (from theorem 11.7 and its proof) For any ǫ > 0 there exists C = C(ǫ, κ) > 0, such that if gij (t) satisfies the assumptions in 11.1, and Mǫ denotes the set of points in M, which are not centers of ǫ-necks, then 1 Mǫ is compact and moreover, diamMǫ ≤ CQ− 2 , and C −1 Q ≤ R(x, 0) ≤ CQ whenever x ∈ Mǫ , where Q = R(x0 , 0) for some x0 ∈ ∂Mǫ . 11.9 Remark. It can be shown that there exists κ0 > 0, such that if an ancient solution on a noncompact three-manifold satisfies the assumptions in 11.1 with some κ > 0, then it would satisfy these assumptions with κ = κ0 . This follows from the arguments in 7.3, 11.2, and the statement (which is not hard to prove) that there are no noncompact three-dimensional gradient 31
shrinking solitons, satisfying 11.1, other than the round cylinder and its Z2 quotients. Furthermore, I believe that there is only one (up to scaling) noncompact three-dimensional κ-noncollapsed ancient solution with bounded positive curvature - the rotationally symmetric gradient steady soliton, studied by R.Bryant. In this direction, I have a plausible, but not quite rigorous argument, showing that any such ancient solution can be made eternal, that is, can be extended for t ∈ (−∞, +∞); also I can prove uniqueness in the class of gradient steady solitons. 11.10* The earlier work on ancient solutions and all that can be found in [H 4, §16 − 22, 25, 26].
12
Almost nonnegative curvature in dimension three
12.1 Let φ be a decreasing function of one variable, tending to zero at infinity. A solution to the Ricci flow is said to have φ-almost nonnegative curvature if it satisfies Rm(x, t) ≥ −φ(R(x, t))R(x, t) for each (x, t). Theorem. Given ǫ > 0, κ > 0 and a function φ as above, one can find r0 > 0 with the following property. If gij (t), 0 ≤ t ≤ T is a solution to the Ricci flow on a closed three-manifold M, which has φ-almost nonnegative curvature and is κ-noncollapsed on scales < r0 , then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) : dist2t0 (x, x0 ) < (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is , after scaling by the factor Q, ǫ-close to the corresponding subset of some ancient solution, satisfying the assumptions in 11.1. Proof. An argument by contradiction. Take a sequence of r0 converging to zero, and consider the solutions gij (t), such that the conclusion does not hold for some (x0 , t0 ); moreover, by tampering with the condition t0 ≥ 1 a little bit, choose among all such (x0 , t0 ), in the solution under consideration, the one with nearly the smallest curvature Q. (More precisely, we can choose (x0 , t0 ) in such a way that the conclusion of the theorem holds for all (x, t), satisfying R(x, t) > 2Q, t0 − HQ−1 ≤ t ≤ t0 , where H → ∞ as r0 → 0) Our goal is to show that the sequence of blow-ups of such solutions at such points with factors Q would converge, along some subsequence of r0 → 0, to an ancient solution, satisfying 11.1.
32
¯ Claim 1. For each (¯ x, t¯) with t0 − HQ−1 ≤ t¯ ≤ t0 we have R(x, t) ≤ 4Q 1 ¯ = Q + R(¯ ¯ −1 ≤ t ≤ t¯ and distt¯(x, x¯) ≤ cQ ¯ − 2 , where Q x, t¯) whenever t¯ − cQ and c = c(κ) > 0 is a small constant. Proof of Claim 1. Use the fact ( following from the choice of (x0 , t0 ) and the description of the ancient solutions) that for each (x, t) with R(x, t) > 2Q and t0 − HQ−1 ≤ t ≤ t0 we have the estimates |Rt (x, t)| ≤ CR2 (x, t), 3 |∇R|(x, t) ≤ CR 2 (x, t). Claim 2. There exists c = c(κ) > 0 and for any A > 0 there exist D = D(A) < ∞, ρ0 = ρ0 (A) > 0, with the following property. Suppose that r0 < ρ0 , and let γ be a shortest geodesic with endpoints x¯, x in gij (t¯), for some t¯ ∈ [t0 − HQ−1 , t0 ], such that R(y, ¯t) > 2Q for each y ∈ γ. Let z ∈ γ satisfy ¯ Then distt¯(¯ ¯ − 12 whenever R(x, t¯) ≥ DQ. ¯ cR(z, t¯) > R(¯ x, t¯) = Q. x, z) ≥ AQ Proof of Claim 2. Note that from the choice of (x0 , t0 ) and the description of the ancient solutions it follows that an appropriate parabolic (backward in time) neighborhood of a point y ∈ γ at t = t¯ is ǫ-close to the evolving round ¯ ≤ R(y, ¯t) ≤ cR(x, t¯) for an appropriate c = c(κ). cylinder, provided c−1 Q Now assume that the conclusion of the claim does not hold, take r0 to zero, ¯ R(x, t¯) - to infinity, and consider the scalings around (¯ x, t¯) with factors Q. We can imagine two possibilities for the behavior of the curvature along γ in the scaled metric: either it stays bounded at bounded distances from x¯, or not. In the first case we can take a limit (for a subsequence) of the scaled metrics along γ and get a nonnegatively curved almost cylindrical metric, with γ going to infinity. Clearly, in this case the curvature at any point of the limit does not exceed c−1 ; therefore, the point z must have escaped to infinity, and the conclusion of the claim stands. In the second case, we can also take a limit along γ; it is a smooth nonnegatively curved manifold near x¯ and has cylindrical shape where curvature is large; the radius of the cylinder goes to zero as we approach the (first) singular point, which is located at finite distance from x¯; the region beyond the first singular point will be ignored. Thus, at t = t¯ we have a metric, which is a smooth metric of nonnegative curvature away from a single singular point o. Since the metric is cylindrical at points close to o, and the radius of the cylinder is at most ǫ times the distance from o, the curvature at o is nonnegative in Aleksandrov sense. Thus, the metric near o must be cone-like. In other words, the scalings of our metric at points xi → o with factors R(xi , t¯) converge to a piece of nonnegatively curved non-flat metric cone. Moreover, using claim 1, we see that we actually have the convergence 33
of the solutions to the Ricci flow on some time interval, and not just metrics at t = t¯. Therefore, we get a contradiction with the strong maximum principle of Hamilton [H 2]. Now continue the proof of theorem, and recall that we are considering scalings at (x0 , t0 ) with factor Q. It follows from claim 2 that at t = t0 the curvature of the scaled metric is bounded at bounded distances from x0 . This allows us to extract a smooth limit at t = t0 (of course, we use the κ-noncollapsing assumption here). The limit has bounded nonnegative curvature (if the curvatures were unbounded, we would have a sequence of cylindrical necks with radii going to zero in a complete manifold of nonnegative curvature). Therefore, by claim 1, we have a limit not only at t = t0 , but also in some interval of times smaller than t0 . We want to show that the limit actually exists for all t < t0 . Assume that this is not the case, and let t′ be the smallest value of time, such that the blowup limit can be taken on (t′ , t0 ]. From the differential Harnack inequality of Hamilton [H 3] we have an estimate Rt (x, t) ≥ −R(x, t)(t−t′ )−1 , therefore, if ˜ denotes the maximum of scalar curvature at t = t0 , then R(x, t) ≤ Q ˜ t0 −t′′ . Q t−t Hence by lemma 8.3(b) dist (x, y) ≤ dist (x, y) + C for all t, where C = t t 0 q ˜ 10n(t0 − t′ ) Q. The next step is needed only if our limit is noncompact. In this case there exists D > 0, such that for any y satisfying d = distt0 (x0 , y) > D, one can find x satisfying distt0 (x, y) = d, distt0 (x, x0 ) > 32 d. We claim that the scalar curvature R(y, t) is uniformly bounded for all such y and all t ∈ (t′ , t0 ]. Indeed, if R(y, t) is large, then the neighborhood of (y, t) is like in an ancient solution; therefore, (long) shortest geodesics γ and γ0 , connecting at time t the point y to x and x0 respectively, make the angle close to 0 or π at y; the former case is ruled out by the assumptions on distances, if D > 10C; in the latter case, x and x0 are separated at time t by a small neighborhood of y, 1 with diameter of order R(y, t)− 2 , hence the same must be true at time t0 , which is impossible if R(y, t) is too large. Thus we have a uniform bound on curvature outside a certain compact set, which has uniformly bounded diameter for all t ∈ (t′ , t0 ]. Then claim 2 gives a uniform bound on curvature everywhere. Hence, by claim 1, we can extend our blow-up limit past t′ - a contradiction. 12.2 Theorem. Given a function φ as above, for any A > 0 there exists K = K(A) < ∞ with the following property. Suppose in dimension three we have a solution to the Ricci flow with φ-almost nonnegative curvature, which 34
satisfies the assumptions of theorem 8.2 with r0 = 1. Then R(x, 1) ≤ K whenever dist1 (x, x0 ) < A. Proof. In the first step of the proof we check the following Claim. There exists K = K(A) < ∞, such that a point (x, 1) satisfies the conclusion of the previous theorem 12.1 (for some fixed small ǫ > 0), whenever R(x, 1) > K and dist1 (x, x0 ) < A. The proof of this statement essentially repeats the proof of the previous theorem (the κ-noncollapsing assumption is ensured by theorem 8.2). The only difference is in the beginning. So let us argue by contradiction, and suppose we have a sequence of solutions and points x with dist1 (x, x0 ) < A and R(x, 1) → ∞, which do not satisfy the conclusion of 12.1. Then an argument, similar to the one proving claims 1,2 in 10.1, delivers points (¯ x, t¯) 1 x, x0 ) < 2A, with Q = R(¯ x, t¯) → ∞, and such that with 2 ≤ t¯ ≤ 1, distt¯(¯ (x, t) satisfies the conclusion of 12.1 whenever R(x, t) > 2Q, t¯ − DQ−1 ≤ t ≤ 1 t¯, distt¯(¯ x, x) < DQ− 2 , where D → ∞. (There is a little subtlety here in the application of lemma 8.3(b); nevertheless, it works, since we need to apply it only when the endpoint other than x0 either satisfies the conclusion of 12.1, or has scalar curvature at most 2Q) After such (¯ x, t¯) are found, the proof of 12.1 applies. Now, having checked the claim, we can prove the theorem by applying the claim 2 of the previous theorem to the appropriate segment of the shortest geodesic, connecting x and x0 . 12.3 Theorem. For any w > 0 there exist τ = τ (w) > 0, K = K(w) < ∞, ρ = ρ(w) > 0 with the following property. Suppose we have a solution gij (t) to the Ricci flow, defined on M × [0, T ), where M is a closed threemanifold, and a point (x0 , t0 ), such that the ball B(x0 , r0 ) at t = t0 has volume ≥ wr0n , and sectional curvatures ≥ −r0−2 at each point. Suppose that gij (t) is φ-almost nonnegatively curved for some function φ as above. Then we have an estimate R(x, t) < Kr0−2 whenever t0 ≥ 4τ r02 , t ∈ [t0 − τ r02 , t0 ], distt (x, x0 ) ≤ 14 r0 , provided that φ(r0−2 ) < ρ. Proof. If we knew that sectional curvatures are ≥ −r0−2 for all t, then we could just apply corollary 11.6(b) (with the remark after its proof) and take τ (w) = τ0 (w)/2, K(w) = C(w) + 2B(w)/τ0 (w). Now fix these values of τ, K, consider a φ-almost nonnegatively curved solution gij (t), a point (x0 , t0 ) and a radius r0 > 0, such that the assumptions of the theorem do hold whereas the conclusion does not. We may assume that any other point (x′ , t′ ) and radius r ′ > 0 with that property has either t′ > t0 or t′ < t0 − 2τ r02 , or 35
2r ′ > r0 . Our goal is to show that φ(r0−2) is bounded away from zero. Let τ ′ > 0 be the largest time interval such that Rm(x, t) ≥ −r0−2 whenever t ∈ [t0 − τ ′ r02 , t0 ], distt (x, x0 ) ≤ r0 . If τ ′ ≥ 2τ, we are done by corollary 11.6(b). Otherwise, by elementary Aleksandrov space theory, we can find at time t′ = t0 − τ ′ r02 a ball B(x′ , r ′ ) ⊂ B(x0 , r0 ) with V olB(x′ , r ′ ) ≥ 12 ωn (r ′ )n , and with radius r ′ ≥ cr0 for some small constant c = c(w) > 0. By the choice of (x0 , t0 ) and r0 , the conclusion of our theorem holds for (x′ , t′ ), r ′ . Thus we have an estimate R(x, t) ≤ K(r ′ )−2 whenever t ∈ [t′ − τ (r ′ )2 , t′ ], distt (x, x′ ) ≤ 1 ′ r . Now we can apply the previous theorem (or rather its scaled version) and 4 get an estimate on R(x, t) whenever t ∈ [t′ − 21 τ (r ′ )2 , t′ ], distt (x′ , x) ≤ 10r0 . Therefore, if r0 > 0 is small enough, we have Rm(x, t) ≥ −r0−2 for those (x, t), which is a contradiction to the choice of τ ′ . 12.4 Corollary (from 12.2 and 12.3) Given a function φ as above, for any w > 0 one can find ρ > 0 such that if gij (t) is a φ-almost nonnegatively curved solution to the Ricci flow, defined on M × [0, T ), where M is a closed three-manifold, and if B(x0 , r0 ) is a metric ball at time t0 ≥ 1, with r0 < ρ, and such that min Rm(x, t0 ) over x ∈ B(x0 , r0 ) is equal to −r0−2 , then V olB(x0 , r0 ) ≤ wr0n .
13
The global picture of the Ricci flow in dimension three
13.1 Let gij (t) be a smooth solution to the Ricci flow on M × [1, ∞), where M is a closed oriented three-manifold. Then, according to [H 6, theorem 4.1], ˜ the normalized curvatures Rm(x, t) = tRm(x, t) satisfy an estimate of the ˜ ˜ ˜ t), where φ behaves at infinity as 1 . This form Rm(x, t) ≥ −φ(R(x, t))R(x, log estimate allows us to apply the results 12.3,12.4, and obtain the following Theorem. For any w > 0 there exist K = K(w) < ∞, ρ = ρ(w) > 0, such that for sufficiently large S times t the manifold M admits a thick-thin decomposition M = Mthick Mthin with the following properties. (a) √For ˜ ≤ K in the ball B(x, ρ(w) t). every x ∈ Mthick we have an estimate |Rm| √ 1 and the volume of this ball is at least 10 w(ρ(w) t)n . (b) For every y ∈ Mthin √ there exists r = r(y), 0 < r < ρ(w) t, such that for all points in the ball B(y, r) we have Rm ≥ −r −2 , and the volume of this ball is < wrn . Now the arguments in [H 6] show that either Mthick is empty for large t, or , for an appropriate sequence of t → 0 and w → 0, it converges to 36
a (possibly, disconnected) complete hyperbolic manifold of finite volume, whose cusps (if there are any) are incompressible in M. On the other hand, collapsing with lower curvature bound in dimension three is understood well enough to claim that, for sufficiently small w > 0, Mthin is homeomorphic to a graph manifold. The natural questions that remain open are whether the normalized curvatures must stay bounded as t → ∞, and whether reducible manifolds and manifolds with finite fundamental group can have metrics which evolve smoothly by the Ricci flow on the infinite time interval. 13.2 Now suppose that gij (t) is defined on M × [1, T ), T < ∞, and goes singular as t → T. Then using 12.1 we see that, as t → T, either the curvature goes to infinity everywhere, and then M is a quotient of either S3 or S2 × R, or the region of high curvature in gij (t) is the union of several necks and capped necks, which in the limit turn into horns (the horns most likely have finite diameter, but at the moment I don’t have a proof of that). Then at the time T we can replace the tips of the horns by smooth caps and continue running the Ricci flow until the solution goes singular for the next time, e.t.c. It turns out that those tips can be chosen in such a way that the need for the surgery will arise only finite number of times on every finite time interval. The proof of this is in the same spirit, as our proof of 12.1; it is technically quite complicated, but requires no essentially new ideas. It is likely that by passing to the limit in this construction one would get a canonically defined Ricci flow through singularities, but at the moment I don’t have a proof of that. (The positive answer to the conjecture in 11.9 on the uniqueness of ancient solutions would help here) Moreover, it can be shown, using an argument based on 12.2, that every maximal horn at any time T, when the solution goes singular, has volume at least cT n ; this easily implies that the solution is smooth (if nonempty) from some finite time on. Thus the topology of the original manifold can be reconstructed as a connected sum of manifolds, admitting a thick-thin decomposition as in 13.1, and quotients of S3 and S2 × R. 13.3* Another differential-geometric approach to the geometrization conjecture is being developed by Anderson [A]; he studies the elliptic equations, arising as Euler-Lagrange equations for certain functionals of the riemannian metric, perturbing the total scalar curvature functional, and one can observe certain parallelism between his work and that of Hamilton, especially taking into account that, as we have shown in 1.1, Ricci flow is the gradient flow for a functional, that closely resembles the total scalar curvature. 37
References [A] M.T.Anderson Scalar curvature and geometrization conjecture for three-manifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997), 49-82. [B-Em] D.Bakry, M.Emery Diffusions hypercontractives. Seminaire de Probabilites XIX, 1983-84, Lecture Notes in Math. 1123 (1985), 177-206. [Cao-C] H.-D. Cao, B.Chow Recent developments on the Ricci flow. Bull. AMS 36 (1999), 59-74. [Ch-Co] J.Cheeger, T.H.Colding On the structure of spaces with Ricci curvature bounded below I. Jour. Diff. Geom. 46 (1997), 406-480. [C] B.Chow Entropy estimate for Ricci flow on compact two-orbifolds. Jour. Diff. Geom. 33 (1991), 597-600. [C-Chu 1] B.Chow, S.-C. Chu A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. Math. Res. Let. 2 (1995), 701-718. [C-Chu 2] B.Chow, S.-C. Chu A geometric approach to the linear trace Harnack inequality for the Ricci flow. Math. Res. Let. 3 (1996), 549-568. [D] E.D’Hoker String theory. Quantum fields and strings: a course for mathematicians (Princeton, 1996-97), 807-1011. [E 1] K.Ecker Logarithmic Sobolev inequalities on submanifolds of euclidean space. Jour. Reine Angew. Mat. 522 (2000), 105-118. [E 2] K.Ecker A local monotonicity formula for mean curvature flow. Ann. Math. 154 (2001), 503-525. [E-Hu] K.Ecker, G.Huisken In terior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), 547-569. [Gaw] K.Gawedzki Lectures on conformal field theory. Quantum fields and strings: a course for mathematicians (Princeton, 1996-97), 727-805. [G] L.Gross Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992) Lecture Notes in Math. 1563 (1993), 54-88. [H 1] R.S.Hamilton Three manifolds with positive Ricci curvature. Jour. Diff. Geom. 17 (1982), 255-306. [H 2] R.S.Hamilton Four manifolds with positive curvature operator. Jour. Diff. Geom. 24 (1986), 153-179. [H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff. Geom. 37 (1993), 225-243. [H 4] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys in Diff. Geom. 2 (1995), 7-136. 38
[H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Commun. Anal. Geom. 5 (1997), 1-92. [H 6] R.S.Hamilton Non-singular solutions of the Ricci flow on threemanifolds. Commun. Anal. Geom. 7 (1999), 695-729. [H 7] R.S.Hamilton A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1 (1993), 113-126. [H 8] R.S.Hamilton Monotonicity formulas for parabolic flows on manifolds. Commun. Anal. Geom. 1 (1993), 127-137. [H 9] R.S.Hamilton A compactness property for solutions of the Ricci flow. Amer. Jour. Math. 117 (1995), 545-572. [H 10] R.S.Hamilton The Ricci flow on surfaces. Contemp. Math. 71 (1988), 237-261. [Hu] G.Huisken Asymptotic behavior for singularities of the mean curvature flow. Jour. Diff. Geom. 31 (1990), 285-299. [I] T.Ivey Ricci solitons on compact three-manifolds. Diff. Geo. Appl. 3 (1993), 301-307. [L-Y] P.Li, S.-T. Yau On the parabolic kernel of the Schrodinger operator. Acta Math. 156 (1986), 153-201. [Lott] J.Lott Some geometric properties of the Bakry-Emery-Ricci tensor. arXiv:math.DG/0211065.
39
arXiv:math/0303109v1 [math.DG] 10 Mar 2003
Ricci flow with surgery on three-manifolds Grisha Perelman∗ February 1, 2008 This is a technical paper, which is a continuation of [I]. Here we verify most of the assertions, made in [I, §13]; the exceptions are (1) the statement that a 3-manifold which collapses with local lower bound for sectional curvature is a graph manifold - this is deferred to a separate paper, as the proof has nothing to do with the Ricci flow, and (2) the claim about the lower bound for the volumes of the maximal horns and the smoothness of the solution from some time on, which turned out to be unjustified, and, on the other hand, irrelevant for the other conclusions. The Ricci flow with surgery was considered by Hamilton [H 5,§4,5]; unfortunately, his argument, as written, contains an unjustified statement (RMAX = Γ, on page 62, lines 7-10 from the bottom), which I was unable to fix. Our approach is somewhat different, and is aimed at eventually constructing a canonical Ricci flow, defined on a largest possible subset of space-time, - a goal, that has not been achieved yet in the present work. For this reason, we consider two scale bounds: the cutoff radius h, which is the radius of the necks, where the surgeries are performed, and the much larger radius r, such that the solution on the scales less than r has standard geometry. The point is to make h arbitrarily small while keeping r bounded away from zero.
Notation and terminology B(x, t, r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P (x, t, r, △t) denotes a parabolic neighborhood, that is the set of all points (x′ , t′ ) with x′ ∈ B(x, t, r) and t′ ∈ [t, t + △t] or t′ ∈ [t + △t, t], depending on the sign of △t. A ball B(x, t, ǫ−1 r) is called an ǫ-neck, if, after scaling the metric with factor −2 r , it is ǫ-close to the standard neck S2 × I, with the product metric, where S2 has constant scalar curvature one, and I has length 2ǫ−1 ; here ǫ-close refers to C N topology, with N > ǫ−1 . A parabolic neighborhood P (x, t, ǫ−1 r, r2 ) is called a strong ǫ-neck, if, after scaling with factor r−2 , it is ǫ-close to the evolving standard neck, which at each ∗ St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email:
[email protected] or
[email protected]
1
time t′ ∈ [−1, 0] has length 2ǫ−1 and scalar curvature (1 − t′ )−1 . A metric on S2 × I, such that each point is contained in some ǫ-neck, is called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, and tends to infinity on both ends, respectively. ¯ 3 , such that each point outside some compact A metric on B3 or RP3 \ B subset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively. We denote by ǫ a fixed small positive constant. In contrast, δ denotes a positive quantity, which is supposed to be as small as needed in each particular argument.
1
Ancient solutions with bounded entropy
1.1 In this section we review some of the results, proved or quoted in [I,§11], correcting a few inaccuracies. We consider smooth solutions gij (t) to the Ricci flow on oriented 3-manifold M , defined for −∞ < t ≤ 0, such that for each t the metric gij (t) is a complete non-flat metric of bounded nonnegative sectional curvature, κ-noncollapsed on all scales for some fixed κ > 0; such solutions will be called ancient κ-solutions for short. By Theorem I.11.7, the set of all such solutions with fixed κ is compact modulo scaling, that is from any sequence α of such solutions (M α , gij (t)) and points (xα , 0) with R(xα , 0) = 1, we can extract a smoothly (pointed) convergent subsequence, and the limit (M, gij (t)) belongs to the same class of solutions. (The assumption in I.11.7. that M α be noncompact was clearly redundant, as it was not used in the proof. Note also that M need not have the same topology as M α .) Moreover, according to Proposition I.11.2, the scalings of any ancient κ-solution gij (t) with factors (−t)−1 about appropriate points converge along a subsequence of t → −∞ to a non-flat gradient shrinking soliton, which will be called an asymptotic soliton of the ancient solution. If the sectional curvature of this asymptotic soliton is not strictly positive, then by Hamilton’s strong maximum principle it admits local metric splitting, and it is easy to see that in this case the soliton is either the round infinite cylinder, or its Z2 quotient, containing one-sided projective plane. If the curvature is strictly positive and the soliton is compact, then it has to be a metric quotient of the round 3-sphere, by [H 1]. The noncompact case is ruled out below. 1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature. Proof. A gradient shrinking soliton gij (t), −∞ < t < 0, satisfies the equation 1 gij = 0 2t Differentiating and switching the order of differentiation, we get ∇i ∇j f + Rij +
∇i R = 2Rij ∇j f 2
(1.1)
(1.2)
Fix some t < 0, say t = −1, and consider a long shortest geodesic γ(s), 0 ≤ s ≤ s¯; let x = γ(0), x ¯ = γ(¯ s), X(s) = γ(s). ˙ Since the curvature is bounded and R s¯ positive, it is clear from the second variation formula that 0 Ric(X, X)ds ≤ R s¯ R s¯ √ const. Therefore, 0 |Ric(X, ·)|2 ds ≤ const, and 0 |Ric(X, Y )|ds ≤ const( s¯ + 1) for any unit vector field Y along γ, orthogonal to X.√Thus by integrating s))| ≤ const( s¯ + 1). We conclude (1.1) we get X · f (γ(¯ s)) ≥ 2s¯ + const, |Y · f (γ(¯ that at large distances from x0 the function f has no critical points, and its gradient makes small angle with the gradient of the distance function from x0 . Now from (1.2) we see that R is increasing along the gradient curves of f, ¯ = lim sup R > 0. If we take a limit of our soliton about points in particular, R α ¯ then we get an ancient κ-solution, which splits (x , −1) where R(xα ) → R, off a line, and it follows from I.11.3, that this solution is the shrinking round ¯ at time t = −1. Now comparing the infinite cylinder with scalar curvature R evolution equations for the scalar curvature on a round cylinder and for the ¯ = 1. asymptotic scalar curvature on a shrinking soliton we conclude that R Hence, R(x) < 1 when the distance from x to x0 is large enough, and R(x) → 1 when this distance tends to infinity. Now let us check that the level surfaces of f, sufficiently distant from x0 , are convex. Indeed, if Y is a unit tangent vector to such a surface, then ∇Y ∇Y f = 1 R 1 2 − Ric(Y, Y ) ≥ 2 − 2 > 0. Therefore, the area of the level surfaces grows as f increases, and is converging to the area of the round sphere of scalar curvature one. On the other hand, the intrinsic scalar curvature of a level surface turns out to be less than one. Indeed, denoting by X the unit normal vector, this intrinsic curvature can be computed as R − 2Ric(X, X) + 2
det(Hessf ) (1 − R + Ric(X, X))2 ≤ R − 2Ric(X, X) + <1 |∇f |2 2|∇f |2
when R is close to one and |∇f | is large. Thus we get a contradiction to the Gauss-Bonnet formula. 1.3 Now, having listed all the asymptotic solitons, we can classify the ancient κ-solutions. If such a solution has a compact asymptotic soliton, then it is itself a metric quotient of the round 3-sphere, because the positive curvature pinching can only improve in time [H 1]. If the asymptotic soliton contains the one-sided projective plane, then the solution has a Z2 cover, whose asymptotic soliton is the round infinite cylinder. Finally, if the asymptotic soliton is the cylinder,then the solution can be either noncompact (the round cylinder itself, or the Bryant soliton, for instance), or compact. The latter possibility, which was overlooked in the first paragraph of [I.11.7], is illustrated by the example below, which also gives the negative answer to the question in the very end of [I.5.1]. 1.4 Example. Consider a solution to the Ricci flow, starting from a metric on S3 that looks like a long round cylinder S2 × I (say, with radius one and length L >> 1), with two spherical caps, smoothly attached to its boundary components. By [H 1] we know that the flow shrinks such a metric to a point in time, comparable to one (because both the lower bound for scalar curvature and the upper bound for sectional curvature are comparable to one) , and after 3
normalization, the flow converges to the round 3-sphere. Scale the initial metric and choose the time parameter in such a way that the flow starts at time t0 = t0 (L) < 0, goes singular at t = 0, and at t = −1 has the ratio of the maximal sectional curvature to the minimal one equal to 1 + ǫ. The argument in [I.7.3] shows that our solutions are κ-noncollapsed for some κ > 0 independent of L. We also claim that t0 (L) → −∞ as L → ∞. Indeed, the Harnack inequality of 0) for t ≤ −1, Hamilton [H 3] implies that Rt ≥ t0R−t , hence R ≤ 2(−1−t t−t0 p d Rmax (t) dist (x, y) ≥ −const and then the distance change estimate t dt from [H 2,§17] implies that the diameter of g (t ) does not exceed −const · t0 , ij 0 √ which is less than L −t0 unless t0 is large enough. Thus, a subsequence of our solutions with L → ∞ converges to an ancient κ-solution on S3 , whose asymptotic soliton can not be anything but the cylinder. 1.5 The important conclusion from the classification above and the proof of Proposition I.11.2 is that there exists κ0 > 0, such that every ancient κ-solution is either κ0 -solution, or a metric quotient of the round sphere. Therefore, the compactness theorem I.11.7 implies the existence of a universal constant η, such that at each point of every ancient κ-solution we have estimates 3
|∇R| < ηR 2 , |Rt | < ηR2
(1.3)
Moreover, for every sufficiently small ǫ > 0 one can find C1,2 = C1,2 (ǫ), such that for each point (x, t) in every ancient κ-solution there is a radius r, 0 < r < 1 C1 R(x, t)− 2 , and a neighborhood B, B(x, t, r) ⊂ B ⊂ B(x, t, 2r), which falls into one of the four categories: (a) B is a strong ǫ-neck (more precisely, the slice of a strong ǫ-neck at its maximal time), or (b) B is an ǫ-cap, or (c) B is a closed manifold, diffeomorphic to S3 or RP3 , or (d) B is a closed manifold of constant positive sectional curvature; furthermore, the scalar curvature in B at time t is between C2−1 R(x, t) and 3 C2 R(x, t), its volume in cases (a),(b),(c) is greater than C2−1 R(x, t)− 2 , and in −1 case (c) the sectional curvature in B at time t is greater than C2 R(x, t).
2
The standard solution
Consider a rotationally symmetric metric on R3 with nonnegative sectional curvature, which splits at infinity as the metric product of a ray and the round 2-sphere of scalar curvature one. At this point we make some choice for the metric on the cap, and will refer to it as the standard cap; unfortunately, the most obvious choice, the round hemisphere, does not fit, because the metric on R3 would not be smooth enough, however we can make our choice as close to it as we like. Take such a metric on R3 as the initial data for a solution gij (t) to the Ricci flow on some time interval [0, T ), which has bounded curvature for each t ∈ [0, T ). Claim 1. The solution is rotationally symmetric for all t. 4
Indeed, if ui is a vector field evolving by uit = △ui + Rji uj , then vij = ∇i uj evolves by (vij )t = △vij + 2Rikjl vkl − Rik vkj − Rkj vik . Therefore, if ui was a Killing field at time zero, it would stay Killing by the maximum principle. It is also clear that the center of the cap, that is the unique maximum point for the Busemann function, and the unique point, where all the Killing fields vanish, retains these properties, and the gradient of the distance function from this point stays orthogonal to all the Killing fields. Thus, the rotational symmetry is preserved. Claim 2. The solution converges at infinity to the standard solution on the round infinite cylinder of scalar curvature one. In particular, T ≤ 1. Claim 3. The solution is unique. Indeed, using Claim 1, we can reduce the linearized Ricci flow equation to the system of two equations on (−∞, +∞) of the following type ft = f ′′ + a1 f ′ + b1 g ′ + c1 f + d1 g, gt = a2 f ′ + b2 g ′ + c2 f + d2 g, where the coefficients and their derivatives are bounded, and the unknowns f, g and their derivatives tend to zero at infinity by Claim 2. So we get uniqueness RA by looking at the integrals −A (f 2 + g 2 ) as A → ∞. Claim 4. The solution can be extended to the time interval [0, 1). Indeed, we can obtain our solution as a limit of the solutions on S3 , starting from the round cylinder S2 × I of length L and scalar curvature one, with two caps attached; the limit is taken about the center p of one of the caps, L → ∞. Assume that our solution goes singular at some time T < 1. Take T1 < T very close to T, T − T1 << 1 − T. By Claim 2, given δ > 0, we can find ¯ D ¯ < ∞, depending on δ and T1 , such that for any point x at distance D ¯ L, ¯ from p at time zero, in the solution with L ≥ L, the ball B(x, T1 , 1) is δ-close to the corresponding ball in the round cylinder of scalar curvature (1 − T1 )−1 . We can also find r = r(δ, T ), independent of T1 , such that the ball B(x, T1 , r) is δ-close to the corresponding euclidean ball. Now we can apply Theorem I.10.1 and get a uniform estimate on the curvature at x as t → T , provided that T − T1 < ǫ2 r(δ, T )2 . Therefore, the t → T limit of our limit solution on the capped infinite cylinder will be smooth near x. Thus, this limit will be a positively curved space with a conical point. However, this leads to a contradiction via a blow-up argument; see the end of the proof of the Claim 2 in I.12.1. The solution constructed above will be called the standard solution. Claim 5. The standard solution satisfies the conclusions of 1.5 , for an appropriate choice of ǫ, η, C1 (ǫ), C2 (ǫ), except that the ǫ-neck neighborhood need not be strong; more precisely, we claim that if (x, t) has neither an ǫ-cap neighborhood as in 1.5(b), nor a strong ǫ-neck neighborhood as in 1.5(a), then x is not in B(p, 0, ǫ−1 ), t < 3/4, and there is an ǫ-neck B(x, t, ǫ−1 r), such that the solution in P (x, t, ǫ−1 r, −t) is, after scaling with factor r−2 , ǫ-close to the appropriate piece of the evolving round infinite cylinder. Moreover, we have an estimate Rmin (t) ≥ const · (1 − t)−1 .
5
Indeed, the statements follow from compactness and Claim 2 on compact subintervals of [0, 1), and from the same arguments as for ancient solutions, when t is close to one.
3
The structure of solutions at the first singular time
Consider a smooth solution gij (t) to the Ricci flow on M × [0, T ), where M is a closed oriented 3-manifold, T < ∞. Assume that curvature of gij (t) does not stay bounded as t → T. Recall that we have a pinching estimate Rm ≥ −φ(R)R for some function φ decreasing to zero at infinity [H 4,§4], and that the solution is κ-noncollapsed on the scales ≤ r for some κ > 0, r > 0 [I, §4].Then by Theorem I.12.1 and the conclusions of 1.5 we can find r = r(ǫ) > 0, such that each point (x, t) with R(x, t) ≥ r−2 satisfies the estimates (1.3) and has a neighborhood, which is either an ǫ-neck, or an ǫ-cap, or a closed positively curved manifold. In the latter case the solution becomes extinct at time T, so we don’t need to consider it any more. If this case does not occur, then let Ω denote the set of all points in M, where curvature stays bounded as t → T. The estimates (1.3) imply that Ω is open and that R(x, t) → ∞ as t → T for each x ∈ M \Ω. If Ω is empty, then the solution becomes extinct at time T and it is entirely covered by ǫ-necks and caps shortly before that time, so it is easy to see that M is diffeomorphic to either S3 , or RP3 , or S2 × S1 , or RP3 ♯ RP3 . Otherwise, if Ω is not empty, we may (using the local derivative estimates due to W.-X.Shi, see [H 2,§13]) consider a smooth metric g¯ij on Ω, which is the limit of gij (t) as t → T. Let Ωρ for some ρ < r denotes the set of points x ∈ Ω, ¯ where the scalar curvature R(x) ≤ ρ−2 . We claim that Ωρ is compact. Indeed, if −2 ¯ R(x) ≤ ρ , then we can estimate the scalar curvature R(x, t) on [T − η −1 ρ2 , T ) using (1.3), and for earlier times by compactness, so x is contained in Ω with a ball of definite size, depending on ρ. Now take any ǫ-neck in (Ω, g¯ij ) and consider a point x on one of its boundary components. If x ∈ Ω\Ωρ , then there is either an ǫ-cap or an ǫ-neck, adjacent to the initial ǫ-neck. In the latter case we can take a point on the boundary of the second ǫ-neck and continue. This procedure can either terminate when we reach a point in Ωρ or an ǫ-cap, or go on indefinitely, producing an ǫ-horn. The same procedure can be repeated for the other boundary component of the initial ǫ-neck. Therefore, taking into account that Ω has no compact components, we conclude that each ǫ-neck of (Ω, g¯ij ) is contained in a subset of Ω of one of the following types: (a) An ǫ-tube with boundary components in Ωρ , or (b) An ǫ-cap with boundary in Ωρ , or (c) An ǫ-horn with boundary in Ωρ , or (d) A capped ǫ-horn, or (e) A double ǫ-horn.
6
Clearly, each ǫ-cap, disjoint from Ωρ , is also contained in one of the subsets above. It is also clear that there is a definite lower bound (depending on ρ) for the volume of subsets of types (a),(b),(c), so there can be only finite number of them. Thus we can conclude that there is only a finite number of components of Ω, containing points of Ωρ , and every such component has a finite number of ends, each being an ǫ-horn. On the other hand, every component of Ω, containing no points of Ωρ , is either a capped ǫ-horn, or a double ǫ-horn. Now, by looking at our solution for times t just before T, it is easy to see that the topology of M can be reconstructed as follows: take the components Ωj , 1 ≤ j ≤ i of Ω which contain points of Ωρ , truncate their ǫ-horns, and glue to the boundary components of truncated Ωj a collection of tubes S2 × I and caps ¯ j , 1 ≤ j ≤ i, B3 or RP3 \B3 . Thus, M is diffeomorphic to a connected sum of Ω 2 1 with a finite number of S × S (which correspond to gluing a tube to two ¯j boundary components of the same Ωj ), and a finite number of RP3 ; here Ω denotes Ωj with each ǫ-horn one point compactified.
4
Ricci flow with cutoff
4.1 Suppose we are given a collection of smooth solutions gij (t) to the Ricci flow, + + k defined on Mk × [t− ¯ij ) be the limits k , tk ), which go singular as t → tk . Let (Ωk , g + of the corresponding solutions as t → tk , as in the previous section. Suppose + k−1 k − also that for each k we have t− ¯ij ) and (Mk , gij (tk )) k = tk−1 , and (Ωk−1 , g contain compact (possibly disconnected) three-dimensional submanifolds with smooth boundary, which are isometric. Then we can identify these isometric submanifolds and talk about the solution to the Ricci flow with surgery on the + union of all [t− k , tk ). Fix a small number ǫ > 0 which is admissible in sections 1,2. In this section we consider only solutions to the Ricci flow with surgery, which satisfy the following a priori assumptions: (pinching) There exists a function φ, decreasing to zero at infinity, such that Rm ≥ −φ(R)R, (canonical neighborhood) There exists r > 0, such that every point where scalar curvature is at least r−2 has a neighborhood, satisfying the conclusions of 1.5. (In particular, this means that if in case (a) the neighborhood in question is B(x0 , t0 , ǫ−1 r0 ), then the solution is required to be defined in the whole P (x0 , t0 , ǫ−1 r0 , −r02 ); however, this does not rule out a surgery in the time interval (t0 − r02 , t0 ), that occurs sufficiently far from x0 .) Recall that from the pinching estimate of Ivey and Hamilton, and Theorem I.12.1, we know that the a priori assumptions above hold for a smooth solution on any finite time interval. For Ricci flow with surgery they will be justified in the next section. 4.2 Claim 1. Suppose we have a solution to the Ricci flow with surgery, satisfying the canonical neighborhood assumption, and let Q = R(x0 , t0 )+r−2 . Then 1 we have estimate R(x, t) ≤ 8Q for those (x, t) ∈ P (x0 , t0 , 21 η −1 Q− 2 , − 81 η −1 Q−1 ), for which the solution is defined. 7
Indeed, this follows from estimates (1.3). Claim 2. For any A < ∞ one can find Q = Q(A) < ∞ and ξ = ξ(A) > 0 with the following property. Suppose we have a solution to the Ricci flow with surgery, satisfying the pinching and the canonical neighborhood assumptions. Let γ be a shortest geodesic in gij (t0 ) with endpoints x0 and x, such that R(y, t0 ) > r−2 for each y ∈ γ, and Q0 = R(x0 , t0 ) is so large that φ(Q0 ) < ξ. Finally, let z ∈ γ be any point satisfying R(z, t0 ) > 10C2 R(x0 , t0 ). −1
Then distt0 (x0 , z) ≥ AQ0 2 whenever R(x, t0 ) > QQ0 . The proof is exactly the same as for Claim 2 in Theorem I.12.1; in the very end of it, when we get a piece of a non-flat metric cone as a blow-up limit, we get a contradiction to the canonical neighborhood assumption, because the canonical neighborhoods of types other than (a) are not close to a piece of metric cone, and type (a) is ruled out by the strong maximum principle, since the ǫ-neck in question is strong. 4.3 Suppose we have a solution to the Ricci flow with surgery, satisfying our a priori assumptions, defined on [0, T ), and going singular at time T. Choose a small δ > 0 and let ρ = δr. As in the previous section, consider the limit (Ω, g¯ij ) of our solution as t → T, and the corresponding compact set Ωρ . Lemma. There exists a radius h, 0 < h < δρ, depending only on δ, ρ and the ¯ − 12 (x) ≤ h in an ǫpinching function φ, such that for each point x with h(x) = R horn of (Ω, g¯ij ) with boundary in Ωρ , the neighborhood P (x, T, δ −1 h(x), −h2 (x)) is a strong δ-neck. Proof. An argument by contradiction. Assuming the contrary, take a seα quence of solutions with limit metrics (Ωα , g¯ij ) and points xα with h(xα ) → 0. α Since x lies deeply inside an ǫ-horn, its canonical neighborhood is a strong ǫ-neck. Now Claim 2 gives the curvature estimate that allows us to take a limit α of appropriate scalings of the metrics gij on [T − h2 (xα ), T ] about xα , for a subsequence of α → ∞. By shifting the time parameter we may assume that the limit is defined on [−1, 0]. Clearly, for each time in this interval, the limit is a complete manifold with nonnegative sectional curvature; moreover, since xα was α contained in an ǫ-horn with boundary in Ωα ρ , and h(x )/ρ → 0, this manifold has two ends. Thus, by Toponogov, it admits a metric splitting S2 × R. This implies that the canonical neighborhood of the point (xα , T − h2 (xα )) is also of type (a), that is a strong ǫ-neck, and we can repeat the procedure to get the limit, defined on [−2, 0], and so on. This argument works for the limit in any finite time interval [−A, 0], because h(xα )/ρ → 0. Therefore, we can construct a limit on [−∞, 0]; hence it is the round cylinder, and we get a contradiction. 4.4 Now we can specialize our surgery and define the Ricci flow with δ-cutoff. Fix δ > 0, compute ρ = δr and determine h from the lemma above. Given a smooth metric gij on a closed manifold, run the Ricci flow until it goes singular at some time t+ ; form the limit (Ω, g¯ij ). If Ωρ is empty, the procedure stops here, and we say that the solution became extinct. Otherwise we remove the components of Ω which contain no points of Ωρ , and in every ǫ-horn of each of the remaining components we find a δ-neck of radius h, cut it along the middle two-sphere, remove the horn-shaped end, and glue in an almost standard cap
8
in such a way that the curvature pinching is preserved and a metric ball of radius (δ ′ )−1 h centered near the center of the cap is, after scaling with factor h−2 , δ ′ -close to the corresponding ball in the standard capped infinite cylinder, considered in section 2. (Here δ ′ is a function of δ alone, which tends to zero with δ.) The possibility of capping a δ-neck preserving a certain pinching condition in dimension four was proved by Hamilton [H 5,§4]; his argument works in our case too (and the estimates are much easier to verify). The point is that we can change our δ-neck metric near the middle of the neck by a conformal factor e−f , where f = f (z) is positive on the part of the neck we want to remove, and zero on the part we want to preserve, and z is the coordinate along I in our parametrization S2 × I of the neck. Then, in the region near the middle of the neck, where f is small, the dominating terms in the formulas for the change of curvature are just positive constant multiples of f ′′ , so the pinching improves, and all the curvatures become positive on the set where f > δ ′ . Now we can continue our solution until it becomes singular for the next time. Note that after the surgery the manifold may become disconnected; in this case, each component should be dealt with separately. Furthermore, let us agree to declare extinct every component which is ǫ-close to a metric quotient of the round sphere; that allows to exclude such components from the list of canonical neighborhoods. Now since every surgery reduces the volume by at least h3 , the sequence of surgery times is discrete, and, taking for granted the a priori assumptions, we can continue our solution indefinitely, not ruling out the possibility that it may become extinct at some finite time. 4.5 In order to justify the canonical neighborhood assumption in the next section, we need to check several assertions. ¯ Lemma. For any A < ∞, 0 < θ < 1, one can find δ¯ = δ(A, θ) with the following property. Suppose we have a solution to the Ricci flow with δ-cutoff, ¯ Suppose we have a satisfying the a priori assumptions on [0, T ], with δ < δ. surgery at time T0 ∈ (0, T ), let p correspond to the center of the standard cap, and let T1 = min(T, T0 + θh2 ). Then either (a) The solution is defined on P (p, T0 , Ah, T1 − T0 ), and is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding subset on the standard solution from section 2, or (b) The assertion (a) holds with T1 replaced by some time t+ ∈ [T0 , T1 ), where + t is a surgery time; moreover, for each point in B(p, T0 , Ah), the solution is defined for t ∈ [T0 , t+ ) and is not defined past t+ . Proof. Let Q be the maximum of the scalar curvature on the standard solution in the time interval [0, θ], let △t = N −1 (T1 − T0 ) < ǫη −1 Q−1 h2 , and let tk = T0 + k△t, k = 0, ..., N. Assume first that for each point in B(p, T0 , A0 h), where A0 = ǫ(δ ′ )−1 , the solution is defined on [t0 , t1 ]. Then by (1.3) and the choice of △t we have a uniform curvature bound on this set for h−2 -scaled metric. Therefore we can define A1 , depending only on A0 and tending to infinity with A0 , such that the solution in P (p, T0 , A1 h, t1 − t0 ) is, after scaling and time shifting, A−1 1 close to the corresponding subset in the standard solution. In particular, the 9
scalar curvature on this subset does not exceed 2Qh−2 . Now if for each point in B(p, T0 , A1 h) the solution is defined on [t1 , t2 ], then we can repeat the procedure, defining A2 etc. Continuing this way, we eventually define AN , and it would remain to choose δ so small, and correspondingly A0 so large, that AN > A. Now assume that for some k, 0 ≤ k < N, and for some x ∈ B(p, T0 , Ak h) the solution is defined on [t0 , tk ] but not on [tk , tk+1 ]. Then we can find a surgery time t+ ∈ [tk , tk+1 ], such that the solution on B(p, T0 , Ak h) is defined on [t0 , t+ ), but for some points of this ball it is not defined past t+ . Clearly, the A−1 k+1 closeness assertion holds on P (p, T0 , Ak+1 h, t+ − T0 ). On the other hand, the solution on B(p, T0 , Ak h) is at least ǫ-close to the standard one for all t ∈ [tk , t+ ), hence no point of this set can be the center of a δ-neck neighborhood at time t+ . However, the surgery is always done along the middle two-sphere of such a neck. It follows that for each point of B(p, T0 , Ak h) the solution terminates at t+ . 4.6 Corollary. For any l < ∞ one can find A = A(l) < ∞ and θ = θ(l), 0 < θ < 1, with the following property. Suppose we are in the situation ¯ of the lemma above, with δ < δ(A, θ). Consider smooth curves γ in the set B(p, T0 , Ah), parametrized by t ∈ [T0 , Tγ ], such that γ(T0 ) ∈ B(p, T0 , Ah/2) and either Tγ = T1 < T , or Tγ < T1 and γ(Tγ ) ∈ ∂B(p, T0 , Ah). Then R Tγ 2 (R(γ(t), t) + |γ(t)| ˙ )dt > l. T0 Proof. Indeed, if Tγ = T1 , then on the standard solution we would have R Tγ Rθ −1 dt = −const · (log(1 − θ))−1 , so by choosing T0 R(γ(t), t)dt ≥ const 0 (1 − t) θ sufficiently close to one we can handle this case. Then we can choose A so large that on the standard solution distt (p, ∂B(p, 0, A)) ≥ 3A/4 for each t ∈ [0, θ]. RT 2 Now if γ(Tγ ) ∈ ∂B(p, T0 , Ah) then T0γ |γ(t)| ˙ dt ≥ A2 /100, so by taking A large enough, we can handle this case as well. 4.7 Corollary. For any Q < ∞ there exists θ = θ(Q), 0 < θ < 1 with the following property. Suppose we are in the situation of the lemma above, ¯ with δ < δ(A, θ), A > ǫ−1 . Suppose that for some point x ∈ B(p, T0 , Ah) the solution is defined at x (at least) on [T0 , Tx ], Tx ≤ T, and satisfies Q−1 R(x, t) ≤ R(x, Tx ) ≤ Q(Tx − T0 )−1 for all t ∈ [T0 , Tx ]. Then Tx ≤ T0 + θh2 . Proof. Indeed, if Tx > T0 + θh2 , then by lemma R(x, T0 + θh2 ) ≥ const · (1 − θ)−1 h−2 , whence R(x, Tx ) ≥ const · Q−1 (1 − θ)−1 h−2 , and Tx − T0 ≤ const · Q2 (1 − θ)h2 < θh2 if θ is close enough to one.
5
Justification of the a priori assumption
5.1 Let us call a riemannian manifold (M, gij ) normalized if M is a closed oriented 3-manifold, the sectional curvatures of gij do not exceed one in absolute value, and the volume of every metric ball of radius one is at least half the volume of the euclidean unit ball. For smooth Ricci flow with normalized initial data we have, by [H 4, 4.1], at any time t > 0 the pinching estimate Rm ≥ −φ(R(t + 1))R,
10
(5.1)
1 . As explained where φ is a decreasing function, which behaves at infinity like log in 4.4, this pinching estimate can be preserved for Ricci flow with δ-cutoff. Justification of the canonical neighborhood assumption requires additional arguments. In fact, we are able to construct solutions satisfying this assumption only allowing r and δ be functions of time rather than constants; clearly, the arguments of the previous section are valid in this case, if we assume that r(t), δ(t) are non-increasing, and bounded away from zero on every finite time interval. Proposition. There exist decreasing sequences 0 < rj < ǫ2 , κj > 0, 0 < ¯ δj < ǫ2 , j = 1, 2, ..., such that for any normalized initial data and any function δ(t), satisfying 0 < δ(t) < δ¯j for t ∈ [2j−1 ǫ, 2j ǫ], the Ricci flow with δ(t)-cutoff is defined for t ∈ [0, +∞] and satisfies the κj -noncollapsing assumption and the canonical neighborhood assumption with parameter rj on the time interval [2j−1 ǫ, 2j ǫ].( Recall that we have excluded from the list of canonical neighborhoods the closed manifolds, ǫ-close to metric quotients of the round sphere. Complete extinction of the solution in finite time is not ruled out.) The proof of the proposition is by induction: having constructed our sequences for 1 ≤ j ≤ i, we make one more step, defining ri+1 , κi+1 , δ¯i+1 , and redefining δ¯i = δ¯i+1 ; each step is analogous to the proof of Theorem I.12.1. First we need to check a κ-noncollapsing condition. 5.2 Lemma. Suppose we have constructed the sequences, satisfying the proposition for 1 ≤ j ≤ i. Then there exists κ > 0, such that for any r, 0 < r < ¯ > 0, which may also depend on the already constructed ǫ2 , one can find δ¯ = δ(r) sequences, with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on a time interval [0, T ], with normalized initial data, satisfying the proposition on [0, 2i ǫ], and the canonical neighborhood assumption with parameter r on [2i ǫ, T ], where 2i ǫ ≤ T ≤ 2i+1 ǫ, 0 < δ(t) < δ¯ f or t ∈ [2i−1 ǫ, T ]. Then it is κ-noncollapsed on all scales less than ǫ. Proof. Consider a neighborhood P (x0 , t0 , r0 , −r02 ), 2i ǫ < t0 ≤ T, 0 < r0 < ǫ, where the solution is defined and satisfies |Rm| ≤ r0−2 . We may assume r0 ≥ r, since otherwise the lower bound for the volume of the ball B(x0 , t0 , r0 ) follows from the canonical neighborhood assumption. If the solution was smooth everywhere, we could estimate from below the volume of the ball B(x0 , t0 , r0 ) using the argument from [I.7.3]: define τ (t) = t0 − t and consider the reduced volume function using the L-exponential map from x0 ; take a point (x, ǫ) where the reduced distance l attains its minimum for τ = t0 − ǫ, l(x, τ ) ≤ 3/2; use it to obtain an upper bound for the reduced distance to the points of B(x, 0, 1), thus getting a lower bound for the reduced volume at τ = t0 , and apply the monotonicity formula. Now if the solution undergoes surgeries, then we still can measure the L-length, but only for admissible curves, which stay in the region, unaffected by surgery. An inspection of the constructions in [I,§7] shows that the argument would go through if we knew that every barely admissible curve, that is a curve on the boundary of the set of admissible curves, has reduced length at least 3/2 + κ′ for some fixed κ′ > 0. Unfortunately, at the moment I don’t see how to ensure that without imposing new restrictions on δ(t) for all t ∈ [0, T ], so we need some additional arguments. Recall that for a curve γ, parametrized by t, with γ(t0 ) = x0 , we have
11
Rt √ 2 L(γ, τ ) = t00−τ t0 − t(R(γ(t), t) + |γ(t)| ˙ )dt. We can also define L+ (γ, τ ) by √ replacing in the previous formula R with R+ = max(R, 0). Then L+ ≤ L+4T T because R ≥ −6 by the maximum principle and normalization. Now suppose we could show that every barely admissible √ curve with endpoints (x0 , t0 ) and (x, t), where t ∈ [2i−1 ǫ, T ), has L+ > 2ǫ−2 T T ; then we could argue that either there √ exists a point (x, t), t ∈ [2i−1 ǫ, 2i ǫ], such that R(x, t) ≤ ri−2 and L+ ≤ ǫ−2 T T , in which case we can take this point in place of (x, ǫ) in the argument of the previous paragraph, and obtain (using Claim 1 in 4.2) an estimate for κ in i−1 terms of r√ ǫ, t0 ], γ(t0 ) = x0 , we have L+ ≥ i , κi , T, or for any γ, defined√on [2 2 i−1 23 −2 −2 −2 min(ǫ T T , 3 (2 ǫ) ri ) > ǫ T T , which is in contradiction with the assumed bound for barely admissible curves and the bound min l(x, t0 −2i−1 ǫ) ≤ 3/2, valid in the smooth case. Thus, to conclude the proof it is sufficient to check the following assertion. ¯ r0 ) > 0 with the 5.3 Lemma. For any L < ∞ one can find δ¯ = δ(L, following property. Suppose that in the situation of the previous lemma we have a curve γ, parametrized by t ∈ [T0 , t0 ], 2i−1 ǫ ≤ T0 < t0 , such that γ(t0 ) = x0 , T0 is a surgery time, and γ(T0 ) ∈ B(p, T0 , ǫ−1 h), where p corresponds to the of the cap, and h is the radius of the δ-neck. Then we have an estimate Rcenter t0 √ 2 t − t(R+ (γ(t), t) + |γ(t)| ˙ )dt ≥ L. 0 T0 Proof. It is clear that if we take △t = ǫr04 L−2 , then either γ satisfies our estimate, or γ stays in P (x0 , t0 , r0 , −△t) for t ∈ [t0 − △t, t0 ]. In the latter 1 case our estimate follows from Corollary 4.6, for l = L(△t)− 2 , since clearly Tγ < t0 − △t when δ is small enough. 5.4 Proof of proposition. Assume the contrary, and let the sequences rα , δ¯αβ be such that rα → 0 as α → ∞, δ¯αβ → 0 as β → ∞ with fixed α, and let αβ (M αβ , gij ) be normalized initial data for solutions to the Ricci flow with δ(t)cutoff, δ(t) < δ¯αβ on [2i−1 ǫ, 2i+1 ǫ], which satisfy the statement on [0, 2i ǫ], but violate the canonical neighborhood assumption with parameter rα on [2i ǫ, 2i+1 ǫ]. Slightly abusing notation, we’ll drop the indices α, β when we consider an individual solution. Let t¯ be the first time when the assumption is violated at some point x¯; clearly such time exists, because it is an open condition. Then by lemma 5.2 we have uniform κ-noncollapsing on [0, t¯]. Claims 1,2 in 4.2 are also valid on [0, t¯]; moreover, since h << r, it follows from Claim 1 that the solution is defined on the whole parabolic neighborhood indicated there in case R(x0 , t0 ) ≤ r−2 . Scale our solution about (¯ x, t¯) with factor R(¯ x, t¯) ≥ r−2 and take a limit for ¯ subsequences of α, β → ∞. At time t, which we’ll shift to zero in the limit, the curvature bounds at finite distances from x ¯ for the scaled metric are ensured by Claim 2 in 4.2. Thus, we get a smooth complete limit of nonnegative sectional curvature, at time zero. Moreover, the curvature of the limit is uniformly bounded, since otherwise it would contain ǫ-necks of arbitrarily small radius. Let Q0 denote the curvature bound. Then, if there was no surgery, we could, using Claim 1 in 4.2, take a limit on the time interval [−ǫη −1 Q−1 0 , 0]. To prevent −1 this, there must exist surgery times T0 ∈ [t¯ − ǫη −1 Q−1 (¯ x, t¯), t¯] and points 0 R x, t¯) uniformly bounded as α, β → ∞, such that the x with dist2T0 (x, x¯)R−1 (¯ 12
solution at x is defined on [T0 , t¯], but not before T0 . Using Claim 2 from 4.2 at time T0 , we see that R(¯ x, t¯)h2 (T0 ) must be bounded away from zero. Therefore, in this case we can apply Corollary 4.7, Lemma 4.5 and Claim 5 in section 2 to show that the point (¯ x, t¯) in fact has a canonical neighborhood, contradicting its choice. (It is not excluded that the strong ǫ-neck neighborhood extends to times before T0 , where it is a part of the strong δ-neck that existed before surgery.) Thus we have a limit on a certain time interval. Let Q1 be the curvature bound for this limit. Then we either can construct a limit on the time interval −1 [−ǫη −1 (Q−1 0 + Q1 ), 0], or there is a surgery, and we get a contradiction as before. We can continue this procedure indefinitely, and the final part of the proof of Theorem I.12.1 shows that the bounds Qk can not go to infinity while the limit is defined on a bounded time interval. Thus we get a limit on (−∞, 0], which is κ-noncollapsed by Lemma 5.2, and this means that (¯ x, t¯) has a canonical neighborhood by the results of section 1 - a contradiction.
6
Long time behavior I
6.1 Let us summarize what we have achieved so far. We have shown the exis¯ (which tence of decreasing (piecewise constant) positive functions r(t) and δ(t) we may assume converging to zero at infinity), such that if (M, gij ) is a normal¯ ized manifold, and 0 < δ(t) < δ(t), then there exists a solution to the Ricci flow with δ(t)-cutoff on the time interval [0, +∞], starting from (M, gij ) and satisfying on each subinterval [0, t] the canonical neighborhood assumption with parameter r(t), as well as the pinching estimate (5.1). In particular, if the initial data has positive scalar curvature, say R ≥ a > 0, 3 , and it follows that M in then the solution becomes extinct in time at most 2a this case is diffeomorphic to a connected sum of several copies of S2 × S1 and metric quotients of round S3 . ( The topological description of 3-manifolds with positive scalar curvature modulo quotients of homotopy spheres was obtained by Schoen-Yau and Gromov-Lawson more than 20 years ago, see [G-L] for instance; in particular, it is well known and easy to check that every manifold that can be decomposed in a connected sum above admits a metric of positive scalar curvature.) Moreover, if the scalar curvature is only nonnegative, then by the strong maximum principle it instantly becomes positive unless the metric is (Ricci-)flat; thus in this case, we need to add to our list the flat manifolds. However, if the scalar curvature is negative somewhere, then we need to work more in order to understand the long tome behavior of the solution. To achieve this we need first to prove versions of Theorems I.12.2 and I.12.3 for solutions with cutoff. 6.2 Correction to Theorem I.12.2. Unfortunately, the statement of Theorem I.12.2 was incorrect. The assertion I had in mind is as follows: Given a function φ as above, for any A < ∞ there exist K = K(A) < ∞ and ρ = ρ(A) > 0 with the following property. Suppose in dimension three we have a solution to the Ricci flow with φ-almost nonnegative curvature, which satisfies the assumptions of theorem 8.2 for some x0 , r0 with φ(r0−2 ) < ρ. Then 13
R(x, r02 ) ≤ Kr0−2 whenever distr02 (x, x0 ) < Ar0 . It is this assertion that was used in the proof of Theorem I.12.3 and Corollary I.12.4. 6.3 Proposition. For any A < ∞ one can find κ = κ(A) > 0, K1 = K1 (A) < ∞, K2 = K2 (A) < ∞, r¯ = r¯(A) > 0, such that for any t0 < ∞ there exists δ¯ = δ¯A (t0 ) > 0, decreasing in t0 , with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on time interval [0, T ], δ(t) < ¯ on [0, T ], δ(t) < δ¯ on [t0 /2, t0 ], with normalized initial data; assume that the δ(t) solution is defined in the whole parabolic neighborhood P (x0 , t0 , r0 , −r02 ), 2r02 < t0 , and satisfies |Rm| ≤ r0−2 there, and that the volume of the ball B(x0 , t0 , r0 ) is at least A−1 r03 . Then (a) The solution is κ-noncollapsed on the scales less than r0 in the ball B(x0 , t0 , Ar0 ). (b) Every point x ∈ B(x0 , t0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighborhood as √ in 4.1. (c) If r0 ≤ r¯ t0 then R ≤ K2 r0−2 in B(x0 , t0 , Ar0 ). Proof. (a) This is an analog of Theorem I.8.2. Clearly we have p κ-noncollapsing on the scales less than r(t0 ), so we may assume r(t0 ) ≤ r0 ≤ t0 /2 , and study the scales ρ, r(t0 ) ≤ ρ ≤ r0 . In particular, for fixed t0 we are interested in the scales, uniformly equivalent to one. So assume that x ∈ B(x0 , t0 , Ar0 ) and the solution is defined in the whole P (x, t0 , ρ, −ρ2 ) and satisfies |Rm| ≤ ρ−2 there. An inspection of the proof of I.8.2 shows that in order to make the argument work it suffices to check that for any barely admissible curve γ, parametrized by t ∈ [tγ , t0 ], t0 − r02 ≤ tγ ≤ t0 , such that γ(t0 ) = x, we have an estimate p 2 t0 − tγ
Z
t0
tγ
√
2 t0 − t(R(γ(t), t) + |γ(t)| ˙ )dt ≥ C(A)r02
(6.1)
for a certain function C(A) that can be made explicit. Now we would like to conclude the proof by using Lemma 5.3. However, unlike the situation in Lemma 5.2, here Lemma 5.3 provides the estimate we need only if t0 − tγ is bounded away from zero, and otherwise we only get an estimate ρ2 in place of C(A)r02 . Therefore we have to return to the proof of I.8.2. Recall that in that proof we scaled the solution to make r0 = 1 and worked on the time interval [1/2, 1]. The maximum principle for the evolution equation of the scalar curvature implies that on this time interval we have R ≥ −3. We ˆ t))L(y, ˆ τ ), where φ is a certain considered a function of the form h(y, t) = φ(d(y, ˆ ˆ τ ) = L(y, ¯ τ ) + 7, cutoff function, τ = 1 − t, d(y, t) = distt (x0 , y) − A(2t − 1), L(y, ¯ ˆ ˆ ¯ τ) + and in [I,(7.15)]. Now we redefine L, taking L(y, τ ) = L(y, √ √ L was defined 2 ˆ 2 τ . Clearly, L > 0 because R ≥ −3 and 2 τ > 4τ for 0 < τ ≤ 1/2. Then the computations and estimates of I.8.2 yield 1 2h ≥ −C(A)h − (6 + √ )φ τ
14
Now denoting by h0 (τ ) the minimum of h(y, 1 − t), we can estimate √ h0 (τ ) 6 τ +1 1 50 d √ − (log( √ )) ≤ C(A) + ≤ C(A) + √ , dτ τ 2τ − 4τ 2 τ 2τ τ whence h0 (τ ) ≤
√
√ τ exp(C(A)τ + 100 τ ),
(6.2)
(6.3)
because the left hand side of (6.2) tends to zero as τ → 0 + . Now we can return to our proof, replace the right hand side of (6.1) by the right hand side of (6.3) times r02 , with τ = r0−2 (t0 − tγ ), and apply Lemma 5.3. (b) Assume the contrary, take a sequence K1α → ∞ and consider the solu−2 tions violating the statement. Clearly, K1α (r0α )−2 < (r(tα , whence tα 0 )) 0 → ∞; When K1 is large enough, we can, arguing as in the proof of Claim 1 in ¯= [I.10.1], find a point (¯ x, t¯), x ∈ B(x0 , t¯, 2Ar0 ), t¯ ∈ [t0 − r02 /2, t0 ], such that Q R(¯ x, t¯) > K1 r0−2 , (¯ x, t¯) does not satisfy the canonical neighborhood assumption, ¯ does, where P¯ is the set of all (x, t) but each point (x, t) ∈ P¯ with R(x, t) ≥ 4Q 1 1 −1 ¯ ¯ − 12 . (Note ≤ t ≤ t¯, distt (x0 , x) ≤ distt¯(x0 , x¯) + K12 Q satisfying t¯ − 4 K1 Q that P¯ is not a parabolic neighborhood.) Clearly we can use (a) with slightly different parameters to ensure κ-noncollapsing in P¯ . Now we apply the argument from 5.4. First, by Claim 2 in 4.2, for any ¯Q ¯ in B(¯ ¯ − 12 ) when K1 is large A¯ < ∞ we have an estimate R ≤ Q(A) x, t¯, A¯Q ¯ about enough; therefore we can take a limit as α → ∞ of scalings with factor Q (¯ x, t¯), shifting the time t¯ to zero; the limit at time zero would be a smooth complete nonnegatively curved manifold. Next we observe that this limit has curvature uniformly bounded, say, by Q0 , and therefore, for each fixed A¯ and for ¯ − 12 , −ǫη −1 Q−1 Q ¯ −1 ) sufficiently large K1 , the parabolic neighborhood P (¯ x, t¯, A¯Q 0 ¯ is contained in P . (Here we use the estimate of distance change, given by Lemma I.8.3(a).) Thus we can take a limit on the interval [−ǫη −1 Q−1 0 , 0]. (The possibility of surgeries is ruled out as in 5.4) Then we repeat the procedure indefinitely, getting an ancient κ-solution in the limit, which means a contradiction. (c) If x ∈ B(x0 , t0 , Ar0 ) has very large curvature, then on the shortest geodesic γ at time t0 , that connects x0 and x, we can find a point y, such that R(y, t0 ) = K1 (A)r0−2 and the curvature is larger at all points of the segment of γ between x and y. Then our statement follows from Claim 2 in 4.2, applied to this segment. ¯ to be min(δ(t), ¯ From now on we redefine the function δ(t) δ¯2t (2t)), so that the proposition above always holds for A = t0 . 6.4 Proposition. There exist τ > 0, r¯ > 0, K < ∞ with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on the time √ interval [0, t0 ], with normalized initial data. Let r0 , t0 satisfy 2C1 h ≤ r0 ≤ r¯ t0 , where h is the maximal cutoff radius for surgeries in [t0 /2, t0 ], and assume that the ball B(x0 , t0 , r0 ) has sectional curvatures at least −r0−2 at each point, and the volume of any subball B(x, t0 , r) ⊂ B(x0 , t0 , r0 ) with any radius r > 0 is at least (1 − ǫ) times the volume of the euclidean ball of the same radius. Then the solution is defined in P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. 15
Proof. Let us first consider the case r0 ≤ r(t0 ). Then clearly R(x0 , t0 ) ≤ C12 r0−2 , since an ǫ-neck of radius r can not contain an almost euclidean ball of radius ≥ r. Thus we can take K = 2C12 , τ = ǫη −1 C1−2 in this case, and since 2 r0 ≥ 2C1 h, the surgeries do not interfere in P (x0 , t0 , r√ 0 /4, −τ r0 ). In order to handle the other case r(t0 ) < r0 ≤ r¯ t0 we need a couple of lemmas. 6.5 Lemma. There exist τ0 > 0 and K0 < ∞, such that if we have a smooth solution to the Ricci flow in P (x0 , 0, 1, −τ ), τ ≤ τ0 , having sectional curvatures at least −1, and the volume of the ball B(x0 , 0, 1) is at least (1 − ǫ) times the volume of the euclidean unit ball, then (a) R ≤ K0 τ −1 in P (x0 , 0, 1/4, −τ /2), and 1 (b) the ball B(x0 , 1/4, −τ ) has volume at least 10 times the volume of the euclidean ball of the same radius. The proof can be extracted from the proof of Lemma I.11.6. 6.6 Lemma. For any w > 0 there exists θ0 = θ0 (w) > 0, such that if B(x, 1) is a metric ball of volume at least w, compactly contained in a manifold without boundary with sectional curvatures at least −1, then there exists a ball B(y, θ0 ) ⊂ B(x, 1), such that every subball B(z, r) ⊂ B(y, θ0 ) of any radius r has volume at least (1 − ǫ) times the volume of the euclidean ball of the same radius. This is an elementary fact from the theory of Aleksandrov spaces. 6.7 Now we continue the proof of the proposition. We claim that one can take τ = min(τ0 /2, ǫη −1 C1−2 ), K = max(2K0 τ −1 , 2C12 ). Indeed, assume the contrary, and take a sequence of r¯α → 0 and solutions, violating our assertion for α α α the chosen τ, K. Let tα 0 be the first time when it is violated, and let B(x0 , t0 , r0 ) α be the counterexample with the smallest radius. Clearly r0 > r(tα 0 ) and −1 (r0α )2 (tα ) → 0 as α → ∞. 0 Consider any ball B(x1 , t0 , r) ⊂ B(x0 , t0 , r0 ), r < r0 . Clearly we can apply our proposition to this ball and get the solution in P (x1 , t0 , r/4, −τ r2 ) with the curvature bound R < Kr−2 . Now if r02 t−1 0 is small enough, then we can apply proposition 6.3(c) to get an estimate R(x, t) ≤ K ′ (A)r−2 for (x, t) satisfying t ∈ [t0 − τ r2 /2, t0 ], distt (x, x1 ) < Ar, for some function K ′ (A) that can be made explicit. Let us choose A = 100r0 r−1 ; then we get the solution with a curvature estimate in P (x0 , t0 , r0 , −△t), where △t = K ′ (A)−1 r2 . Now the pinching estimate implies Rm ≥ −r0−2 on this set, if r02 t−1 is small enough 0 while rr0−1 is bounded away from zero. Thus we can use lemma 6.5(b) to 1 estimate the volume of the ball B(x0 , t0 − △t, r0 /4) by at least 10 of the volume of the euclidean ball of the same radius, and then by lemma 6.6 we can find a 1 )r0 /4), satisfying the assumptions of our proposition. subball B(x2 , t0 −△t, θ0 ( 10 1 Therefore, if we put r = θ0 ( 10 )r0 /4, then we can repeat our procedure as many times as we like, until we reach the time t0 − τ0 r02 , when the lemma 6.5(b) stops working. But once we reach this time, we can apply lemma 6.5(a) and get the required curvature estimate, which is a contradiction. 6.8 Corollary. For any w > 0 one can find τ = τ (w) > 0, K = K(w) < ∞, r¯ = r¯(w) > 0, θ = θ(w) > 0 with the following property. Suppose we have a solution to the Ricci flow with δ(t)-cutoff on the time interval [0, t0 ], with 16
√ normalized initial data. Let t0 , r0 satisfy θ−1 (w)h ≤ r0 ≤ r¯ t0 , and assume that the ball B(x0 , t0 , r0 ) has sectional curvatures at least −r02 at each point, and volume at least wr03 . Then the solution is defined in P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Indeed, we can apply proposition 6.4 to a smaller ball, provided by lemma 6.6, and then use proposition 6.3(c).
7
Long time behavior II
In this section we adapt the arguments of Hamilton [H 4] to a more general setting. Hamilton considered smooth Ricci flow with bounded normalized curvature; we drop both these assumptions. In the end of [I,13.2] I claimed that the volumes of the maximal horns can be effectively bounded below, which would imply that the solution must be smooth from some time on; however, the argument I had in mind seems to be faulty. On the other hand, as we’ll see below, the presence of surgeries does not lead to any substantial problems. From now on we assume that our initial manifold does not admit a metric with nonnegative scalar curvature, and that once we get a component with nonnegative scalar curvature, it is immediately removed. 7.1 (cf. [H 4,§2,7]) Recall that for a solution to the smooth Ricci flow the scalar curvature satisfies the evolution equation 2 d R = △R + 2|Ric|2 = △R + 2|Ric◦ |2 + R2 , dt 3 where Ric◦ is the trace-free part of Ric. Then Rmin (t) satisfies whence 1 3 Rmin (t) ≥ − 2 t + 1/4
d dt Rmin
(7.1) 2 ≥ 32 Rmin ,
(7.2)
for a solution withR normalized initial data. The evolution equation for the d volume is dt V = − RdV, in particular d V ≤ −Rmin V, dt
(7.3)
3 whence by (7.2) the function V (t)(t+1/4)− 2 is non-increasing in t. Let V¯ denote its limit as t → ∞. ˆ = Rmin V 23 satisfies Now the scale invariant quantity R Z d ˆ 2 ˆ −1 R(t) ≥ RV (Rmin − R)dV, (7.4) dt 3
which is nonnegative whenever Rmin ≤ 0, which we have assumed from the ¯ denote the limit of R(t) ˆ beginning of the section. Let R as t → ∞. Assume for a moment that V¯ > 0. Then it follows from (7.2) and (7.3) that 3 ¯ V¯ − 23 = − 3 . Now the inequality Rmin (t) is asymptotic to − 2t ; in other words, R 2 (7.4) implies that whenever we have a sequence of parabolic neighborhoods 17
√ P (xα , tα , r tα , −r2 tα ), for tα → ∞ and some fixed small r > 0, such that the scalings of our solution with factor tα smoothly converge to some limit solution, defined in an abstract parabolic neighborhood P (¯ x, 1, r, −r2 ), then the scalar curvature of this limit solution is independent of the space variables and equals 3 at time t ∈ [1 − r2 , 1]; moreover, the strong maximum principle for (7.1) − 2t implies that the sectional curvature of the limit at time t is constant and equals 1 − 4t . This conclusion is also valid without the a priori assumption that V¯ > 0, since otherwise it is vacuous. Clearly the inequalities and conclusions above hold for the solutions to the Ricci flow with δ(t)-cutoff, defined in the previous sections. From now on we assume that we are given such a solution, so the estimates below may depend on it. 7.2 Lemma. (a) Given w > 0,√ r > 0, ξ > 0 one can find T = T (w, r, ξ) < ∞, such that if the ball B(x0 , t0 , r t0 ) at some time t0 ≥ T has volume at least wr3 and sectional curvature at least −r−2 t−1 0 , then curvature at x0 at time t = t0 satisfies |2tRij + gij | < ξ. (7.5) (b) Given in addition A < ∞ and √ allowing T to depend on A, we can ensure (7.5) for all points in B(x0 , t0 , Ar t0 ). √ (c) The same is true for P (x0 , t0 , Ar t0 , Ar2 t0 ). Proof. (a) If T is large enough √ then we can apply corollary 6.8 to the ball B(x0 , t0 , r0 ) for r0 = min(r, r¯(w)) t0 ; then use the conclusion of 7.1. (b) The curvature control in P (x0 , t0 , r0 /4, −τ r02 ), provided by corollary 6.8, allows us to apply proposition 6.3 (a),(b) to a controllably smaller neighborhood √ P (x0 , t0 , r0′ , −(r0′ )2 ). Thus by 6.3(b) we know that each point in B(x0 , t0 , Ar t0 ) −2 ′ with scalar curvature at least Q = K1 (A)r0 has a canonical neighborhood. This implies that for T large enough such points do not exist, since if there was a point with R larger than Q, there would be a point having a canonical neighborhood with R = Q in the same ball, and that contradicts the already proved assertion (a). Therefore we have curvature control in the ball in question, and applying 6.3(a) we also get volume control there, so our assertion has been reduced to (a). √ (c) If ξ is small enough, then the solution in the ball B(x0 , t0 , Ar t0 ) would 2 stay almost homothetic to itself on the time interval [t0 , t0 + Ar t0 ] until (7.5) is violated at some (first) time t′ in this interval. However, if T is large enough, then this violation could not happen, because we can apply the already proved assertion (b) at time t′ for somewhat larger A. 7.3 Let ρ(x, t) denote the radius ρ of the ball B(x, t, ρ) where inf Rm = −ρ−2 . It follows from corollary 6.8, proposition 6.3(c), and the pinching estimate √ (5.1) that for any w > 0 we can find ρ¯ = ρ¯(w) > 0, such that if ρ(x, t) < ρ¯ t, then V ol B(x, t, ρ(x, t)) < wρ3 (x, t), (7.6) provided that t is large enough (depending on w). Let M − (w, t) denote the thin part of M, that is the set of x ∈ M where (7.6) holds at time t, and let M + (w, t) be its complement. Then for t large enough 18
(depending on w) every point of M + satisfies the assumptions of lemma 7.2. Assume first that for some w > 0 the set M + (w, t) is not empty for a sequence of t → ∞. Then the arguments of Hamilton [H 4,§8-12] work in our situation. In particular, if we take a sequence of points xα ∈ M + (w, tα ), tα → α ∞, then the scalings of gij about xα with factors (tα )−1 converge, along a subsequence of α → ∞, to a complete hyperbolic manifold of finite volume. The limits may be different for different choices of (xα , tα ). If none of the limits is closed, and H1 is such a limit with the least number of cusps, then, by an argument in [H 4,§8-10], based on hyperbolic rigidity, for all sufficiently small w′ , 0 < w′ < w(H ¯ 1 ), there exists a standard truncation H1 (w′ ) of H1 , such that, for t large enough, M + (w′ /2, t) contains an almost isometric copy of H1 (w′ ), which in turn contains a component of M + (w′ , t); moreover, this embedded copy of H1 (w′ ) moves by isotopy as t increases to infinity. If for some w > 0 the complement M + (w, t) \ H1 (w) is not empty for a sequence of t → ∞, then we can repeat the argument and get another complete hyperbolic manifold H2 , etc., until we find a finite collection of Hj , 1 ≤ j ≤ i, such that for each sufficiently small w > 0 the embeddings of Hj (w) cover M + (w, t) for all sufficiently large t. Furthermore, the boundary tori of Hj (w) are incompressible in M. This is proved [H 4,§11,12] by a minimal surface argument, using a result of Meeks and Yau. This argument does not use the uniform bound on the normalized curvature, and goes through even in the presence of surgeries, because the area of the least area disk in question can only decrease when we make a surgery. 7.4 Let us redefine the thin part in case the thick one isn’t empty, M − (w, t) = M \(H1 (w)∪...∪Hi (w)). Then, for sufficiently small w > 0 and sufficiently large t, M − (w, t) is diffeomorphic to a graph manifold, as implied by the following general result on collapsing with local lower curvature bound, applied to the metrics t−1 gij (t). α Theorem. Suppose (M α , gij ) is a sequence of compact oriented riemannian 3-manifolds, closed or with convex boundary, and wα → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα (x), 0 < ρ < 1, not exceeding the diameter of the manifold, such that the ball B(x, ρ) in the metric α gij has volume at most wα ρ3 and sectional curvatures at least −ρ−2 ; (2) each component of the boundary of M α has diameter at most wα , and has a (topologically trivial) collar of length one, where the sectional curvatures are between −1/4 − ǫ and −1/4 + ǫ; (3) For every w′ > 0 there exist r¯ = r¯(w′ ) > 0 and Km = Km (w′ ) < ∞, m = 0, 1, 2..., such that if α is large enough, 0 < r ≤ r¯, and the ball B(x, r) α in gij has volume at least w′ r3 and sectional curvatures at least −r2 , then the curvature and its m-th order covariant derivatives at x, m = 1, 2..., are bounded by K0 r−2 and Km r−m−2 respectively. Then M α for sufficiently large α are diffeomorphic to graph manifolds. Indeed, there is only one exceptional case, not covered by the theorem above, namely, when M = M − (w, t), and ρ(x, t), for some x ∈ M, is much larger than the diameter d(t) of the manifold, whereas the ratio V (t)/d3 (t) is bounded away from zero. In this case, since by the observation after formula (7.3) the
19
3
2 volume V (t) can not √ grow faster than const · t , the diameter does not grow faster than const · t, hence if we scale our metrics gij (t) to keep the diameter equal to one, the scaled metrics would satisfy the assumption (3) of the theorem above and have the minimum of sectional curvatures tending to zero. Thus we can take a limit and get a smooth solution to the Ricci flow with nonnegative sectional curvature, but not strictly positive scalar curvature. Therefore, in this exceptional case M is diffeomorphic to a flat manifold. The proof of the theorem above will be given in a separate paper; it has nothing to do with the Ricci flow; its main tool is the critical point theory for distance functions and maps, see [P,§2] and references therein. The assumption (3) is in fact redundant; however, it allows to simplify the proof quite a bit, by avoiding 3-dimensional Aleksandrov spaces, and in particular, the nonelementary Stability Theorem. Summarizing, we have shown that for large t every component of the solution is either diffeomorphic to a graph manifold, or to a closed hyperbolic manifold, or can be split by a finite collection of disjoint incompressible tori into parts, each being diffeomorphic to either a graph manifold or to a complete noncompact hyperbolic manifold of finite volume. The topology of graph manifolds is well understood [W]; in particular, every graph manifold can be decomposed in a connected sum of irreducible graph manifolds, and each irreducible one can in turn be split by a finite collection of disjoint incompressible tori into Seifert fibered manifolds.
8
On the first eigenvalue of the operator −4△+R
8.1 Recall from [I,§1,2] that Ricci flow is the gradient flow for the first eigenvalue 2 d λ of the operator −4△ + R; moreover, dt λ(t) ≥ 23 λ2 (t) and λ(t)V 3 (t) is nondecreasing whenever it is nonpositive. We would like to extend these inequalities to the case of Ricci flow with δ(t)-cutoff. Recall that we immediately remove components with nonnegative scalar curvature. Lemma. Given any positive continuous function ξ(t) one can chose δ(t) in such a way that for any solution to the Ricci flow with δ(t)-cutoff, with normalized initial data, and any surgery time T0 , after which there is at least one component, where the scalar curvature is not strictly positive, we have an estimate λ+ (T0 )−λ− (T0 ) ≥ ξ(T0 )(V + (T0 )−V − (T0 )), where V − , V + and λ− , λ+ are the volumes and the first eigenvalues of −4△+R before and after the surgery respectively. Proof. Consider the minimizer a for the functional Z (4|∇a|2 + Ra2 ) (8.1) R under normalization a2 = 1, for the metric after the surgery on a component where scalar curvature is not strictly positive. Clearly is satisfies the equation 4△a = Ra − λ− a 20
(8.2)
Observe that since the metric contains an ǫ-neck of radius about r(T0 ), we can estimate λ− (T0 ) from above by about r(T0 )−2 . Let Mcap denote the cap, added by the surgery. It is attached to a long tube, consisting of ǫ-necks of various radii. Let us restrict our attention to a maximal subtube, on which the scalar curvature at each point is at least 2λ− (T0 ). Choose any ǫ-neck in this subtube, say, with radius r0 , and consider the distance function with range [0, 2ǫ−1 r0 ], whose level sets Mz are almost round two-spheres; let Mz+ ⊃ Mcap be the part of M, chopped off by Mz . Then Z Z Z −4aaz = (4|∇a|2 + Ra2 − λ− a2 ) > r0−2 /2 a2 Mz+
Mz
On the other hand, Z |
Mz
Z 2aaz − (
Mz+
2
Mz
a )z | ≤ const ·
These two inequalities easily imply that Z Z a2 ≥ exp(ǫ−1 /10) M0+
Z
Mz
ǫr0−1 a2
a2
M +−1 ǫ
r0
Now the chosen subtube contains at least about −ǫ−1 log(λ− (T0 )h2 (T0 )) disjoint ǫ-necks, where h denotes the cutoff radius, as before. Since h tends to zero with δ, whereas r(T0 ), that occurs in the bound for λ− , is independent of δ, we that the number of necks is greater then log h, and therefore, R can 2ensure 6 a < h , say. Then standard estimates for the equation (8.2) show that Mcap 2 2 |∇a| and Ra are bounded by const · h on Mcap , which makes it possible to extend a to the metric before surgery in such a way that the functional (8.1) is preserved up to const · h4 . However, the loss of volume in the surgery is at least h3 , so it suffices to take δ so small that h is much smaller than ξ. 8.2 The arguments above lead to the following result (a) If (M, gij ) has λ > 0, then, for an appropriate choice of the cutoff parameter, the solution becomes extinct in finite time. Thus, if M admits a metric with λ > 0 then it is diffeomorphic to a connected sum of a finite collection of S2 × S1 and metric quotients of the round S3 . Conversely, every such connected sum admits a metric with R > 0, hence with λ > 0. ¯ denote the (b) Suppose M does not admit any metric with λ > 0, and let λ 2 ¯ = 0 implies that supremum of λV 3 over all metrics on this manifold. Then λ ¯ < 0. M is a graph manifold. Conversely, a graph manifold can not have λ ¯ 32 . Then V¯ is the minimum of V, such ¯ < 0 and let V¯ = (− 2 λ) (c) Suppose λ 3 that M can be decomposed in connected sum of a finite collection of S2 × S1 , metric quotients of the round S3 , and some other components, the union of which will be denoted by M ′ , and there exists a (possibly disconnected) complete hyperbolic manifold, with sectional curvature −1/4 and volume V, which can be embedded in M ′ in such a way that the complement (if not empty) is a graph
21
manifold. Moreover, if such a hyperbolic manifold has volume V¯ , then its cusps (if any) are incompressible in M ′ . For the proof one needs in addition easily verifiable statements that one can put metrics on connected sums preserving the lower bound for scalar curvature [G-L], that one can put metrics on graph manifolds with scalar curvature bounded below and volume tending to zero [C-G], and that one can close a compressible cusp, preserving the lower bound for scalar curvature and reducing the volume, cf. [A,5.2]. Notice that using these results we can avoid the hyperbolic rigidity and minimal surface arguments, quoted in 7.3, which, however, have the advantage of not requiring any a priori topological information about the complement of the hyperbolic piece. The results above are exact analogs of the conjectures for the Sigma constant, formulated by Anderson [A], at least in the nonpositive case.
References [I] G.Perelman The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 v1 [A] M.T.Anderson Scalar curvature and geometrization conjecture for threemanifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997), 49-82. [C-G] J.Cheeger, M.Gromov Collapsing Riemannian manifolds while keeping their curvature bounded I. Jour. Diff. Geom. 23 (1986), 309-346. [G-L] M.Gromov, H.B.Lawson Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58 (1983), 83-196. [H 1] R.S.Hamilton Three-manifolds with positive Ricci curvature. Jour. Diff. Geom. 17 (1982), 255-306. [H 2] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys in Diff. Geom. 2 (1995), 7-136. [H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff. Geom. 37 (1993), 225-243. [H 4] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7 (1999), 695-729. [H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Commun. Anal. Geom. 5 (1997), 1-92. G.Perelman Spaces with curvature bounded below. Proceedings of ICM1994, 517-525. F.Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I,II. Invent. Math. 3 (1967), 308-333 and 4 (1967), 87-117.
22
arXiv:math/0307245v1 [math.DG] 17 Jul 2003
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds Grisha Perelman∗ February 1, 2008 In our previous paper we constructed complete solutions to the Ricci flow with surgery for arbitrary initial riemannian metric on a (closed, oriented) three-manifold [P,6.1], and used the behavior of such solutions to classify threemanifolds into three types [P,8.2]. In particular, the first type consisted of those manifolds, whose prime factors are diffeomorphic copies of spherical space forms and S2 × S1 ; they were characterized by the property that they admit metrics, that give rise to solutions to the Ricci flow with surgery, which become extinct in finite time. While this classification was sufficient to answer topological questions, an analytical question of significant independent interest remained open, namely, whether the solution becomes extinct in finite time for every initial metric on a manifold of this type. In this note we prove that this is indeed the case. Our argument (in conjunction with [P,§1-5]) also gives a direct proof of the so called ”elliptization conjecture”. It turns out that it does not require any substantially new ideas: we use only a version of the least area disk argument from [H,§11] and a regularization of the curve shortening flow from [A-G].
1
Finite time extinction
1.1 Theorem. Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. Then for any initial metric on M the solution to the Ricci flow with surgery becomes extinct in finite time. Proof for irreducible M . Let ΛM denote the space of all contractible loops in C 1 (S1 → M ). Given a riemannian metric g on M and c ∈ ΛM, define A(c, g) to be the infimum of the areas of all lipschitz maps from D2 to M, whose restriction to ∂D2 = S1 is c. For a family Γ ⊂ ΛM let A(Γ, g) be the supremum of A(c, g) over all c ∈ Γ. Finally, for a nontrivial homotopy class α ∈ π∗ (ΛM, M ) let A(α, g) be the infimum of A(Γ, g) over all Γ ∈ α. Since M is not aspherical, it follows from a classical (and elementary) result of Serre that such a nontrivial homotopy class exists. ∗ St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191023, Russia. Email:
[email protected] or
[email protected]
1
1.2 Lemma. (cf. [H,§11]) If g t is a smooth solution to the Ricci flow, then for any α the rate of change of the function At = A(α, g t ) satisfies the estimate 1 t d t A ≤ −2π − Rmin At dt 2 t (in the sense of the lim sup of the forward difference quotients), where Rmin t denotes the minimum of the scalar curvature of the metric g . A rigorous proof of this lemma will be given in §3, but the idea is simple and can be explained here. Let us assume that at time t the value At is attained by the family Γ, such that the loops c ∈ Γ where A(c, g t ) is close to At are embedded and sufficiently smooth. For each such c consider the minimal disk Dc with boundary c and with area A(c, g t ). Now let the metric evolve by the Ricci flow and let the curves c evolve by the curve shortening flow (which moves every point of the curve in the direction of its curvature vector at this point) with the same time parameter. Then the rate of change of the area of Dc can be computed as Z Z T (−Tr(Ric )) + (−kg ) Dc
c
T
where Ric is the Ricci tensor of M restricted to the tangent plane of Dc , and kg is the geodesic curvature of c with respect to Dc (cf. [A-G, Lemma 3.2]). In three dimensions the first integrand equals − 21 R − (K − det II), where K is the intrinsic curvature of Dc and det II, the determinant of the second fundamental form, is nonpositive, because Dc is minimal. Thus, the rate of change of the area of Dc can be estimated from above by Z Z Z 1 1 (− R) − 2π (− R − K) + (−kg ) = 2 2 c Dc Dc by the Gauss-Bonnet theorem, and the statement of the lemma follows. The problem with this argument is that if Γ contains curves, which are not immersed (for instance, a curve could pass an arc once in one direction and then make an about turn and pass the same arc in the opposite direction), then it is not clear how to define curve shortening flow so that it would be continuous both in the time parameter and in the family parameter. In §3 we’ll explain how to circumvent this difficulty, essentially by adding one dimension to the ambient manifold. This regularization of the curve shortening flow has been worked out by Altschuler and Grayson [A-G] (who were interested in approximating the singular curve shortening flow on the plane and obtained for that case more precise results than what we need). 1.3 Now consider the solution to the Ricci flow with surgery. Since M is assumed irreducible, the surgeries are topologically trivial, that is one of the components of the post-surgery manifold is diffeomorphic to the pre-surgery manifold, and all the others are spheres. Moreover, by the construction of the surgery [P,4.4], the diffeomorphism from the pre-surgery manifold to the post-surgery one can be chosen to be distance non-increasing ( more precisely, (1 + ξ)-lipschitz, where ξ > 0 can be made as small as we like). It follows that 2
the conclusion of the lemma above holds for the solutions to the Ricci flow with surgery as well. Now recall that the evolution equation for the scalar curvature 2 d R = △R + 2|Ric|2 = △R + R2 + 2|Ric◦ |2 dt 3 1 At t implies the estimate Rmin ≥ − 23 t+const . It follows that Aˆt = t+const satisfies 2π d ˆt A ≤ − , which implies finite extinction time since the right hand side dt t+const t ˆ is non-integrable at infinity whereas A can not become negative. 1.4 Remark. The finite time extinction result for irreducible non-aspherical manifolds already implies (in conjuction with the work in [P,§1-5] and the Kneser finiteness theorem) the so called ”elliptization conjecture”, claiming that a closed manifold with finite fundamental group is diffeomorphic to a spherical space form. The analysis of the long time behavior in [P,§6-8] is not needed in this case; moreover the argument in [P,§5] can be slightly simplified, replacing ¯ since we already have an upper the sequences rj , κj , δ¯j by single values r, κ, δ, bound on the extinction time in terms of the initial metric. In fact, we can even avoid the use of the Kneser theorem. Indeed, if we start from an initial metric on a homotopy sphere (not assumed irreducible), then at each surgery time we have (almost) distance non-increasing homotopy equivalences from the pre-surgery manifold to each of the post-surgery components, and this is enough to keep track of the nontrivial relative homotopy class of the loop space. 1.5 Proof of theorem 1.1 for general M . The Kneser theorem implies that our solution undergoes only finitely many topologically nontrivial surgeries, so from some time T on all the surgeries are trivial. Moreover, by the Milnor uniqueness theorem, each component at time T satisfies the assumption of the theorem. Since we already know from 1.4 that there can not be any simply connected prime factors, it follows that every such component is either irreducible, or has nontrivial π2 ; in either case the proof in 1.1-1.3 works.
2
Preliminaries on the curve shortening flow
In this section we rather closely follow [A-G]. 2.1 Let M be a closed n-dimensional manifold, n ≥ 3, and let g t be a smooth family of riemannian metrics on M evolving by the Ricci flow on a finite time interval [t0 , t1 ]. It is known [B] that g t for t > t0 are real analytic. Let ct be a solution to the curve shortening flow in (M, g t ), that is ct satisfies the equation d t t 1 t dt c (x) = H (x), where x is the parameter on S , and H is the curvature vector field of ct with respect to g t . It is known [G-H] that for any smoothly immersed initial curve c the solution ct exists on some time interval [t0 , t′1 ), each ct for t > t0 is an analytic immersed curve, and either t′1 = t1 , or the curvature 1 k t = g t (H t , H t ) 2 is unbounded when t → t′1 .
3
1
Denote by X t the tangent vector field to ct , and let S t = g t (X t , X t )− 2 X t be the unit tangent vector field; then H = ∇S S (from now on we drop the superscript t except where this omission can cause confusion). We compute d g(X, X) = −2Ric(X, X) − 2g(X, X)k 2 , dt
(1)
[H, S] = (k 2 + Ric(S, S))S
(2)
d 2 k = (k 2 )′′ − 2g((∇S H)⊥ , (∇S H)⊥ ) + 2k 4 + ..., dt
(3)
which implies Now we can compute
where primes denote differentiation with respect to the arclength parameter s, and where dots stand for the terms containing the curvature tensor of g, which can be estimated in absolute value by const · (k 2 + k). Thus the curvature k satisfies d k ≤ k ′′ + k 3 + const · (k + 1) (4) dt Now Rit follows from (1) and (4) that the length L and the total curvature Θ = kds satisfy Z d L ≤ (const − k 2 )ds, (5) dt Z d Θ ≤ const · (k + 1)ds (6) dt
In particular, both quantities can grow at most exponentially in t (they would be non-increasing in a flat manifold). 2.2 In general the curvature of ct may concentrate near certain points, creating singularities. However, if we know that this does not happen at some time t∗ , then we can estimate the curvature and higher derivatives at times shortly thereafter. More precisely, there exist constants ǫ, C1 , C2 , ... (which may depend on the curvatures of the ambient space and their derivatives, but are independent of ct ), such that if at time t∗ for some r > 0 the length of ct is at least r and the total curvature of each arc of length r does not exceed ǫ, then for every t ∈ (t∗ , t∗ + ǫr2 ) the curvature k and higher derivatives satisfy the estimates k 2 = g(H, H) ≤ C0 (t − t∗ )−1 , g(∇S H, ∇S H) ≤ C1 (t − t∗ )−2 , ... This can be proved by adapting the arguments of Ecker and Huisken [E-Hu]; see also [A-G,§4]. ¯ , g¯t ) × S1 , 2.3 Now suppose that our manifold (M, g t ) is a metric product (M λ where the second factor is the circle of constant length λ; let U denote the unit tangent vector field to this factor. Then u = g(S, U ) satisfies the evolution equation d u = u′′ + (k 2 + Ric(S, S))u (7) dt
4
Assume that u was strictly positive everywhere at time t0 (in this case the curve is called a ramp). Then it will remain positive and bounded away from zero as long as the solution exists. Now combining (4) and (7) we can estimate the right hand side of the evolution equation for the ratio uk and conclude that this ratio, and hence the curvature k, stays bounded (see [A-G,§2]). It follows that ct is defined on the whole interval [t0 , t1 ]. 2.4 Assume now that we have two ramp solutions ct1 , ct2 , each winding once around the S1λ factor. Let µt be the infimum of the areas of the annuli with boundary ct1 ∪ ct2 . Then d t µ ≤ (2n − 1)|Rmt |µt , dt
(8)
where |Rmt | denotes a bound on the absolute value of sectional curvatures of g t . Indeed, the curves ct1 and ct2 , being ramps, are embedded and without substantial loss of generality we may assume them to be disjoint. In this case the results of Morrey [M] and Hildebrandt [Hi] yield an analytic minimal annulus A, immersed, except at most finitely many branch points, with prescribed boundary and with area µ. The rate of change of the area of A can be computed as Z Z Z (−Tr(RicT )) + (−kg ) ≤ (−Tr(RicT ) + K) A
∂A
≤
Z
A
A
(−Tr(RicT ) + RmT ) ≤ (2n − 1)|Rm|µ,
where the first inequality comes from the Gauss-Bonnet theorem, with possible contribution of the branch points, and the second one is due to the fact that a minimal surface has nonpositive extrinsic curvature with respect to any normal vector. 2.5 The estimate (8) implies that µt can grow at most exponentially; in particular, if ct1 and ct2 were very close at time t0 , then they would be close for all t ∈ [t0 , t1 ] in the sense of minimal annulus area. In general this does not imply that the lengths of the curves are also close. However, an elementary argument shows that if ǫ > 0 is small then, given any r > 0, one can find µ ¯, depending only on r and on upper bound for sectional curvatures of the ambient space, such that if the length of ct1 is at least r, each arc of ct1 with length r has total curvature at most ǫ, and µt ≤ µ ¯ , then L(ct2 ) ≥ (1 − 100ǫ)L(ct1).
3
Proof of lemma 1.2
3.1 In this section we prove the following statement Let M be a closed three-manifold, and let (M, g t ) be a smooth solution to the Ricci flow on a finite time interval [t0 , t1 ]. Suppose that Γ ⊂ ΛM is a compact family. Then for any ξ > 0 one can construct a continuous deformation Γt , t ∈ [t0 , t1 ], Γt0 = Γ, such that for each curve c ∈ Γ either the value A(ct1 , g t1 ) is bounded from above by ξ plus the value at t = t1 of the solution to the ODE 5
t = −2π − 21 Rmin w(t) with the initial data w(t0 ) = A(ct0 , g t0 ), or L(c ) ≤ ξ; moreover, if c was a constant map, then all ct are constant maps. It is clear that our statement implies lemma 1.2, because a family consisting of very short loops can not represent a nontrivial relative homotopy class. 3.2 As a first step of the proof of the statement we can replace Γ by a family, which consists of piecewise geodesic loops with some large fixed number of vertices and with each segment reparametrized in some standard way to make the parametrizations of the whole curves twice continuously differentiable. Now consider the manifold Mλ = M × S1λ , 0 < λ < 1, and for each c ∈ Γ consider the smooth embedded closed curve cλ such that p1 cλ (x) = c(x) and p2 cλ (x) = λx mod λ, where p1 and p2 are projections of Mλ to the first and second factor respectively, and x is the parameter of the curve c on the standard circle of length one. Using 2.3 we can construct a solution ctλ , t ∈ [t0 , t1 ] to the curve shortening flow with initial data cλ . The required deformation will be obtained as Γt = p1 Γtλ (where Γtλ denotes the family consisting of ctλ ) for certain sufficiently small λ > 0. We’ll verify that an appropriate λ can be found for each individual curve c, or for any finite number of them, and then show that if our λ works for all elements of a µ-net in Γ, for sufficiently small µ > 0, then it works for all elements of Γ. 3.3 In the following estimates we shall denote by C large constants that may depend on metrics g t , family Γ and ξ, but are independent of λ, µ and a particular curve c. The first step in 3.2 implies that the lengths and total curvatures of cλ are uniformly bounded, so by 2.1 the same is true for all ctλ . It follows that the area swept by ctλ , t ∈ [t′ , t′′ ] ⊂ [t0 , t1 ] is bounded above by C(t′′ − t′ ), and therefore ′′ ′′ ′ ′ we have the estimates A(p1 ctλ , g t ) ≤ C, A(p1 ctλ , g t ) − A(p1 ctλ , g t ) ≤ C(t′′ − t′ ). R t1 R 2 3.4 It follows from (5) that t0 k dsdt ≤ C for any ctλ . Fix some large constant B, to be chosen later. ThenR there is a subset RIB (cλ ) ⊂ [t0 , t1 ] of measure at least t1 − t0 − CB −1 where k 2 ds ≤ B, hence kds ≤ ǫ on any arc of length ≤ ǫ2 B −1 . Assuming that ctλ are at least that long, we can apply 2.2 and construct another subset JB (cλ ) ⊂ [t0 , t1 ] of measure at least t1 −t0 −CB −1 , consisting of finitely many intervals of measure at least C −1 B −2 each, such that for any t ∈ JB (cλ ) we have pointwise estimates on ctλ for curvature and higher derivatives, of the form k ≤ CB, ... Now fix c, B, and consider any sequence of λ → 0. Assume again that the lengths of ctλ are bounded below by ǫ2 B −1 , at least for t ∈ [t0 , t2 ], where t2 = t1 − B −1 . Then an elementary argument shows that we can find a subsequence Λc and a subset JB (c) ⊂ [t0 , t2 ] of measure at least t1 − t0 − CB −1 , consisting of finitely many intervals, such that JB (c) ⊂ JB (cλ ) for all λ ∈ Λc . It follows that on every interval of JB (c) the curve shortening flows ctλ smoothly converge (as λ → 0 in some subsequence of Λc ) to a curve shortening flow in M. d t Let wc (t) be the solution of the ODE dt wc (t) = −2π − 12 Rmin wc (t) with t0 initial data wc (t0 ) = A(c, g ). Then for sufficiently small λ ∈ Λc we have A(p1 ctλ , g t ) ≤ wc (t) + 12 ξ provided that B > Cξ −1 . Indeed, on the intervals of JB (c) we can estimate the change of A for the limit flow using the minimal d dt w(t) t1
6
disk argument as in 1.2, and this implies the corresponding estimate for p1 ctλ if λ ∈ Λc is small enough, whereas for the intervals of the complement of JB (c) we can use the estimate in 3.3. On the other hand, if our assumption on the lower bound for lengths does not hold, then it follows from (5) that L(ctλ2 ) ≤ CB −1 ≤ 21 ξ. 3.5 Now apply the previous argument to all elements of some finite µ-net ˆ ⊂ Γ for small µ > 0 to be determined later. We get a λ > 0 such that for each Γ ˆ either A(p1 cˆt1 , g t1 ) ≤ wcˆ(t1 ) + 1 ξ or L(ˆ cˆ ∈ Γ ctλ2 ) ≤ 12 ξ. Now for any curve c ∈ Γ λ 2 ˆ µ-close to c, and apply the result of 2.4. It follows that if pick a curve cˆ ∈ Γ, t1 t1 A(p1 cˆλ , g ) ≤ wcˆ(t1 ) + 21 ξ and µ ≤ C −1 ξ, then A(p1 ctλ1 , g t1 ) ≤ wc (t1 ) + ξ. On the other hand, if L(ˆ ctλ2 ) ≤ 12 ξ, then we can conclude that L(ctλ1 ) ≤ ξ provided that µ > 0 is small enough in comparison with ξ and B −1 . Indeed, if L(ctλ1 ) > ξ, then L(ctλ ) > 34 ξ for Rall t ∈ [t2 , t1 ]; on the other hand, using (5) we can find a t ∈ [t2 , t1 ], such that k 2 ds ≤ CB for ctλ ; hence, applying 2.5, we get L(ˆ ctλ ) > 32 ξ t2 1 for this t, which is incompatible with L(ˆ cλ ) ≤ 2 ξ. The proof of the statement 3.1 is complete.
References [A-G] S.Altschuler, M.Grayson Shortening space curves and flow through singularities. Jour. Diff. Geom. 35 (1992), 283-298. [B] S.Bando Real analyticity of solutions of Hamilton’s equation. Math. Zeit. 195 (1987), 93-97. [E-Hu] K.Ecker, G.Huisken Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), 547-569. [G-H] M.Gage, R.S.Hamilton The heat equation shrinking convex plane curves. Jour. Diff. Geom. 23 (1986), 69-96. [H] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7 (1999), 695-729. [Hi] S.Hildebrandt Boundary behavior of minimal surfaces. Arch. Rat. Mech. Anal. 35 (1969), 47-82. [M] C.B.Morrey The problem of Plateau on a riemannian manifold. Ann. Math. 49 (1948), 807-851. [P] G.Perelman Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 v1
7
c 2006 International Press
ASIAN J. MATH. Vol. 10, No. 2, pp. 165–492, June 2006
001
´ AND A COMPLETE PROOF OF THE POINCARE GEOMETRIZATION CONJECTURES – APPLICATION OF THE HAMILTON-PERELMAN THEORY OF THE RICCI FLOW∗ HUAI-DONG CAO† AND XI-PING ZHU‡ Abstract. In this paper, we give a complete proof of the Poincar´ e and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow. Key words. Ricci flow, Ricci flow with surgery, Hamilton-Perelman theory, Poincar´ e Conjecture, geometrization of 3-manifolds AMS subject classifications. 53C21, 53C44
CONTENTS Introduction
167
1 Evolution Equations 1.1 The Ricci Flow . . . . . . . . . . . . . . . . . 1.2 Short-time Existence and Uniqueness . . . . . 1.3 Evolution of Curvatures . . . . . . . . . . . . 1.4 Derivative Estimates . . . . . . . . . . . . . . 1.5 Variational Structure and Dynamic Property
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
172 172 177 183 190 199
2 Maximum Principle and Li-Yau-Hamilton Inequalities 2.1 Preserving Positive Curvature . . . . . . . . . . . . . . . . 2.2 Strong Maximum Principle . . . . . . . . . . . . . . . . . 2.3 Advanced Maximum Principle for Tensors . . . . . . . . . 2.4 Hamilton-Ivey Curvature Pinching Estimate . . . . . . . . 2.5 Li-Yau-Hamilton Estimates . . . . . . . . . . . . . . . . . 2.6 Perelman’s Estimate for Conjugate Heat Equations . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
210 210 213 217 223 226 234
3 Perelman’s Reduced Volume 3.1 Riemannian Formalism in Potentially Infinite Dimensions 3.2 Comparison Theorems for Perelman’s Reduced Volume . . 3.3 No Local Collapsing Theorem I . . . . . . . . . . . . . . . 3.4 No Local Collapsing Theorem II . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
239 239 243 255 261
4 Formation of Singularities 4.1 Cheeger Type Compactness 4.2 Injectivity Radius Estimates 4.3 Limiting Singularity Models 4.4 Ricci Solitons . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
267 267 286 291 302
. . . .
. . . .
. . . .
. . . .
∗ Received
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . . .
. . . .
. . . . .
. . . .
. . . . .
. . . .
. . . . .
. . . .
. . . . .
. . . .
. . . . .
. . . .
. . . .
December 12, 2005; accepted for publication April 16, 2006. of Mathematics, Lehigh University, Bethlehem, PA 18015, USA (
[email protected]). ‡ Department of Mathematics, Zhongshan University, Guangzhou 510275, P. R. China (stszxp@ zsu.edu.cn). † Department
165
166
H.-D. CAO AND X.-P. ZHU
5 Long Time Behaviors 307 5.1 The Ricci Flow on Two-manifolds . . . . . . . . . . . . . . . . . . . . 308 5.2 Differentiable Sphere Theorems in 3-D and 4-D . . . . . . . . . . . . . 321 5.3 Nonsingular Solutions on Three-manifolds . . . . . . . . . . . . . . . . 336 6 Ancient κ-solutions 6.1 Preliminaries . . . . . . . . . . . . . . . . 6.2 Asymptotic Shrinking Solitons . . . . . . 6.3 Curvature Estimates via Volume Growth . 6.4 Ancient κ-solutions on Three-manifolds .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
357 357 364 373 384
7 Ricci Flow on Three-manifolds 7.1 Canonical Neighborhood Structures . . . . . . . . . . . . 7.2 Curvature Estimates for Smooth Solutions . . . . . . . . . 7.3 Ricci Flow with Surgery . . . . . . . . . . . . . . . . . . . 7.4 Justification of the Canonical Neighborhood Assumptions 7.5 Curvature Estimates for Surgically Modified Solutions . . 7.6 Long Time Behavior . . . . . . . . . . . . . . . . . . . . . 7.7 Geometrization of Three-manifolds . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
398 398 405 413 432 452 468 481
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
References
486
Index
491
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
167
Introduction. In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincar´e conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman. An important problem in differential geometry is to find a canonical metric on a given manifold. In turn, the existence of a canonical metric often has profound topological implications. A good example is the classical uniformization theorem in two dimensions which, on one hand, provides a complete topological classification for compact surfaces, and on the other hand shows that every compact surface has a canonical geometric structure: a metric of constant curvature. How to formulate and generalize this two-dimensional result to three and higher dimensional manifolds has been one of the most important and challenging topics in modern mathematics. In 1977, W. Thurston [122], based on ideas about Riemann surfaces, Haken’s work and Mostow’s rigidity theorem, etc, formulated a geometrization conjecture for three-manifolds which, roughly speaking, states that every compact orientable three-manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure. In particular, Thurston’s conjecture contains, as a special case, the Poincar´e conjecture: A closed three-manifold with trivial fundamental group is necessarily homeomorphic to the 3-sphere S3 . In the past thirty years, many mathematicians have contributed to the understanding of this conjecture of Thurston. While Thurston’s theory is based on beautiful combination of techniques from geometry and topology, there has been a powerful development of geometric analysis in the past thirty years, lead by S.-T. Yau, R. Schoen, C. Taubes, K. Uhlenbeck, and S. Donaldson, on the construction of canonical geometric structures based on nonlinear PDEs (see, e.g., Yau’s survey papers [129, 130]). Such canonical geometric structures include K¨ahler-Einstein metrics, constant scalar curvature metrics, and self-dual metrics, among others. However, the most important contribution for geometric analysis on three-manifolds is due to Hamilton. In 1982, Hamilton [58] introduced the Ricci flow ∂gij = −2Rij ∂t to study compact three-manifolds with positive Ricci curvature. The Ricci flow, which evolves a Riemannian metric by its Ricci curvature, is a natural analogue of the heat equation for metrics. As a consequence, the curvature tensors evolve by a system of diffusion equations which tends to distribute the curvature uniformly over the manifold. Hence, one expects that the initial metric should be improved and evolve into a canonical metric, thereby leading to a better understanding of the topology of the underlying manifold. In the celebrated paper [58], Hamilton showed that on a compact three-manifold with an initial metric having positive Ricci curvature, the Ricci flow converges, after rescaling to keep constant volume, to a metric of positive constant sectional curvature, proving the manifold is diffeomorphic to the three-sphere S3 or a quotient of the three-sphere S3 by a linear group of isometries. Shortly after, Yau suggested that the Ricci flow should be the best way to prove the structure theorem for general three-manifolds. In the past two decades, Hamilton proved many important and remarkable theorems for the Ricci flow, and laid the foundation for the program to approach the Poincar´e conjecture and Thurston’s geometrization conjecture via the Ricci flow.
168
H.-D. CAO AND X.-P. ZHU
The basic idea of Hamilton’s program can be briefly described as follows. For any given compact three-manifold, one endows it with an arbitrary (but can be suitably normalized by scaling) initial Riemannian metric on the manifold and then studies the behavior of the solution to the Ricci flow. If the Ricci flow develops singularities, then one tries to find out the structures of singularities so that one can perform (geometric) surgery by cutting off the singularities, and then continue the Ricci flow after the surgery. If the Ricci flow develops singularities again, one repeats the process of performing surgery and continuing the Ricci flow. If one can prove there are only a finite number of surgeries during any finite time interval and if the long-time behavior of solutions of the Ricci flow with surgery is well understood, then one would recognize the topological structure of the initial manifold. Thus Hamilton’s program, when carried out successfully, will give a proof of the Poincar´e conjecture and Thurston’s geometrization conjecture. However, there were obstacles, most notably the verification of the so called “Little Loop Lemma” conjectured by Hamilton [63] (see also [17]) which is a certain local injectivity radius estimate, and the verification of the discreteness of surgery times. In the fall of 2002 and the spring of 2003, Perelman [103, 104] brought in fresh new ideas to figure out important steps to overcome the main obstacles that remained in the program of Hamilton. (Indeed, in page 3 of [103], Perelman said “the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds” and “In this paper we carry out some details of Hamilton program”.) Perelman’s breakthrough on the Ricci flow excited the entire mathematics community. His work has since been examined to see whether the proof of the Poincar´e conjecture and geometrization program, based on the combination of Hamilton’s fundamental ideas and Perelman’s new ideas, holds together. The present paper grew out of such an effort. Now we describe the three main parts of Hamilton’s program in more detail. (i) Determine the structures of singularities Given any compact three-manifold M with an arbitrary Riemannian metric, one evolves the metric by the Ricci flow. Then, as Hamilton showed in [58], the solution g(t) to the Ricci flow exists for a short time and is unique (also see Theorem 1.2.1). In fact, Hamilton [58] showed that the solution g(t) will exist on a maximal time interval [0, T ), where either T = ∞, or 0 < T < ∞ and the curvature becomes unbounded as t tends to T . We call such a solution g(t) a maximal solution of the Ricci flow. If T < ∞ and the curvature becomes unbounded as t tends to T , we say the maximal solution develops singularities as t tends to T and T is the singular time. In the early 1990s, Hamilton systematically developed methods to understand the structure of singularities. In [61], based on suggestion by Yau, he proved the fundamental Li-Yau [82] type differential Harnack estimate (the Li-Yau-Hamilton estimate) for the Ricci flow with nonnegative curvature operator in all dimensions. With the help of Shi’s interior derivative estimate [114], he [62] established a compactness theorem for smooth solutions to the Ricci flow with uniformly bounded curvatures and uniformly bounded injectivity radii at the marked points. By imposing an injectivity radius condition, he rescaled the solution to show that each singularity is asymptotic to one of the three types of singularity models [63]. In [63] he discovered (also independently by Ivey [73]) an amazing curvature pinching estimate for the Ricci flow on three-manifolds. This pinching estimate implies that any three-dimensional singularity model must have nonnegative curvature. Thus in dimension three, one only needs to obtain a complete classification for nonnegatively curved singularity models.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
169
For Type I singularities in dimension three, Hamilton [63] established an isoperimetric ratio estimate to verify the injectivity radius condition and obtained spherical or necklike structures for any Type I singularity model. Based on the Li-Yau-Hamilton estimate, he showed that any Type II singularity model with nonnegative curvature is either a steady Ricci soliton with positive sectional curvature or the product of the so called cigar soliton with the real line [66]. (Characterization for nonnegatively curved Type III models was obtained in [30].) Furthermore, he developed a dimension reduction argument to understand the geometry of steady Ricci solitons [63]. In the three-dimensional case, he showed that each steady Ricci soliton with positive curvature has some necklike structure. Hence Hamilton had basically obtained a canonical neighborhood structure at points where the curvature is comparable to the maximal curvature for solutions to the three-dimensional Ricci flow. However two obstacles remained: (a) the verification of the imposed injectivity radius condition in general; and (b) the possibility of forming a singularity modelled on the product of the cigar soliton with a real line which could not be removed by surgery. The recent spectacular work of Perelman [103] removed these obstacles by establishing a local injectivity radius estimate, which is valid for the Ricci flow on compact manifolds in all dimensions. More precisely, Perelman proved two versions of “no local collapsing” property (Theorem 3.3.3 and Theorem 3.3.2), one with an entropy functional he introduced in [103], which is monotone under the Ricci flow, and the other with a space-time distance function obtained by path integral, analogous to what Li-Yau did in [82], which gives rise to a monotone volume-type (called reduced volume by Perelman) estimate. By combining Perelman’s no local collapsing theorem I′ (Theorem 3.3.3) with the injectivity radius estimate of Cheng-Li-Yau (Theorem 4.2.2), one immediately obtains the desired injectivity radius estimate, or the Little Loop Lemma (Theorem 4.2.4) conjectured by Hamilton. Furthermore, Perelman [103] developed a refined rescaling argument (by considering local limits and weak limits in Alexandrov spaces) for singularities of the Ricci flow on three-manifolds to obtain a uniform and global version of the canonical neighborhood structure theorem. We would like to point out that our proof of the singularity structure theorem (Theorem 7.1.1) is different from that of Perelman in two aspects: (1) we avoid using his crucial estimate in Claim 2 in Section 12.1 of [103]; (2) we give a new approach to extend the limit backward in time to an ancient solution. These differences are due to the difficulties in understanding Perelman’s arguments at these points. (ii) Geometric surgeries and the discreteness of surgery times After obtaining the canonical neighborhoods (consisting of spherical, necklike and caplike regions) for the singularities, one would like to perform geometric surgery and then continue the Ricci flow. In [64], Hamilton initiated such a surgery procedure for the Ricci flow on four-manifolds with positive isotropic curvature and presented a concrete method for performing the geometric surgery. His surgery procedures can be roughly described as follows: cutting the neck-like regions, gluing back caps, and removing the spherical regions. As will be seen in Section 7.3 of this paper, Hamilton’s geometric surgery method also works for the Ricci flow on compact orientable threemanifolds. Now an important challenge is to prevent surgery times from accumulating and make sure one performs only a finite number of surgeries on each finite time interval. The problem is that, when one performs the surgeries with a given accuracy at each surgery time, it is possible that the errors may add up to a certain amount which
170
H.-D. CAO AND X.-P. ZHU
could cause the surgery times to accumulate. To prevent this from happening, as time goes on, successive surgeries must be performed with increasing accuracy. In [104], Perelman introduced some brilliant ideas which allow one to find “fine” necks, glue “fine” caps, and use rescaling to prove that the surgery times are discrete. When using the rescaling argument for surgically modified solutions of the Ricci flow, one encounters the difficulty of how to apply Hamilton’s compactness theorem (Theorem 4.1.5), which works only for smooth solutions. The idea to overcome this difficulty consists of two parts. The first part, due to Perelman [104], is to choose the cutoff radius in neck-like regions small enough to push the surgical regions far away in space. The second part, due to the authors and Chen-Zhu [34], is to show that the surgically modified solutions are smooth on some uniform (small) time intervals (on compact subsets) so that Hamilton’s compactness theorem can still be applied. To do so, we establish three time-extension results (see Step 2 in the proof of Proposition 7.4.1.). Perhaps, this second part is more crucial. Without it, Shi’s interior derivative estimate (Theorem 1.4.2) may not applicable, and hence one cannot be certain that Hamilton’s compactness theorem holds when only having the uniform C 0 bound on curvatures. We remark that in our proof of this second part, as can be seen in the proof of Proposition 7.4.1, we require a deep comprehension of the prolongation of the gluing “fine” caps for which we will use the recent uniqueness theorem of BingLong Chen and the second author [33] for solutions of the Ricci flow on noncompact manifolds. Once surgeries are known to be discrete in time, one can complete the classification, started by Schoen-Yau [109, 110], for compact orientable three-manifolds with positive scalar curvature. More importantly, for simply connected three-manifolds, if one can show that solutions to the Ricci flow with surgery become extinct in finite time, then the Poincar´e conjecture would follow. Such a finite extinction time result was proposed by Perelman [105], and a proof also appears in Colding-Minicozzi [42]. Thus, the combination of Theorem 7.4.3 (i) and the finite extinction time result provides a complete proof to the Poincar´e conjecture. (iii) The long-time behavior of surgically modified solutions. To approach the structure theorem for general three-manifolds, one still needs to analyze the long-time behavior of surgically modified solutions to the Ricci flow. In [65], Hamilton studied the long time behavior of the Ricci flow on compact threemanifolds for a special class of (smooth) solutions, the so called nonsingular solutions. These are the solutions that, after rescaling to keep constant volume, have (uniformly) bounded curvature for all time. Hamilton [65] proved that any three-dimensional nonsingular solution either collapses or subsequently converges to a metric of constant curvature on the compact manifold or, at large time, admits a thick-thin decomposition where the thick part consists of a finite number of hyperbolic pieces and the thin part collapses. Moreover, by adapting Schoen-Yau’s minimal surface arguments in [110] and using a result of Meeks-Yau [86], Hamilton showed that the boundary of hyperbolic pieces are incompressible tori. Consequently, when combined with the collapsing results of Cheeger-Gromov [24, 25], this shows that any nonsingular solution to the Ricci flow is geometrizable in the sense of Thurston [122]. Even though the nonsingular assumption seems very restrictive and there are few conditions known so far which can guarantee a solution to be nonsingular, nevertheless the ideas and arguments of Hamilton’s work [65] are extremely important. In [104], Perelman modified Hamilton’s arguments to analyze the long-time be-
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
171
havior of arbitrary smooth solutions to the Ricci flow and solutions with surgery to the Ricci flow in dimension three. Perelman also argued that the proof of Thurston’s geometrization conjecture could be based on a thick-thin decomposition, but he could only show the thin part will only have a (local) lower bound on the sectional curvature. For the thick part, based on the Li-Yau-Hamilton estimate, Perelman [104] established a crucial elliptic type estimate, which allowed him to conclude that the thick part consists of hyperbolic pieces. For the thin part, he announced in [104] a new collapsing result which states that if a three-manifold collapses with (local) lower bound on the sectional curvature, then it is a graph manifold. Assuming this new collapsing result, Perelman [104] claimed that the solutions to the Ricci flow with surgery have the same long-time behavior as nonsingular solutions in Hamilton’s work, a conclusion which would imply a proof of Thurston’s geometrization conjecture. Although the proof of this new collapsing result promised by Perelman in [104] is still not available in literature, Shioya-Yamaguchi [118] has published a proof of the collapsing result in the special case when the manifold is closed. In the last section of this paper (see Theorem 7.7.1), we will provide a proof of Thurston’s geometrization conjecture by only using Shioya-Yamaguchi’s collapsing result. In particular, this gives another proof of the Poincar´e conjecture. We would like to point out that Perelman [104] did not quite give an explicit statement of the thick-thin decomposition for surgical solutions. When we were trying to write down an explicit statement, we needed to add a restriction on the relation between the accuracy parameter ε and the collapsing parameter w. Nevertheless, we are still able to obtain a weaker version of the thick-thin decomposition (Theorem 7.6.3) that is sufficient to deduce the geometrization result. In this paper, we shall give complete and detailed proofs of what we outlined above, especially of Perelman’s work in his second paper [104] in which many key ideas of the proofs are sketched or outlined but complete details of the proofs are often missing. As we pointed out before, we have to substitute several key arguments of Perelman by new approaches based on our study, because we were unable to comprehend these original arguments of Perelman which are essential to the completion of the geometrization program. Our paper is aimed at both graduate students and researchers who want to learn Hamilton’s Ricci flow and to understand the Hamilton-Perelman theory and its application to the geometrization of three-manifolds. For this purpose, we have made the paper to be essentially self-contained so that the proof of the geometrization is accessible to those who are familiar with basics of Riemannian geometry and elliptic and parabolic partial differential equations. The reader may find some original papers, particularly those of Hamilton’s on the Ricci flow, before the appearance of Perelman’s preprints in the book “Collected Papers on Ricci Flow” [17]. For introductory materials to the Hamilton-Perelman theory of Ricci flow, we also refer the reader to the recent book by B. Chow and D. Knopf [39] and the forthcoming book by B. Chow, P. Lu and L. Ni [41]. We remark that there have also appeared several sets of notes on Perelman’s work, including the one written by B. Kleiner and J. Lott [78], which cover part of the materials that are needed for the geometrization program. There also have appeared several survey articles by Cao-Chow [16], Milnor [91], Anderson [4] and Morgan [95] for the geometrization of three-manifolds via the Ricci flow. We are very grateful to Professor S.-T. Yau, who suggested us to write this paper based on our notes, for introducing us to the wonderland of the Ricci flow. His vision and strong belief in the Ricci flow encouraged us to persevere. We also thank
172
H.-D. CAO AND X.-P. ZHU
him for his many suggestions and constant encouragement. Without him, it would be impossible for us to finish this paper. We are enormously indebted to Professor Richard Hamilton for creating the Ricci flow and developing the entire program to approach the geometrization of three-manifolds. His work on the Ricci flow and other geometric flows has influenced on virtually everyone in the field. The first author especially would like to thank Professor Hamilton for teaching him so much about the subject over the past twenty years, and for his constant encouragement and friendship. We are indebted to Dr. Bing-Long Chen, who contributed a great deal in the process of writing this paper. We benefited a lot from constant discussions with him on the subjects of geometric flows and geometric analysis. He also contributed many ideas in various proofs in the paper. We would like to thank Ms. Huiling Gu, a Ph.D student of the second author, for spending many months of going through the entire paper and checking the proofs. Without both of them, it would take much longer time for us to finish this paper. The first author would like to express his gratitude to the John Simon Guggenheim Memorial Foundation, the National Science Foundation (grants DMS-0354621 and DMS-0506084), and the Outstanding Overseas Young Scholar Fund of Chinese National Science Foundation for their support for the research in this paper. He also would like to thank Tsinghua University in Beijing for its hospitality and support while he was working there. The second author wishes to thank his wife, Danlin Liu, for her understanding and support over all these years. The second author is also indebted to the National Science Foundation of China for the support in his work on geometric flows, some of which has been incorporated in this paper. The last part of the work in this paper was done and the material in Chapter 3, Chapter 6 and Chapter 7 was presented while the second author was visiting the Harvard Mathematics Department in the fall semester of 2005 and the early spring semester of 2006. He wants to especially thank Professor Shing-Tung Yau, Professor Cliff Taubes and Professor Daniel W. Stroock for the enlightening comments and encouragement during the lectures. Also he gratefully acknowledges the hospitality and the financial support of Harvard University. 1. Evolution Equations. In this chapter, we introduce Hamilton’s Ricci flow and derive evolution equations of curvatures. The short time existence and uniqueness theorem of the Ricci flow on a compact manifold is proved in Section 1.2. In Section 1.4, we prove Shi’s local derivative estimate, which plays an important role in the Ricci flow. Perelman’s two functionals and their monotonicity properties are discussed in Section 1.5. 1.1. The Ricci Flow. Let M be an n-dimensional complete Riemannian manifold with the Riemannian metric gij . The Levi-Civita connection is given by the Christoffel symbols ∂gil ∂gij 1 kl ∂gjl k + − Γij = g 2 ∂xi ∂xj ∂xl where g ij is the inverse of gij . The summation convention of summing over repeated indices is used here and throughout the book. The Riemannian curvature tensor is given by k Rijl =
∂Γkjl ∂xi
−
∂Γkil + Γkip Γpjl − Γkjp Γpil . ∂xj
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
173
We lower the index to the third position, so that p Rijkl = gkp Rijl .
The curvature tensor Rijkl is anti-symmetric in the pairs i, j and k, l and symmetric in their interchange: Rijkl = −Rjikl = −Rijlk = Rklij . Also the first Bianchi identity holds (1.1.1)
Rijkl + Rjkil + Rkijl = 0.
The Ricci tensor is the contraction Rik = g jl Rijkl , and the scalar curvature is R = g ij Rij . ∂ We denote the covariant derivative of a vector field v = v j ∂x j by
∇i v j =
∂v j + Γjik v k ∂xi
∇i vj =
∂vj − Γkij vk . ∂xi
and of a 1-form by
These definitions extend uniquely to tensors so as to preserve the product rule and contractions. For the exchange of two covariant derivatives, we have (1.1.2) (1.1.3)
l ∇i ∇j v l − ∇j ∇i v l = Rijk vk ,
∇i ∇j vk − ∇j ∇i vk = Rijkl g lm vm ,
and similar formulas for more complicated tensors. The second Bianchi identity is given by (1.1.4)
∇m Rijkl + ∇i Rjmkl + ∇j Rmikl = 0.
i For any tensor T = Tjk we define its length by i 2 i l |Tjk | = gil g jm g kp Tjk Tmp ,
and we define its Laplacian by i i , ∆Tjk = g pq ∇p ∇q Tjk
the trace of the second iterated covariant derivatives. Similar definitions hold for more general tensors. The Ricci flow of Hamilton [58] is the evolution equation (1.1.5)
∂gij = −2Rij ∂t
174
H.-D. CAO AND X.-P. ZHU
for a family of Riemannian metrics gij (t) on M . It is a nonlinear system of second order partial differential equations on metrics. In order to get a feel for the Ricci flow (1.1.5) we first present some examples of specific solutions. (1) Einstein metrics A Riemannian metric gij is called Einstein if Rij = λgij for some constant λ. A smooth manifold M with an Einstein metric is called an Einstein manifold. If the initial metric is Ricci flat, so that Rij = 0, then clearly the metric does not change under (1.1.5). Hence any Ricci flat metric is a stationary solution of the Ricci flow. This happens, for example, on a flat torus or on any K3-surface with a Calabi-Yau metric. If the initial metric is Einstein with positive scalar curvature, then the metric will shrink under the Ricci flow by a time-dependent factor. Indeed, since the initial metric is Einstein, we have Rij (x, 0) = λgij (x, 0),
∀x ∈ M
and some λ > 0. Let gij (x, t) = ρ2 (t)gij (x, 0). From the definition of the Ricci tensor, one sees that Rij (x, t) = Rij (x, 0) = λgij (x, 0). Thus the equation (1.1.5) corresponds to ∂(ρ2 (t)gij (x, 0)) = −2λgij (x, 0). ∂t This gives the ODE (1.1.6)
dρ λ =− , dt ρ
whose solution is given by ρ2 (t) = 1 − 2λt. Thus the evolving metric gij (x, t) shrinks homothetically to a point as t → T = 1/2λ. Note that as t → T , the scalar curvature becomes infinite like 1/(T − t). By contrast, if the initial metric is an Einstein metric of negative scalar curvature, the metric will expand homothetically for all times. Indeed if Rij (x, 0) = −λgij (x, 0) with λ > 0 and gij (x, t) = ρ2 (t)gij (x, 0).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
175
Then ρ(t) satisfies the ODE (1.1.7)
λ dρ = , dt ρ
with the solution ρ2 (t) = 1 + 2λt. Hence the evolving metric gij (x, t) = ρ2 (t)gij (x, 0) exists and expands homothetically for all times, and the curvature will fall back to zero like −1/t. Note that now the evolving metric gij (x, t) only goes back in time to −1/2λ, when the metric explodes out of a single point in a “big bang”. (2) Ricci Solitons We will call a solution to an evolution equation which moves under a oneparameter subgroup of the symmetry group of the equation a steady soliton. The symmetry group of the Ricci flow contains the full diffeomorphism group. Thus a solution to the Ricci flow (1.1.5) which moves by a one-parameter group of diffeomorphisms ϕt is called a steady Ricci soliton. If ϕt is a one-parameter group of diffeomorphisms generated by a vector field V on M , then the Ricci soliton is given by (1.1.8)
gij (x, t) = ϕ∗t gij (x, 0)
which implies that the Ricci term −2Ric on the RHS of (1.1.5) is equal to the Lie derivative LV g of the evolving metric g. In particular, the initial metric gij (x, 0) satisfies the following steady Ricci soliton equation (1.1.9)
2Rij + gik ∇j V k + gjk ∇i V k = 0.
If the vector field V is the gradient of a function f then the soliton is called a steady gradient Ricci soliton. Thus (1.1.10)
Rij + ∇i ∇j f = 0,
or Ric + ∇2 f = 0,
is the steady gradient Ricci soliton equation. Conversely, it is clear that a metric gij satisfying (1.1.10) generates a steady gradient Ricci soliton gij (t) given by (1.1.8). For this reason we also often call such a metric gij a steady gradient Ricci soliton and do not necessarily distinguish it with the solution gij (t) it generates. More generally, we can consider a solution to the Ricci flow (1.1.5) which moves by diffeomorphisms and also shrinks or expands by a (time-dependent) factor at the same time. Such a solution is called a homothetically shrinking or homothetically expanding Ricci soliton. The equation for a homothetic Ricci soliton is (1.1.11)
2Rij + gik ∇j V k + gjk ∇i V k − 2λgij = 0,
or for a homothetic gradient Ricci soliton, (1.1.12)
Rij + ∇i ∇j f − λgij = 0,
where λ is the homothetic constant. For λ > 0 the soliton is shrinking, for λ < 0 it is expanding. The case λ = 0 is a steady Ricci soliton, the case V = 0 (or f being
176
H.-D. CAO AND X.-P. ZHU
a constant function) is an Einstein metric. Thus Ricci solitons can be considered as natural extensions of Einstein metrics. In fact, the following result states that there are no nontrivial gradient steady or expanding Ricci solitons on any compact manifold. We remark that if the underlying manifold M is a complex manifold and the initial metric is K¨ahler, then it is well known (see, e.g., [62, 11]) that the solution metric to the Ricci flow (1.1.5) remains K¨ahler. For this reason, the Ricci flow on a K¨ahler manifold is called the K¨ ahler-Ricci flow. A (steady, or shrinking, or expanding) Ricci soliton to the K¨ahler-Ricci flow is called a (steady, or shrinking, or expanding repectively) K¨ ahler-Ricci soliton. Proposition 1.1.1. On a compact n-dimensional manifold M , a gradient steady or expanding Ricci soliton is necessarily an Einstein metric. Proof. We shall only prove the steady case and leave the expanding case as an exercise. Our argument here follows that of Hamilton [63]. Let gij be a complete steady gradient Ricci soliton on a manifold M so that Rij + ∇i ∇j f = 0. Taking the trace, we get (1.1.13)
R + ∆f = 0.
Also, taking the covariant derivatives of the Ricci soliton equation, we have ∇i ∇j ∇k f − ∇j ∇i ∇k f = ∇j Rik − ∇i Rjk . On the other hand, by using the commutating formula (1.1.3), we otain ∇i ∇j ∇k f − ∇j ∇i ∇k f = Rijkl ∇l f. Thus ∇i Rjk − ∇j Rik + Rijkl ∇l f = 0. Taking the trace on j and k, and using the contracted second Bianchi identity (1.1.14)
∇j Rij =
1 ∇i R, 2
we get ∇i R − 2Rij ∇j f = 0. Then ∇i (|∇f |2 + R) = 2∇j f (∇i ∇j f + Rij ) = 0. Therefore (1.1.15) for some constant C.
R + |∇f |2 = C
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
177
Taking the difference of (1.1.13) and (1.1.15), we get ∆f − |∇f |2 = −C.
(1.1.16)
We claim C = 0 when M is compact. Indeed, this follows either from Z Z (1.1.17) 0=− ∆(e−f )dV = (∆f − |∇f |2 )e−f dV, M
M
or from considering (1.1.16) at both the maximum point and minimum point of f . Then, by integrating (1.1.16) we obtain Z |∇f |2 dV = 0. M
Therefore f is a constant and gij is Ricci flat. Remark 1.1.2. By contrast, there do exist nontrivial compact gradient shrinking Ricci solitons (see Koiso [80], Cao [13] and Wang-Zhu [127] ). Also, there exist complete noncompact steady gradient Ricci solitons that are not Ricci flat. In two dimensions Hamilton [60] wrote down the first such example on R2 , called the cigar soliton, where the metric is given by (1.1.18)
ds2 =
dx2 + dy 2 , 1 + x2 + y 2
and the vector field is radial, given by V = −∂/∂r = −(x∂/∂x + y∂/∂y). This metric has positive curvature and is asymptotic to a cylinder of finite circumference 2π at ∞. Higher dimensional examples were found by Robert Bryant [10] on Rn in the Riemannian case, and by the first author [13] on Cn in the K¨ahler case. These examples are complete, rotationally symmetric, of positive curvature and found by solving certain nonlinear ODE (system). Noncompact expanding solitons were also constructed by the first author [13]. More recently, Feldman, Ilmanen and Knopf [46] constructed new examples of noncompact shrinking and expanding K¨ahler-Ricci solitons. 1.2. Short-time Existence and Uniqueness. In this section we establish the short-time existence and uniqueness result for the Ricci flow (1.1.5) on a compact ndimensional manifold M . We will see that the Ricci flow is a system of second order nonlinear weakly parabolic partial differential equations. In fact, the degeneracy of the system is caused by the diffeomorphism group of M which acts as the gauge group of the Ricci flow. For any diffeomorphism ϕ of M , we have Ric (ϕ∗ (g)) = ϕ∗ (Ric (g)). Thus, if g(t) is a solution to the Ricci flow (1.1.5), so is ϕ∗ (g(t)). Because the Ricci flow (1.1.5) is only weakly parabolic, even the existence and uniqueness result on a compact manifold does not follow from standard PDE theory. The short-time existence and uniqueness result in the compact case is first proved by Hamilton [58] using the Nash-Moser implicit function theorem. Shortly after Denis De Turck [43] gave a much simpler proof using the gauge fixing idea which we will present here. In the noncompact case, the short-time existence was established by Shi [114] in 1989, but the uniqueness result has been proved only very recently by Bing-Long Chen and the second author. These results will be presented at the end of this section.
178
H.-D. CAO AND X.-P. ZHU
Let M be a compact n-dimensional Riemannian manifold. The Ricci flow equation is a second order nonlinear partial differential system ∂ gij = E(gij ), ∂t
(1.2.1)
for a family of Riemannian metrics gij (·, t) on M , where E(gij ) = −2Rij ∂ k ∂ k p k k p = −2 Γ − Γ + Γ Γ − Γ Γ kp ij ip kj ∂xk ij ∂xi kj ∂ ∂ ∂ ∂ ∂ kl ∂ kl = g g − g g + g − g kl jl il ij ∂xi ∂xj ∂xk ∂xi ∂xj ∂xl + 2Γkip Γpkj − 2Γkkp Γpij .
The linearization of this system is ∂˜ gij = DE(gij )˜ gij ∂t where g˜ij is the variation in gij and DE is the derivative of E given by DE(gij )˜ gij = g
kl
∂ 2 g˜kl ∂ 2 g˜jl ∂ 2 g˜il ∂ 2 g˜ij − − + ∂xi ∂xj ∂xi ∂xk ∂xj ∂xk ∂xk ∂xl
+ (lower order terms).
We now compute the symbol of DE. This is to take the highest order derivatives and ∂ replace ∂x i by the Fourier transform variable ζi . The symbol of the linear differential operator DE(gij ) in the direction ζ = (ζ1 , . . . , ζn ) is σDE(gij )(ζ)˜ gij = g kl (ζi ζj g˜kl + ζk ζl g˜ij − ζi ζk g˜jl − ζj ζk g˜il ). To see what the symbol does, we can always assume ζ has length 1 and choose coordinates at a point such that gij = δij , Then
ζ = (1, 0, . . . , 0).
(σDE(gij )(ζ))(˜ gij ) = g˜ij + δi1 δj1 (˜ g11 + · · · + g˜nn ) − δi1 g˜1j − δj1 g˜1i ,
i.e., [σDE(gij )(ζ)(˜ gij )]11 = g˜22 + · · · + g˜nn ,
[σDE(gij )(ζ)(˜ gij )]1k = 0, [σDE(gij )(ζ)(˜ gij )]kl = g˜kl ,
if k 6= 1, if k = 6 1, l 6= 1.
179
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
In particular
(˜ gij ) =
∗ ∗ .. . ∗
∗ ··· 0 ··· .. . . . . 0 ···
∗ 0 .. . 0
are zero eigenvectors of the symbol. The presence of the zero eigenvalue shows that the system can not be strictly parabolic. Therefore, instead of considering the system (1.2.1) (or the Ricci flow equation (1.1.5)) we will follow a trick of De Turck[43] to consider a modified evolution equation, which turns out to be strictly parabolic, so that we can apply the standard theory of parabolic equations. Suppose gˆij (x, t) is a solution of the Ricci flow (1.1.5), and ϕt : M → M is a family of diffeomorphisms of M . Let gij (x, t) = ϕ∗t gˆij (x, t) be the pull-back metrics. We now want to find the evolution equation for the metrics gij (x, t). Denote by y(x, t) = ϕt (x) = {y 1 (x, t), y 2 (x, t), . . . , y n (x, t)} in local coordinates. Then (1.2.2)
gij (x, t) =
∂y α ∂y β gˆαβ (y, t), ∂xi ∂xj
and ∂ ∂y α ∂y β ∂ gij (x, t) = gˆαβ (y, t) ∂t ∂t ∂xi ∂xj =
α β ∂y α ∂y β ∂ ∂ ∂y ∂y g ˆ (y, t) + gˆαβ (y, t) αβ i j i ∂x ∂x ∂t ∂x ∂t ∂xj β ∂y α ∂ ∂y + gˆαβ (y, t). ∂xi ∂xj ∂t
Let us choose a normal coordinate {xi } around a fixed point p ∈ M such that at p. Since gαβ ∂y γ ∂ ˆ αβ (y, t) + ∂ˆ gˆαβ (y, t) = −2R , ∂t ∂y γ ∂t
∂gij ∂xk
=0
180
H.-D. CAO AND X.-P. ZHU
we have in the normal coordinate, ∂ gij (x, t) ∂t ∂y α ∂y β ∂ˆ gαβ ∂y γ ∂y α ∂y β ˆ Rαβ (y, t) + = −2 i j i j ∂x ∂x ∂x ∂x ∂y γ ∂t α β β α ∂ ∂y ∂y ∂ ∂y ∂y + g ˆ (y, t) + gˆαβ (y, t) αβ ∂xi ∂t ∂xj ∂xj ∂t ∂xi α k ∂y α ∂y β ∂ˆ gαβ ∂y γ ∂ ∂y ∂x = −2Rij (x, t) + + i gjk ∂xi ∂xj ∂y γ ∂t ∂x ∂t ∂y α β k ∂y ∂x ∂ gik + ∂xj ∂t ∂y β α k ∂y α ∂y β ∂ˆ gαβ ∂y γ ∂ ∂y ∂x = −2Rij (x, t) + + i gjk ∂xi ∂xj ∂y γ ∂t ∂x ∂t ∂y α β k k k ∂ ∂y ∂x ∂y α ∂ ∂x ∂y β ∂ ∂x + g − g − gik . ik jk j β i α j ∂x ∂t ∂y ∂t ∂x ∂y ∂t ∂x ∂y β The second term on the RHS gives, in the normal coordinate, k gαβ ∂y α ∂y β ∂y γ ∂ ∂x ∂xl ∂y α ∂y β ∂y γ ∂ˆ = g kl ∂xi ∂xj ∂t ∂y γ ∂xi ∂xj ∂t ∂y γ ∂y α ∂y β k ∂y α ∂y γ ∂ ∂xk ∂y β ∂y γ ∂ ∂x = g + gik jk ∂xi ∂t ∂y γ ∂y α ∂xj ∂t ∂y γ ∂y β ∂y α ∂ 2 xk ∂y β gjk + ∂t ∂y α ∂y β ∂xi k ∂x ∂y α ∂ gjk + = ∂t ∂xi ∂y α =
∂y β ∂t ∂y β ∂t
∂ 2 xk ∂y α gik ∂y α ∂y β ∂xj k ∂ ∂x gik . ∂xj ∂y β
So we get (1.2.3)
∂ gij (x, t) ∂t = −2Rij (x, t) + ∇i
∂y α ∂xk gjk ∂t ∂y α
+ ∇j
If we define y(x, t) = ϕt (x) by the equations ok ∂yα = ∂yαk g jl (Γk − Γ ), jl jl ∂t ∂x (1.2.4) α y (x, 0) = xα , ok
∂y β ∂xk gik . ∂t ∂y β
and Vi = gik g jl (Γkjl − Γjl ), we get the following evolution equation for the pull-back metric ∂ ∂t gij (x, t) = −2Rij (x, t) + ∇i Vj + ∇j Vi , (1.2.5) o gij (x, 0) =gij (x),
181
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW ok
o
where g ij (x) is the initial metric and Γjl is the connection of the initial metric. Lemma 1.2.1. The modified evolution equation (1.2.5) is a strictly parabolic system. Proof. The RHS of the equation (1.2.5) is given by − 2Rij (x, t) + ∇i Vj + ∇j Vi ∂ ∂ ∂gjl ∂gil ∂gij kl ∂gkl kl = g − g + − ∂xi ∂xj ∂xk ∂xi ∂xj ∂xl 1 kl ∂gpl ∂gql ∂gpq ∂ g + − + gjk g pq i ∂x 2 ∂xq ∂xp ∂xl 1 kl ∂gpl ∂gql ∂gpq pq ∂ + gik g g + − ∂xj 2 ∂xq ∂xp ∂xl + (lower order terms) 2 ∂ 2 gjl ∂ 2 gil ∂ 2 gij ∂ gkl kl − i k − + k l =g ∂xi ∂xj ∂x ∂x ∂xj ∂xk ∂x ∂x 2 2 2 1 pq ∂ gpj ∂ gqj ∂ gpq + g + − i j 2 ∂xi ∂xq ∂xi ∂xp ∂x ∂x 2 ∂ gpi ∂ 2 gqi ∂ 2 gpq 1 pq + j p− i j + g 2 ∂xj ∂xq ∂x ∂x ∂x ∂x + (lower order terms)
= g kl
∂ 2 gij + (lower order terms). ∂xk ∂xl
Thus its symbol is (g kl ζk ζl )˜ gij . Hence the equation in (1.2.5) is strictly parabolic. Now since the equation (1.2.5) is strictly parabolic and the manifold M is compact, it follows from the standard theory of parabolic equations (see for example [81]) that (1.2.5) has a solution for a short time. From the solution of (1.2.5) we can obtain a solution of the Ricci flow from (1.2.4) and (1.2.2). This shows existence. Now we argue the uniqueness of the solution. Since Γkjl =
∂xk ∂ 2 y α ∂y α ∂y β ∂xk ˆ γ Γ + , ∂xj ∂xl ∂y γ αβ ∂y α ∂xj ∂xl
the initial value problem (1.2.4) can be written as o k ∂y α α ∂ 2 yα jl ˆ α ∂yβ ∂y − Γ j l jl ∂xk + Γγβ ∂xj ∂t = g ∂x ∂x (1.2.6) α y (x, 0) = xα .
∂y γ ∂xl
,
(1)
(2)
This is clearly a strictly parabolic system. For any two solutions gˆij (·, t) and gˆij (·, t) of the Ricci flow (1.1.5) with the same initial data, we can solve the initial value (1) (2) problem (1.2.6) (or equivalently, (1.2.4)) to get two families ϕt and ϕt of dif(1) (1) ∗ (1) feomorphisms of M . Thus we get two solutions, gij (·, t) = (ϕt ) gˆij (·, t) and (2)
(2)
(2)
gij (·, t) = (ϕt )∗ gˆij (·, t), to the modified evolution equation (1.2.5) with the same
182
H.-D. CAO AND X.-P. ZHU
initial metric. The uniqueness result for the strictly parabolic equation (1.2.5) implies (1) (2) that gij = gij . Then by equation (1.2.4) and the standard uniqueness result of ODE (1)
(2)
systems, the corresponding solutions ϕt and ϕt of (1.2.4) (or equivalently (1.2.6)) (1) (2) must agree. Consequently the metrics gˆij and gˆij must agree also. Thus we have proved the following result. Theorem 1.2.2 (Hamilton [58], De Turck [43]). Let (M, gij (x)) be a compact Riemannian manifold. Then there exists a constant T > 0 such that the initial value problem ∂ gij (x, t) = −2Rij (x, t) ∂t g (x, 0) = g (x) ij ij
has a unique smooth solution gij (x, t) on M × [0, T ). The case of a noncompact manifold is much more complicated and involves a huge amount of techniques from the theory of partial differential equations. Here we will only state the existence and uniqueness results and refer the reader to the cited references for the proofs. The following existence result was obtained by Shi [114] in his thesis published in 1989. Theorem 1.2.3 (Shi [114]). Let (M, gij (x)) be a complete noncompact Riemannian manifold of dimension n with bounded curvature. Then there exists a constant T > 0 such that the initial value problem ∂ gij (x, t) = −2Rij (x, t) ∂t g (x, 0) = g (x) ij ij
has a smooth solution gij (x, t) on M × [0, T ] with uniformly bounded curvature. The Ricci flow is a heat type equation. It is well-known that the uniqueness of a heat equation on a complete noncompact manifold is not always held if there are no further restrictions on the growth of the solutions. For example, the heat equation on Euclidean space with zero initial data has a nontrivial solution which grows faster than exp(a|x|2 ) for any a > 0 whenever t > 0. This implies that even for the standard linear heat equation on Euclidean space, in order to ensure the uniqueness one can only allow the solution to grow at most as exp(C|x|2 ) for some constant C > 0. Note that 2 on a K¨ahler manifold, the Ricci curvature is given by Rαβ¯ = − ∂zα∂∂ z¯β log det(gγ δ¯). So the reasonable growth rate for the uniqueness of the Ricci flow to hold is that the solution has bounded curvature. Thus the following uniqueness result of Bing-Long Chen and the second author [33] is essentially the best one can hope for. Theorem 1.2.4 (Chen-Zhu [33]). Let (M, gˆij ) be a complete noncompact Riemannian manifold of dimension n with bounded curvature. Let gij (x, t) and g¯ij (x, t) be two solutions, defined on M × [0, T ], to the Ricci flow (1.1.5) with gˆij as initial data and with bounded curvatures. Then gij (x, t) ≡ g¯ij (x, t) on M × [0, T ].
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
183
1.3. Evolution of Curvatures. The Ricci flow is an evolution equation on the metric. The evolution for the metric implies a nonlinear heat equation for the Riemannian curvature tensor Rijkl which we will now derive. Proposition 1.3.1 (Hamilton [58]). Under the Ricci flow (1.1.5), the curvature tensor satisfies the evolution equation ∂ Rijkl = ∆Rijkl + 2(Bijkl − Bijlk − Biljk + Bikjl ) ∂t − g pq (Rpjkl Rqi + Ripkl Rqj + Rijpl Rqk + Rijkp Rql ) where Bijkl = g pr g qs Rpiqj Rrksl and ∆ is the Laplacian with respect to the evolving metric. Proof. Choose {x1 , . . . , xm } to be a normal coordinate system at a fixed point. At this point, we compute 1 hm ∂ ∂ ∂ ∂ ∂ ∂ ∂ h Γ = g glm + l gjm − m gjl ∂t jl 2 ∂xj ∂t ∂x ∂t ∂x ∂t 1 = g hm (∇j (−2Rlm ) + ∇l (−2Rjm ) − ∇m (−2Rjl )), 2 ∂ ∂ h ∂ ∂ h ∂ h R = Γ − j Γ , ∂t ijl ∂xi ∂t jl ∂x ∂t il ∂ ∂ h ∂ghk h Rijkl = ghk Rijl + R . ∂t ∂t ∂t ijl Combining these identities we get ∂ 1 hm Rijkl = ghk ∇i [g (∇j (−2Rlm ) + ∇l (−2Rjm ) − ∇m (−2Rjl ))] ∂t 2 1 hm ∇j [g (∇i (−2Rlm ) + ∇l (−2Rim ) − ∇m (−2Ril ))] − 2 h − 2Rhk Rijl
= ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik − Rijlp g pq Rqk − Rijkp g pq Rql − 2Rijpl g pq Rqk
= ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik − g pq (Rijkp Rql + Rijpl Rqk ).
Here we have used the exchanging formula (1.1.3). Now it remains to check the following identity, which is analogous to the Simon′ s identity in extrinsic geometry, (1.3.1)
∆Rijkl + 2(Bijkl − Bijlk − Biljk + Bikjl ) = ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik + g pq (Rpjkl Rqi + Ripkl Rqj ).
Indeed, from the second Bianchi identity (1.1.4), we have ∆Rijkl = g pq ∇p ∇q Rijkl = g pq ∇p ∇i Rqjkl − g pq ∇p ∇j Rqikl .
184
H.-D. CAO AND X.-P. ZHU
Let us examine the first term on the RHS. By using the exchanging formula (1.1.3) and the first Bianchi identity (1.1.1), we have g pq ∇p ∇i Rqjkl − g pq ∇i ∇p Rqjkl
= g pq g mn (Rpiqm Rnjkl + Rpijm Rqnkl + Rpikm Rqjnl + Rpilm Rqjkn ) = Rim g mn Rnjkl + g pq g mn Rpimj (Rqkln + Rqlnk ) + g pq g mn Rpikm Rqjnl + g pq g mn Rpilm Rqjkn = Rim g mn Rnjkl − Bijkl + Bijlk − Bikjl + Biljk ,
while using the contracted second Bianchi identity g pq ∇p Rqjkl = ∇k Rjl − ∇l Rjk ,
(1.3.2) we have
g pq ∇i ∇p Rqjkl = ∇i ∇k Rjl − ∇i ∇l Rjk . Thus g pq ∇p ∇i Rqjkl
= ∇i ∇k Rjl − ∇i ∇l Rjk − (Bijkl − Bijlk − Biljk + Bikjl ) + g pq Rpjkl Rqi .
Therefore we obtain ∆Rijkl = g pq ∇p ∇i Rqjkl − g pq ∇p ∇j Rqikl = ∇i ∇k Rjl − ∇i ∇l Rjk − (Bijkl − Bijlk − Biljk + Bikjl ) + g pq Rpjkl Rqi
− ∇j ∇k Ril + ∇j ∇l Rik + (Bjikl − Bjilk − Bjlik + Bjkil ) − g pq Rpikl Rqj = ∇i ∇k Rjl − ∇i ∇l Rjk − ∇j ∇k Ril + ∇j ∇l Rik + g pq (Rpjkl Rqi + Ripkl Rqj ) − 2(Bijkl − Bijlk − Biljk + Bikjl )
as desired, where in the last step we used the symmetries (1.3.3)
Bijkl = Bklij = Bjilk .
Corollary 1.3.2. The Ricci curvature satisfies the evolution equation ∂ Rik = ∆Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk . ∂t Proof. ∂ ∂ Rik = g jl Rijkl + ∂t ∂t
∂ jl g Rijkl ∂t
= g jl [∆Rijkl + 2(Bijkl − Bijlk − Biljk + Bikjl ) − g pq (Rpjkl Rqi + Ripkl Rqj + Rijpl Rqk + Rijkp Rql )] ∂ − g jp gpq g ql Rijkl ∂t
= ∆Rik + 2g jl (Bijkl − 2Bijlk ) + 2g pr g qs Rpiqk Rrs − 2g pq Rpk Rqi .
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
185
We claim that g jl (Bijkl − 2Bijlk ) = 0. Indeed by using the first Bianchi identity, we have g jl Bijkl = g jl g pr g qs Rpiqj Rrksl = g jl g pr g qs Rpqij Rrskl = g jl g pr g qs (Rpiqj − Rpjqi )(Rrksl − Rrlsk ) = 2g jl (Bijkl − Bijlk )
as desired. Thus we obtain ∂ Rik = ∆Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk . ∂t
Corollary 1.3.3. The scalar curvature satisfies the evolution equation ∂R = ∆R + 2|Ric |2 . ∂t
Proof. ∂R ik ∂Rik ip ∂gpq qk =g + −g g Rik ∂t ∂t ∂t
= g ik (∆Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk ) + 2Rpq Rik g ip g qk = ∆R + 2|Ric |2 .
To simplify the evolution equations of curvatures, we will represent the curvature tensors in an orthonormal frame and evolve the frame so that it remains orthonormal. More precisely, let us pick an abstract vector bundle V over M isomorphic to the tangent bundle T M . Locally, the frame F = {F1 , . . . , Fa , . . . , Fn } of V is ∂ i i given by Fa = Fai ∂x i with the isomorphism {Fa }. Choose {Fa } at t = 0 such that F = {F1 , . . . , Fa , . . . , Fn } is an orthonormal frame at t = 0, and evolve {Fai } by the equation ∂ i F = g ij Rjk Fak . ∂t a Then the frame F = {F1 , . . . , Fa , . . . , Fn } will remain orthonormal for all times since the pull back metric on V hab = gij Fai Fbj remains constant in time. In the following we will use indices a, b, . . . on a tensor to denote its components in the evolving orthonormal frame. In this frame we have the
186
H.-D. CAO AND X.-P. ZHU
following: Rabcd = Fai Fbj Fck Fdl Rijkl , Γajb = Fia
∂Fbi + Γijk Fia Fbk , ∂xj
((Fia ) = (Fai )−1 )
∂ a V + Γaib V b , ∂xi ∇b V a = Fbi ∇i V a , ∇i V a =
where Γajb is the metric connection of the vector bundle V with the metric hab . Indeed, by direct computations, ∂Fbj + Fbk Γjik − Fcj Γcib ∂xi k ∂Fbj k j j c ∂Fb l c k + Fb Γik − Fc Fk + Γik Fl Fb = ∂xi ∂xi
∇i Fbj =
= 0,
∇i hab = ∇i (gij Fai Fbj ) = 0. So ∇a Vb = Fai Fbj ∇i V j , and ∆Rabcd = ∇l ∇l Rabcd
= g ij ∇i ∇j Rabcd
= g ij Fak Fbl Fcm Fdn ∇i ∇j Rklmn . In an orthonormal frame F = {F1 , . . . , Fa , . . . , Fn }, the evolution equations of curvature tensors become (1.3.4) (1.3.5) (1.3.6)
∂ Rabcd = ∆Rabcd + 2(Babcd − Babdc − Badbc + Bacbd ) ∂t ∂ Rab = ∆Rab + 2Racbd Rcd ∂t ∂ R = ∆R + 2|Ric |2 ∂t
where Babcd = Raebf Rcedf . Equation (1.3.4) is a reaction-diffusion equation. We can understand the quadratic terms of this equation better if we think of the curvature tensor Rabcd as a symmetric bilinear form on the two-forms Λ2 (V ) given by the formula Rm(ϕ, ψ) = Rabcd ϕab ψcd ,
for ϕ, ψ ∈ Λ2 (V ).
A two-form ϕ ∈ Λ2 (V ) can be regarded as an element of the Lie algebra so(n) (i.e. the skew-symmetric matrix (ϕab )n×n ), where the metric on Λ2 (V ) is given by hϕ, ψi = ϕab ψab
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
187
and the Lie bracket is given by [ϕ, ψ]ab = ϕac ψbc − ψac ϕbc . Choose an orthonormal basis of Λ2 (V ) Φ = {ϕ1 , . . . , ϕα , . . . , ϕ
n(n−1) 2
}
where ϕα = {ϕα ab }. The Lie bracket is given by [ϕα , ϕβ ] = Cγαβ ϕγ , where C αβγ = Cσαβ δ σγ = h[ϕα , ϕβ ], ϕγ i are the Lie structure constants. β Write Rabcd = Mαβ ϕα ab ϕcd . We now claim that the first part of the quadratic terms in (1.3.4) is given by β 2(Babcd − Babdc ) = Mαγ Mβγ ϕα ab ϕcd .
(1.3.7)
Indeed, by the first Bianchi identity, Babcd − Babdc = Raebf Rcedf − Raebf Rdecf = Raebf (−Rcef d − Rcf de ) = Raebf Rcdef .
On the other hand, Raebf Rcdef = (−Rabf e − Raf eb )Rcdef
= Rabef Rcdef − Raf eb Rcdef = Rabef Rcdef − Raf be Rcdf e
which implies Raebf Rcdef = 12 Rabef Rcdef . Thus we obtain β 2(Babcd − Babdc ) = Rabef Rcdef = Mαγ Mβγ ϕα ab ϕcd .
We next consider the last part of the quadratic terms: 2(Bacbd − Badbc )
= 2(Raecf Rbedf − Raedf Rbecf )
= 2(Mγδ ϕγae ϕδcf Mηθ ϕηbe ϕθdf − Mγθ ϕγae ϕθdf Mηδ ϕηbe ϕδcf )
γ δ δ θ η θ = 2[Mγδ (ϕηae ϕγbe + Cαγη ϕα ab )ϕcf Mηθ ϕdf − Mηθ ϕae ϕdf Mγδ ϕbe ϕcf ]
= 2Mγδ ϕδcf Mηθ ϕθdf Cαγη ϕα ab . But Mγδ ϕδcf Mηθ ϕθdf Cαγη ϕα ab
θ δ δθ β = Mγδ Mηθ Cαγη ϕα ab [ϕcf ϕdf + Cβ ϕcd ] γη δθ α β δ θ = −Mηθ Mγδ Cαγη ϕα ab ϕcf ϕdf + Mγδ Mηθ Cα Cβ ϕab ϕcd
γη δθ α β δ θ = −Mηθ Mγδ Cαγη ϕα ab ϕcf ϕdf + (Cα Cβ Mγδ Mηθ )ϕab ϕcd
188
H.-D. CAO AND X.-P. ZHU
which implies Mγδ ϕδcf Mηθ ϕθdf Cαγη ϕα ab =
1 γη δθ β (C C Mγδ Mηθ )ϕα ab ϕcd . 2 α β
Then we have β 2(Bacbd − Badbc ) = (Cαγη Cβδθ Mγδ Mηθ )ϕα ab ϕcd .
(1.3.8)
Therefore, combining (1.3.7) and (1.3.8), we can reformulate the curvature evolution equation (1.3.4) as follows. β Proposition 1.3.4 (Hamilton [59]). Let Rabcd = Mαβ ϕα ab ϕcd . Then under the Ricci flow (1.1.5), Mαβ satisfies the evolution equation
∂Mαβ # 2 = ∆Mαβ + Mαβ + Mαβ ∂t
(1.3.9)
# 2 where Mαβ = Mαγ Mβγ is the operator square and Mαβ = (Cαγη Cβδθ Mγδ Mηθ ) is the Lie algebra square. # Let us now consider the operator Mαβ in dimensions 3 and 4 in more detail. In dimension 3, let ω1 , ω2 , ω3 be a positively oriented orthonormal basis for oneforms. Then √ √ √ ϕ1 = 2ω1 ∧ ω2 , ϕ2 = 2ω2 ∧ ω3 , ϕ3 = 2ω3 ∧ ω1
form an orthonormal basis for two-forms Λ2 . Write ϕα = {ϕα ab }, α = 1, 2, 3, as √ 2 0 0 0 0 0√ √ 2 2 0√ (ϕ1ab ) = − 2 , (ϕ2ab ) = 0 0 0 2 , 2 0 − 22 0 0 0 0 √ 0 0 − 22 3 0 , (ϕab ) = √0 0 2 0 0 2
then
√ √ 2 0 0 0√ 0 0 √0 0 − 0 0√ 22 0 2 √ [ϕ1 , ϕ2 ] = − 2 0 0 0 √0 − 22 − 0 0√ 22 2 0 0 2 2 2 2 0 0 0 0 0 0 0 0 2 0− 2 0 0 0 − 21 = 00 0 1 2 0 0 √ 2 3 ϕ . = 2
So C 123 = h[ϕ1 , ϕ2 ], ϕ3 i =
√ 2 2 ,
in particular ( √ ± 22 , if α 6= β 6= γ, αβγ C = 0, otherwise.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
189
# Hence the matrix M # = (Mαβ ) is just the adjoint matrix of M = (Mαβ ):
M # = det M · t M −1 .
(1.3.10)
In dimension 4, we can use the Hodge star operator to decompose the space of two-forms Λ2 as Λ2 = Λ2+ ⊕Λ2− where Λ2+ (resp. Λ2− ) is the eigenspace of the star operator with eigenvalue +1 (resp. −1). Let ω1 , ω2 , ω3 , ω4 be a positively oriented orthonormal basis for one-forms. A basis for Λ2+ is then given by ϕ1 = ω1 ∧ ω2 + ω3 ∧ ω4 ,
ϕ2 = ω1 ∧ ω3 + ω4 ∧ ω2 ,
ϕ3 = ω1 ∧ ω4 + ω2 ∧ ω3 ,
while a basis for Λ2− is given by ψ 1 = ω1 ∧ ω2 − ω3 ∧ ω4 ,
ψ 2 = ω1 ∧ ω3 − ω4 ∧ ω2 ,
ψ 3 = ω1 ∧ ω4 − ω2 ∧ ω3 .
In particular, {ϕ1 , ϕ2 , ϕ3 , ψ 1 , ψ 2 , ψ 3 } forms an orthonormal basis for the space of two-forms Λ2 . By using this basis we obtain a block decomposition of the curvature operator matrix M as A B M = (Mαβ ) = . t B C Here A, B and C are 3 × 3 matrices with A and C being symmetric. Then we can write each element of the basis as a skew-symmetric 4 × 4 matrix and compute as above to get # A B# # (1.3.11) M # = (Mαβ )=2 t # , B C# where A# , B # , C # are the adjoint of 3 × 3 submatrices as before. For later applications in Chapter 5, we now give some computations for the entries of the matrices A, C and B as follows. First for the matrices A and C, we have A11 = Rm(ϕ1 , ϕ1 ) = R1212 + R3434 + 2R1234 A22 = Rm(ϕ2 , ϕ2 ) = R1313 + R4242 + 2R1342 A33 = Rm(ϕ3 , ϕ3 ) = R1414 + R2323 + 2R1423 and C11 = Rm(ψ 1 , ψ 1 ) = R1212 + R3434 − 2R1234 C22 = Rm(ψ 2 , ψ 2 ) = R1313 + R4242 − 2R1342 C33 = Rm(ψ 3 , ψ 3 ) = R1414 + R2323 − 2R1423 .
190
H.-D. CAO AND X.-P. ZHU
By the Bianchi identity R1234 + R1342 + R1423 = 0, so we have trA = trC =
1 R. 2
Next for the entries of the matrix B, we have B11 = Rm(ϕ1 , ψ 1 ) = R1212 − R3434 B22 = Rm(ϕ2 , ψ 2 ) = R1313 − R4242 B33 = Rm(ϕ3 , ψ 3 ) = R1414 − R2323 and B12 = Rm(ϕ1 , ψ 2 ) = R1213 + R3413 − R1242 − R3442
etc.
Thus the entries of B can be written as B11 =
1 (R11 + R22 − R33 − R44 ) 2
B22 =
1 (R11 + R33 − R44 − R22 ) 2
B33 =
1 (R11 + R44 − R22 − R33 ) 2
and B12 = R23 − R14
etc.
If we choose the frame {ω1 , ω2 , ω3 , ω4 } so that the Ricci tensor is diagonal, then the matrix B is also diagonal. In particular, the matrix B is identically zero when the four-manifold is Einstein. 1.4. Derivative Estimates. In the previous section we have seen that the curvatures satisfy nonlinear heat equations with quadratic growth terms. The parabolic nature will give us a bound on the derivatives of the curvatures at any time t > 0 in terms of a bound of the curvatures. We begin with the global version of the derivative estimates. Theorem 1.4.1 (Shi [114]). There exist constants Cm , m = 1, 2, . . . , such that if the curvature of a complete solution to Ricci flow is bounded by |Rijkl | ≤ M up to time t with 0 < t ≤ by
1 M,
then the covariant derivative of the curvature is bounded √ |∇Rijkl | ≤ C1 M/ t
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
191
and the mth covariant derivative of the curvature is bounded by |∇m Rijkl | ≤ Cm M/t 2 . m
Here the norms are taken with respect to the evolving metric. Proof. We shall only give the proof for the compact case. The noncompact case can be deduced from the next local derivative estimate theorem. Let us denote the curvature tensor by Rm and denote by A ∗ B any tensor product of two tensors A and B when we do not need the precise expression. We have from Proposition 1.3.1 that ∂ Rm = ∆Rm + Rm ∗ Rm. ∂t
(1.4.1) Since
∂ i ∂gkl 1 il ∂gjl ∂gjk Γ = g ∇j + ∇k − ∇l ∂t jk 2 ∂t ∂t ∂t = ∇Rm,
it follows that (1.4.2)
∂ (∇Rm) = ∆(∇Rm) + Rm ∗ (∇Rm). ∂t
Thus ∂ |Rm|2 ≤ ∆|Rm|2 − 2|∇Rm|2 + C|Rm|3 , ∂t ∂ |∇Rm|2 ≤ ∆|∇Rm|2 − 2|∇2 Rm|2 + C|Rm| · |∇Rm|2 , ∂t for some constant C depending only on the dimension n. Let A >0 be a constant (to be determined) and set F = t|∇Rm|2 + A|Rm|2 . We compute ∂ ∂ ∂F = |∇Rm|2 + t |∇Rm|2 + A |Rm|2 ∂t ∂t ∂t ≤ ∆(t|∇Rm|2 + A|Rm|2 ) + |∇Rm|2 (1 + tC|Rm| − 2A) + CA|Rm|3 . Taking A ≥ C + 1, we get ∂F ¯ 3 ≤ ∆F + CM ∂t for some constant C¯ depending only on the dimension n. We then obtain 2 ¯ 3 t ≤ (A + C)M ¯ F ≤ F (0) + CM ,
and then 2 ¯ |∇Rm|2 ≤ (A + C)M /t.
192
H.-D. CAO AND X.-P. ZHU
The general case follows in the same way. If we have bounds |∇k Rm| ≤ Ck M/t 2 , k
we know from (1.4.1) and (1.4.2) that ∂ k CM 3 |∇ Rm|2 ≤ ∆|∇k Rm|2 − 2|∇k+1 Rm|2 + k , ∂t t and CM 3 ∂ k+1 |∇ Rm|2 ≤ ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 + k+1 . ∂t t Let Ak > 0 be a constant (to be determined) and set Fk = tk+2 |∇k+1 Rm|2 + Ak tk+1 |∇k Rm|2 . Then ∂ ∂ Fk = (k + 2)tk+1 |∇k+1 Rm|2 + tk+2 |∇k+1 Rm|2 ∂t ∂t ∂ + Ak (k + 1)tk |∇k Rm|2 + Ak tk+1 |∇k Rm|2 ∂t ≤ (k + 2)tk+1 |∇k+1 Rm|2 CM 3 + tk+2 ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 + k+1 t + Ak (k + 1)tk |∇k Rm|2 CM 3 + Ak tk+1 ∆|∇k Rm|2 − 2|∇k+1 Rm|2 + k t
≤ ∆Fk + Ck+1 M 2
for some positive constant Ck+1 , by choosing Ak large enough. This implies that |∇k+1 Rm| ≤
Ck+1 M t
k+1 2
.
The above derivative estimate is a somewhat standard Bernstein estimate in PDEs. By using a cutoff argument, we will derive the following local version, which is called Shi’s derivative estimate. The following proof is adapted from Hamilton [63]. Theorem 1.4.2 (Shi [114]). There exist positive constants θ, Ck , k = 1, 2, . . . , depending only on the dimension with the following property. Suppose that the curvature of a solution to the Ricci flow is bounded θ |Rm| ≤ M, on U × 0, M where U is an open set of the manifold. Assume that the closed ball B0 (p, r), centered at p of radius r with respect to the metric at t = 0, is contained in U and the time t ≤ θ/M . Then we can estimate the covariant derivatives of the curvature at (p, t) by 1 1 + + M , |∇Rm(p, t)|2 ≤ C1 M 2 r2 t
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
193
and the k th covariant derivative of the curvature at (p, t) by 1 1 k 2 2 k |∇ Rm(p, t)| ≤ Ck M + k +M . r2k t √ Proof. Without loss of generality, we may assume r ≤ θ/ M and the exponential map at p at time t = 0 is injective on the ball of radius r (by passing to a local cover if necessary, and pulling back the local solution of the Ricci flow to the ball of radius r in the tangent space at p at time t = 0). Recall ∂ |Rm|2 ≤ ∆|Rm|2 − 2|∇Rm|2 + C|Rm|3 , ∂t ∂ |∇Rm|2 ≤ ∆|∇Rm|2 − 2|∇2 Rm|2 + C|Rm| · |∇Rm|2 . ∂t Define S = (BM 2 + |Rm|2 )|∇Rm|2 where B is a positive constant to be determined. By choosing B ≥ C 2 /4 and using the Cauchy inequality, we have ∂ S ≤ ∆S − 2BM 2 |∇2 Rm|2 − 2|∇Rm|4 ∂t + CM |∇Rm|2 · |∇2 Rm| + CBM 3 |∇Rm|2 ≤ ∆S − |∇Rm|4 + CB 2 M 6
≤ ∆S −
S2 + CB 2 M 6 . (B + 1)2 M 4
If we take F = b(BM 2 + |Rm|2 )|∇Rm|2 /M 4 = bS/M 4 , and b ≤ min{1/(B + 1)2 , 1/CB 2 }, we get (1.4.3)
∂F ≤ ∆F − F 2 + M 2 . ∂t
We now want to choose a cutoff function ϕ with the support in the ball B0 (p, r) such that at t = 0, ϕ(p) = r,
0 ≤ ϕ ≤ Ar,
and |∇ϕ| ≤ A,
|∇2 ϕ| ≤
A r
for some positive constant A depending only on the dimension. Indeed, let g : (−∞, +∞) → [0, +∞) be a smooth, nonnegative function satisfying ( 1, u ∈ (− 21 , 12 ), g(u) = 0, outside (−1, 1).
194
H.-D. CAO AND X.-P. ZHU
Set ϕ = rg
s2 r2
,
where s is the geodesic distance function from p with respect to the metric at t = 0. Then 2 s 1 · 2s∇s ∇ϕ = g ′ r r2 and hence |∇ϕ| ≤ 2C1 . Also, ∇2 ϕ =
1 ′′ g r
s2 r2
1 2 1 4s ∇s · ∇s + g ′ 2 r r
s2 r2
1 2∇s · ∇s + g ′ r
s2 r2
· 2s∇2 s.
Thus, by using the standard Hessian comparison, C1 C1 + |s∇2 s| r r C2 √ C1 1+s + M ≤ r s C3 . ≤ r
|∇2 ϕ| ≤
Here C1 , C2 and C3 are positive constants depending only on the dimension. θ Now extend ϕ to U × [0, M ] by letting ϕ to be zero outside B0 (p, r) and independent of time. Introduce the barrier function √ (12 + 4 n)A2 1 (1.4.4) H= + +M 2 ϕ t which is defined and smooth on the set {ϕ > 0} × (0, T ]. As the metric evolves, we will still have 0 ≤ ϕ ≤ Ar (since ϕ is independent of time t); but |∇ϕ|2 and ϕ|∇2 ϕ| may increase. By continuity it will be a while before they double. Claim 1. As long as |∇ϕ|2 ≤ 2A2 , ϕ|∇2 ϕ| ≤ 2A2 , we have ∂H > ∆H − H 2 + M 2 . ∂t Indeed, by the definition of H, we have √ 1 (12 + 4 n)2 A4 2 + 2 + M 2, H > 4 ϕ t ∂H 1 = − 2, ∂t t
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
195
and √ 1 2 ∆H = (12 + 4 n)A ∆ ϕ2 √ 6|∇ϕ|2 − 2ϕ∆ϕ 2 = (12 + 4 n)A ϕ4 √ 2 √ 12A + 4 nA2 2 ≤ (12 + 4 n)A ϕ4 √ 2 4 (12 + 4 n) A . = ϕ4 Therefore, H 2 > ∆H −
∂H + M 2. ∂t
Claim 2. If the constant θ > 0 is small enough √ compared to b, B and A, then we have the following property: as long as r ≤ θ/ M , t ≤ θ/M and F ≤ H, we will have |∇ϕ|2 ≤ 2A2
and
ϕ|∇2 ϕ| ≤ 2A2 .
Indeed, by considering the evolution of ∇ϕ, we have ∂ ∂ ∇a ϕ = (Fai ∇i ϕ) ∂t ∂t ∂ϕ = Fai ∇i + ∇i ϕRki Fak ∂t = Rab ∇b ϕ
which implies ∂ |∇ϕ|2 ≤ CM |∇ϕ|2 , ∂t and then |∇ϕ|2 ≤ A2 eCMt ≤ 2A2 , provided t ≤ θ/M with θ ≤ log 2/C. By considering the evolution of ∇2 ϕ, we have ∂ ∂ (∇a ∇b ϕ) = (Fai Fbj ∇i ∇j ϕ) ∂t ∂t 2 ∂ ∂ ϕ i j k ∂ϕ = Fa Fb − Γij k ∂t ∂xi ∂xj ∂x ∂ϕ + Rac ∇b ∇c ϕ + Rbc ∇a ∇c ϕ = ∇a ∇b ∂t + (∇c Rab − ∇a Rbc − ∇b Rac )∇c ϕ
196
H.-D. CAO AND X.-P. ZHU
which implies ∂ 2 |∇ ϕ| ≤ C|Rm| · |∇2 ϕ| + C|∇Rm| · |∇ϕ|. ∂t
(1.4.5)
By assumption F ≤ H, we have √ 1 2M 2 (12 + 4 n)A2 2 + , (1.4.6) |∇Rm| ≤ bB ϕ2 t
for t ≤ θ/M.
Thus by noting ϕ independent of t and ϕ ≤ Ar, we get from (1.4.5) and (1.4.6) that ∂ r 2 2 √ (ϕ|∇ ϕ|) ≤ CM ϕ|∇ ϕ| + 1 + ∂t t which implies Z t r ϕ|∇2 ϕ| ≤ eCMt (ϕ|∇2 ϕ|)|t=0 + CM 1 + √ dt t 0 h √ i CMt 2 ≤e A + CM (t + 2r t) ≤ 2A2
√ provided r ≤ θ/ M , and t ≤ θ/M with θ small enough. Therefore we have obtained Claim 2. The combination of Claim 1 and Claim 2 gives us
√ as long as r ≤ θ/ M ,
∂H > ∆H − H 2 + M 2 ∂t t ≤ θ/M and F ≤ H. And (1.4.3) tells us ∂F ≤ ∆F − F 2 + M 2 . ∂t
Then the standard maximum principle immediately gives the estimate θ i 1 1 + + M on {ϕ > 0} × 0, , |∇Rm|2 ≤ CM 2 ϕ2 t M
which implies the first order derivative estimate. The higher order derivative estimates can be obtained in the same way by induction. Suppose we have the bounds 1 1 k |∇k Rm|2 ≤ Ck M 2 + + M . r2k tk As before, by (1.4.1) and (1.4.2), we have ∂ k |∇ Rm|2 ≤ ∆|∇k Rm|2 − 2|∇k+1 Rm|2 + CM 3 ∂t
1 1 k + k +M , r2k t
and ∂ k+1 |∇ Rm|2 ≤ ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 ∂t k+1
+ CM |∇
2
Rm| + CM
3
1
r2(k+1)
+
1 tk+1
+M
k+1
.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
197
Here and in the following we denote by C various positive constants depending only on Ck and the dimension. Define
Sk = Bk M
2
1 1 + k + Mk 2k r t
k
2
+ |∇ Rm|
· |∇k+1 Rm|2
where Bk is a positive constant to be determined. By choosing Bk large enough and Cauchy inequality, we have ∂ Sk ≤ ∂t
k + ∆|∇k Rm|2 − 2|∇k+1 Rm|2 tk+1 1 1 k + + M · |∇k+1 Rm|2 + CM 3 r2k tk 1 1 k k 2 + Bk M 2 + |∇ Rm| + + M r2k tk · ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 1 1 k+1 + + CM 3 + M tk+1 r2(k+1) k ≤ ∆Sk + 8|∇k Rm| · |∇k+1 Rm|2 · |∇k+2 Rm| − k+1 |∇k+1 Rm|2 t 1 1 k+1 4 3 k+1 2 k − 2|∇ Rm| + CM |∇ Rm| + + M r2k tk 1 1 k k 2 + + M + |∇ Rm| − 2|∇k+2 Rm|2 Bk M 2 r2k tk 1 1 k k 2 + CM |∇k+1 Rm|2 Bk M 2 + |∇ Rm| + + M r2k tk 1 1 + CM 3 + k+1 + M k+1 t r2(k+1) 1 1 k k 2 · Bk M 2 + + M + |∇ Rm| r2k tk 1 1 2k + + M ≤ ∆Sk − |∇k+1 Rm|4 + CBk2 M 6 r4k t2k 1 1 2k+1 + CBk M 5 + + M t2k+1 r2(2k+1) 1 1 2k+1 + M ≤ ∆Sk − |∇k+1 Rm|4 + CBk2 M 5 + t2k+1 r2(2k+1) Sk ≤ ∆Sk − 2 (B + 1)2 M 4 r12k + t1k + M k 1 1 2 5 2k+1 + 2k+1 + M + CBk M . t r2(2k+1) −
198
H.-D. CAO AND X.-P. ZHU
Let u = 1/r2 + 1/t + M and set Fk = bSk /uk . Then ∂Fk Fk2 ≤ ∆Fk − + bCBk2 M 5 uk+1 + kFk u ∂t b(Bk + 1)2 M 4 uk Fk2 + b(C + 2k 2 )(Bk + 1)2 M 4 uk+2 . ≤ ∆Fk − 2b(Bk + 1)2 M 4 uk By choosing b ≤ 1/(2(C + 2k 2 )(Bk + 1)2 M 4 ), we get 1 ∂Fk ≤ ∆Fk − k Fk2 + uk+2 . ∂t u Introduce Hk = 5(k + 1)(2(k + 1) + 1 +
√ n)A2 ϕ−2(k+1) + Lt−(k+1) + M k+1 ,
where L ≥ k + 2. Then by using Claim 1 and Claim 2, we have ∂Hk = −(k + 1)Lt−(k+2) , ∂t ∆Hk ≤ 20(k + 1)2 (2(k + 1) + 1 +
√ n)A4 ϕ−2(k+2)
and Hk2 > 25(k + 1)2 (2(k + 1) + 1 +
√
n)A4 ϕ−4(k+1) + L2 t−2(k+1) + M 2(k+1) .
These imply 1 ∂Hk > ∆Hk − k Hk2 + uk+2 . ∂t u Then the maximum principle immediately gives the estimate Fk ≤ Hk . In particular, b Bk M 2 uk
1 1 + k + Mk 2k r t
· |∇k+1 Rm|2 √ ≤ 5(k + 1) 2(k + 1) + 1 + n A2 ϕ−2(k+1) + Lt−(k+1) + M k+1 .
So by the definition of u and the choosing of b, we obtain the desired estimate |∇k+1 Rm|2 ≤ Ck+1 M 2
1 r2(k+1)
+
1 tk+1
+ M k+1 .
Therefore we have completed the proof of the theorem.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
199
1.5. Variational Structure and Dynamic Property. In this section, we introduce two functionals of Perelman [103], F and W, and discuss their relations with the Ricci flow. It was not known whether the Ricci flow is a gradient flow until Perelman [103] showed that the Ricci flow is, in a certain sense, the gradient flow of the functional F . If we consider the Ricci flow as a dynamical system on the space of Riemannian metrics, then these two functionals are of Lyapunov type for this dynamical system. Obviously, Ricci flat metrics are fixed points of the dynamical system. When we consider the space of Riemannian metrics modulo diffeomorphism and scaling, fixed points of the Ricci flow dynamical system correspond to steady, or shrinking, or expanding Ricci solitons. The following concept corresponds to a periodic orbit.
Definition 1.5.1. A metric gij (t) evolving by the Ricci flow is called a breather if for some t1 < t2 and α > 0 the metrics αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism; the case α = 1, α < 1, α > 1 correspond to steady, shrinking and expanding breathers, respectively.
Clearly, (steady, shrinking or expanding) Ricci solitons are trivial breathers for which the metrics gij (t1 ) and gij (t2 ) differ only by diffeomorphism and scaling for every pair t1 and t2 . We always assume M is a compact n-dimensional manifold in this section. Let us first consider the functional
F (gij , f ) =
(1.5.1)
Z
M
(R + |∇f |2 )e−f dV
of Perelman [103] defined on the space of Riemannian metrics, and smooth functions on M . Here R is the scalar curvature of gij .
Lemma 1.5.2 (Perelman [103]). If δgij = vij and δf = h are variations of gij and f respectively, then the first variation of F is given by Z δF (vij , h) =
h v i −vij (Rij + ∇i ∇j f ) + − h (2∆f − |∇f |2 + R) e−f dV 2 M
where v = g ij vij .
200
H.-D. CAO AND X.-P. ZHU
Proof. In any normal coordinates at a fixed point, we have ∂ ∂ (δΓh ) − (δΓhil ) ∂xi jl ∂xj ∂ 1 hm = g (∇j vlm + ∇l vjm − ∇m vjl ) ∂xi 2 ∂ 1 hm − j g (∇i vlm + ∇l vim − ∇m vil ) , ∂x 2 ∂ 1 im g (∇j vlm + ∇l vjm − ∇m vjl ) δRjl = ∂xi 2 ∂ 1 im − j g (∇i vlm + ∇l vim − ∇m vil ) ∂x 2 1 ∂ 1 ∂ [∇j vli + ∇l vji − ∇i vjl ] − [∇l v], = 2 ∂xi 2 ∂xj δR = δ(g jl Rjl )
h δRijl =
= −vjl Rjl + g jl δRjl 1 ∂ 1 ∂ = −vjl Rjl + [∇l vli + ∇l v il − ∇i v] − [∇j v] i 2 ∂x 2 ∂xj = −vjl Rjl + ∇i ∇l vil − ∆v. Thus δR(vij ) = −∆v + ∇i ∇j vij − vij Rij .
(1.5.2)
The first variation of the functional F (gij , f ) is (1.5.3)
(R + |∇f |2 )e−f dV M Z = [δR(vij ) + δ(g ij ∇i f ∇j f )]e−f dV
δ
Z
M
h v i + (R + |∇f |2 ) −he−f dV + e−f dV 2 Z = − ∆v + ∇i ∇j vij − Rij vij − vij ∇i f ∇j f M v + 2h∇f, ∇hi + (R + |∇f |2 ) − h e−f dV. 2 On the other hand, Z
M
(∇i ∇j vij − vij ∇i f ∇j f )e
Z
(∇i f ∇j vij − vij ∇i f ∇j f )e−f dV Z (∇i ∇j f )vij e−f dV, =− ZM Z Z −f 2h∇f, ∇hie dV = −2 h∆f e−f dV + 2 |∇f |2 he−f dV, M
−f
dV =
M
M
M
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
201
and Z
(−∆v)e
M
−f
Z
dV = − h∇f, ∇vie−f dV Z M Z = v∆f e−f dV − |∇f |2 ve−f dV. M
M
Plugging these identities into (1.5.3) the first variation formula follows. Now let us study the functional F when the metric evolves under the Ricci flow and the function evolves by a backward heat equation. Proposition 1.5.3 (Perelman [103]). Let gij (t) and f (t) evolve according to the coupled flow ( ∂gij ∂t
∂f ∂t
= −2Rij , = −∆f + |∇f |2 − R.
Then d F (gij (t), f (t)) = 2 dt
Z
M
|Rij + ∇i ∇j f |2 e−f dV
R and M e−f dV is constant. In particular F (gij (t), f (t)) is nondecreasing in time and the monotonicity is strict unless we are on a steady gradient soliton. Proof. Under the coupled flow and using the first variation formula in Lemma 1.5.2, we have d F (gij (t), f (t)) dt Z =
− (−2Rij )(Rij + ∇i ∇j f ) ∂f 1 (−2R) − (2∆f − |∇f |2 + R) e−f dV + 2 ∂t
M
=
Z
M
[2Rij (Rij + ∇i ∇j f ) + (∆f − |∇f |2 )(2∆f − |∇f |2 + R)]e−f dV.
Now Z
(∆f − |∇f |2 )(2∆f − |∇f |2 )e−f dV M Z = −∇i f ∇i (2∆f − |∇f |2 )e−f dV M Z = −∇i f (2∇j (∇i ∇j f ) − 2Rij ∇j f − 2h∇f, ∇i ∇f i)e−f dV M Z = −2 [(∇i f ∇j f −∇i ∇j f )∇i ∇j f −Rij ∇i f ∇j f −h∇f, ∇i ∇f i∇i f ]e−f dV Z M =2 [|∇i ∇j f |2 + Rij ∇i f ∇j f ]e−f dV, M
202
H.-D. CAO AND X.-P. ZHU
and Z
(∆f − |∇f |2 )Re−f dV M Z = −∇i f ∇i Re−f dV M Z Z −f =2 ∇i ∇j f Rij e dV − 2 M
M
∇i f ∇j f Rij e−f dV.
Here we have used the contracted second Bianchi identity. Therefore we obtain d F (gij (t), f (t)) dt Z =
M
=2
Z
[2Rij (Rij + ∇i ∇j f ) + 2(∇i ∇j f )(∇i ∇j f + Rij )]e−f dV
M
|Rij + ∇i ∇j f |2 e−f dV.
R It remains to show M e−f dV is a constant. Note that the volume element dV = p detgij dx evolves under the Ricci flow by ∂ ∂ p dV = ( det gij )dx ∂t ∂t 1 ∂ = log(det gij ) dV 2 ∂t 1 ij ∂ gij )dV = (g 2 ∂t = −RdV.
(1.5.4)
Hence (1.5.5)
∂f − − R dV ∂t
∂ −f e dV = e−f ∂t
= (∆f − |∇f |2 )e−f dV = −∆(e−f )dV.
It then follows that d dt
Z
M
e
−f
dV = −
Z
∆(e−f )dV = 0.
M
This finishes the proof of the proposition. Next we define the associated energy Z (1.5.6) λ(gij ) = inf F (gij , f ) | f ∈ C ∞ (M ),
M
e−f dV = 1 .
If we set u = e−f /2 , then the functional F can be expressed in terms of u as Z F= (Ru2 + 4|∇u|2 )dV, M
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
203
R R and the constraint M e−f dV = 1 becomes M u2 dV = 1. Therefore λ(gij ) is just the first eigenvalue of the operator −4∆ + R. Let u0 > 0 be a first eigenfunction of the operator −4∆ + R satisfying −4∆u0 + Ru0 = λ(gij )u0 . The f0 = −2 log u0 is a minimizer: λ(gij ) = F (gij , f0 ). Note that f0 satisfies the equation (1.5.7)
−2∆f0 + |∇f0 |2 − R = −λ(gij ).
Observe that the evolution equation ∂f = −∆f + |∇f |2 − R ∂t can be rewritten as the following linear equation ∂ −f (e ) = −∆(e−f ) + R(e−f ). ∂t Thus we can always solve the evolution equation for f backwards inR time. Suppose at t = t0 , the infimum λ(gij ) is achieved by some function f0 with M e−f0 dV = 1. We solve the backward heat equation ( ∂f ∂t
= −∆f + |∇f |2 − R
f |t=t0 = f0
to obtain a solution f (t) for t ≤ t0 which satisfies Proposition 1.5.3 that
R
M
e−f dV = 1. It then follows from
λ(gij (t)) ≤ F(gij (t), f (t)) ≤ F(gij (t0 ), f (t0 )) = λ(gij (t0 )). Also note λ(gij ) is invariant under diffeomorphism. Thus we have proved Corollary 1.5.4. (i) λ(gij (t)) is nondecreasing along the Ricci flow and the monotonicity is strict unless we are on a steady gradient soliton; (ii) A steady breather is necessarily a steady gradient soliton. To deal with the expanding case we consider a scale invariant version ¯ ij ) = λ(gij )V n2 (gij ). λ(g Here V = V ol(gij ) denotes the volume of M with respect to the metric gij .
204
H.-D. CAO AND X.-P. ZHU
Corollary 1.5.5. ¯ ij ) is nondecreasing along the Ricci flow whenever it is nonpositive; more(i) λ(g over, the monotonicity is strict unless we are on a gradient expanding soliton; (ii) An expanding breather is necessarily an expanding gradient soliton. Proof. Let f0 be a minimizer of λ(gij (t)) at t = t0 and solve the backward heat equation ∂f = −∆f + |∇f |2 − R ∂t
R to obtain f (t), t ≤ t0 , with M e−f (t) dV = 1. We compute the derivative (understood in the barrier sense) at t = t0 , d¯ λ(gij (t)) dt 2 d ≥ (F (gij (t), f (t)) · V n (gij (t))) dt Z 2 n
2|Rij + ∇i ∇j f |2 e−f dV M Z Z 2 2−n (−R)dV · (R + |∇f |2 )e−f dV + V n n M M 2 Z 2 Rij + ∇i ∇j f − 1 (R + ∆f )gij e−f dV = 2V n n M Z Z Z 1 1 1 (R+∆f )2 e−f dV + − (R+|∇f |2 )e−f dV RdV , + n M n V M M
=V
where we have used the formula (1.5.4) in the computation of dV /dt. Suppose λ(gij (t0 )) ≤ 0, then the last term on the RHS is given by, Z Z 1 1 2 −f − (R + |∇f | e dV ) RdV n V MZ Z M 1 2 −f 2 −f − (R + |∇f | )e dV (R + |∇f | )e dV ≥ n M M Z 2 1 =− (R + ∆f )e−f dV . n M Thus at t = t0 , (1.5.8)
d¯ λ(gij (t)) dt Z ≥ 2V +
2 n
1 n
1 (R + ∆f )gij |2 e−f dV n M Z 2 ! Z 2 −f −f (R + ∆f ) e dV − (R + ∆f )e dV ≥0 M
|Rij + ∇i ∇j f −
M
by the Cauchy-Schwarz inequality. Thus we have proved statement (i). We note that on an expanding breather on [t1 , t2 ] with αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism for some α > 1, it would necessary have dV > 0, for some t ∈ [t1 , t2 ]. dt
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
205
On the other hand, for every t, −
d 1 log V = dt V
Z
M
RdV ≥ λ(gij (t))
by the definition of λ(gij (t)). It follows that on an expanding breather on [t1 , t2 ], ¯ ij (t)) = λ(gij (t))V n2 (gij (t)) < 0 λ(g for some t ∈ [t1 , t2 ]. Then by using statement (i), it implies ¯ ij (t1 )) < λ(g ¯ ij (t2 )) λ(g ¯ ij (t)) is invariant unless we are on an expanding gradient soliton. We also note that λ(g under diffeomorphism and scaling which implies ¯ ij (t1 )) = λ(g ¯ ij (t2 )). λ(g Therefore the breather must be an expanding gradient soliton. In particular part (ii) of Corollaries 1.5.4 and 1.5.5 imply that all compact steady or expanding Ricci solitons are gradient ones. Combining this fact with Proposition (1.1.1), we immediately get Proposition 1.5.6. On a compact manifold, a steady or expanding breather is necessarily an Einstein metric. In order to handle the shrinking case, we introduce the following important functional, also due to Perelman [103], Z n (1.5.9) W(gij , f, τ ) = [τ (R + |∇f |2 ) + f − n](4πτ )− 2 e−f dV M
where gij is a Riemannian metric, f is a smooth function on M , and τ is a positive scale parameter. Clearly the functional W is invariant under simultaneous scaling of τ and gij (or equivalently the parabolic scaling), and invariant under diffeomorphism. Namely, for any positive number a and any diffeomorphism ϕ (1.5.10)
W(aϕ∗ gij , ϕ∗ f, aτ ) = W(gij , f, τ ).
Similar to Lemma 1.5.2, we have the following first variation formula for W. Lemma 1.5.7 (Perelman [103]). If vij = δgij , h = δf, and η = δτ , then δW(vij , h, η) Z n 1 gij (4πτ )− 2 e−f dV = −τ vij Rij + ∇i ∇j f − 2τ M Z n n v −h− η [τ (R + 2∆f − |∇f |2 ) + f − n − 1](4πτ )− 2 e−f dV + 2 2τ ZM n n + η R + |∇f |2 − (4πτ )− 2 e−f dV. 2τ M Here v = g ij vij as before.
206
H.-D. CAO AND X.-P. ZHU
Proof. Arguing as in the proof of Lemma 1.5.2, the first variation of the functional W can be computed as follows, δW (vij , h, η) Z = [η(R + |∇f |2 ) + τ (−∆v + ∇i ∇j vij − Rij vij − vij ∇i f ∇j f M
n
+ 2h∇f, ∇hi) + h](4πτ )− 2 e−f dV Z h nη i v n + (τ (R + |∇f |2 ) + f − n) − + − h (4πτ )− 2 e−f dV 2τ 2 M Z n = [η(R + |∇f |2 ) + h](4πτ )− 2 e−f dV M Z n + [−τ vij (Rij + ∇i ∇j f ) + τ (v − 2h)(∆f − |∇f |2 )](4πτ )− 2 e−f dV ZM h nη i n v + (τ (R + |∇f |2 ) + f − n) − + − h (4πτ )− 2 e−f dV 2τ 2 ZM 1 n =− τ vij Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV 2τ ZM n v −h− η [τ (R + |∇f |2 ) + 2τ M 2
+ f − n + 2τ (∆f − |∇f |2 )](4πτ )− 2 e−f dV Z h n n v n i + η R + |∇f |2 − + h− + η (4πτ )− 2 e−f dV 2τ 2 2τ Z M 1 n = −τ vij Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV 2τ M Z v n n + −h− η [τ (R + 2∆f − |∇f |2 ) + f − n − 1](4πτ )− 2 e−f dV 2 2τ ZM n n + η R + |∇f |2 − (4πτ )− 2 e−f dV. 2τ M n
The following result is analogous to Proposition 1.5.3. Proposition 1.5.8. If gij (t), f (t) and τ (t) evolve according to the system ∂gij = −2Rij , ∂t ∂f n = −∆f + |∇f |2 − R + , ∂t 2τ ∂τ = −1, ∂t
then we have the identity
d W(gij (t), f (t), τ (t)) = dt
2 1 n 2τ Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV 2τ M
Z
R n and M (4πτ )− 2 e−f dV is constant. In particular W(gij (t), f (t), τ (t)) is nondecreasing in time and the monotonicity is strict unless we are on a shrinking gradient soliton.
207
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
Proof. Using Lemma 1.5.7, we have (1.5.11)
d W(gij (t), f (t), τ (t)) dt Z =
2τ Rij
M
+ −
Z
1 n gij (4πτ )− 2 e−f dV 2τ
(∆f − |∇f |2 )[τ (R + 2∆f − |∇f |2 ) + f ](4πτ )− 2 e−f dV
ZM M
Rij + ∇i ∇j f −
n
R + |∇f |2 −
n n (4πτ )− 2 e−f dV. 2τ
R Here we have used the fact that M (∆f − |∇f |2 )e−f dV = 0. The second term on the RHS of (1.5.11) is Z n (∆f − |∇f |2 )[τ (R + 2∆f − |∇f |2 ) + f ](4πτ )− 2 e−f dV M Z n = (∆f − |∇f |2 )(2τ ∆f − τ |∇f |2 )(4πτ )− 2 e−f dV M Z Z n −f 2 −n 2 (−∇i f )(∇i R)(4πτ )− 2 e−f dV − |∇f | (4πτ ) e dV + τ M ZM n 2 =τ (−∇i f )(∇i (2∆f − |∇f | ))(4πτ )− 2 e−f dV Z ZM n n ∇i f ∇j Rij (4πτ )− 2 e−f dV − ∆f (4πτ )− 2 e−f dV − 2τ M M Z n = −2τ (∇i f )(∇i ∆f − h∇f, ∇i ∇f i)(4πτ )− 2 e−f dV ZM n + 2τ [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV ZM 1 n − gij (∇i ∇j f )(4πτ )− 2 e−f dV + 2τ 2τ ZM = −2τ [(∇i f ∇j f − ∇i ∇j f )∇i ∇j f − Rij ∇i f ∇j f M
n
− ∇i ∇j f ∇i f ∇j f ](4πτ )− 2 e−f dV Z n + 2τ [(∇i ∇j f )Rij − ∇i f ∇j f Rij ](4πτ )− 2 e−f dV ZM 1 n + 2τ − gij (∇i ∇j f )(4πτ )− 2 e−f dV 2τ M Z n 1 gij (4πτ )− 2 e−f dV. = 2τ (∇i ∇j f ) ∇i ∇j f + Rij − 2τ M
Also the third term on the RHS of (1.5.11) is Z n n (4πτ )− 2 e−f dV − R + |∇f |2 − 2τ M Z n n = − R + ∆f − (4πτ )− 2 e−f dV 2τ M Z −1 1 n = 2τ gij Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV. 2τ 2τ M
208
H.-D. CAO AND X.-P. ZHU
Therefore, by combining the above identities, we obtain 2 Z d Rij + ∇i ∇j f − 1 gij (4πτ )− n2 e−f dV. W(gij (t), f (t), τ (t)) = 2τ dt 2τ M
Finally, by using the computations in (1.5.5) and the evolution equations of f and τ , we have ∂ n n −f n ∂ −f (4πτ )− 2 e−f dV = (4πτ )− 2 (e dV ) + e dV ∂t ∂t 2τ n
= −(4πτ )− 2 ∆(e−f )dV.
Hence d dt
Z
n
M
n
(4πτ )− 2 e−f dV = −(4πτ )− 2
Z
∆(e−f )dV = 0.
M
Now we set µ(gij , τ ) = inf W(gij , f, τ ) | f ∈ C ∞ (M ),
(1.5.12)
1 (4πτ )n/2
Z
e−f dV = 1
M
and ν(gij ) = inf W(g, f, τ ) | f ∈ C ∞ (M ), τ > 0,
1 (4πτ )n/2
Z
e−f dV = 1 .
Note that if we let u = e−f /2 , then the functional W can be expressed as Z n W(gij , f, τ ) = [τ (Ru2 + 4|∇u|2 ) − u2 log u2 − nu2 ](4πτ )− 2 dV M
R
R n n and the constraint M (4πτ )− 2 e−f dV = 1 becomes M u2 (4πτ )− 2 dV = 1. Thus µ(gij , τ ) corresponds to the best constant of a logarithmic Sobolev inequality. Since the nonquadratic term is subcritical (in view of Sobolev exponent), it is rather straightforward to show that Z Z n 2 2 2 2 2 −n 2 − inf [τ (4|∇u| + Ru ) − u log u − nu ](4πτ ) 2 dV u (4πτ ) 2 dV = 1 M
M
is achieved by some nonnegative function u ∈ H 1 (M ) which satisfies the EulerLagrange equation τ (−4∆u + Ru) − 2u log u − nu = µ(gij , τ )u.
One can further show that u is positive (see [108]). Then the standard regularity theory of elliptic PDEs shows that u is smooth. We refer the reader to Rothaus [108] for more details. It follows that µ(gij , τ ) is achieved by a minimizer f satisfying the nonlinear equation (1.5.13)
τ (2∆f − |∇f |2 + R) + f − n = µ(gij , τ ).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
209
Corollary 1.5.9. (i) µ(gij (t), τ −t) is nondecreasing along the Ricci flow; moveover, the monotonicity is strict unless we are on a shrinking gradient soliton; (ii) A shrinking breather is necessarily a shrinking gradient soliton. Proof. Fix any time t0 , let f0 be a minimizer of µ(gij (t0 ), τ − t0 ). Note that the backward heat equation ∂f n = −∆f + |∇f |2 − R + ∂t 2τ is equivalent to the linear equation n n n ∂ ((4πτ )− 2 e−f ) = −∆((4πτ )− 2 e−f ) + R((4πτ )− 2 e−f ). ∂t
Thus we can solve the backward heat equation of f with f |t=t0 = f0 to obtain R n f (t), t ≤ t0 , with M (4πτ )− 2 e−f (t) dV = 1. It then follows from Proposition 1.5.8 that µ(gij (t), τ − t) ≤ W(gij (t), f (t), τ − t) ≤ W(gij (t0 ), f (t0 ), τ − t0 ) = µ(gij (t0 ), τ − t0 )
for t ≤ t0 and the second inequality is strict unless we are on a shrinking gradient soliton. This proves statement (i). Consider a shrinking breather on [t1 , t2 ] with αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism for some α < 1. Recall that the functional W is invariant under simultaneous scaling of τ and gij and invariant under diffeomorphism. Then for τ > 0 to be determined, µ(gij (t1 ), τ − t1 ) = µ(αgij (t1 ), α(τ − t1 )) = µ(gij (t2 ), α(τ − t1 )) and by the monotonicity of µ(gij (t), τ − t), µ(gij (t1 ), τ − t1 ) ≤ µ(gij (t2 ), τ − t2 ). Now take τ > 0 such that α(τ − t1 ) = τ − t2 , i.e., τ=
t2 − αt1 . 1−α
This shows the equality holds in the monotonicity of µ(gij (t), τ − t). So the shrinking breather must be a shrinking gradient soliton. Finally, we remark that Hamilton, Ilmanen and the first author [18] have obtained the second variation formulas for both λ-energy and ν-energy. We refer the reader to their paper [18] for more details and related stability questions.
210
H.-D. CAO AND X.-P. ZHU
2. Maximum Principle and Li-Yau-Hamilton Inequalities. The maximum principle is a fundamental tool in the study of parabolic equations in general. In this chapter, we present various maximum principles for tensors developed by Hamilton in the Ricci flow. As an immediate consequence, the Ricci flow preserves the nonnegativity of the curvature operator. We also present the two crucial estimates in the Ricci flow: the Hamilton-Ivey curvature pinching estimate (when dimension n = 3), and the Li-Yau-Hamilton estimate from which one obtains the Harnack inequality for the evolved scalar curvature via a Li-Yau path integral. Finally, we describe Perelman’s Li-Yau type estimate for solutions to the conjugate heat equation and show how Li-Yau type path integral leads to a space-time distance function (i.e., what Perelman called the reduced distance). 2.1. Preserving Positive Curvature. Let M be an n-dimensional complete manifold. Consider a family of smooth metrics gij (t) evolving by the Ricci flow with uniformly bounded curvature for t ∈ [0, T ] with T < +∞. Denote by dt (x, y) the distance between two points x, y ∈ M with respect to the metric gij (t). Lemma 2.1.1. There exists a smooth function f on M such that f ≥ 1 everywhere, f (x) → +∞ as d0 (x, x0 ) → +∞ (for some fixed x0 ∈ M ), |∇f |gij (t) ≤ C
and
|∇2 f |gij (t) ≤ C
on M × [0, T ] for some positive constant C. Proof. Let ϕ(v) be a smooth function on Rn which is nonnegative, rotationRally symmetric and has compact support in a small ball centered at the origin with Rn ϕ(v)dv = 1. For each x ∈ M , set Z f (x) = ϕ(v)(d0 (x0 , expx (v)) + 1)dv, Rn
where the integral is taken over the tangent space Tx M at x which we have identified with Rn . If the size of the support of ϕ(v) is small compared to the maximum curvature, then it is well known that this defines a smooth function f on M with f (x) → +∞ as d0 (x, x0 ) → +∞, while the bounds on the first and second covariant derivatives of f with respect to the metric gij (·, 0) follow from the Hessian comparison theorem. Thus it remains to show these bounds hold with respect to the evolving metric gij (t). We compute, using the frame {Fai ∇i f } introduced in Section 1.3, ∂ ∂ ∇a f = (Fai ∇i f ) = Rab ∇b f. ∂t ∂t Hence |∇f | ≤ C1 · eC2 t , where C1 , C2 are some positive constants depending only on the dimension. Also 2 ∂ ∂ ∂ f i j k ∂f (∇a ∇b f ) = Fa Fb − Γij k ∂t ∂t ∂xi ∂xj ∂x = Rac ∇b ∇c f + Rbc ∇a ∇c f + (∇c Rab − ∇a Rbc − ∇b Rac )∇c f.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
211
Then by Shi’s derivative estimate (Theorem 1.4.1), we have ∂ 2 C3 |∇ f | ≤ C3 |∇2 f | + √ , ∂t t which implies Z |∇2 f |gij (t) ≤ eC3 t |∇2 f |gij (0) +
t
0
C √3 e−C3 τ dτ τ
for some positive constants C3 depending only on the dimension and the curvature bound. We now use the weak maximum principle to derive the following Proposition 2.1.2. If the scalar curvature R of the solution gij (t), 0 ≤ t ≤ T , to the Ricci flow is nonnegative at t = 0, then it remains so on 0 ≤ t ≤ T . Proof. Let f be the function constructed in Lemma 2.1.1 and recall ∂R = ∆R + 2|Ric |2 . ∂t For any small constant ε > 0 and large constant A > 0, we have ∂R ∂ (R + εeAt f ) = + εAeAt f ∂t ∂t = ∆(R + εeAt f ) + 2|Ric|2 + εeAt (Af − ∆f ) > ∆(R + εeAt f )
by choosing A large enough. We claim that R + εeAt f > 0
on M × [0, T ].
Suppose not, then there exist a first time t0 > 0 and a point x0 ∈ M such that (R + εeAt f )(x0 , t0 ) = 0, ∇(R + εeAt f )(x0 , t0 ) = 0,
and
∆(R + εeAt f )(x0 , t0 ) ≥ 0, ∂ (R + εeAt f )(x0 , t0 ) ≤ 0. ∂t
Then 0≥
∂ (R + εeAt f )(x0 , t0 ) > ∆(R + εeAt f )(x0 , t0 ) ≥ 0, ∂t
which is a contradiction. So we have proved that R + εeAt f > 0
on M × [0, T ].
Letting ε → 0, we get R≥0
on M × [0, T ].
212
H.-D. CAO AND X.-P. ZHU
This finishes the proof of the proposition. Next we derive a maximum principle of Hamilton for tensors. Let M be a complete manifold with a metric g = {gij }, V a vector bundle over M with a metric h = {hαβ } and a connection ∇ = {Γα iβ } compatible with h, and suppose h is fixed but g and ∇ may vary smoothly with time t. Let Γ(V ) be the vector space of C ∞ sections of V . The Laplacian ∆ acting on a section σ ∈ Γ(V ) is defined by ∆σ = g ij ∇i ∇j σ. Let Mαβ be a symmetric bilinear form on V . We say Mαβ ≥ 0 if Mαβ v α v β ≥ 0 for all vectors v = {v α }. Assume Nαβ = P(Mαβ , hαβ ) is a polynomial in Mαβ formed by contracting products of Mαβ with itself using the metric h = {hαβ }. Assume that the tensor Mαβ is uniformly bounded in space-time and let gij evolve by the Ricci flow with bounded curvature. Lemma 2.1.3. Suppose that on 0 ≤ t ≤ T , ∂ Mαβ = ∆Mαβ + ui ∇i Mαβ + Nαβ ∂t where ui (t) is a time-dependent vector field on M with uniform bound and Nαβ = P(Mαβ , hαβ ) satisfies Nαβ v α v β ≥ 0
whenever
Mαβ v β = 0.
If Mαβ ≥ 0 at t = 0, then it remains so on 0 ≤ t ≤ T . Proof. Set ˜ αβ = Mαβ + εeAt f hαβ , M where A > 0 is a suitably large constant (to be chosen later) and f is the function constructed in Lemma 2.1.1. ˜ αβ > 0 on M × [0, T ] for every ε > 0. If not, then for some ε > 0, We claim that M ˜ αβ acquires a null vector v α of unit length there will be a first time t0 > 0 where M at some point x0 ∈ M . At (x0 , t0 ), ˜αβ v α v β Nαβ v α v β ≥ Nαβ v α v β − N ≥ −CεeAt0 f (x0 ),
˜αβ = P(M ˜ αβ , hαβ ), and C is a positive constant (depending on the bound of where N Mαβ , but independent of A). Let us extend v α to a local vector field in a neighborhood of x0 by parallel translating v α along geodesics (with respect to the metric gij (t0 )) emanating radially out of x0 , with v α independent of t. Then, at (x0 , t0 ), we have ∂ ˜ (Mαβ v α v β ) ≤ 0, ∂t ˜ αβ v α v β ) = 0, ∇(M
˜ αβ v α v β ) ≥ 0. and ∆(M
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
213
But 0≥
∂ ˜ ∂ (Mαβ v α v β ) = (Mαβ v α v β + εeAt f ), ∂t ∂t ˜ αβ v α v β ) − ∆(εeAt f ) + ui ∇i (M ˜ αβ v α v β ) = ∆(M − ui ∇i (εeAt f ) + Nαβ v α v β + εAeAt0 f (x0 )
≥ −CεeAt0 f (x0 ) + εAeAt0 f (x0 ) > 0 when A is chosen sufficiently large. This is a contradiction. By applying Lemma 2.1.3 to the evolution equation ∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ ∂t
of the curvature operator Mαβ , we immediately obtain the following important result. Proposition 2.1.4 (Hamilton [59]). Nonnegativity of the curvature operator Mαβ is preserved by the Ricci flow. In the K¨ahler case, the nonnegativity of the holomorpic bisectional curvature is preserved under the K¨ahler-Ricci flow. This result is proved by Bando [5] for complex dimension n = 3 and by Mok [92] for general dimension n when the manifold is compact, and by Shi [115] when the manifold is noncompact. Proposition 2.1.5. Under the K¨ ahler-Ricci flow if the initial metric has positive (nonnegative) holomorphic bisectional curvature then the evolved metric also has positive (nonnegative) holomorphic bisectional curvature. 2.2. Strong Maximum Principle. Let Ω be a bounded, connected open set of a complete n-dimensional manifold M , and let gij (x, t) be a smooth solution to the Ricci flow on Ω × [0, T ]. Consider a vector bundle V over Ω with a fixed metric hαβ (independent of time), and a connection ∇ = {Γα iβ } which is compatible with hαβ and may vary with time t. Let Γ(V ) be the vector space of C ∞ sections of V over Ω. The Laplacian ∆ acting on a section σ ∈ Γ(V ) is defined by ∆σ = g ij (x, t)∇i ∇j σ. Consider a family of smooth symmetric bilinear forms Mαβ evolving by (2.2.1)
∂ Mαβ = ∆Mαβ + Nαβ , ∂t
on Ω × [0, T ],
where Nαβ = P (Mαβ , hαβ ) is a polynomial in Mαβ formed by contracting products of Mαβ with itself using the metric hαβ and satisfies Nαβ ≥ 0, whenever Mαβ ≥ 0. The following result, due to Hamilton [59], shows that the solution of (2.2.1) satisfies a strong maximum principle. Theorem 2.2.1 (Hamilton’s strong maximum principle). Let Mαβ be a smooth solution of the equation (2.2.1). Suppose Mαβ ≥ 0 on Ω × [0, T ]. Then there exists a positive constant 0 < δ ≤ T such that on Ω × (0, δ), the rank of Mαβ is constant, and
214
H.-D. CAO AND X.-P. ZHU
the null space of Mαβ is invariant under parallel translation and invariant in time and also lies in the null space of Nαβ . Proof. Set l = max{rank of Mαβ (x, 0)}. x∈Ω
Then we can find a nonnegative smooth function ρ(x), which is positive somewhere and has compact support in Ω, so that at every point x ∈ Ω, n−l+1 X i=1
Mαβ (x, 0)viα viβ ≥ ρ(x)
for any (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at x. Let us evolve ρ(x) by the heat equation ∂ ρ = ∆ρ ∂t with the Dirichlet condition ρ|∂Ω = 0 to get a smooth function ρ(x, t) defined on Ω×[0, T ]. By the standard strong maximum principle, we know that ρ(x, t) is positive everywhere in Ω for all t ∈ (0, T ]. For every ε > 0, we claim that at every point (x, t) ∈ Ω × [0, T ], there holds n−l+1 X
Mαβ (x, t)viα viβ + εet > ρ(x, t)
i=1
for any (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at x. We argue by contradiction. Suppose not, then for some ε > 0, there will be a first time t0 > 0 and some (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at some point x0 ∈ Ω so that n−l+1 X
Mαβ (x0 , t0 )viα viβ + εet0 = ρ(x0 , t0 )
i=1
Let us extend each vi (i = 1, . . . , n − l + 1) to a local vector field, independent of t, in a neighborhood of x0 by parallel translation along geodesics (with respect to the metric gij (t0 )) emanating radially out of x0 . Clearly {v1 , . . . , vn−l+1 } remain orthogonal unit vectors in the neighborhood. Then, at (x0 , t0 ), we have ∂ ∂t and ∆
n−l+1 X
Mαβ viα viβ
+ εe − ρ
Mαβ viα viβ
t
i=1
n−l+1 X i=1
t
!
+ εe − ρ
!
≤ 0, ≥ 0.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
But, since Nαβ ≥ 0 by our assumption, we have n−l+1 X
∂ 0≥ ∂t =
i=1
=
t
+ εe − ρ
!
n−l+1 X i=1
≥
Mαβ viα viβ
215
(∆Mαβ + Nαβ )viα viβ + εet − ∆ρ
n−l+1 X i=1
n−l+1 X i=1
∆(Mαβ viα viβ ) + εet − ∆ρ ∆(Mαβ viα viβ + εet − ρ) + εet
≥ εet > 0. This is a contradiction. Thus by letting ε → 0, we prove that n−l+1 X i=1
Mαβ (x, t)viα viβ ≥ ρ(x, t)
for any (n − l + 1) orthogonal unit vectors {v1 , . . . , vn−l+1 } at x ∈ Ω and t ∈ [0, T ]. Hence Mαβ has at least rank l everywhere in the open set Ω for all t ∈ (0, T ]. Therefore we can find a positive constant δ(≤ T ) such that the rank Mαβ is constant over Ω × (0, δ). Next we proceed to analyze the null space of Mαβ . Let v be any smooth section of V in the null of Mαβ on 0 < t < δ. Then ∂ (Mαβ v α v β ) ∂t ∂ ∂v β = Mαβ v α v β + 2Mαβ v α ∂t ∂t ∂ Mαβ v α v β , = ∂t
0=
and 0 = ∆(Mαβ v α v β ) = (∆Mαβ )v α v β + 4g kl ∇k Mαβ · v α ∇l v β
+ 2Mαβ g kl ∇k v α · ∇l v β + 2Mαβ v α ∆v β
= (∆Mαβ )v α v β + 4g kl ∇k Mαβ · v α ∇l v β + 2Mαβ g kl ∇k v α · ∇l v β . By noting that 0 = ∇k (Mαβ v β ) = (∇k Mαβ )v α + Mαβ ∇k v α and using the evolution equation (2.2.1), we get Nαβ v α v β + 2Mαβ g kl ∇k v α · ∇l v β = 0.
216
H.-D. CAO AND X.-P. ZHU
Since Mαβ ≥ 0 and Nαβ ≥ 0, we must have v ∈ null (Nαβ )
and ∇i v ∈ null (Mαβ ),
for all i.
The first inclusion shows that null (Mαβ ) ⊂ null (Nαβ ), and the second inclusion shows that null (Mαβ ) is invariant under parallel translation. To see null (Mαβ ) is also invariant in time, we first note that ∆v = ∇i (∇i v) ∈ null (Mαβ ) and then g kl ∇k Mαβ · ∇l v α = g kl ∇k (Mαβ ∇l v α ) − Mαβ ∆v α = 0. Thus we have 0 = ∆(Mαβ v α ) = (∆Mαβ )v α + 2g kl ∇k Mαβ · ∇l v α + Mαβ ∆v α
= (∆Mαβ )v α , and hence 0=
∂ (Mαβ v α ) ∂t
= (∆Mαβ + Nαβ )v α + Mαβ = Mαβ
∂v α . ∂t
∂v α ∂t
This shows that ∂v ∈ null (Mαβ ), ∂t so the null space of Mαβ is invariant in time. We now apply Hamilton’s strong maximum principle to the evolution equation of the curvature operator Mαβ . Recall ∂Mαβ # 2 = ∆Mαβ + Mαβ + Mαβ ∂t # where Mαβ = Cαξγ Cβηθ Mξη Mγθ . Suppose we have a solution to the Ricci flow with nonnegative curvature operator. Then by Theorem 2.2.1, the null space of the curvature operator Mαβ of the solution has constant rank and is invariant in time and under parallel translation over some time interval 0 < t < δ . Moreover the null space # of Mαβ must also lie in the null space of Mαβ . Denote by (n − k) the rank of Mαβ on 0 < t < δ. Let us diagonalize Mαβ so that # Mαα = 0 if α ≤ k and Mαα > 0 if α > k. Then we have Mαα = 0 also for α ≤ k from the evolution equation of Mαα . Since # 0 = Mαα = Cαξγ Cαηθ Mξη Mγθ ,
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
217
it follows that Cαξγ = hv α , [v ξ , v γ ]i = 0, if α ≤ k and ξ, γ > k. This says that the image of Mαβ is a Lie subalgebra (in fact it is the subalgebra of the restricted holonomy group by using the Ambrose-Singer holonomy theorem [3]). This proves the following result. Theorem 2.2.2 (Hamilton [59]). Suppose the curvature operator Mαβ of the initial metric is nonnegative. Then, under the Ricci flow, for some interval 0 < t < δ the image of Mαβ is a Lie subalgebra of so(n) which has constant rank and is invariant under parallel translation and invariant in time. 2.3. Advanced Maximum Principle for Tensors. In this section we present Hamilton’s advanced maximum principle for tensors which generalizes Lemma 2.1.3 and shows how a tensor evolving by a nonlinear heat equation may be controlled by a system of ODEs. An important application of the advanced maximum principle is the Hamilton-Ivey curvature pinching estimate for the Ricci flow on three-manifolds given in the next section. More applications will be given in Chapter 5. Let M be a complete manifold equipped with a one-parameter family of Riemannian metrics gij (t), 0 ≤ t ≤ T , with T < +∞. Let V → M be a vector bundle with a time-independent bundle metric hab and Γ(V ) be the vector space of C ∞ sections of V . Let ∇t :
Γ(V ) → Γ(V ⊗ T ∗ M ),
t ∈ [0, T ]
be a smooth family of time-dependent connections compatible with hab , i.e. ∆
(∇t )i hab = (∇t )
∂ ∂xi
hab = 0,
for any local coordinate { ∂x∂ 1 , . . . , ∂x∂ n }. The Laplacian ∆t acting on a section σ ∈ Γ(V ) is defined by ∆t σ = g ij (x, t)(∇t )i (∇t )j σ. For the application to the Ricci flow, we will always assume that the metrics gij (·, t) evolve by the Ricci flow. Since M may be noncompact, we assume that, for the sake of simplicity, the curvature of gij (t) is uniformly bounded on M × [0, T ]. Let N : V × [0, T ] → V be a fiber preserving map, i.e., N (x, σ, t) is a timedependent vector field defined on the bundle V and tangent to the fibers. We assume that N (x, σ, t) is continuous in x, t and satisfies |N (x, σ1 , t) − N (x, σ2 , t)| ≤ CB |σ1 − σ2 | for all x ∈ M , t ∈ [0, T ] and |σ1 | ≤ B, |σ2 | ≤ B, where CB is a positive constant depending only on B. Then we can form the nonlinear heat equation (PDE)
∂ σ(x, t) = ∆t σ(x, t) + ui (∇t )i σ(x, t) + N (x, σ(x, t), t) ∂t
where ui = ui (t) is a time-dependent vector field on M which is uniformly bounded on M × [0, T ]. Let K be a closed subset of V . One important question is under what conditions will solutions of the PDE which start in K remain in K. To answer this question, Hamilton [59] imposed the following two conditions on K:
218
H.-D. CAO AND X.-P. ZHU
(H1) K is invariant under parallel translation defined by the connection ∇t for each t ∈ [0, T ]; ∆ (H2) in each fiber Vx , the set Kx = Vx ∩ K is closed and convex. Then one can judge the behavior of the PDE by comparing to that of the following ODE dσx = N (x, σx , t) dt
(ODE) for σx = σx (t) in each fiber Vx .
Theorem 2.3.1 (Hamilton’s advanced maximum principle [59]). Let K be a closed subset of V satisfying the hypothesis (H1) and (H2). Suppose that for any x ∈ M and any initial time t0 ∈ [0, T ), any solution σx (t) of the (ODE) which starts in Kx at t0 will remain in Kx for all later times. Then for any initial time t0 ∈ [0, T ) the solution σ(x, t) of the (PDE) will remain in K for all later times provided σ(x, t) starts in K at time t0 and σ(x, t) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ]. We remark that Lemma 2.1.3 is a special case of the above theorem where V is given by a symmetric tensor product of a vector bundle and K corresponds to the convex set consisting of all nonnegative symmetric bilinear forms. We also remark that Hamilton [59] established the above theorem for a general evolving metric gij (x, t) which does not necessarily satisfy the Ricci flow. Before proving Theorem 2.3.1, we need to establish three lemmas. Let ϕ : [a, b] → R be a Lipschitz function. We consider dϕ dt (t) at t ∈ [a, b) in the sense of limsup of the forward difference quotients, i.e., dϕ ϕ(t + h) − ϕ(t) (t) = lim sup . dt h h→0+ Lemma 2.3.2. Suppose ϕ : [a, b] → R is Lipschitz continuous and suppose for some constant C < +∞, d ϕ(t) ≤ Cϕ(t), dt and
whenever ϕ(t) ≥ 0 on [a, b),
ϕ(a) ≤ 0.
Then ϕ(t) ≤ 0 on [a, b]. Proof. By replacing ϕ by e−Ct ϕ, we may assume
and
d ϕ(t) ≤ 0, dt ϕ(a) ≤ 0.
whenever ϕ(t) ≥ 0 on [a, b),
For arbitrary ε > 0, we shall show ϕ(t) ≤ ε(t − a) on [a, b]. Clearly we may assume ϕ(a) = 0. Since lim sup h→0+
ϕ(a + h) − ϕ(a) ≤ 0, h
there must be some interval a ≤ t < δ on which ϕ(t) ≤ ε(t − a).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
219
Let a ≤ t < c be the largest interval with c ≤ b such that ϕ(t) ≤ ε(t − a) on [a, c). Then by continuity ϕ(t) ≤ ε(t − a) on the closed interval [a, c]. We claim that c = b. Suppose not, then we can find δ > 0 such that ϕ(t) ≤ ε(t − a) on [a, c + δ] since lim sup h→0+
ϕ(c + h) − ϕ(c) ≤ 0. h
This contradicts the choice of the largest interval [a, c). Therefore, since ε > 0 can be arbitrary small, we have proved ϕ(t) ≤ 0 on [a, b].
The second lemma below is a general principle on the derivative of a sup-function which will bridge solutions between ODEs and PDEs. Let X be a complete smooth manifold and Y be a compact subset of X. Let ψ(x, t) be a smooth function on X × [a, b] and let ϕ(t) = sup{ψ(y, t) | y ∈ Y }. Then it is clear that ϕ(t) is Lipschitz continuous. We have the following useful estimate on its derivative. Lemma 2.3.3. d ϕ(t) ≤ sup dt
∂ψ (y, t) | y ∈ Y satisfies ψ(y, t) = ϕ(t) . ∂t
Proof. Choose a sequence of times {tj } decreasing to t for which lim
tj →t
dϕ(t) ϕ(tj ) − ϕ(t) = . tj − t dt
Since Y is compact, we can choose yj ∈ Y with ϕ(tj ) = ψ(yj , tj ). By passing to a subsequence, we can assume yj → y for some y ∈ Y . By continuity, we have ϕ(t) = ψ(y, t). It follows that ψ(yj , t) ≤ ψ(y, t), and then ϕ(tj ) − ϕ(t) ≤ ψ(yj , tj ) − ψ(yj , t) =
∂ ψ(yj , t˜j ) · (tj − t) ∂t
for some t˜j ∈ [t, tj ] by the mean value theorem. Thus we have lim
tj →t
ϕ(tj ) − ϕ(t) ∂ ≤ ψ(y, t). tj − t ∂t
This proves the result. We remark that the above two lemmas are somewhat standard facts in the theory of PDEs and we have implicitly used them in the previous sections when we apply the maximum principle. The third lemma gives a characterization of when a system of ODEs preserve closed convex sets in Euclidean space. Let Z ⊂ Rn be a closed convex subset. We define the tangent cone Tϕ Z to the closed convex set Z at a point ϕ ∈ ∂Z as the smallest closed convex cone with vertex at ϕ which contains Z. Lemma 2.3.4. Let U ⊂ Rn be an open set and Z ⊂ U be a closed convex subset. Consider the ODE (2.3.1)
dϕ = N (ϕ, t) dt
where N : U × [0, T ] → Rn is continuous and Lipschitz in ϕ. Then the following two statements are equivalent.
220
H.-D. CAO AND X.-P. ZHU
(i) For any initial time t0 ∈ [0, T ], any solution of the ODE (2.3.1) which starts in Z at t0 will remain in Z for all later times; (ii) ϕ + N (ϕ, t) ∈ Tϕ Z for all ϕ ∈ ∂Z and t ∈ [0, T ). Proof. We say that a linear function l on Rn is a support function for Z at ϕ ∈ ∂Z and write l ∈ Sϕ Z if |l| = 1 and l(ϕ) ≥ l(η) for all η ∈ Z. Then ϕ+N (ϕ, t) ∈ Tϕ Z if and only if l(N (ϕ, t)) ≤ 0 for all l ∈ Sϕ Z. Suppose l(N (ϕ, t)) > 0 for some ϕ ∈ ∂Z and some l ∈ Sϕ Z. Then d dϕ l(ϕ) = l = l(N (ϕ, t)) > 0, dt dt so l(ϕ) is strictly increasing and the solution ϕ(t) of the ODE (2.3.1) cannot remain in Z. To see the converse, first note that we may assume Z is compact. This is because we can modify the vector field N (ϕ, t) by multiplying a cutoff function which is everywhere nonnegative, equals one on a large ball and equals zero on the complement of a larger ball. The paths of solutions of the ODE are unchanged inside the first large ball, so we can intersect Z with the second ball to make Z convex and compact. If there were a counterexample before the modification there would still be one after as we chose the first ball large enough. Let s(ϕ) be the distance from ϕ to Z in Rn . Clearly s(ϕ) = 0 if ϕ ∈ Z. Then s(ϕ) = sup{l(ϕ − η) | η ∈ ∂Z and l ∈ Sη Z}. The sup is taken over a compact subset of Rn × Rn . Hence by Lemma 2.3.3 d s(ϕ) ≤ sup{l(N (ϕ, t)) | η ∈ ∂Z, l ∈ Sη Z and s(ϕ) = l(ϕ − η)}. dt It is clear that the sup on the RHS of the above inequality can be takeen only when η is the unique closest point in Z to ϕ and l is the linear function of length one with gradient in the direction of ϕ − η. Since N (ϕ, t) is Lipschitz in ϕ and continuous in t, we have |N (ϕ, t) − N (η, t)| ≤ C|ϕ − η| for some constant C and all ϕ and η in the compact set Z. By hypothesis (ii), l(N (η, t)) ≤ 0, and for the unique η, the closest point in Z to ϕ, |ϕ − η| = s(ϕ). Thus d s(ϕ) ≤ sup dt
(
≤ Cs(ϕ).
l(N (η, t)) + |l(N (ϕ, t)) − l(N (η, t))| l ∈ Sη Z,
| η ∈ ∂Z,
and s(ϕ) = l(ϕ − η)
)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
221
Since s(ϕ) = 0 to start at t0 , it follows from Lemma 2.3.2 that s(ϕ) = 0 for t ∈ [t0 , T ]. This proves the lemma. We are now ready to prove Theorem 2.3.1. Proof of Theorem 2.3.1. Since the solution σ(x, t) of the (PDE) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ] by hypothesis, we may assume that K is contained in a tubular neighborhood V (r) of the zero section in V whose intersection with each fiber Vx is a ball of radius r around the origin measured by the bundle metric hab for some large r > 0. Recall that gij (·, t), t ∈ [0, T ], is a smooth solution to the Ricci flow with uniformly bounded curvature on M × [0, T ]. From Lemma 2.1.1, we have a smooth function f such that f ≥ 1 everywhere, f (x) → +∞ as d0 (x, x0 ) → +∞ for some fixed point x0 ∈ M , and the first and second covariant derivatives with respect to the metrics gij (·, t) are uniformly bounded on M × [0, T ]. Using the metric hab in each fiber Vx and writing |ϕ − η| for the distance between ϕ ∈ Vx and η ∈ Vx , we set ∆
s(t) = sup {inf{|σ(x, t) − η| | η ∈ Kx = K ∩ Vx } − ǫeAt f (x)} x∈M
where ǫ is an arbitrarily small positive number and A is a positive constant to be determined. We rewrite the function s(t) as s(t) = sup{l(σ(x, t) − η) − ǫeAt f (x) | x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx }. By the construction of the function f , we see that the sup is taken in a compact subset of M × V × V ∗ for all t. Then by Lemma 2.3.3, ∂ ds(t) ≤ sup l(σ(x, t) − η) − ǫAeAt f (x) (2.3.2) dt ∂t where the sup is over all x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx such that l(σ(x, t) − η) − ǫeAt f (x) = s(t); in particular we have |σ(x, t) − η| = l(σ(x, t) − η), where η is the unique closest point in Kx to σ(x, t), and l is the linear function of length one on the fiber Vx with gradient in the direction of η to σ(x, t). We compute at these (x, η, l), (2.3.3)
∂ l(σ(x, t) − η) − ǫAeAt f (x) ∂t ∂σ(x, t) − ǫAeAt f (x) =l ∂t
= l(∆t σ(x, t)) + l(ui (x, t)(∇t )i σ(x, t)) + l(N (x, σ(x, t), t)) − ǫAeAt f (x).
By the assumption and Lemma 2.3.4 we have η + N (x, η, t) ∈ Tη Kx . Hence, for those (x, η, l), l(N (x, η, t)) ≤ 0 and then (2.3.4)
l(N (x, σ(x, t), t)) ≤ l(N (x, η, t)) + |N (x, σ(x, t), t) − N (x, η, t)| ≤ C|σ(x, t) − η| = C(s(t) + ǫeAt f (x))
222
H.-D. CAO AND X.-P. ZHU
for some positive constant C by the assumption that N (x, σ, t) is Lipschitz in σ and the fact that the sup is taken on a compact set. Thus the combination of (2.3.2)–(2.3.4) gives (2.3.5)
ds(t) ≤ l(∆t σ(x, t)) + l(ui (x, t)(∇t )i σ(x, t)) + Cs(t) + ǫ(C − A)eAt f (x) dt
for those x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx such that l(σ(x, t) − η) − ǫeAt f (x) = s(t). Next we estimate the first two terms of (2.3.5). As we extend a vector in a bundle from a point x by parallel translation along geodesics emanating radially out of x, we will get a smooth section of the bundle in some small neighborhood of x such that all the symmetrized covariant derivatives at x are zero. Now let us extend η ∈ Vx and l ∈ Vx∗ in this manner. Clearly, we continue to have |l|(·) = 1. Since K is invariant under parallel translations, we continue to have η(·) ∈ ∂K and l(·) as a support function for K at η(·). Therefore l(σ(·, t) − η(·)) − ǫeAt f (·) ≤ s(t) in the neighborhood. It follows that the function l(σ(·, t) − η(·)) − ǫeAt f (·) has a local maximum at x, so at x (∇t )i (l(σ(x, t) − η) − ǫeAt f (x)) = 0,
and ∆t (l(σ(x, t) − η) − ǫeAt f (x)) ≤ 0. Hence at x l((∇t )i σ(x, t)) − ǫeAt (∇t )i f (x) = 0,
and l(∆t σ(x, t)) − ǫeAt ∆t f (x) ≤ 0. Therefore by combining with (2.3.5), we have
d s(t) ≤ Cs(t) + ǫ(∆t f (x) + ui (∇t )i f (x) + (C − A)f (x))eAt dt ≤ Cs(t) for A > 0 large enough, since f (x) ≥ 1 and the first and second covariant derivatives of f are uniformly bounded on M × [0, T ]. So by applying Lemma 2.3.2 and the arbitrariness of ǫ, we have completed the proof of Theorem 2.3.1. Finally, we would like to state a useful generalization of Theorem 2.3.1 by Chow and Lu in [40] which allows the set K to depend on time. One can consult the paper [40] for the proof. Theorem 2.3.5 (Chow and Lu [40]). Let K(t) ⊂ V , t ∈ [0, T ] be closed subsets which satisfy the following hypotheses (H3) K(t) is invariant under parallel translation defined by the connection ∇t for each t ∈ [0, T ]; ∆ (H4) in each fiber Vx , the set Kx (t) = K(t) ∩ Vx is nonempty, closed and convex for each t ∈ [0, T ]; S (H5) the space-time track (∂K(t) × {t}) is a closed subset of V × [0, T ]. t∈[0,T ]
223
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
Suppose that, for any x ∈ M and any initial time t0 ∈ [0, T ), and for any solution σx (t) of the (ODE) which starts in Kx (t0 ), the solution σx (t) will remain in Kx (t) for all later times. Then for any initial time t0 ∈ [0, T ) the solution σ(x, t) of the (PDE) will remain in K(t) for all later times if σ(x, t) starts in K(t0 ) at time t0 and the solution σ(x, t) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ]. 2.4. Hamilton-Ivey Curvature Pinching Estimate. The Hamilton-Ivey curvature pinching estimate roughly says that if a solution to the Ricci flow on a three-manifold becomes singular (i.e., the curvature goes to infinity) as time t approaches the maximal time T , then the most negative sectional curvature will be small compared to the most positive sectional curvature. This pinching estimate plays a crucial role in analyzing the formation of singularities in the Ricci flow on three-manifolds. Consider a complete solution to the Ricci flow ∂ gij = −2Rij ∂t on a complete three-manifold with bounded curvature in space for each time t ≥ 0. Recall from Section 1.3 that the evolution equation of the curvature operator Mαβ is given by (2.4.1)
∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ ∂t
2 where Mαβ is the operator square 2 Mαβ = Mαγ Mβγ # and Mαβ is the Lie algebra so(n) square # Mαβ = Cαγζ Cβηθ Mγη Mζθ . # In dimension n = 3, we know that Mαβ is the adjoint matrix of Mαβ . If we diagonalize Mαβ with eigenvalues λ ≥ µ ≥ ν so that λ , µ (Mαβ ) = ν # 2 then Mαβ and Mαβ are also diagonal, with 2 λ µν 2 and (M # ) = µ2 (Mαβ )= αβ ν2
λν λµ
.
Thus the ODE corresponding to PDE (2.4.1) for Mαβ (in the space of 3 × 3 matrices) is given by the following system d 2 dt λ = λ + µν, d 2 (2.4.2) dt µ = µ + λν, d 2 dt ν = ν + λµ.
224
H.-D. CAO AND X.-P. ZHU
Let P be the principal bundle of the manifold and form the associated bundle V = P ×G E, where G = O(3) and E is the vector space of symmetric bilinear forms on so(3). The curvature operator Mαβ is a smooth section of V = P ×G E. According to Theorem 2.3.1, any closed convex set of curvature operator matrices Mαβ which is O(3)-invariant (and hence invariant under parallel translation) and preserved by ODE (2.4.2) is also preserved by the Ricci flow. We are now ready to state and prove the Hamilton-Ivey pinching estimate . Theorem 2.4.1 (Hamilton [63], Ivey [73]). Suppose we have a solution to the Ricci flow on a three-manifold which is complete with bounded curvature for each t ≥ 0. Assume at t = 0 the eigenvalues λ ≥ µ ≥ ν of the curvature operator at each point are bounded below by ν ≥ −1. The scalar curvature R = λ + µ + ν is their sum. Then at all points and all times t ≥ 0 we have the pinching estimate R ≥ (−ν)[log(−ν) − 3], whenever ν < 0. Proof. Consider the function y = f (x) = x(log x − 3) defined on e2 ≤ x < +∞. It is easy to check that f is increasing and convex with range −e2 ≤ y < +∞. Let f −1 (y) = x be the inverse function, which is also increasing but concave and satisfies (2.4.3)
f −1 (y) =0 y→∞ y lim
Consider also the set K of matrices Mαβ defined by the inequalities λ + µ + ν ≥ −3, (2.4.4) K: ν + f −1 (λ + µ + ν) ≥ 0.
By Theorem 2.3.1 and the assumptions in Theorem 2.4.1 at t = 0, we only need to check that the set K defined above is closed, convex and preserved by the ODE (2.4.2). Clearly K is closed because f −1 is continuous. λ + µ + ν is just the trace function of 3 × 3 matrices which is a linear function. Hence the first inequality in (2.4.4) defines a linear half-space, which is convex. The function ν is the least eigenvalue function, which is concave. Also note that f −1 is concave. Thus the second inequality in (2.4.4) defines a convex set as well. Therefore we proved K is closed and convex. Under the ODE (2.4.2) d (λ + µ + ν) = λ2 + µ2 + ν 2 + λµ + λν + µν dt 1 = [(λ + µ)2 + (λ + ν)2 + (µ + ν)2 ] 2 ≥ 0. Thus the first inequality in (2.4.4) is preserved by the ODE.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
225
The second inequality in (2.4.4) can be written as whenever ν ≤ −e2 ,
λ + µ + ν ≥ f (−ν), which becomes (2.4.5)
whenever ν ≤ −e2 .
λ + µ ≥ (−ν)[log(−ν) − 2],
To show the inequality is preserved we only need to look at points on the boundary of the set. If ν + f −1 (λ + µ + ν) = 0 then ν = −f −1 (λ + µ + ν) ≤ −e2 since f −1 (y) ≥ e2 . Hence the RHS of (2.4.5) is nonnegative. We thus have λ ≥ 0 because λ ≥ µ. But µ may have either sign. We split our consideration into two cases: Case (i): µ ≥ 0. We need to verify dλ dµ d(−ν) + ≥ (log(−ν) − 1) dt dt dt when λ + µ = (−ν)[log(−ν) − 2]. Solving for log(−ν) − 2 =
λ+µ (−ν)
and substituting above, we must show 2
2
λ + µν + µ + λν ≥
λ+µ + 1 (−ν 2 − λµ) (−ν)
which is equivalent to (λ2 + µ2 )(−ν) + λµ(λ + µ + (−ν)) + (−ν)3 ≥ 0. Since λ, µ and (−ν) are all nonnegative we are done in the first case. Case (ii): µ < 0. We need to verify dλ d(−µ) d(−ν) ≥ + (log(−ν) − 1) dt dt dt when λ = (−µ) + (−ν)[log(−ν) − 2]. Solving for log(−ν) − 2 =
λ − (−µ) (−ν)
and substituting above, we need to show λ − (−µ) 2 2 λ + µν ≥ −µ − λν + + 1 (−ν 2 − λµ) (−ν) or λ2 + (−µ)(−ν) ≥ λ(−ν) − (−µ)2 +
λ − (−µ) + 1 (λ(−µ) − (−ν)2 ) (−ν)
226
H.-D. CAO AND X.-P. ZHU
which reduces to λ2 (−ν) + λ(−µ)2 + (−µ)2 (−ν) + (−ν)3 ≥ λ2 (−µ) + λ(−µ)(−ν) or equivalently (λ2 − λ(−µ) + (−µ)2 )((−ν) − (−µ)) + (−µ)3 + (−ν)3 ≥ 0. Since λ2 − λ(−µ) + (−µ)2 ≥ 0 and (−ν) − (−µ) ≥ 0 we are also done in the second case. Therefore the proof is completed. 2.5. Li-Yau-Hamilton Estimates. In [82], Li-Yau developed a fundamental gradient estimate, now called Li-Yau estimate, for positive solutions to the heat equation on a complete Riemannian manifold with nonnegative Ricci curvature. They used it to derive the Harnack inequality for such solutions by path integration. Then based on the suggestion of Yau, Hamilton [60] developed a similar estimate for the scalar curvature of solutions to the Ricci flow on a Riemann surface with positive curvature, and later obtained a matrix version of the Li-Yau estimate for solutions to the Ricci flow with positive curvature operator in all dimensions. This matrix version of the Li-Yau estimate is the Li-Yau-Hamilton estimate, which we will present in this section. The Li-Yau-Hamilton estimate plays a central role in the analysis of formation of singularities and the application of the Ricci flow to three-manifold topology. We have seen that in the Ricci flow the curvature tensor satisfies a nonlinear heat equation, and the nonnegativity of the curvature operator is preserved by the Ricci flow. Roughly speaking the Li-Yau-Hamilton estimate says the nonnegativity of a certain combination of the derivatives of the curvature up to second order is also preserved by the Ricci flow. Let us begin by describing the Li-Yau estimate for positive solutions to the heat equation on a complete Riemannian manifold with nonnegative Ricci curvature. Theorem 2.5.1 (Li-Yau [82]). Let (M, gij ) be an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature. Let u(x, t) be any positive solution to the heat equation ∂u = ∆u ∂t
on M × [0, ∞).
Then we have (2.5.1)
∂u |∇u|2 n − + u≥0 ∂t u 2t
on
M × (0, ∞).
We remark that one can in fact prove the following quadratic version that for any vector field V i on M , (2.5.2)
n ∂u + 2∇u · V + u|V |2 + u ≥ 0. ∂t 2t
If we take the optimal vector field V = −∇u/u, we recover the inequality (2.5.1). Now we consider the Ricci flow on a Riemann surface. Since in dimension two the Ricci curvature is given by Rij =
1 Rgij , 2
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
227
the Ricci flow (1.1.5) becomes ∂gij = −Rgij . ∂t
(2.5.3)
Now let gij (x, t) be a complete solution of the Ricci flow (2.5.3) on a Riemann surface M and 0 ≤ t < T . Then the scalar curvature R(x, t) evolves by the semilinear equation ∂R = △R + R2 ∂t on M × [0, T ). Suppose the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. Then it follows from Proposition 2.1.2 that the scalar curvature R(x, t) of the evolving metric remains nonnegative. Moreover, from the standard strong maximum principle (which works in each local coordinate neighborhood), the scalar curvature is positive everywhere for t > 0. In [60], Hamilton obtained the following Li-Yau estimate for the scalar curvature R(x, t). Theorem 2.5.2 (Hamilton [60]). Let gij (x, t) be a complete solution of the Ricci flow on a surface M . Assume the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. Then the scalar curvature R(x, t) satisfies the Li-Yau estimate R ∂R |∇R|2 − + ≥ 0. ∂t R t
(2.5.4)
Proof. By the above discussion, we know R(x, t) > 0 for t > 0. If we set L = log R(x, t)
for t > 0,
then 1 ∂ L = (△R + R2 ) ∂t R = △L + |∇L|2 + R and (2.5.4) is equivalent to ∂L 1 1 − |∇L|2 + = △L + R + ≥ 0. ∂t t t Following Li-Yau [82] in the linear heat equation case, we consider the quantity (2.5.5)
Q=
∂L − |∇L|2 = △L + R. ∂t
Then by a direct computation, ∂Q ∂ = (△L + R) ∂t ∂t ∂L ∂R =△ + R△L + ∂t ∂t
= △Q + 2∇L · ∇Q + 2|∇2 L|2 + 2R(△L) + R2 ≥ △Q + 2∇L · ∇Q + Q2 .
228
H.-D. CAO AND X.-P. ZHU
So we get ∂ 1 1 1 1 1 Q+ ≥△ Q+ + 2∇L · ∇ Q + + Q− Q+ . ∂t t t t t t Hence by a similar maximum principle argument as in the proof of Lemma 2.1.3, we obtain Q+
1 ≥ 0. t
This proves the theorem. As an immediate consequence, we obtain the following Harnack inequality for the scalar curvature R by taking the Li-Yau type path integral as in [82]. Corollary 2.5.3 (Hamilton [60]). Let gij (x, t) be a complete solution of the Ricci flow on a surface with bounded and nonnegative scalar curvature. Then for any points x1 , x2 ∈ M , and 0 < t1 < t2 , we have R(x2 , t2 ) ≥
t1 −dt (x1 ,x2 )2 /4(t2 −t1 ) e 1 R(x1 , t1 ). t2
Proof. Take the geodesic path γ(τ ), τ ∈ [t1 , t2 ], from x1 to x2 at time t1 with constant velocity dt1 (x1 , x2 )/(t2 − t1 ). Consider the space-time path η(τ ) = (γ(τ ), τ ), τ ∈ [t1 , t2 ]. We compute log
R(x2 , t2 ) = R(x1 , t1 ) =
Z
Z
t2
t1 t2
t1
≥
Z
t2
t1
d L(γ(τ ), τ )dτ dτ dγ 1 ∂R + ∇R · dτ R ∂τ dτ ! 2 ∂L 1 dγ dτ. − |∇L|2gij (τ ) − ∂τ 4 dτ gij (τ )
Then by Theorem 2.5.2 and the fact that the metric is shrinking (since the scalar curvature is nonnegative), we have ! 2 Z t2 1 1 dγ R(x2 , t2 ) dτ ≥ − − log R(x1 , t1 ) τ 4 dτ gij (τ ) t1 = log
dt (x1 , x2 )2 t1 − 1 t2 4(t2 − t1 )
After exponentiating above, we obtain the desired Harnack inequality. To prove a similar inequality as (2.5.4) for the scalar curvature of solutions to the Ricci flow in higher dimensions is not so simple. First of all, we will need to require nonnegativity of the curvature operator (which we know is preserved under the Ricci flow). Secondly, one does not get inequality (2.5.4) directly, but rather indirectly as the trace of certain matrix estimate. The key ingredient in formulating this matrix version is to derive some identities from the soliton solutions and prove an elliptic
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
229
inequality based on these quantities. Hamilton found such a general principle which was based on the idea of Li-Yau [82] when an identity is checked on the heat kernel before an inequality was found. To illustrate this point, let us first examine the heat equation case. Consider the heat kernel u(x, t) = (4πt)−n/2 e−|x|
2
/4t
for the standard heat equation on Rn which can be considered as an expanding soliton solution. Differentiating the function u, we get xj or ∇j u + uVj = 0, (2.5.6) ∇j u = −u 2t where xj ∇j u Vj = =− . 2t u Differentiating (2.5.6), we have (2.5.7)
∇i ∇j u + ∇i uVj +
u δij = 0. 2t
To make the expression in (2.5.7) symmetric in i, j, we multiply Vi to (2.5.6) and add to (2.5.7) and obtain u (2.5.8) ∇i ∇j u + ∇i uVj + ∇j uVi + uVi Vj + δij = 0. 2t Taking the trace in (2.5.8) and using the equation ∂u/∂t = ∆u, we arrive at ∂u n + 2∇u · V + u|V |2 + u = 0, ∂t 2t which shows that the Li-Yau inequality (2.5.1) becomes an equality on our expanding soliton solution u! Moreover, we even have the matrix identity (2.5.8). Based on the above observation and using a similar process, Hamilton found a matrix quantity, which vanishes on expanding gradient Ricci solitons and is nonnegative for any solution to the Ricci flow with nonnegative curvature operator. Now we describe the process of finding the Li-Yau-Hamilton quadratic for the Ricci flow in arbitrary dimension. Consider a homothetically expanding gradient soliton g, we have 1 gab = ∇a Vb 2t in the orthonormal frame coordinate chosen as in Section 1.3. Here Vb = ∇b f for some function f . Differentiating (2.5.9) and commuting give the first order relations
(2.5.9)
(2.5.10)
Rab +
∇a Rbc − ∇b Rac = ∇a ∇b Vc − ∇b ∇a Vc = Rabcd Vd ,
and differentiating again, we get ∇a ∇b Rcd − ∇a ∇c Rbd = ∇a (Rbcde Ve ) = ∇a Rbcde Ve + Rbcde ∇a Ve = ∇a Rbcde Ve + Rae Rbcde +
1 Rbcda . 2t
230
H.-D. CAO AND X.-P. ZHU
We further take the trace of this on a and b to get ∆Rcd − ∇a ∇c Rad − Rae Racde +
1 Rcd − ∇a Racde Ve = 0, 2t
and then by commuting the derivatives and second Bianchi identity, 1 1 ∆Rcd − ∇c ∇d R + 2Rcade Rae − Rce Rde + Rcd + (∇e Rcd − ∇d Rce )Ve = 0. 2 2t Let us define 1 1 Mab = ∆Rab − ∇a ∇b R + 2Racbd Rcd − Rac Rbc + Rab , 2 2t Pabc = ∇a Rbc − ∇b Rac . Then (2.5.11)
Mab + Pcba Vc = 0,
We rewrite (2.5.10) as Pabc = Rabcd Vd and then (2.5.12)
Pcab Vc + Racbd Vc Vd = 0.
Adding (2.5.11) and (2.5.12) we have Mab + (Pcab + Pcba )Vc + Racbd Vc Vd = 0 and then Mab Wa Wb + (Pcab + Pcba )Wa Wb Vc + Racbd Wa Vc Wb Vd = 0. If we write Uab =
1 (Va Wb − Vb Wa ) = V ∧ W, 2
then the above identity can be rearranged as (2.5.13)
∆
Q = Mab Wa Wb + 2Pabc Uab Wc + Rabcd Uab Ucd = 0.
This is the Li-Yau-Hamilton quadratic we look for. Note that the proof of the LiYau-Hamilton estimate below does not depend on the existence of such an expanding gradient Ricci soliton. It is only used as inspiration. Now we are ready to state the remarkable Li-Yau-Hamilton estimate for the Ricci flow. Theorem 2.5.4 (Hamilton [61]). Let gij (x, t) be a complete solution with bounded curvature to the Ricci flow on a manifold M for t in some time interval (0, T ) and suppose the curvature operator of gij (x, t) is nonnegative. Then for any one-form Wa and any two-form Uab we have Mab Wa Wb + 2Pabc Uab Wc + Rabcd Uab Ucd ≥ 0
231
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
on M × (0, T ). The proof of this theorem requires some rather intense calculations. Here we only give a sketch of the proof. For more details, we refer the reader to Hamilton’s original paper [61]. Sketch of the Proof. Let gij (x, t) be the complete solution with bounded and nonnegative curvature operator. Recall that in the orthonormal frame coordinate system, the curvatures evolve by ∂ ∂t Rabcd = ∆Rabcd + 2(Babcd − Babdc − Badbc + Bacbd ), ∂ ∂t Rab = ∆Rab + 2Racbd Rcd , ∂ 2 ∂t R = ∆R + 2|Ric| , where Babcd = Raebf Rcedf . By a long but straightforward computation from these evolution equations, one can get ∂ − ∆ Pabc = 2Radbe Pdec + 2Radce Pdbe + 2RbdcePade − 2Rde ∇d Rabce ∂t and
∂ − ∆ Mab = 2Racbd Mcd + 2Rcd (∇c Pdab + ∇c Pdba ) ∂t + 2Pacd Pbcd − 4Pacd Pbdc + 2Rcd Rce Radbe −
1 Rab . 2t2
Now consider ∆
Q = Mab Wa Wb + 2Pabc Uab Wc + Rabcd Uab Ucd . At a point where
(2.5.14)
1 ∂ ∂ − ∆ Wa = Wa , − ∆ Uab = 0, ∂t t ∂t
and (2.5.15)
∇a Wb = 0, ∇a Ubc =
1 1 (Rab Wc − Rac Wb ) + (gab Wc − gac Wb ), 2 4t
we have (2.5.16)
∂ − ∆ Q = 2Racbd Mcd Wa Wb − 2Pacd Pbdc Wa Wb ∂t
+ 8RadcePdbe Uab Wc + 4Raecf Rbedf Uab Ucd
+ (Pabc Wc + Rabcd Ucd )(Pabe We + Rabef Uef ). For simplicity we assume the manifold is compact and the curvature operator is strictly positive. (For the general case we shall mess the formula up a bit to sneak in the term ǫeAt f , as done in Lemma 2.1.3). Suppose not; then there will be a first time
232
H.-D. CAO AND X.-P. ZHU
when the quantity Q is zero, and a point where this happens, and a choice of U and W giving the null eigenvectors. We can extend U and W any way we like in space and time and still have Q ≥0, up to the critical time. In particular we can make the first derivatives in space and time to be anything we like, so we can extend first in space to make (2.5.15) hold at that point. And then, knowing ∆Wa and ∆Uab , we can extend in time to make (2.5.14) hold at that point and that moment. Thus we have (2.5.16) at the point. In the RHS of (2.5.16) the quadratic term (Pabc Wc + Rabcd Ucd )(Pabe We + Rabef Uef ) is clearly nonnegative. By similar argument as in the proof of Lemma 2.1.3, to get a contradiction we only need to show the remaining part in the RHS of (2.5.16) is also nonnegative. A nonnegative quadratic form can always be written as a sum of squares of linear forms. This is equivalent to diagonalizing a symmetric matrix and writing each nonnegative eigenvalue as a square. Write X n(n − 1) k Q= (Xak Wa + Yab Uab )2 , 1≤k ≤n+ . 2 k
This makes Mab =
X
Xak Xbk ,
Pabc =
k
X
k Yab Xck
k
and Rabcd =
X
k k Yab Ycd .
k
It is then easy to compute 2Racbd Mcd Wa Wb − 2Pacd Pbdc Wa Wb + 8Radce Pdbe Uab We + 4Raecf Rbedf Uab Ucd ! ! X X k k l l =2 Yac Ybd Xa Yc Wa Wb k
−2 +8
X
k Xdk Yac
k
X
k k Yad Yce
k
+4 =
X k,l
X k
k k Yae Ycf
!
!
!
l
X
l Ybd Xcl
l
X l
X l
l Ydb Xel
!
!
l Ybe Ydfl
Wa Wb
!
Uab Wc Uab Ucd
k l k l (Yac Xcl Wa − Yac Xck Wa − 2Yac Ybc Uab )2
≥ 0.
This says that the remaining part in the RHS of (2.5.16) is also nonnegative. Therefore we have completed the sketch of the proof.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
233
By taking Uab = 12 (Va Wb − Vb Wa ) and tracing over Wa , we immediately get Corollary 2.5.5 (Hamilton [61]). For any one-form Va we have ∂R R + + 2∇a R · Va + 2Rab Va Vb ≥ 0. ∂t t In particular by taking V ≡ 0, we see that the function tR(x, t) is pointwise nondecreasing in time. By combining this property with the local derivative estimate of curvature, we have the following elliptic type estimate. Corollary 2.5.6. Suppose we have a solution to the Ricci flow for t > 0 which is complete with bounded curvature, and has nonnegative curvature operator. Suppose also that at some time t > 0 we have the scalar curvature R ≤ M for some constant M in the ball of radius r around some point p. Then for k = 1, 2, . . ., the k th order derivatives of the curvature at p at the time t satisfy a bound 1 1 k k 2 2 + k +M |∇ Rm(p, t)| ≤ Ck M r2k t for some constant Ck depending only on the dimension and k. Proof. Since tR is nondecreasing in time, we get a bound R ≤ 2M in the given region for times between t/2 and t. The nonnegative curvature hypothesis tells us the metric is shrinking. So we can apply the local derivative estimate in Theorem 1.4.2 to deduce the result. By a similar argument as in Corollary 2.5.3, one readily has the following Harnack inequality. Corollary 2.5.7. Let gij (x, t) be a complete solution of the Ricci flow on a manifold with bounded and nonnegative curvature operator, and let x1 , x2 ∈ M, 0 < t1 < t2 . Then the following inequality holds R(x2 , t2 ) ≥
t1 −dt (x1 ,x2 )2 /2(t2 −t1 ) e 1 · R(x1 , t1 ). t2
In the above discussion, we assumed that the solution to the Ricci flow exists on 0 ≤ t < T , and we derived the Li-Yau-Hamilton estimate with terms 1/t in it. When the solution happens to be ancient, i.e., defined on −∞ < t < T , Hamilton [61] found an interesting and simple procedure for getting rid of them. Suppose we have a solution on α < t < T we can replace t by t − α in the Li-Yau-Hamilton estimate. If we let α → −∞, then the expression 1/(t − α) → 0 and disappears! In particular the trace Li-Yau-Hamilton estimate in Corollary 2.5.5 becomes (2.5.17)
∂R + 2∇a R · Va + 2Rab Va Vb ≥ 0. ∂t
By taking V = 0, we see that
∂R ∂t
≥ 0. Thus, we have the following
Corollary 2.5.8 (Hamilton [61]). Let gij (x, t) be a complete ancient solution of the Ricci flow on M × (−∞, T ) with bounded and nonnegative curvature operator, then the scalar curvature R(x, t) is pointwise nondecreasing in time t.
234
H.-D. CAO AND X.-P. ZHU
Corollary 2.5.8 will be very useful later on when we study ancient κ-solutions in Chapter 6, especially combined with Shi’s derivative estimate. We end this section by stating the Li-Yau-Hamilton estimate for the K¨ahler-Ricci flow, due to the first author [12], under the weaker curvature assumption of nonnegative holomorphic bisectional curvature. Note that the following Li-Yau-Hamilton estimate in the K¨ahler case is really a Li-Yau-Hamilton estimate for the Ricci tensor of the evolving metric, so not only can we derive an estimate on the scalar curvature, which is the trace of the Ricci curvature, similar to Corollary 2.5.5 but also an estimate on the determinant of the Ricci curvature as well. Theorem 2.5.9 (Cao [12]). Let gαβ¯(x, t) be a complete solution to the K¨ ahlerRicci flow on a complex manifold M with bounded curvature and nonnegtive bisectional curvature and 0 ≤ t < T . For any point x ∈ M and any vector V in the holomorphic tangent space Tx1,0 M , let Qαβ¯ =
∂ 1 γ δ¯ R ¯ + Rα¯γ Rγ β¯ + ∇γ Rαβ¯ V γ + ∇γ¯ Rαβ¯ V γ¯ + Rαβγ ¯ δ¯V V + Rαβ¯ . ∂t αβ t
Then we have ¯
Qαβ¯W α W β ≥ 0 for all x ∈ M , V, W ∈ Tx1,0 M , and t > 0. Corollary 2.5.10 (Cao [12]). Under the assumptions of Theorem 2.5.9, we have (i) the scalar curvature R satisfies the estimate R ∂R |∇R|2 − + ≥ 0, ∂t R t and (ii) assuming Rαβ¯ > 0, the determinant φ = det(Rαβ¯ )/ det(gαβ¯) of the Ricci curvature satisfies the estimate ∂φ |∇φ|2 nφ − + ≥0 ∂t nφ t for all x ∈ M and t > 0. 2.6. Perelman’s Estimate for Conjugate Heat Equations. In [103] Perelman obtained a Li-Yau type estimate for fundamental solutions of the conjugate heat equation, which is a backward heat equation, when the metric evolves by the Ricci flow. In this section we shall describe how to get this estimate along the same line as in the previous section. More importantly, we shall show how the Li-Yau path integral, when applied to Perelman’s Li-Yau type estimate, leads to an important space-time distance function introduced by Perelman [103]. We learned from Hamilton [67] this idea of looking at Perelman’s Li-Yau estimate. We saw in the previous section that the Li-Yau quantity and the Li-Yau-Hamilton quantity vanish on expanding solutions. Note that when we consider a backward heat equation, shrinking solitons can be viewed as expanding backward in time. So we start by looking at shrinking gradient Ricci solitons.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
235
Suppose we have a shrinking gradient Ricci soliton gij with potential function f on manifold M and −∞ < t < 0 so that, for τ = −t, (2.6.1)
Rij + ∇j ∇i f −
1 gij = 0. 2τ
Then, by taking the trace, we have (2.6.2)
R + ∆f −
n = 0. 2τ
Also, by similar calculations as in deriving (1.1.15), we get (2.6.3)
R + |∇f |2 −
f =C τ
where C is a constant which we can set to be zero. Moreover, observe ∂f = |∇f |2 ∂t
(2.6.4)
because f evolves in time with the rate of change given by the Lie derivative in the direction of ∇f generating the one-parameter family of diffeomorphisms. Combining (2.6.2) with (2.6.4), we see f satisfies the backward heat equation (2.6.5)
∂f n = −∆f + |∇f |2 − R + , ∂t 2τ
or equivalently (2.6.6)
n ∂f = ∆f − |∇f |2 + R − . ∂τ 2τ
Recall the Li-Yau-Hamilton quadratic is a certain combination of the second order space derivative (or first order time derivative), first order space derivatives and zero orders. Multiplying (2.6.2) by a factor of 2 and subtracting (2.6.3) yields 2∆f − |∇f |2 + R +
n 1 f− =0 τ τ
valid for our potential function f of the shrinking gradient Ricci soliton. The quantity on the LHS of the above identity is precisely the Li-Yau-Hamilton type quadratic found by Perelman [103]. Note that a function f satisfies the backward heat equation (2.6.6) if and only if the function n
u = (4πτ )− 2 e−f satisfies the so called conjugate heat equation (2.6.7)
∗ u ,
∂u − ∆u + Ru = 0. ∂τ
Lemma 2.6.1 (Perelman [103]). Let gij (x, t), 0 ≤ t < T , be a complete solution n to the Ricci flow on an n-dimensional manifold M and let u = (4πτ )− 2 e−f be a solution to the conjugate equation (2.6.7) with τ = T − t. Set H = 2∆f − |∇f |2 + R +
f −n τ
236
H.-D. CAO AND X.-P. ZHU
and v = τ Hu = τ (R + 2∆f − |∇f |2 ) + f − n u. Then we have
and
2 1 1 ∂H = ∆H − 2∇f · ∇H − H − 2 Rij + ∇i ∇j f − gij , ∂τ τ 2τ 2 ∂v 1 = ∆v − Rv − 2τ u Rij + ∇i ∇j f − gij . ∂τ 2τ Proof. By direct computations, we have f −n 2 2△f − |∇f | + R + τ ∂f ∂f − 2h2Rij , fij i − 2 ∇f, ∇ + 2Ric (∇f, ∇f ) = 2△ ∂τ ∂τ 1 ∂ f −n ∂ R+ f− + ∂τ τ ∂τ τ2 n = 2△ △f − |∇f |2 + R − − 4hRij , fij i + 2Ric (∇f, ∇f ) 2τ D n E − 2 ∇f, ∇ △f − |∇f |2 + R − − △R − 2|Rij |2 2τ n f −n 1 △f − |∇f |2 + R − − , + τ 2τ τ 2 f −n ∇H = ∇ 2△f − |∇f |2 + R + τ 1 = 2∇(△f ) − 2h∇∇i f, ∇i f i + ∇R + ∇f, τ f −n △H = △ 2△f − |∇f |2 + R + τ 1 = 2△(△f ) − △(|∇f |2 ) + △R + △f, τ
∂ ∂ H= ∂τ ∂τ
and 1 ∇f, ∇f i τ 2 = 2 [h2∇(△f ), ∇f i − 2hfij , fi fj i + h∇R, ∇f i] + |∇f |2 . τ
2∇H · ∇f = 2h2∇(△f ) − 2h∇∇i f, ∇i f i + ∇R +
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
237
Thus we get 1 ∂ H − △H + 2∇f · ∇H + H ∂τ τ = −4hRij , fij i + 2Ric (∇f, ∇f ) − 2|Rij |2 − △(|∇f |2 ) + 2h∇(△f ), ∇f i 2 2 n + △f + R − 2 τ τ 2τ n R 1 2 2 = −2 |Rij | + |fij | + 2 + 2hRij , fij i − − △f 4τ τ τ 2 1 gij , = −2 Rij + ∇i ∇j f − 2τ
and
∂ ∂ −△+R v = − △ + R (τ Hu) ∂τ ∂τ ∂ − △ (τ H) · u − 2h∇(τ H), ∇ui = ∂τ ∂ = − △ (τ H) − 2h∇(τ H), ∇f i u ∂τ 1 ∂H − △H + 2∇f · ∇H + H u =τ ∂τ τ 2 1 = −2τ u Rij + ∇i ∇j f − gij . 2τ
Note that, since f satisfies the equation (2.6.6), we can rewrite H as (2.6.8)
H=2
1 ∂f + |∇f |2 − R + f. ∂τ τ
Then, by Lemma 2.6.1, we have 2 ∂ 1 (τ H) = ∆(τ H) − 2∇f · ∇(τ H) − 2τ Ric + ∇2 f − g . ∂τ 2τ
So by the maximum principle, we find max(τ H) is nonincreasing as τ increasing. When u is chosen to be a fundamental solution to (2.6.7), one can show that limτ →0 τ H ≤ 0 and hence H ≤ 0 on M for all τ ∈ (0, T ] (see, for example, [99]). Since this fact is not used in later chapters and will be only used in the rest of the section to introduce a space-time distance via Li-Yau path integral, we omit the details of the proof. Once we have Perelman’s Li-Yau type estimate H ≤ 0, we can apply the Li-Yau path integral as in [82] to estimate the above solution u (i.e., a heat kernel estimate for the conjugate heat equation, see also the earlier work of Cheeger-Yau [28]). Let p, q ∈ M be two points and γ(τ ), τ ∈ [0, τ¯], be a curve joining p and q, with γ(0) = p
238
H.-D. CAO AND X.-P. ZHU
and γ(¯ τ ) = q. Then along the space-time path (γ(τ ), τ ), τ ∈ [0, τ¯], we have √ ∂f d √ 1 2 τ f (γ(τ ), τ ) = 2 τ + ∇f · γ(τ ˙ ) +√ f dτ ∂τ τ √ √ 2 ˙ ) + τR ≤ τ − |∇f |gij (τ ) + 2∇f · γ(τ √ √ = − τ |∇f − γ(τ ˙ )|2gij (τ ) + τ (R + |γ(τ ˙ )|2gij (τ ) ) √ ≤ τ (R + |γ(τ ˙ )|2gij (τ ) ) where we have used the fact that H ≤ 0 and the expression for H in (2.6.8). Integrating the above inequality from τ = 0 to τ = τ¯, we obtain Z τ¯ √ √ τ (R + |γ(τ ˙ )|2gij (τ ) )dτ, 2 τ¯f (q, τ¯) ≤ 0
or
1 f (q, τ¯) ≤ √ L(γ), 2 τ¯ where L(γ) ,
(2.6.9) Denote by
Z
τ¯
0
√
τ (R + |γ(τ ˙ )|2gij (τ ) )dτ.
1 l(q, τ¯) , inf √ L(γ), γ 2 τ ¯
(2.6.10)
where the inf is taken over all space curves γ(τ ), 0 ≤ τ ≤ τ¯, joining p and q. The space-time distance function l(q, τ¯) obtained by the above Li-Yau path integral argument is first introduced by Perelman in [103] and is what Perelman calls reduced distance. Since Perelman pointed out in page 19 of [103] that “an even closer reference is [82], where they use ‘length’, associated to a linear parabolic equation, which is pretty much the same as in our case”, it is natural to call l(q, τ¯) the Li-YauPerelman distance. See Chapter 3 for much more detailed discussions. Finally, we conclude this section by relating the quantity H (or v) and the Wfunctional of Perelman defined in (1.5.9). Observe that v happens to be the integrand of the W-functional, Z W(gij (t), f, τ ) = vdV. M
Hence, when M is compact,
∂ v + Rv dV ∂τ M 2 Z Ric + ∇2 f − 1 g udV = −2τ 2τ M
d W= dτ
Z
≤ 0,
or equivalently,
d W(gij (t), f (t), τ (t)) = dt
2 n 1 2τ Rij + ∇i ∇j f − gij (4πτ )− 2 e−f dV, 2τ M
Z
which is the same as stated in Proposition 1.5.8.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
239
3. Perelman’s Reduced Volume. In Section 1.5 we introduced the F functional and the W-functional of Perelman and proved their monotonicity properties under the Ricci flow. In the last section of the previous chapter we have defined the Li-Yau-Perelman distance. The main purpose of this chapter is to use the Li-YauPerelman distance to define the Perelman’s reduced volume, which was introduced by Perelman in [103], and prove the monotonicity property of the reduced volume under the Ricci flow. This new monotonicity formula is more useful for local considerations, especially when we consider the formation of singularities in Chapter 6 and work on the Ricci flow with surgery in Chapter 7. As first applications we will present two no local collapsing theorems of Perelman [103] in this chapter. More applications can be found in Chapter 6 and 7. 3.1. Riemannian Formalism in Potentially Infinite Dimensions. In Section 2.6, from an analytic view point, we saw how the Li-Yau path integral of Perelman’s estimate for fundamental solutions to the conjugate heat equation leads to the Li-Yau-Perelman distance. In this section we present, from a geometric view point, another motivation why one is lead to the consideration of the Li-Yau-Perelman distance function, as well as a reduced volume concept. Interestingly enough, the LiYau-Hamilton quadratic introduced in Section 2.5 appears again in this geometric consideration. We consider the Ricci flow ∂ gij = −2Rij ∂t on a manifold M where we assume that gij (·, t) are complete and have uniformly bounded curvatures. Recall from Section 2.5 that the Li-Yau-Hamilton quadratic introduced in [61] is Q = Mij Wi Wj + 2Pijk Uij Wk + Rijkl Uij Ukl where 1 1 Mij = ∆Rij − ∇i ∇j R + 2Rikjl Rkl − Rik Rjk + Rij , 2 2t Pijk = ∇i Rjk − ∇j Rik and Uij is any two-form and Wi is any 1-form. Here and throughout this chapter we do not always bother to raise indices; repeated indices is short hand for contraction with respect to the metric. In [63], Hamilton predicted that the Li-Yau-Hamilton quadratic is some sort of jet extension of positive curvature operator on some larger space. Such an interpretation of the Li-Yau-Hamilton quadratic as a curvature operator on the space M × R+ was found by Chow and Chu [38] where a potentially degenerate Riemannian metric on M × R+ was constructed. The degenerate Riemannian metric on M × R+ is the limit of the following two-parameter family of Riemannian metrics gN,δ (x, t) = g(x, t) + (R(x, t) +
N )dt2 2(t + δ)
as N tends to infinity and δ tends to zero, where g(x, t) is the solution of the Ricci flow on M and t ∈ R+ .
240
H.-D. CAO AND X.-P. ZHU
˜ = M ×SN ×R+ To avoid the degeneracy, Perelman [103] considers the manifold M with the following metric: g˜ij = gij , g˜αβ = τ gαβ , N + R, g˜oo = 2τ g˜iα = g˜io = g˜αo = 0, where i, j are coordinate indices on M ; α, β are coordinate indices on SN ; and the coordinate τ on R+ has index o. Let τ = T − t for some fixed constant T . Then gij ∂ will evolve with τ by the backward Ricci flow ∂τ gij = 2Rij . The metric gαβ on SN is 1 a metric with constant sectional curvature 2N . We remark that the metric g˜αβ on SN is chosen so that the product metric (˜ gij , g˜αβ ) on M × SN evolves by the Ricci flow, while the component g˜oo is just the ˜ = M × SN × R+ is scalar curvature of (˜ gij , g˜αβ ). Thus the metric g˜ defined on M exactly a “regularization” of Chow-Chu’s degenerate metric on M × R+ . Proposition 3.1.1. The components of the curvature tensor of the metric g˜ coincide (modulo N −1 ) with the components of the Li-Yau-Hamilton quadratic.
Proof. By definition, the Christoffel symbols of the metric g˜ are given by the following list: ˜ kij = Γkij , Γ ˜ γ = 0, ˜ k = 0 and Γ Γ iβ ij k ˜ ˜ Γαβ = 0 and Γγiβ = 0, ˜ o = −˜ ˜ k = g kl Rli and Γ g oo Rij , Γ io ij ˜ o = 1 g˜oo ∂ R, ˜ k = − 1 g kl ∂ R and Γ Γ io oo 2 ∂xl 2 ∂xi γ o k ˜ iβ = 0, Γ ˜ oβ = 0 and Γ ˜ = 0, Γ oj γ γ ˜ Γ =Γ , αβ
˜γ Γ αo ˜ γoo Γ ˜o Γ oo
αβ
1 γ ˜ o = − 1 g˜oo gαβ , = δα and Γ αβ 2τ 2 o ˜ = 0 and Γoβ = 0, 1 oo N ∂ = g˜ − 2+ R . 2 2τ ∂τ
Fix a point (p, s, τ ) ∈ M × SN × R+ and choose normal coordinates around p ∈ M and normal coordinates around s ∈ SN such that Γkij (p) = 0 and Γγαβ (s) = 0 for all ˜ of the metric g˜ at the point i, j, k and α, β, γ. We compute the curvature tensor Rm
241
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
as follows:
˜ oil = Rijkl + O ˜ ijkl = Rijkl + Γ ˜ kio Γ ˜ ojl − Γ ˜ kjo Γ R ˜ ijkδ = 0, R ˜ ijγδ = 0 R
1 N
,
˜ iβkδ = Γ ˜ kio Γ ˜ oβδ − Γ ˜ kβo Γ ˜ oiδ = − 1 g˜oo gβδ g kl Rli = O and R 2
˜ iβγδ = 0, R
1 N
,
˜ kio Γ ˜ ojo − Γ ˜ kjo Γ ˜ oio = Pijk + O 1 , ˜ ijko = ∂ Rjk − ∂ Rik + Γ R ∂xi ∂xj N 2 1 ∂ ∂ ˜ ioko = − ˜ kio Γ ˜ ooo − Γ ˜ koj Γ ˜j − Γ ˜ koo Γ ˜ oio R R− (Ril g lk ) + Γ io 2 ∂xi ∂xk ∂τ ∂ 1 1 1 ∇ ∇ R − R + 2R R − R − R R + O =− i k ik ik lk ik ij jk 2 ∂τ 2τ N 1 = Mik + O( ), N ˜ ijγo = 0 and R ˜ iγjo = 0, R 1 γ ˜o ˜ ˜ ˜ ioγδ = 0, Riβγo = −τ Γβo Γio = O and R N ˜ ioγo = 0, R
˜ αβγo = 0, R 1 1 γ ˜γ ˜o γ ˜β ˜ ˜ δα + Γαo Γoo − Γoβ Γαo τ = O , Rαoγo = 2 2τ N ˜ αβγδ = O 1 . R N
Thus the components of the curvature tensor of the metric g˜ coincide (modulo N −1 ) with the components of the Li-Yau-Hamilton quadratic.
The following observation due to Perelman [103] gives an important motivation to define Perelman’s reduced volume.
Corollary 3.1.2. All components of the Ricci tensor of g˜ are zero (modulo N −1 ).
Proof. From the list of the components of the curvature tensor of g˜ given above,
242
H.-D. CAO AND X.-P. ZHU
we have ˜ ij = g˜kl R ˜ ijkl + g˜αβ R ˜ iαjβ + g˜oo R ˜ iojo R 1 1 1 αβ oo αβ oo g g˜ g Rij + g˜ Mij − Rij + O = Rij − 2τ 2τ N N oo 1 = Rij − g˜ Rij + O 2τ N 1 , =O N ˜ iγ = g˜kl R ˜ ikγl + g˜αβ R ˜ iαγβ + g˜oo R ˜ ioγo = 0, R ˜ io = g˜kl R ˜ ikol + g˜αβ R ˜ iαoβ + g˜oo R ˜ iooo R 1 , = −g kl Pikl + O N ˜ αβ = g˜kl R ˜ αkβl + g˜γδ R ˜ αγβδ + g˜oo R ˜ αoβo R 1 =O , N ˜ αo = g˜kl R ˜ αkol + g˜βγ R ˜ αβoγ + g˜oo R ˜ αooo = 0, R ˜ oo = g˜kl R ˜ okol + g˜αβ R ˜ oαoβ + g˜oo R ˜ oooo R 1 1 +O . = g kl Mkl + O N N Since g˜oo is of order N −1 , we see that the norm of the Ricci tensor is given by 1 ˜ |Ric|g˜ = O . N This proves the result. We now use the Ricci-flatness of the metric g˜ to interpret the Bishop-Gromov relative volume comparison theorem which will motivate another monotonicity formula for the Ricci flow. The argument in the following will not be rigorous. However it ˜ , g˜) gives an intuitive picture of what one may expect. Consider a metric ball in (M N ˜ centered at some point (p, s, 0) ∈ M . Note that the metric of the sphere S at τ = 0 degenerates and it shrinks to a point. Then the shortest geodesic γ(τ ) between (p, s, 0) ˜ is always orthogonal to the SN fibre. The length and an arbitrary point (q, s¯, τ¯) ∈ M of γ(τ ) can be computed as s
N + R + |γ(τ ˙ )|2gij (τ ) dτ 2τ 0 Z τ¯ √ √ 3 1 τ (R + |γ(τ ˙ )|2gij )dτ + O(N − 2 ). = 2N τ¯ + √ 2N 0
Z
τ¯
Thus a shortest geodesic should minimize L(γ) =
Z
0
τ¯
√
τ (R + |γ(τ ˙ )|2gij )dτ.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
243
Let minimum. We claim that a metric sphere √ L(q, τ¯) denote the corresponding √ ˜ of radius 2N τ¯ centered at (p, s, 0) is O(N −1 )-close to the hyperSM˜ ( 2N τ¯) in M √ surface {τ = τ¯}. Indeed, if (x, s′ , τ (x)) lies on the metric sphere SM˜ ( 2N τ¯), then the distance between (x, s′ , τ (x)) and (p, s, 0) is p √ 3 1 L(x, τ (x)) + O N − 2 2N τ¯ = 2N τ (x) + √ 2N which can be written as p √ 1 L(x, τ (x)) + O(N −2 ) = O(N −1 ). τ (x) − τ¯ = − 2N √ This shows that the metric sphere SM˜ ( 2N τ¯) is O(N −1 )-close to the hypersurface 1 {τ = τ¯}. Note that the metric gαβ on SN has constant sectional curvature 2N . Thus √ 2N τ¯ Vol SM˜ Z Z ≈ dVτ (x)gαβ dVgij (x) SN ZM N = (τ (x)) 2 Vol (S N )dVM M
N √ 1 −2 ≈ (2N ) ωN L(x, τ (x)) + O(N ) dVM τ¯ − 2N M N Z √ N 1 ≈ (2N ) 2 ωN τ¯ − L(x, τ¯) + o(N −1 ) dVM , 2N M N 2
Z
where ωN is the volume of √ the standard N -dimensional sphere. Now the volume of Euclidean sphere of radius 2N τ¯ in Rn+N +1 is √ N +n Vol (SRn+N +1 ( 2N τ¯)) = (2N τ¯) 2 ωn+N . Thus we have √ Z Vol (SM˜ ( 2N τ¯)) 1 −n −n 2 2 √ · (¯ τ) exp − √ L(x, τ¯) dVM . ≈ const · N 2 τ¯ Vol (SRn+N +1 ( 2N τ¯)) M ˜ is zero (modulo N −1 ), the Bishop-Gromov volume Since the Ricci curvature of M comparison theorem then suggests that the integral Z 1 ∆ −n ˜ 2 exp − √ L(x, τ¯) dVM , V (¯ τ) = (4π¯ τ) 2 τ¯ M which we will call Perelman’s reduced volume, should be nonincreasing in τ¯. A rigorous proof of this monotonicity property will be given in the next section. One should note the analog of reduced volume with the heat kernel and there is a parallel calculation for the heat kernel of the Shr¨odinger equation in the paper of Li-Yau [82]. 3.2. Comparison Theorems for Perelman’s Reduced Volume. In this section we will write the Ricci flow in the backward version ∂ gij = 2Rij ∂τ
244
H.-D. CAO AND X.-P. ZHU
on a manifold M with τ = τ (t) satisfying dτ /dt = −1 (in practice we often take τ = t0 − t for some fixed time t0 ). We always assume that either M is compact or gij (τ ) are complete and have uniformly bounded curvature. To each (smooth) space curve γ(τ ), 0 < τ1 ≤ τ ≤ τ2 , in M , we define its L-length as Z τ2 √ L(γ) = τ (R(γ(τ ), τ ) + |γ(τ ˙ )|2gij (τ ) )dτ. τ1
Let X(τ ) = γ(τ ˙ ), and let Y (τ ) be any (smooth) vector field along γ(τ ). First of all, we compute the first variation formula for L-length. Lemma 3.2.1 (First variation formula). Z τ2 √ √ 1 δY (L) = 2 τ hX, Y i|ττ21 + τ Y, ∇R − 2∇X X − 4Ric (·, X) − X dτ τ τ1 where h·, ·i denotes the inner product with respect to the metric gij (τ ). Proof. By direct computations, Z τ2 √ δY (L) = τ (h∇R, Y i + 2hX, ∇Y Xi)dτ τ1 Z τ2 √ = τ (h∇R, Y i + 2hX, ∇X Y i)dτ τ1 Z τ2 √ d τ h∇R, Y i + 2 hX, Y i − 2h∇X X, Y i − 4Ric (X, Y ) dτ = dτ τ1 Z τ2 √ √ 1 = 2 τ hX, Y i|ττ21 + τ Y, ∇R − 2∇X X − 4Ric (·, X) − X dτ. τ τ1 A smooth curve γ(τ ) in M is called an L-geodesic if it satisfies the following L-geodesic equation (3.2.1)
1 1 ∇X X − ∇R + X + 2Ric (X, ·) = 0. 2 2τ
Given any two points p, q ∈ M and τ2 > τ1 > 0, there always exists an L-shortest curve (or shortest L-geodesic) γ(τ ): [τ1 , τ2 ] → M connecting p to q which satisfies √ the above L-geodesic equation. Multiplying the L-geodesic equation (3.2.1) by τ , we get √ √ √ τ ∇X ( τ X) = ∇R − 2 τ Ric (X, ·) on [τ1 , τ2 ], 2 or equivalently √ √ d √ τ ( τ X) = ∇R − 2Ric ( τ X, ·) on [τ1 , τ2 ]. dτ 2 Thus if a continuous curve, defined on [0, τ2 ], satisfies the L-geodesic equation on √ every subinterval 0 < τ1 ≤ τ ≤ τ2 , then τ1 X(τ1 ) has a limit as τ1 → 0+ . This allows us to extend the definition of the L-length to include the case τ1 = 0 for
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
245
all those (continuous) curves γ : [0, τ2 ] → M which are smooth on (0, τ2 ] and have √ limits lim+ τ γ(τ ˙ ). Clearly, there still exists an L-shortest curve γ(τ ) : [0, τ2 ] → M τ →0
connecting arbitrary two points p, q ∈ M and satisfying the L-geodesic equation (3.2.1) on (0, τ2 ]. Moreover, √ for any vector v ∈ Tp M , we can find an L-geodesic γ(τ ) ˙ ) = v. starting at p with lim τ γ(τ τ →0+
From now on, we fix a point p ∈ M and set τ1 = 0. The L-distance function on the space-time M × R+ is denoted by L(q, τ¯) and defined to be the L-length of the L-shortest curve γ(τ ) connecting p and q with 0 ≤ τ ≤ τ¯. Consider a shortest L-geodesic γ : [0, τ¯] → M connecting p to q. In the computations below we pretend that L-shortest geodesics between p and q are unique for all pairs (q, τ¯); if this is not the case, the inequalities that we obtain are still valid, by a standard barrier argument, when understood in the sense of distributions (see, for example, [112]). The first variation formula in Lemma 3.2.1 implies that E D √ τ ), Y (¯ τ) . ∇Y L(q, τ¯) = 2 τ¯X(¯ Thus
√ τ ), ∇L(q, τ¯) = 2 τ¯X(¯ and |∇L|2 = 4¯ τ |X|2 = −4¯ τ R + 4¯ τ (R + |X|2 ).
(3.2.2) We also compute (3.2.3)
d L(γ(τ ), τ )|τ =¯τ − h∇L, Xi dτ √ √ = τ¯(R + |X|2 ) − 2 τ¯|X|2 √ √ = 2 τ¯R − τ¯(R + |X|2 ).
Lτ¯ (γ(¯ τ ), τ¯) =
To evaluate R + |X|2 , we compute by using (3.2.1), d (R(γ(τ ), τ ) + |X(τ )|2gij (τ ) ) dτ = Rτ + h∇R, Xi + 2h∇X X, Xi + 2Ric (X, X) 1 1 = Rτ + R + 2h∇R, Xi − 2Ric (X, X) − (R + |X|2 ) τ τ 1 2 = −Q(X) − (R + |X| ), τ where Q(X) = −Rτ −
R − 2h∇R, Xi + 2Ric (X, X) τ
is the trace Li-Yau-Hamilton quadratic in Corollary 2.5.5. Hence 3 1√ d 3 (τ 2 (R + |X|2 ))|τ =¯τ = τ¯(R + |X|2 ) − τ¯ 2 Q(X) dτ 2 3 1 d = L(γ(τ ), τ )|τ =¯τ − τ¯ 2 Q(X). 2 dτ
246
H.-D. CAO AND X.-P. ZHU
Therefore, 3
τ¯ 2 (R + |X|2 ) =
(3.2.4)
1 L(q, τ¯) − K, 2
where (3.2.5)
K=
Z
τ¯
3
τ 2 Q(X)dτ. 0
Combining (3.2.2) with (3.2.3), we obtain 2 4 |∇L|2 = −4¯ τR + √ L − √ K τ¯ τ¯
(3.2.6) and
√ 1 1 Lτ¯ = 2 τ¯R − L + K. 2¯ τ τ¯
(3.2.7)
Next we compute the second variation of an L-geodesic. Lemma 3.2.2 (Second variation formula). For any L-geodesic γ, we have Z τ¯ √ √ τ [2|∇X Y |2 + 2hR(Y, X)Y, Xi δY2 (L) = 2 τ h∇Y Y, Xi|τ0¯ + 0
+ ∇Y ∇Y R + 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X)]dτ.
Proof. We compute δY2
Z
τ¯
√
τ (Y (R) + 2h∇Y X, Xi)dτ (L) = Y 0 Z τ¯ √ = τ (Y (Y (R)) + 2h∇Y ∇Y X, Xi + 2|∇Y X|2 )dτ 0 Z τ¯ √ = τ (Y (Y (R)) + 2h∇Y ∇X Y, Xi + 2|∇X Y |2 )dτ 0
and 2h∇Y ∇X Y, Xi = 2h∇X ∇Y Y, Xi + 2hR(Y, X)Y, Xi d = 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) − 2h∇Y Y, ∇X Xi dτ d ∇Y Y, X − 2h∇X ∇Y Y, Xi + 2hR(Y, X)Y, Xi − 2 dτ d = 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) − 2h∇Y Y, ∇X Xi dτ ∂ i j kl − 2 Y Y (g (∇i Rlj + ∇j Rli − ∇l Rij )) k , X + 2hR(Y, X)Y, Xi ∂x d = 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) − 2h∇Y Y, ∇X Xi − 4∇Y Ric (X, Y ) dτ + 2∇X Ric (Y, Y ) + 2hR(Y, X)Y, Xi,
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
247
where we have used the computation ∂ k Γ = g kl (∇i Rlj + ∇j Rli − ∇l Rij ). ∂τ ij Thus by using the L-geodesic equation (3.2.1), we get Z τ¯ √ d δY2 (L) = τ Y (Y (R)) + 2 h∇Y Y, Xi − 4Ric (∇Y Y, X) dτ 0
− 2h∇Y Y, ∇X Xi − 4∇Y Ric (X, Y ) + 2∇X Ric (Y, Y ) 2 + 2hR(Y, X)Y, Xi + 2|∇X Y | dτ Z τ¯ √ d 1 = 2 τ h∇Y Y, Xi + √ h∇Y Y, Xi dτ dτ τ 0 Z τ¯ √ + τ [Y (Y (R)) − h∇Y Y, ∇Ri − 4∇Y Ric (X, Y ) 0
+ 2∇X Ric (Y, Y ) + 2hR(Y, X)Y, Xi + 2|∇X Y |2 ]dτ Z τ¯ √ √ = 2 τ h∇Y Y, Xi|τ0¯ + τ [2|∇X Y |2 + 2hR(Y, X)Y, Xi 0
+ ∇Y ∇Y R − 4∇Y Ric (X, Y ) + 2∇X Ric (Y, Y )]dτ.
We now use the above second variation formula to estimate the Hessian of the L-distance function. Let γ(τ ) : [0, τ¯] → M be an L-shortest curve connecting p and q so that the L-distance function L = L(q, τ¯) is given by the L-length of γ. We fix a vector Y at τ = τ¯ with |Y |gij (¯τ ) = 1, and extend Y along the L-shortest geodesic γ on [0, τ¯] by solving the following ODE ∇X Y = −Ric (Y, ·) +
(3.2.8)
1 Y. 2τ
This is similar to the usual parallel translation and multiplication with proportional parameter. Indeed, suppose {Y1 , . . . , Yn } is an orthonormal basis at τ = τ¯ (with respect to the metric gij (¯ τ )) and extend this basis along the L-shortest geodesic γ by solving the above ODE (3.2.8). Then d hYi , Yj i = 2Ric (Yi , Yj ) + h∇X Yi , Yj i + hYi , ∇X Yj i dτ 1 = hYi , Yj i τ for all i, j. Hence, (3.2.9)
hYi (τ ), Yj (τ )i =
τ δij τ¯
and {Y1 (τ ), . . . , Yn (τ )} remains orthogonal on [0, τ¯] with Yi (0) = 0, i = 1, . . . , n. Proposition 3.2.3. Given any unit vector Y at any point q ∈ M with τ = τ¯, consider an L-shortest geodesic γ connecting p to q and extend Y along γ by solving
248
H.-D. CAO AND X.-P. ZHU
the ODE (3.2.8). Then the Hessian of the L-distance function L on M with τ = τ¯ satisfies √ 1 HessL (Y, Y ) ≤ √ − 2 τ¯Ric (Y, Y ) − τ¯
Z
τ¯
√ τ Q(X, Y )dτ
0
in the sense of distributions, where Q(X, Y ) = −∇Y ∇Y R − 2hR(Y, X)Y, Xi − 4∇X Ric (Y, Y ) + 4∇Y Ric (Y, X) 1 − 2Ric τ (Y, Y ) + 2|Ric (Y, ·)|2 − Ric (Y, Y ) τ is the Li-Yau-Hamilton quadratic. Moreover the equality holds if and only if the vector field Y (τ ), τ ∈ [0, τ¯], is an L-Jacobian field (i.e., Y is the derivative of a variation of γ by L-geodesics). Proof. As said before, we pretend that the shortest L-geodesics between p and q are unique so that L(q, τ¯) is smooth. Otherwise, the inequality is still valid, by a standard barrier argument, when understood in the sense of distributions (see, for example, [112]). √ √ Recall that ∇L(q, τ¯) = 2 τ¯X. Then h∇Y Y, ∇Li = 2 τ¯h∇Y Y, Xi. We compute by using Lemma 3.2.2, (3.2.8) and (3.2.9), HessL (Y, Y ) = Y (Y (L))(¯ τ ) − h∇Y Y, ∇Li(¯ τ) √ 2 τ) ≤ δY (L) − 2 τ¯h∇Y Y, Xi(¯ Z τ¯ √ = τ [2|∇X Y |2 + 2hR(Y, X)Y, Xi + ∇Y ∇Y R 0
+ 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X)]dτ 2 Z τ¯ √ 1 Y + 2hR(Y, X)Y, Xi + ∇Y ∇Y R = τ 2 − Ric (Y, ·) + 2τ 0 + 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X) dτ Z τ¯ √ 1 2 = + 2hR(Y, X)Y, Xi τ 2|Ric (Y, ·)|2 − Ric (Y, Y ) + τ 2τ τ¯ 0 + ∇Y ∇Y R + 2∇X Ric (Y, Y ) − 4∇Y Ric (Y, X) dτ.
Since d Ric (Y, Y ) = Ric τ (Y, Y ) + ∇X Ric (Y, Y ) + 2Ric (∇X Y, Y ) dτ = Ric τ (Y, Y ) + ∇X Ric (Y, Y ) − 2|Ric (Y, ·)|2 +
1 Ric (Y, Y ), τ
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
249
we have HessL (Y, Y ) Z τ¯ √ 1 2 + 2hR(Y, X)Y, Xi ≤ τ 2|Ric (Y, ·)|2 − Ric (Y, Y ) + τ 2τ τ¯ 0 d + ∇Y ∇Y R − 4(∇Y Ric )(X, Y ) − 2 Ric (Y, Y ) − 2Ric τ (Y, Y ) dτ 2 2 + 4|Ric (Y, ·)| − Ric (Y, Y ) + 4∇X Ric (Y, Y ) dτ τ Z τ¯ Z τ¯ √ d 1 1 1 √ dτ =− 2 τ Ric (Y, Y ) + √ Ric (Y, Y ) dτ + dτ 2¯ τ 0 τ τ 0 Z τ¯ √ 1 τ 2hR(Y, X)Y, Xi + ∇Y ∇Y R + Ric (Y, Y ) + τ 0
+ 4(∇X Ric (Y, Y ) − ∇Y Ric (X, Y )) + 2Ric τ (Y, Y ) − 2|Ric (Y, ·)|2 dτ
√ 1 = √ − 2 τ¯Ric (Y, Y ) − τ¯
Z
τ¯
√
τ Q(X, Y )dτ.
0
This proves the inequality. As usual, the quadratic form I(V, V ) =
Z
0
τ¯
√
τ [2|∇X V |2 + 2hR(V, X)V, Xi + ∇V ∇V R
+2∇X Ric (V, V ) − 4∇V Ric (V, X)]dτ, for any vector field V along γ, is called the index form. Since γ is shortest, the standard Dirichlet principle for I(V, V ) implies that the equality holds if and only if the vector field Y is the derivative of a variation of γ by L-geodesics. Corollary 3.2.4. We have √ 1 n ∆L ≤ √ − 2 τ¯R − K τ¯ τ¯ in the sense of distribution. Moreover, the equality holds if and only if we are on a gradient shrinking soliton with 1 1 gij . Rij + √ ∇i ∇j L = 2¯ τ 2 τ¯
Proof. Choose an orthonormal basis {Y1 , . . . , Yn } at τ = τ¯ and extend them along the shortest L-geodesic γ to get vector fields Yi (τ ), i = 1, . . . , n, by solving the ODE (3.2.8), with hYi (τ ), Yj (τ )i = ττ¯ δij on [0, τ¯]. Taking Y = Yi in Proposition 3.2.3 and
250
H.-D. CAO AND X.-P. ZHU
summing over i, we get n Z τ¯ X √ √ n τ Q(X, Yi )dτ ∆L ≤ √ − 2 τ¯R − τ¯ i=1 0 Z τ¯ √ √ τ n = √ − 2 τ¯R − τ Q(X)dτ τ¯ τ¯ 0 √ 1 n = √ − 2 τ¯R − K. τ¯ τ¯
(3.2.10)
Moreover, by Proposition 3.2.3, the equality in (3.2.10) holds everywhere if and only if for each (q, τ¯) and any shortest L-geodesic γ on [0, τ¯] connecting p and q, and for any unit vector Y at τ = τ¯, the extended vector field Y (τ ) along γ by the ODE (3.2.8) must be an L-Jacobian field. When Yi (τ ), i = 1, . . . , n are L-Jacobian fields along γ, we have d hYi (τ ), Yj (τ )i dτ = 2Ric (Yi , Yj ) + h∇X Yi , Yj i + hYi , ∇X Yj i 1 1 √ ∇L , Yj + Yi , ∇Yj √ ∇L = 2Ric (Yi , Yj ) + ∇Yi 2 τ¯ 2 τ¯ 1 = 2Ric (Yi , Yj ) + √ HessL (Yi , Yj ) τ¯ and then by (3.2.9), 1 1 2Ric (Yi , Yj ) + √ HessL (Yi , Yj ) = δij , τ¯ τ¯
at τ = τ¯.
Therefore the equality in (3.2.10) holds everywhere if and only if we are on a gradient shrinking soliton with 1 1 Rij + √ ∇i ∇j L = gij . 2¯ τ 2 τ¯
In summary, from (3.2.6), (3.2.7) and Corollary 3.2.4, we have
∂L ∂ τ¯
√ = 2 τ¯R −
L 2¯ τ
+
K τ¯ ,
|∇L|2 = −4¯ τ R + √2τ¯ L − √4τ¯ K, ∆L ≤ −2√τ¯R + √n − K , τ¯ τ¯
in the sense of distributions. Now the Li-Yau-Perelman distance l = l(q, τ¯) is defined by √ l(q, τ¯) = L(q, τ¯)/2 τ¯. We thus have the following
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
251
Lemma 3.2.5. For the Li-Yau-Perelman distance l(q, τ¯) defined above, we have l 1 ∂l = − + R + 3/2 K, ∂ τ¯ τ¯ 2¯ τ 1 l 2 |∇l| = −R + − 3/2 K, τ¯ τ¯ n 1 ∆l ≤ −R + − 3/2 K, 2¯ τ 2¯ τ
(3.2.11) (3.2.12) (3.2.13)
in the sense of distributions. Moreover, the equality in (3.2.13) holds if and only if we are on a gradient shrinking soliton. As the first consequence, we derive the following upper bound on the minimum of l(·, τ ) for every τ which will be useful in proving the no local collapsing theorem in the next section. Corollary 3.2.6. Let gij (τ ), τ ≥ 0, be a family of metrics evolving by the Ricci ∂ gij = 2Rij on a compact n-dimensional manifold M . Fix a point p in M and flow ∂τ let l(q, τ ) be the Li-Yau-Perelman distance from (p, 0). Then for all τ , min{l(q, τ ) | q ∈ M } ≤
n . 2
Proof. Let ¯ τ ) = 4τ l(q, τ ). L(q, Then, it follows from (3.2.11) and (3.2.13) that ¯ 2K ∂L = 4τ R + √ , ∂τ τ and ¯ ≤ −4τ R + 2n − 2K √ . ∆L τ Hence ¯ ∂L ¯ ≤ 2n. + ∆L ∂τ ¯ τ ) − 2nτ | q ∈ M } is Thus, by a standard maximum principle argument, min{L(q, ¯ nonincreasing and therefore min{L(q, τ )| q ∈ M } ≤ 2nτ . As another consequence of Lemma 3.2.5, we obtain ∂l n − ∆l + |∇l|2 − R + ≥ 0. ∂ τ¯ 2¯ τ or equivalently
∂ −∆+R ∂ τ¯
n (4π¯ τ )− 2 exp(−l) ≤ 0.
252
H.-D. CAO AND X.-P. ZHU
If M is compact, we define Perelman’s reduced volume by V˜ (τ ) =
Z
n
(4πτ )− 2 exp(−l(q, τ ))dVτ (q),
M
where dVτ denotes the volume element with respect to the metric gij (τ ). Note that Perelman’s reduced volume resembles the expression in Huisken’s monotonicity formula for the mean curvature flow [72]. It follows, from the above computation, that d d¯ τ
Z
n
(4π¯ τ )− 2 exp(−l(q, τ¯))dVτ¯ (q) Z n n ∂ ((4π¯ τ )− 2 exp(−l(q, τ¯))) + R(4π¯ = τ )− 2 exp(−l(q, τ¯)) dVτ¯ (q) ∂ τ¯ ZM n ≤ ∆((4π¯ τ )− 2 exp(−l(q, τ¯)))dVτ¯ (q) M
M
= 0.
This says that if M is compact, then Perelman’s reduced volume V˜ (τ ) is nonincreasing in τ ; moreover, the monotonicity is strict unless we are on a gradient shrinking soliton. In order to define and to obtain the monotonicity of Perelman’s reduced volume for a complete noncompact manifold, we need to formulate the monotonicity of Perelman’s reduced volume in a local version. This local version is very important and will play a crucial role in the analysis of the Ricci flow with surgery in Chapter 7. We define the L-exponential map (with parameter τ¯) L exp(¯ τ ) : Tp M → M as follows: for any X ∈ Tp M , we set L expX (¯ τ ) = γ(¯ τ)
√ where γ(τ ) is the L-geodesic, starting at p and having X as the limit of τ γ(τ ˙ ) as τ → 0+ . The associated Jacobian of the L-exponential map is called L-Jacobian. We denote by J (τ ) the L-Jacobian of L exp(τ ) : Tp M → M . We can now deduce an estimate for the L-Jacobian as follows. Let q =√L expX (¯ τ ) and γ(τ ), τ ∈ [0, τ¯], be the shortest L-geodesic connecting p ˙ ) → X as τ → 0+ . For any vector v ∈ Tp M , we consider the family and q with τ γ(τ of L-geodesics: γs (τ ) = L exp(X+sv) (τ ),
0 ≤ τ ≤ τ¯, s ∈ (−ǫ, ǫ).
The associated variation vector field V (τ ), 0 ≤ τ ≤ τ¯, is an L-Jacobian field with V (0) = 0 and V (τ ) = (L expX (τ ))∗ (v). Let v1 , . . . , vn be n linearly independent vectors in Tp M . Then Vi (τ ) = (L expX (τ ))∗ (vi ),
i = 1, 2, . . . , n,
are n L-Jacobian fields along γ(τ ), τ ∈ [0, τ¯]. The L-Jacobian J (τ ) is given by J (τ ) = |V1 (τ ) ∧ · · · ∧ Vn (τ )|gij (τ ) /|v1 ∧ · · · ∧ vn |.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
253
Now for fixed b ∈ (0, τ¯), we can choose linearly independent vectors v1 , . . . , vn ∈ Tp M such that hVi (b), Vj (b)igij (b) = δij . We compute d 2 J dτ =
n X 2 hV1 ∧ · · · ∧ ∇X Vj ∧ · · · ∧ Vn , V1 ∧ · · · ∧ Vn igij (τ ) |v1 ∧ · · · ∧ vn |2 j=1
+
n X 2 hV1 ∧ · · · ∧ Ric (Vj , ·) ∧ · · · ∧ Vn , V1 ∧ · · · ∧ Vn igij (τ ) . |v1 ∧ · · · ∧ vn |2 j=1
At τ = b, n X d 2 2 J (b) = (h∇X Vj , Vj igij (b) + Ric (Vj , Vj )). dτ |v1 ∧ · · · ∧ vn |2 j=1
Thus, n
X d log J (b) = (h∇X Vj , Vj igij (b) + Ric (Vj , Vj )) dτ j=1 ! n X 1 √ ∇L , Vj = ∇Vj + Ric (Vj , Vj ) 2 b gij (b) j=1 n X 1 HessL (Vj , Vj ) + R = √ 2 b j=1 1 = √ ∆L + R. 2 b
Therefore, in view of Corollary 3.2.4, we obtain the following estimate for L-Jacobian: (3.2.14)
d n 1 log J (τ ) ≤ − 3/2 K dτ 2τ 2τ
on [0, τ¯].
On the other hand, by the definition of the Li-Yau-Perelman distance and (3.2.4), we have (3.2.15)
d 1 1 d l(τ ) = − l + √ L dτ 2τ 2 τ dτ 1 √ 1 = − l + √ ( τ (R + |X|2 )) 2τ 2 τ 1 = − 3/2 K. 2τ
Here and in the following we denote by l(τ ) = l(γ(τ ), τ ). Now the combination of (3.2.14) and (3.2.15) implies the following important Jacobian comparison theorem of Perelman [103]. Theorem 3.2.7 (Perelman’s Jacobian comparison theorem). Let gij (τ ) be a ∂ family of complete solutions to the Ricci flow ∂τ gij = 2Rij on a manifold M with
254
H.-D. CAO AND X.-P. ZHU
bounded curvature. Let γ : [0, τ¯] → M be a shortest L-geodesic starting from a fixed point p. Then Perelman’s reduced volume element n
(4πτ )− 2 exp(−l(τ ))J (τ ) is nonincreasing in τ along γ. We now show how to integrate Perelman’s reduced volume element over Tp M to deduce the following monotonicity result of Perelman [103]. Theorem 3.2.8 (Monotonicity of Perelman’s reduced volume). Let gij be a ∂ family of complete metrics evolving by the Ricci flow ∂τ gij = 2Rij on a manifold M with bounded curvature. Fix a point p in M and let l(q, τ ) be the reduced distance from (p, 0). Then (i) Perelman’s reduced volume Z n ˜ V (τ ) = (4πτ )− 2 exp(−l(q, τ ))dVτ (q) M
is finite and nonincreasing in τ ; (ii) the monotonicity is strict unless we are on a gradient shrinking soliton. Proof. For any v ∈ Tp M we can find an L-geodesic γ(τ ), starting at p, with √ lim+ τ γ(τ ˙ ) = v. Recall that γ(τ ) satisfies the L-geodesic equation
τ →0
1 1 ∇γ(τ ˙ ) − ∇R + γ(τ ˙ ) + 2Ric (γ(τ ˙ ), ·) = 0. ˙ ) γ(τ 2 2τ √ Multiplying this equation by τ , we get (3.2.16)
√ d √ 1√ ˙ − τ ∇R + 2Ric ( τ γ(τ ˙ ), ·) = 0. ( τ γ) dτ 2
Since the curvature of the metric gij (τ ) is bounded, it follows from Shi’s derivative estimate (Theorem 1.4.1) that |∇R| is also bounded for small τ > 0. Thus by integrating (3.2.16), we have √ (3.2.17) | τ γ(τ ˙ ) − v| ≤ Cτ (|v| + 1) for τ small enough and for some positive constant C depending only the curvature bound. Let v1 , . . . , vn be n linearly independent vectors in Tp M and let Vi (τ ) = (L expv (τ ))∗ (vi ) =
d |s=0 L exp(v+svi ) (τ ), i = 1, . . . , n. ds
The L-Jacobian J (τ ) is given by J (τ ) = |V1 (τ ) ∧ · · · ∧ Vn (τ )|gij (τ ) /|v1 ∧ · · · ∧ vn | By (3.2.17), we see that √ d τ L exp(v+sv ) (τ ) − (v + svi ) ≤ Cτ (|v| + |vi | + 1) i dτ
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
255
for τ small enough and all s ∈ (−ǫ, ǫ) (for some ǫ > 0 small) and i = 1, . . . , n. This implies that √ lim+ τ V˙ i (τ ) = vi , i = 1, . . . , n, τ →0
so we deduce that (3.2.18)
n
lim τ − 2 J (τ ) = 1.
τ →0+
Meanwhile, by using (3.2.17), we have Z τ √ 1 l(τ ) = √ τ (R + |γ(τ ˙ )|2 )dτ 2 τ 0 → |v|2 as τ → 0+ .
Thus l(0) = |v|2 .
(3.2.19)
Combining (3.2.18) and (3.2.19) with Theorem 3.2.7, we get Z n V˜ (τ ) = (4πτ )− 2 exp(−l(q, τ ))dVτ (q) ZM n ≤ (4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dv Tp M Z n exp(−|v|2 )dv = (4π)− 2 Rn
< +∞.
This proves that Perelman’s reduced volume is always finite and hence well defined. Now the monotonicity assertion in (i) follows directly from Theorem 3.2.7. For the assertion (ii), we note that the equality in (3.2.13) holds everywhere when the monotonicity of Perelman’s reduced volume is not strict. Therefore we have completed the proof of the theorem. 3.3. No Local Collapsing Theorem I. In this section we apply the monotonicity of Perelman’s reduced volume in Theorem 3.2.8 to prove Perelman’s no local collapsing theorem I, which is extremely important not only because it gives a local injectivity radius estimate in terms of local curvature bound but also it will survive the surgeries in Chapter 7. Definition 3.3.1. Let κ, r be two positive constants and let gij (t), 0 ≤ t < T, be a solution to the Ricci flow on an n-dimensional manifold M . We call the solution gij (t) κ-noncollapsed at (x0 , t0 ) ∈ M ×[0, T ) on the scale r if it satisfies the following property: whenever |Rm|(x, t) ≤ r−2 for all x ∈ Bt0 (x0 , r) and t ∈ [t0 − r2 , t0 ], we have V olt0 (Bt0 (x0 , r)) ≥ κrn .
256
H.-D. CAO AND X.-P. ZHU
Here Bt0 (x0 , r) is the geodesic ball centered at x0 ∈ M and of radius r with respect to the metric gij (t0 ). Now we are ready to state the no local collapsing theorem I of Perelman [103]. Theorem 3.3.2 (No local collapsing theorem I). Given any metric gij on an n-dimensional compact manifold M . Let gij (t) be the solution to the Ricci flow on [0, T ), with T < +∞, starting at gij . Then there exist positive constants κ and ρ0 such that for any t0 ∈ [0, T ) and any point x0 ∈ M , the solution gij (t) is κ-noncollapsed at (x0 , t0 ) on all scales less than ρ0 . Proof. We argue by contradiction. Suppose that there are sequences pk ∈ M , tk ∈ [0, T ) and rk → 0 such that |Rm|(x, t) ≤ rk−2
(3.3.1)
for x ∈ Bk = Btk (pk , rk ) and tk − rk2 ≤ t ≤ tk , but (3.3.2)
1
ǫk = rk−1 V oltk (Bk ) n → 0 as k → ∞.
Without loss of generality, we may assume that tk → T as k → +∞.
Let τ¯(t) = tk − t, p = pk and Z n ˜ Vk (¯ τ) = (4π¯ τ )− 2 exp(−l(q, τ¯))dVtk −¯τ (q), M
where l(q, τ¯) is the Li-Yau-Perelman distance with respect to p = pk . Step 1. We first want to show that for k large enough, n
V˜k (ǫk rk2 ) ≤ 2ǫk2 . For any v ∈ Tp M we can find an L-geodesic γ(τ ) starting at p with lim
τ →0
√
τ γ(τ ˙ )
= v. Recall that γ(τ ) satisfies the equation (3.2.16). It follows from assumption (3.3.1) and Shi’s local derivative estimate (Theorem 1.4.2) that |∇R| has a bound in the order of 1/rk3 for t ∈ [tk − ǫk rk2 , tk ]. Thus by integrating (3.2.16) we see that for τ ≤ ǫk rk2 satisfying the property that γ(σ) ∈ Bk as long as σ < τ , there holds √ ˙ ) − v| ≤ Cǫk (|v| + 1) (3.3.3) | τ γ(τ where C is some positive constant depending only on the dimension. Here we have implicitly used the fact that the metric gij (t) is equivalent for x ∈ Bk and t ∈ [tk − ∂g ǫk rk2 , tk ]. In fact since ∂tij = −2Rij and |Rm| ≤ rk−2 on Bk × [tk − rk2 , tk ], we have (3.3.4)
e−2ǫk gij (x, tk ) ≤ gij (x, t) ≤ e2ǫk gij (x, tk ),
for x ∈ Bk and t ∈ [tk − ǫk rk2 , tk ]. −1
Suppose v ∈ Tp M with |v| ≤ 14 ǫk 2 . Let τ ≤ ǫk rk2 such √ that γ(σ) ∈ Bk as long as ˙ ) = v. Then, by (3.3.3) σ < τ , where γ is the L-geodesic starting at p with lim τ γ(τ τ →0
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
257
and (3.3.4), for k large enough, dtk (pk , γ(τ )) ≤
Z
τ
|γ(σ)| ˙ gij (tk ) dσ Z 1 − 1 τ dσ √ < ǫk 2 2 σ 0 − 12 √ = ǫk τ 0
≤ rk . This shows that for k large enough, L exp{|v|≤ 1 ǫ−1/2 } (ǫk rk2 ) ⊂ Bk = Btk (pk , rk ).
(3.3.5)
4 k
We now estimate the integral of V˜k (ǫk rk2 ) as follows, (3.3.6) V˜k (ǫk rk2 ) =
Z
(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) M Z n (4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) = n
2 −1/2 (ǫk rk ) } {|v|≤ 1 ǫ 4 k
L exp
Z
+
(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q). n
2 −1/2 (ǫk rk ) } {|v|≤ 1 ǫ 4 k
M\L exp
We observe that for each q ∈ Bk , Z
L(q, ǫk rk2 ) =
2 ǫ k rk
0
3 √ 3 τ (R + |γ| ˙ 2 )dτ ≥ −C(n)rk−2 (ǫk rk2 ) 2 = −C(n)ǫk2 rk ,
hence l(q, ǫk rk2 ) ≥ −C(n)ǫk . Thus, the first term on the RHS of (3.3.6) can be estimated by Z
(3.3.7)
(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) n
2 −1/2 (ǫk rk ) } {|v|≤ 1 ǫ 4 k
L exp
≤e
nǫk
Z
(4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk (q) n
Bk −n 2
≤ enǫk (4π)− 2 · eC(n)ǫk · ǫk n
n
= e(n+C(n))ǫk (4π)− 2 · ǫk2 , n
· (rk−n Vol tk (Bk ))
where we have also used (3.3.5) and (3.3.4). Meanwhile, by using (3.2.18), (3.2.19) and the Jacobian Comparison Theorem
258
H.-D. CAO AND X.-P. ZHU
3.2.7, the second term on the RHS of (3.3.6) can be estimated as follows Z n (4πǫk rk2 )− 2 exp(−l(q, ǫk rk2 ))dVtk −ǫk rk2 (q) (3.3.8) M\L exp
−1 {|v|≤ 1 ǫ 2 } 4 k
Z
≤
2) (ǫk rk
n
(4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dv −1 2
{|v|> 41 ǫk
−n 2
= (4π)
}
Z
exp(−|v|2 )dv −1 2
{|v|> 41 ǫk
}
n 2
≤ ǫk , for k sufficiently large. Combining (3.3.6)-(3.3.8), we finish the proof of Step 1. Step 2. We next want to show n V˜k (tk ) = (4πtk )− 2
Z
exp(−l(q, tk ))dV0 (q) > C ′ M
for all k, where C ′ is some positive constant independent of k. It suffices to show the Li-Yau-Perelman distance l(·, tk ) is uniformly bounded from above on M . By Corollary 3.2.6 we know that the minimum of l(·, τ ) does not exceed T n 2 for each τ > 0. Choose qk ∈ M such that the minimum of l(·, tk − 2 ) is attained at qk . We now construct a path γ : [0, tk ] → M connecting pk to any given point q ∈ M as follows: the first half path γ|[0,tk − T ] connects pk to qk so that 2
1 T = q l qk , tk − 2 2 tk −
T 2
Z
tk − T2 0
√
τ (R + |γ(τ ˙ )|2 )dτ ≤
n 2
and the second half path γ|[tk − T ,tk ] is a shortest geodesic connecting qk to q with 2 respect to the initial metric gij (0). Then, for any q ∈ M n , 1 l(q, tk ) = √ L(q, tk ) 2 tk Z tk − T2 Z tk ! √ 1 ≤ √ + τ (R + |γ(τ ˙ )|2 )dτ 2 tk 0 tk − T2 ! r Z tk √ T 1 τ (R + |γ(τ ˙ )|2 )dτ n tk − + ≤ √ 2 2 tk tk − T2 ≤C for some constant C > 0, since all geometric quantities in gij are uniformly bounded when t ∈ [0, T2 ] (or equivalently, τ ∈ [tk − T2 , tk ]). Combining Step 1 with Step 2, and using the monotonicity of V˜k (τ ), we get n
C ′ < V˜k (tk ) ≤ V˜k (ǫk rk2 ) ≤ 2ǫk2 → 0
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
259
as k → ∞. This gives the desired contradiction. Therefore we have proved the theorem. The above no local collapsing theorem I says that if |Rm| ≤ r−2 on the parabolic ball {(x, t) | dt0 (x, x0 ) ≤ r, t0 − r2 ≤ t ≤ t0 }, then the volume of the geodesic ball Bt0 (x0 , r) (with respect to the metric gij (t0 )) is bounded from below by κrn . In [103], Perelman used the monotonicity of the W-functional (defined by (1.5.9)) to obtain a stronger version of the no local collapsing theorem, where the curvature bound assumption on the parabolic ball is replaced by that on the geodesic ball Bt0 (x0 , r). The following result, called no local collapsing theorem I′ , gives a further extension where the bound on the curvature tensor is replaced by the bound on the scalar curvature only. We now follow a clever argument by Bing-Long Chen. Theorem 3.3.3 (No local collapsing theorem I′ ). Suppose M is a compact Riemannian manifold, and gij (t), 0 ≤ t < T < +∞, is a solution to the Ricci flow. Then there exists a positive constant κ depending only the initial metric and T such that for any (x0 , t0 ) ∈ M × (0, T ) if R(x, t0 ) ≤ r−2 ,
√ with 0 < r ≤ T , then we have
∀x ∈ Bt0 (x0 , r)
Volt0 (Bt0 (x0 , r)) ≥ κrn . Proof. We will prove the assertion Volt0 (Bt0 (x0 , a)) ≥ κan
(∗)a
for all 0 < a ≤ r. Recall that Z −n −f 2 (4πτ ) e dV = 1 . µ(gij , τ ) = inf W(gij , f, τ ) M
Set
µ0 =
inf
0≤τ ≤2T
µ(gij (0), τ ) > −∞.
By Corollary 1.5.9, we have (3.3.9)
µ(gij (t0 ), b) ≥ µ(gij (0), t0 + b) ≥ µ0
for 0 < b ≤ r2 . Let 0 < ζ ≤ 1 be a positive smooth function on R where ζ(s) = 1 for |s| ≤ 21 , |ζ ′ |2 /ζ ≤ 20 everywhere, and ζ(s) is very close to zero for |s| ≥ 1. Define a function f on M by n dt0 (x, x0 ) n , (4πr2 )− 2 e−f (x) = e−c (4πr2 )− 2 ζ r R n where the constant c is chosen so that M (4πr2 )− 2 e−f dVt0 = 1. Then it follows from (3.3.9) that Z n (3.3.10) W(gij (t0 ), f, r2 ) = [r2 (|∇f |2 + R) + f − n](4πr2 )− 2 e−f dVt0 M
≥ µ0 .
260
H.-D. CAO AND X.-P. ZHU
Note that 1= ≥
Z
2 −n 2 −c
(4πr )
ZM
e
ζ
dt0 (x, x0 ) r
dVt0
(4πr2 )− 2 e−c dVt0 n
Bt0 (x0 , r2 )
n r . = (4πr2 )− 2 e−c Vol t0 Bt0 x0 , 2 By combining with (3.3.10) and the scalar curvature bound, we have Z ′ 2 (ζ ) n c≥− − log ζ · ζ e−c (4πr2 )− 2 dVt0 + (n − 1) + µ0 ζ M
≥ −2(20 + e−1 )e−c (4πr2 )− 2 Vol t0 (Bt0 (x0 , r)) + (n − 1) + µ0 n
≥ −2(20 + e−1 )
Volt0 (Bt0 (x0 , r)) + (n − 1) + µ0 , Vol t0 (Bt0 (x0 , 2r ))
where we used the fact that ζ(s) is very close to zero for |s| ≥ 1. Note also that Z Z n n 2 e−c (4πr2 )− 2 dVt0 ≥ (4πr2 )− 2 e−f dVt0 = 1. Bt0 (x0 ,r)
Let us set κ = min
M
1 1 exp(−2(20 + e−1 )3−n + (n − 1) + µ0 ), αn 2 2
where αn is the volume of the unit ball in Rn . Then we obtain n 1 Volt0 (Bt0 (x0 , r)) ≥ ec (4πr2 ) 2 2 n 1 ≥ (4π) 2 exp(−2(20 + e−1 )3−n + (n − 1) + µ0 ) · rn 2 ≥ κrn
provided Volt0 (Bt0 (x0 , r2 )) ≥ 3−n V olt0 (Bt0 (x0 , r)). Note that the above argument also works for any smaller radius a ≤ r. Thus we have proved the following assertion: (3.3.11)
Volt0 (Bt0 (x0 , a)) ≥ κan
whenever a ∈ (0, r] and Volt0 (Bt0 (x0 , 2a )) ≥ 3−n Volt0 (Bt0 (x0 , a)). Now we argue by contradiction to prove the assertion (∗)a for any a ∈ (0, r]. Suppose (∗)a fails for some a ∈ (0, r]. Then by (3.3.11) we have a Vol t0 (Bt0 (x0 , )) < 3−n Vol t0 (Bt0 (x0 , a)) 2 < 3−n κan a n . <κ 2 This says that (∗) a2 would also fail. By induction, we deduce that a n a <κ k for all k ≥ 1. Vol t0 Bt0 x0 , k 2 2 n This is a contradiction since lim Vol t0 Bt0 x0 , 2ak / 2ak = αn . k→∞
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
261
3.4. No Local Collapsing Theorem II. By inspecting the arguments in the previous section, one can see that if the injectivity radius of the initial metric is uniformly bounded from below, then the no local collapsing theorem I also holds for complete solutions with bounded curvature on a complete noncompact manifold. In this section we will use a cut-off argument to extend the no local collapsing theorem to any complete solution with bounded curvature. In some sense, the second no local collapsing theorem gives a good relative estimate of the volume element for the Ricci flow. We first need the following useful lemma which contains two assertions. The first one is a parabolic version of the Laplacian comparison theorem (where the curvature sign restriction in the ordinary Laplacian comparison is essentially removed in the Ricci flow). The second one is a generalization of a result of Hamilton (Theorem 17.2 in [63]), where it was derived by an integral version of Bonnet-Myers’ theorem. Lemma 3.4.1 (Perelman [103]). Let gij (x, t) be a solution to the Ricci flow on an n-dimensional manifold M and denote by dt (x, x0 ) the distance between x and x0 with respect to the metric gij (t). (i) Suppose Ric (·, t0 ) ≤ (n − 1)K on Bt0 (x0 , r0 ) for some x0 ∈ M and some positive constants K and r0 . Then the distance function d(x, t) = dt (x, x0 ) satisfies, at t = t0 and outside Bt0 (x0 , r0 ), the differential inequality: 2 ∂ −1 d − ∆d ≥ −(n − 1) Kr0 + r0 . ∂t 3 S (ii) Suppose Ric (·, t0 ) ≤ (n − 1)K on Bt0 (x0 , r0 ) Bt0 (x1 , r0 ) for some x0 , x1 ∈ M and some positive constants K and r0 . Then, at t = t0 , 2 d dt (x0 , x1 ) ≥ −2(n − 1) Kr0 + r0−1 . dt 3 Proof. Let γ : [0, d(x, t0 )] → M be a shortest normal geodesic from x0 to x with respect to the metric gij (t0 ). As usual, we may assume that x and x0 are not conjugate to each other in the metric gij (t0 ), otherwise we can understand the differential inequality in the barrier sense. Let X = γ(0) ˙ and let {X, e1, . . . , en−1 } be an orthonormal basis of Tx0 M . Extend this basis parallel along γ to form a parallel orthonormal basis {X(s), e1 (s), . . . , en−1 (s)} along γ. (i) Let Xi (s), i = 1, . . . , n − 1, be the Jacobian fields along γ such that Xi (0) = 0 and Xi (d(x, t0 )) = ei (d(x, t0 )) for i = 1, . . . , n − 1. Then it is well-known that (see for example [112]) ∆dt0 (x, x0 ) =
n−1 X Z d(x,t0 ) i=1
0
(|X˙ i |2 − R(X, Xi , X, Xi ))ds
(in Proposition 3.2.3 we actually did this for the more complicated L-distance function). Define vector fields Yi , i = 1, . . . , n − 1, along γ as follows: ( s ei (s), if s ∈ [0, r0 ], Yi (s) = r0 ei (s), if s ∈ [r0 , d(x, t0 )].
262
H.-D. CAO AND X.-P. ZHU
which have the same value as the corresponding Jacobian fields Xi (s) at the two end points of γ. Then by using the standard index comparison theorem (see for example [22]) we have ∆dt0 (x, x0 ) = ≤
n−1 X Z d(x,t0 ) 0
(|X˙ i |2 − R(X, Xi , X, Xi ))ds
0
(|Y˙ i |2 − R(X, Yi , X, Yi ))ds
i=1 n−1 X Z d(x,t0 ) i=1 r0
Z
Z d(x,t0 ) 1 2 (n − 1 − s Ric (X, X))ds + (−Ric (X, X))ds r02 0 r0 Z Z r0 s2 (n − 1) + 1 − Ric (X, X) ds = − Ric (X, X) + r02 r02 γ 0 Z 2 −1 Kr0 + r0 . ≤ − Ric (X, X) + (n − 1) 3 γ =
On the other hand, Z d(x,t0 ) q ∂ ∂ gij X i X j ds dt (x, x0 ) = ∂t ∂t 0 Z = − Ric (X, X)ds. γ
Hence we obtain the desired differential inequality. (ii) The proof is divided into three cases. Case (1): dt0 (x0 , x1 ) ≥ 2r0 . Define vector fields Yi , i = 1, . . . , n − 1, along γ as follows: s if s ∈ [0, r0 ], r0 ei (s), Yi (s) = ei (s), if s ∈ [r0 , d(x1 , t0 )], d(x1 ,t0 )−s e (s), if s ∈ [d(x , t ) − r , d(x , t )]. i 1 0 0 1 0 r0
Then by the second variation formula, we have n−1 X Z d(x1 ,t0 ) i=1
0
R(X, Yi , X, Yi )ds ≤
n−1 X Z d(x1 ,t0 ) i=1
0
|Y˙ i |2 ds,
which implies Z
Z d(x,t0 )−r0 s2 Ric (X, X)ds + Ric (X, X)ds r02 0 r0 2 Z d(x1 ,t0 ) d(x1 , t0 ) − s 2(n − 1) + Ric (X, X)ds ≤ . r r0 0 d(x1 ,t0 )−r0 r0
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
263
Thus d (dt (x0 , x1 )) dt Z r0 s2 ≥− 1 − 2 Ric (X, X)ds r0 0 2 ! Z d(x1 ,t0 ) d(x1 , t0 ) − s 2(n − 1) − 1− Ric (X, X)ds − r r0 0 d(x1 ,t0 )−r0 2 Kr0 + r0−1 . ≥ −2(n − 1) 3 Case (2): √22K ≤ dt0 (x0 , x1 ) ≤ 2r0 . 3
get
In this case, letting r1 = √12K and applying case (1) with r0 replaced by r1 , we 3
2 −1 Kr1 + r1 3 2 −1 Kr0 + r0 . ≥ −2(n − 1) 3
d (dt (x0 , x1 )) ≥ −2(n − 1) dt
n o Case (3): dt0 (x0 , x1 ) ≤ min √22K , 2r0 . 3
In this case, Z
d(x1 ,t0 )
2 Ric (X, X)ds ≤ (n − 1)K q
2K 3
0
and
2(n − 1)
2 Kr0 + r0−1 3
√ = (n − 1) 6K,
r
≥ (n − 1)
32 K. 3
This proves the lemma. The following result, called the no local collapsing theorem II, was obtained by Perelman in [103]. Theorem 3.4.2 (No local collapsing theorem II). For any A > 0 there exists κ = κ(A) > 0 with the following property: if gij (t) is a complete solution to the Ricci flow on 0 ≤ t ≤ r02 with bounded curvature and satifying |Rm|(x, t) ≤ r0−2
on B0 (x0 , r0 ) × [0, r02 ]
and Vol 0 (B0 (x0 , r0 )) ≥ A−1 r0n , then gij (t) is κ-noncollapsed on all scales less than r0 at every point (x, r02 ) with dr02 (x, x0 ) ≤ Ar0 .
264
H.-D. CAO AND X.-P. ZHU
Proof. From the evolution equation of the Ricci flow, we know that the metrics gij (·, t) are equivalent to each other on B0 (x0 , r0 ) × [0, r02 ]. Thus, without loss of generality, we may assume that the curvature of the solution is uniformly bounded for all t ∈ [0, r02 ] and all points in Bt (x0 , r0 ). Fix a point (x, r02 ) ∈ M × {r02 }. By scaling we may assume r0 = 1. We may also assume d1 (x, x0 ) = A. Let p = x, τ¯ = 1 − t, and consider Perelman’s reduced volume Z n ˜ V (¯ τ) = (4π¯ τ )− 2 exp(−l(q, τ¯))dV1−¯τ (q), M
where l(q, τ¯) = inf
1 √ 2 τ¯
Z
0
τ¯
√ τ (R + |γ| ˙ 2 )dτ | γ : [0, τ¯] → M with γ(0) = p, γ(¯ τ) = q
is the Li-Yau-Perelman distance. We argue by contradiction. Suppose for some 0 < r < 1 we have |Rm|(y, t) ≤ r−2 1
whenever y ∈ B1 (x, r) and 1 − r2 ≤ t ≤ 1, but ǫ = r−1 Vol 1 (B1 (x, r)) n is very small. Then arguing as in the proof of the no local collapsing theorem I (Theorem 3.3.2), we see that Perelman’s reduced volume n V˜ (ǫr2 ) ≤ 2ǫ 2 .
On the other hand, from the monotonicity of Perelman’s reduced volume we have Z −n 2 exp(−l(q, 1))dV0 (q) = V˜ (1) ≤ V˜ (ǫr2 ). (4π) M
Thus once we bound the function l(q, 1) over B0 (x0 , 1) from above, we will get the desired contradiction and will prove the theorem. For any q ∈ B0 (x0 , 1), exactly as in the proof of the no local collapsing theorem 1 I, we choose a path γ : [0, 1] → M with γ(0) = x, γ(1) = q, γ( 21 ) = y ∈ B 12 (x0 , 10 ) 1 and γ(τ ) ∈ B1−τ (x0 , 1) for τ ∈ [ 2 , 1] such that L(γ|[0, 21 ] ) = 2
r
1 1 1 l y, = L y, . 2 2 2
R1√ Now L(γ|[ 12 ,1] ) = 1 τ (R(γ(τ ), 1 − τ ) + |γ(τ ˙ )|2gij (1−τ ) )dτ is bounded from above by 2 a uniform constant since all geometric quantities in gij are uniformly bounded on {(y, t) | t ∈ [0, 1/2], y ∈ Bt (x0 , 1)} (where t ∈ [0, 1/2] is equivalent to τ ∈ [1/2, 1]). ¯ 1) = Thus all we need is to estimate the minimum of l(·, 12 ), or equivalently L(·, 2 1 1 1 4 2 l(·, 2 ), in the ball B 12 (x0 , 10 ). ¯ satisfies the differential inequality Recall that L (3.4.1)
¯ ∂L ¯ ≤ 2n. + ∆L ∂τ
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
265
We will use this in a maximum principle argument. Let us define ¯ 1 − t) + 2n + 1) h(y, t) = φ(d(y, t) − A(2t − 1)) · (L(y, 1 ), where d(y, t) = dt (y, x0 ), and φ is a function of one variable, equal to 1 on (−∞, 20 1 1 and rapidly increasing to infinity on ( 20 , 10 ) in such a way that:
(3.4.2)
2
(φ′ )2 − φ′′ ≥ (2A + 100n)φ′ − C(A)φ φ
for some constant C(A) < +∞. The existence of such a function φ can be justified ′ as follows: put v = φφ , then the condition (3.4.2) for φ can be written as 3v 2 − v ′ ≥ (2A + 100n)v − C(A) which can be solved for v. Since the scalar curvature R evolves by ∂R 2 = ∆R + 2|Rc|2 ≥ ∆R + R2 , ∂t n we can apply the maximum principle as in Chapter 2 to deduce R(x, t) ≥ −
n for t ∈ (0, 1] and x ∈ M. 2t
Thus for τ¯ = 1 − t ∈ [0, 21 ], √ ¯ τ¯) = 2 τ¯ L(·,
Z
τ¯
√
τ (R + |γ| ˙ 2 )dτ Z τ¯ √ √ n dτ ≥ 2 τ¯ τ − 2(1 − τ ) 0 Z τ¯ √ √ ≥ 2 τ¯ τ (−n)dτ 0
0
> −2n.
That is 1 ¯ L(·, 1 − t) + 2n + 1 ≥ 1, for t ∈ ,1 . 2
(3.4.3)
1 ) and Clearly min h(y, 12 ) is achieved by some y ∈ B 21 (x0 , 10 y∈M
(3.4.4)
min h(y, 1) ≤ h(x, 1) = 2n + 1.
y∈M
266
H.-D. CAO AND X.-P. ZHU
We compute ∂ ∂ ¯ 1 − t) + 2n + 1) −∆ h= − ∆ φ · (L(y, ∂t ∂t ∂ ¯ 1 − t) − 2h∇φ, ∇L(y, ¯ 1 − t)i +φ· − ∆ L(y, ∂t ∂ ′′ 2 ′ ¯ + 2n + 1) − ∆ d − 2A − φ |∇d| · (L = φ ∂t ∂ ¯ − 2h∇φ, ∇Li ¯ +φ· − −∆ L ∂τ ∂ ′′ ′ ¯ + 2n + 1) − ∆ d − 2A − φ · (L ≥ φ ∂t ¯ − 2nφ − 2h∇φ, ∇Li by using (3.4.1). At a minimizing point of h we have ¯ ∇L ∇φ =− ¯ . φ (L + 2n + 1) Hence ¯ =2 −2h∇φ, ∇Li
(φ′ )2 ¯ |∇φ|2 ¯ (L + 2n + 1) = 2 (L + 2n + 1). φ φ
Then at the minimizing point of h, we compute ∂ ∂ ¯ + 2n + 1) − ∆ h ≥ φ′ − ∆ d − 2A − φ′′ · (L ∂t ∂t (φ′ )2 ¯ (L + 2n + 1) − 2nφ + 2 φ ∂ ¯ + 2n + 1) − ∆ d − 2A − φ′′ · (L ≥ φ′ ∂t (φ′ )2 ¯ (L + 2n + 1) − 2nh + 2 φ for t ∈ [ 21 , 1] and ∆h ≥ 0. Let us denote by hmin (t) = min h(y, t). By applying Lemma 3.4.1(i) to the set where y∈M
φ′ 6= 0, we further obtain
(φ′ )2 d ′ ′′ ¯ hmin ≥ (L + 2n + 1) · φ (−100n − 2A) − φ + 2 − 2nhmin dt φ 1 ≥ −(2n + C(A))hmin , for t ∈ [ , 1]. 2
This implies that hmin (t) cannot decrease too fast. By combining (3.4.3) and (3.4.4) ¯ 1 ) in the ball B 1 (x0 , 1 ). we get the required estimate for the minimum L(·, 2 10 2 Therefore we have completed the proof of the theorem.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
267
4. Formation of Singularities. Let gij (x, t) be a solution to the Ricci flow on M × [0, T ) and suppose [0, T ), T ≤ ∞, is the maximal time interval. If T < +∞, then the short time existence theorem tells us the curvature of the solution becomes unbounded as t → T . We then say the solution develops a singularity as t → T . As in the minimal surface theory and harmonic map theory, one usually tries to understand the structure of a singularity of the Ricci flow by rescaling the solution (or blow up) to obtain a sequence of solutions to the Ricci flow with uniformly bounded curvature on compact subsets and looking at its limit. The main purpose of this chapter is to establish a convergence theorem for a sequence of solutions to the Ricci flow with uniform bounded curvature on compact subsets and to use the convergence theorem to give a rough classification for singularities of solutions to the Ricci flow. Further studies on the structures of singularities of the Ricci flow will be given in Chapter 6 and 7. ∞ 4.1. Cheeger Type Compactness. We begin with the concept of Cloc convergence of tensors on a given manifold M . Let Ti be a sequence of tensors on M . We ∞ say that Ti converges to a tensor T in the Cloc topology if we can find a covering {(Us , ϕs )}, ϕs : Us → Rn , of C ∞ coordinate charts so that for every compact set K ⊂ M , the components of Ti converge in the C ∞ topology to the components of T in the intersections of K with these coordinate charts, considered as functions on ϕs (Us ) ⊂ Rn . Consider a Riemannian manifold (M, g). A marking on M is a choice of a point p ∈ M which we call the origin. We will refer to such a triple (M, g, p) as a marked Riemannian manifold.
Definition 4.1.1. Let (Mk , gk , pk ) be a sequence of marked complete Riemannian manifolds, with metrics gk and marked points pk ∈ Mk . Let B(pk , sk ) ⊂ Mk denote the geodesic ball centered at pk ∈ Mk and of radius sk (0 < sk ≤ +∞). We say a sequence of marked geodesic balls (B(pk , sk ), gk , pk ) with sk → s∞ (≤ +∞) ∞ converges in the Cloc topology to a marked (maybe noncomplete) manifold (B∞ , g∞ , p∞ ), which is an open geodesic ball centered at p∞ ∈ B∞ and of radius s∞ with respect to the metric g∞ , if we can find a sequence of exhausting open sets Uk in B∞ containing p∞ and a sequence of diffeomorphisms fk of the sets Uk in B∞ to open sets Vk in B(pk , sk ) ⊂ Mk mapping p∞ to pk such that the pull-back metrics g˜k = (fk )∗ gk converge in C ∞ topology to g∞ on every compact subset of B∞ . ∞ We remark that this concept of Cloc -convergence of a sequence of marked mani∞ folds (Mk , gk , pk ) is not the same as that of Cloc -convergence of metric tensors on a given manifold, even when we are considering the sequence of Riemannian metric gk on the same space M . This is because one can have a sequence of diffeomorphisms ∞ fk : M → M such that (fk )∗ gk converges in Cloc topology while gk itself does not converge. There have been a lot of work in Riemannian geometry on the convergence of a sequence of compact manifolds with bounded curvature, diameter and injectivity radius (see for example Gromov [53], Peters [106], and Greene and Wu [51]). The following theorem, which is a slight generalization of Hamilton’s convergence theorem [62], modifies these results in three aspects: the first one is to allow noncompact limits and then to avoid any diameter bound; the second one is to avoid having to assume a uniform lower bound for the injectivity radius over the whole manifold, a hypothesis which is much harder to satisfy in applications; the last one is to avoid a uniform curvature bound over the whole manifold so that we can take a local limit.
Theorem 4.1.2 (Hamilton [62]). Let (Mk , gk , pk ) be a sequence of marked
268
H.-D. CAO AND X.-P. ZHU
complete Riemannian manifolds of dimension n. Consider a sequence of geodesic balls B(pk , sk ) ⊂ Mk of radius sk (0 < sk ≤ ∞), with sk → s∞ (≤ ∞), around the base point pk of Mk in the metric gk . Suppose (a) for every radius r < s∞ and every integer l ≥ 0 there exists a constant Bl,r , independent of k, and positive integer k(r, l) < +∞ such that as k ≥ k(r, l), the curvature tensors Rm(gk ) of the metrics gk and their lth -covariant derivatives satisfy the bounds |∇l Rm(gk )| ≤ Bl,r on the balls B(pk , r) of radius r around pk in the metrics gk ; and (b) there exists a constant δ > 0 independent of k such that the injectivity radii inj (Mk , pk ) of Mk at pk in the metric gk satisfy the bound inj (Mk , pk ) ≥ δ. Then there exists a subsequence of the marked geodesic balls (B(pk , sk ), gk , pk ) which converges to a marked geodesic ball ∞ (B(p∞ , s∞ ), g∞ , p∞ ) in Cloc topology. Moreover the limit is complete if s∞ = +∞. Proof. In [62] and Theorem 16.1 of [63], Hamilton proved this convergence theorem for the case s∞ = +∞. In the following we only need to modify Hamilton’s argument to prove the remaining case of s∞ < +∞. Suppose we are given a sequence of geodesic balls (B(pk , sk ), gk , pk ) ⊂ (Mk , gk , pk ), with sk → s∞ (< +∞), satisfying the assumptions of Theorem 4.1.2. We will split the proof into three steps. Step 1: Picking the subsequence. By the local injectivity radius estimate (4.2.2) in Corollary 4.2.3 of the next section, we can find a positive decreasing C 1 function ρ(r), 0 ≤ r < s∞ , independent of k such that (4.1.1) (4.1.2)
1 (s∞ − r), 100 1 0 ≥ ρ′ (r) ≥ − , 1000
ρ(r) <
and a sequence of positive constants εk → 0 so that the injectivity radius at any point x ∈ B(pk , sk ) with rk = dk (x, pk ) < s∞ − εk is bounded from below by (4.1.3)
inj (Mk , x) ≥ 500ρ(rk (x)),
where rk (x) = dk (x, pk ) is the distance from x to pk in the metric gk of Mk . We define ρ˜(r) = ρ(r + 20ρ(r)),
ρ˜˜(r) = ρ˜(r + 20ρ˜(r)).
By (4.1.2) we know that both ρ˜(r) and ρ˜˜(r) are nonincreasing positive functions on [0, s∞ ). In each B(pk , s∞ ) we choose inductively a sequence of points xα k for α = 0, 1, 2, . . . in the following way. First we let x0k = pk . Once xα are chosen for α = 0, 1, 2, . . . , σ, we k σ+1 σ+1 pick xσ+1 closest to p so that r = r (x ) is as small as possible, subject to the k k k k k ˜˜σ+1 ) around xσ+1 of radius ρ˜˜σ+1 is disjoint requirement that the open ball B(xσ+1 , ρ k k k k
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
269
˜˜α ) for α = 0, 1, 2, . . . , σ, where ρ˜˜α = ρ˜˜(rα ) and rα = rk (xα ). from the balls B(xα k,ρ k k k k k ˜˜α ), α = 0, 1, 2, . . ., are all disjoint. We claim the In particular, the open balls B(xα k,ρ k ˜˜α ) cover B(pk , s∞ −εk ) and moreover for any r, 0 < r < s∞ −εk , we can balls B(xα k , 2ρ k ˜˜α ) find λ(r) independent of k such that for k large enough, the geodesic balls B(xα k , 2ρ k for α ≤ λ(r) cover the ball B(pk , r). To see this, let x ∈ B(pk , s∞ − εk ) and let r(x) be the distance from x to pk and let ρ˜˜ = ρ˜˜(r(x)). Consider those α with rkα ≤ r(x) < s∞ − εk . Then ρ˜˜ ≤ ρ˜˜α k. Now the given point x must lie in one of the balls B(xα ˜˜α k , 2ρ k ). If not, we could choose β ˜˜ ≤ 2ρ˜˜α the ball the next point in the sequence of xk to be x instead, for since ρ˜˜α k +ρ k α ˜α α ˜ B(x, ρ˜) would miss B(xk , ρ˜k ) with rk ≤ r(x). But this is a contradiction. Moreover for any r, 0 < r < s∞ − εk , using the curvature bound and the injectivity radius ˜˜α ) with rα ≤ r has volume at least ǫ(r)ρ˜˜n where ǫ(r) > 0 is bound, each ball B(xα k,ρ k k some constant depending on r but independent of k. Now these balls are all disjoint and contained in the ball B(pk , (r + s∞ )/2). On the other hand, for large enough k, we can estimate the volume of this ball from above, again using the curvature bound, by a positive function of r that is independent of k. Thus there is a k ′ (r) > 0 such that for each k ≥ k ′ (r), there holds #{α | rkα ≤ r} ≤ λ(r)
(4.1.4)
˜˜α ) for some positive constant λ(r) depending only on r, and the geodesic balls B(xα k , 2ρ k for α ≤ λ(r) cover the ball B(pk , r). By the way, since rkα ≤ rkα−1 + ρ˜˜α−1 + ρ˜˜α k k ≤ rα−1 + 2ρ˜˜α−1 , k
k
and by (4.1.1) 1 (s∞ − rkα−1 ), ρ˜˜kα−1 ≤ 100 we get by induction 1 49 α−1 r + s∞ 50 k 50 α α−1 ! 49 1 49 49 0 ≤ rk + 1+ + ···+ s∞ 50 50 50 50 α 49 = 1− s∞ . 50
rkα ≤
So for each α, with α ≤ λ(r) (r < s∞ ), there holds λ(r) ! 49 α s∞ (4.1.5) rk ≤ 1 − 50 for all k. And by passing to a subsequence (using a diagonalization argument) we α may assume that rkα converges to some rα for each α. Then ρ˜˜α ˜α k (respectively ρ k , ρk ) α α α α α α converges to ρ˜˜ = ρ˜˜(r ) (respectively ρ˜ = ρ˜(r ), ρ = ρ(r )).
270
H.-D. CAO AND X.-P. ZHU
Hence for all α we can find k(α) such that 1 ˜α ˜α ρ˜ ≤ ρ˜k ≤ 2ρ˜˜α 2 1 α ρ˜ ≤ ρ˜α ˜α k ≤ 2ρ 2
and
1 α α ρ ≤ ρα k ≤ 2ρ 2
˜˜α are comparable when k is large enough whenever k ≥ k(α). Thus for all α, ρ˜˜α k and ρ so we can work with balls of a uniform size, and the same is true for ρ˜α ˜α , and k and ρ α α α α α α α α α α α ˆ ˆα. ˜ ˜ ˜ ˜ ˜ ρk and ρ . Let Bk = B(xk , 4ρ˜ ), then ρ˜k ≤ 2ρ˜ and B(xk , 2ρ˜k ) ⊂ B(xk , 4ρ˜ ) = B k So for every r if we let k(r) = max{k(α) | α ≤ λ(r)} then when k ≥ k(r), the balls ˆ α for α ≤ λ(r) cover the ball B(pk , r) as well. Suppose that B ˆ α and B ˆ β meet for B k k k β k ≥ k(α) and k ≥ k(β), and suppose rk ≤ rkα . Then, by the triangle inequality, we must have rkα ≤ rkβ + 4ρ˜˜α + 4ρ˜˜β ≤ rkβ + 8ρ˜˜β < rkβ + 16ρ˜βk . This then implies ρ˜˜βk = ρ˜˜(rkβ ) = ρ˜(rkβ + 20ρ˜(rkβ )) < ρ˜(rkα ) = ρ˜α k and hence ρ˜˜β ≤ 4ρ˜α . ˆ β ⊂ B(xα , 36ρ˜α ) whenever B ˆ α and B ˆ β meet and k ≥ max{k(α), k(β)}. Therefore B k k k k ˜ α = B(xα , ρ˜˜α /2). Note that B ˜α ˜˜α ) and B Next we define the balls Bkα = B(xα , 5 ρ k k k k α α α α α α ˜ ⊂ B(x , ρ˜˜ ). Since B ˆ ⊂ B , the balls B cover B(pk , r) for are disjoint since B k k k k k k α ≤ λ(r) as before. If Bkα and Bkβ meet for k ≥ k(α) and k ≥ k(β) and rkβ ≤ rkα , then by the triangle inequality we get rkα ≤ rkβ + 10ρ˜˜β < rkβ + 20ρ˜βk , and hence ρ˜˜β ≤ 4ρ˜α again. Similarly, ρ˜βk = ρ˜(rkβ ) = ρ(rkβ + 20ρ(rkβ )) < ρ(rkα ) = ρα k. This makes ρ˜β ≤ 4ρα . Now any point in Bkβ has distance at most 5ρ˜˜α + 5ρ˜˜β + 5ρ˜˜β ≤ 45ρ˜α β α from xα ˜α ). Likewise, whenever Bkα and Bkβ meet for k ≥ k(α) k , so Bk ⊂ B(xk , 45ρ and k ≥ k(β), any point in the larger ball B(xβk , 45ρ˜β ) has distance at most
5ρ˜˜α + 5ρ˜˜β + 45ρ˜β ≤ 205ρα
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
271
β α α ¯α from xα ˜β ) ⊂ B(xα ˜α ) k and hence B(xk , 45ρ k , 205ρ ). Now we define Bk = B(xk , 45ρ α ¯ = B(xα , 205ρα). Then the above discussion says that whenever B α and B β and B k k k k meet for k ≥ k(α) and k ≥ k(β), we have
(4.1.6)
¯β ⊂ B ¯α and B k. k
¯kα Bkβ ⊂ B
¯ α is still a nice embedded ball since, by (4.1.3), 205ρα ≤ 410ρα < Note that B k k inj(Mk , xα k ). We claim there exist positive numbers N (r) and k ′′ (r) such that for any given α with rα < r, as k ≥ k ′′ (r), there holds (4.1.7)
#{β | Bkα ∩ Bkβ 6= φ} ≤ N (r).
Indeed, if Bkα meets Bkβ then there is a positive k ′′ (α) such that as k ≥ k ′′ (α), rkβ ≤ rkα + 10ρ˜˜α k ≤ r + 20ρ(r) 1 ≤ r + (s∞ − r), 5 where we used (4.1.2) in the third inequality. Set k ′′ (r) = max{k ′′ (α), k ′ (r) | α ≤ λ(r)} and
1 N (r) = λ r + (s∞ − r) . 5 Then by combining with (4.1.4), these give the desired estimate (4.1.7) Next we observe that by passing to another subsequence we can guarantee that for any pair α and β we can find a number k(α, β) such that if k ≥ k(α, β) then either Bkα always meets Bkβ or it never does. Hence by setting ¯ k(r) = max k(α, β), k(α), k(β), k ′′ (r) | α ≤ λ(r) and
1 , β ≤ λ r + (s∞ − r) 5
¯ we have shown the following results: for every r < s∞ , if k ≥ k(r), we have α (i) the ball B(pk , r) in Mk is covered by the balls Bk for α ≤ λ(r), (ii) whenever Bkα and Bkβ meet for α ≤ λ(r), we have ¯kα Bkβ ⊂ B
¯ α, ¯β ⊂ B and B k k
(iii) for each α ≤ λ(r), there no more than N (r) balls ever meet Bkα , and ¯ (iv) for any α ≤ λ(r) and any β, either Bkα meets Bkβ for all k ≥ k(r) or none for ¯ all k ≥ k(r).
272
H.-D. CAO AND X.-P. ZHU
ˆ α, Eα, E ¯ α , and E¯ α be the balls of radii 4ρ˜˜α , 5ρ˜˜α , 45ρ˜α , and 205ρα Now we let E around the origin in Euclidean space Rn . At each point xα k ∈ B(pk , sk ) we define coordinate charts Hkα : E α → Bkα as the composition of a linear isometry of Rn to the tangent space Txαk Mk with the exponential map expxαk at xα k . We also get maps α α α ¯ ¯ ¯ α α α ¯ : E¯ → B ¯ and H k : E → B k in the same way. Note that (4.1.3) implies that H k k these maps are all well defined. We denote by gkα (and g¯kα and g¯α k ) the pull-backs of α ¯ α α ¯ the metric gk by Hk (and Hk and H k ). We also consider the coordinate transition ¯ α defined by ¯ α and J¯αβ : E ¯β → E functions Jkαβ : E β → E k ¯ kα )−1 H β Jkαβ = (H k
−1 ¯ β ¯α and J¯kαβ = (H Hk . k)
Clearly J¯kαβ Jkβα = I. Moreover Jkαβ is an isometry from gkβ to g¯kα and J¯kαβ from g¯kβ to g¯α k. Now for each fixed α, the metrics gkα are in geodesic coordinates and have their curvatures and their covariant derivatives uniformly bounded. Claim 1. By passing to another subsequence we can guarantee that for each α (and indeed all α by diagonalization) the metrics gkα (or g¯kα or g¯α k ) converge uniformly α ¯ α ) which α α α ¯ with their derivatives to a smooth metric g (or g¯ or g ) on E (or E¯ α or E is also in geodesic coordinates. Look now at any pair α, β for which the balls Bkα and Bkβ always meet for large k, and thus the maps Jkαβ (and J¯kαβ and Jkβα and J¯kβα ) are always defined for large k. Claim 2. The isometries Jkαβ (and J¯kαβ and Jkβα and J¯kβα ) always have a convergent subsequence. So by passing to another subsequence we may assume Jkαβ → J αβ (and J¯kαβ → ¯ α and J and Jkβα → J βα and J¯kβα → J¯βα ). The limit maps J αβ : E β → E α ¯ are isometries in the limit metrics g β and g α . Moreover ¯β → E J¯αβ : E ¯αβ
J αβ J¯βα = I. We are now done picking subsequences, except we still owe the reader the proofs of Claim 1 and Claim 2. Step 2: Finding local diffeomorphisms which are approximate isometries. Take the subsequence (B(pk , sk ), gk , pk ) chosen in Step 1 above. We claim that for every r < s∞ and every (ǫ1 , ǫ2 , . . . , ǫp ), and for all k and l sufficiently large in comparison, we can find a diffeomorphism Fkl of a neighborhood of the ball B(pk , r) ⊂ B(pk , sk ) into an open set in B(pl , sl ) which is an (ǫ1 , ǫ2 , . . . , ǫp ) approximate isometry in the sense that |t ∇Fkl ∇Fkl − I| < ǫ1 and |∇2 Fkl | < ǫ2 , . . . , |∇p Fkl | < ǫp where ∇p Fkl is the pth covariant derivative of Fkl . The idea (following Peters [106] or Greene and Wu [51]) of proving the claim is α α ¯ α ◦ (H ¯ α )−1 , resp.) from B α to B α to define the map Fkℓ = Hlα ◦ (Hkα )−1 (or F¯kℓ =H l k k ℓ
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
273
¯ α to B ¯ α , resp.) for k and ℓ large compared to α so as to be the identity map (or B k ℓ α ¯ α and H ¯ α , resp.), on E (or E¯ α , resp.) in the coordinate charts Hkα and Hℓα (or H k ℓ ¯ and then to define Fkℓ on a neighborhood of B(pk , r) for k, ℓ ≥ k(r) be averaging β the maps F¯kℓ for β ≤ λ(r + 51 (s∞ − r)). To describe the averaging process on Bkα with α ≤ λ(r) we only need to consider those Bkβ which meet Bkα ; there are never more than N (r) of them and each β ≤ λ(r + 15 (s∞ − r)), and they are the same for k ¯ and ℓ when k, ℓ ≥ k(r). The averaging process is defined by taking Fkℓ (x) to be the β center of mass of the F¯kℓ (x) for x ∈ Bkα averaging over those β where Bkβ meets Bkα β using weights µk (x) defined by a partition of unity. The center of mass of the points β y β = Fkℓ (x) with weights µβ is defined to be the point y such that X expy V β = y β and µβ V β = 0. When the points y β are all close and the weights µβ satisfy 0 ≤ µβ ≤ 1 then there will be a unique solution y close to y β which depends smoothly on the y β and the µβ (see [51] for the details). The point y is found by the inverse function theorem, which also provides bounds on all the derivatives of y as a function of the y β and the µβ . ¯ α , the map F¯ β = H ¯ β and B ¯β ⊆ B ¯ β ◦ (H ¯ β )−1 can be represented Since Bkα ⊆ B ℓ k ℓ kℓ l k in local coordinates by the map αβ ¯α Pkℓ : Eα → E
defined by αβ Pkℓ = J¯ℓαβ ◦ Jkβα .
Since Jkβα → J βα as k → ∞ and J¯ℓαβ → J¯αβ as ℓ → ∞ and J¯αβ ◦ J βα = I, we see αβ that the maps Pkℓ → I as k, ℓ → ∞ for each choice of α and β. The weights µβk are defined in the following way. We pick for each β a smooth function ψ β which equals ˆ β and equals 0 outside E β . We then transfer ψ β to a function ψ β on Mk by 1 on E k ¯ β (i.e. ψ β = ψ β ◦ (H ¯ β )−1 ). Then let the coordinate map H k k k .X µβk = ψkβ ψkγ γ
as usual. In the coordinate chart E α the function ψkβ looks like the composition of Jkβα with ψ β . Call this function ψkαβ = ψ β ◦ Jkβα . Then as k → ∞, ψkαβ → ψ αβ where ψ αβ = ψ β ◦ J βα .
In the coordinate chart E α the function µβk looks like .X αβ µαβ ψkαγ k = ψk γ
αβ and µαβ as k → ∞ where k → µ
µαβ = ψ αβ
.X γ
ψ αγ .
274
H.-D. CAO AND X.-P. ZHU
ˆ α cover B(pk , r), it follows that P ψ γ ≥ 1 on this set and by Since the sets B k γ k combining with (4.1.5) and (4.1.7) there is no problem bounding all these functions and their derivatives. There is a small problem in that we want to guarantee that the 0 averaged map still takes pk to pℓ . This is true at least for the map Fkℓ . Therefore it α will suffice to guarantee that µk = 0 in a neighborhood of pk if α 6= 0. This happens if the same is true for ψkα . If not, we can always replace ψkα by ψ˜kα = (1 − ψk0 )ψkα which P still leaves ψ˜kα ≥ 21 ψkα or ψk0 ≥ 12 everywhere, and this is sufficient to make γ ψ˜kγ ≥ 12 everywhere. αβ Now in the local coordinate E α we are averaging maps Pkℓ which converge to αβ the identity with respect to weights µk which converge. It follows that the averaged map converges to the identity in these coordinates. Thus Fkℓ can be made to be an (ǫ1 , ǫ2 , . . . , ǫp ) approximate isometry on B(pk , r) when k and ℓ are suitably large. At least the estimates |t ∇Fkℓ · ∇Fkℓ − I| < ǫ1 and |∇2 Fkℓ | < ǫ2 , . . . , |∇p Fkℓ | < ǫp on B(pk , r) follow from the local coordinates. We still need to check that Fkℓ is a diffeomorphism on a neighborhood of B(pk , r). This, however, follows quickly enough from the fact that we also get a map Fℓk on a slightly larger ball B(pℓ , r′ ) which contains the image of Fkℓ on B(pk , r) if we take r′ = (1 + ǫ1 )r, and Fℓk also satisfies the above estimates. Also Fkℓ and Fℓk fix the markings, so the composition Fℓk ◦ Fkℓ satisfies the same sort of estimates and fixes the origin pk . αβ αβ Since the maps Pkℓ and Pℓk converge to the identity as k, ℓ tend to infinity, Fℓk ◦Fkℓ must be very close to the identity on B(pk , r). It follows that Fkℓ is invertible. This finishes the proof of the claim and the Step 2. Step 3: Constructing the limit geodesic ball (B∞ , g∞ , p∞ ). We now know the geodesic balls (B(pk , sk ), gk , pk ) are nearly isometric for large k. We are now going to construct the limit B∞ . For a sequence of positive numbers rj ր s∞ with each rj < sj , we choose the numbers (ǫ1 (rj ), . . . , ǫj (rj )) so small that when we choose k(rj ) large in comparison and find the maps Fk(rj ),k(rj+1 ) constructed above on neighborhoods of B(pk(rj ) , rj ), in Mk(rj ) into Mk(rj+1 ) the image always lies in B(pk(rj+1 ) , rj+1 ) and the composition of Fk(rj ),k(rj+1 ) with Fk(rj+1 ),k(rj+2 ) and · · · and Fk(rs−1 ),k(rs ) for any s > j is still an (η1 (rj ), . . . , ηj (rj )) isometry for any choice of ηi (rj ), say ηi (rj ) = 1/j for 1 ≤ i ≤ j. Now we simplify the notation by writing Mj in place of Mk(rj ) and Fj in place of Fk(rj ),k(rj+1 ) . Then Fj : B(pj , rj ) → B(pj+1 , rj+1 ) is a diffeomorphism map from B(pj , rj ) into B(pj+1 , rj+1 ), and the composition Fs−1 ◦ · · · ◦ Fj : B(pj , j) → B(ps , s) is always an (η1 (rj ), . . . , ηj (rj )) approximate isometry. We now construct the limit B∞ as a topological space by identifying the balls B(pj , rj ) with each other using the homeomorphisms Fj . Given any two points x and y in B∞ , we have x ∈ B(pj , rj ) and y ∈ B(ps , rs ) for some j and s. If j ≤ s then x ∈ B(ps , rs ) also, by identification. A set in B∞ is open if and only if it intersects each B(pj , rj ) in an open set. Then choosing disjoint neighborhoods of x and y in
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
275
B(ps , rs ) gives disjoint neighborhoods of x and y in B∞ . Thus B∞ is a Hausdorff space. Any smooth chart on B(pj , rj ) also gives a smooth chart on B(ps , rs ) for all s > j. The union of all such charts gives a smooth atlas on B∞ . It is fairly easy to see the metrics gj on B(pj , rj ), converge to a smooth metric g∞ on B∞ uniformly together with all derivatives on compact sets. For since the Fs−1 ◦ · · · ◦ Fj are very good approximate isometries, the gj are very close to each other, and hence form a Cauchy sequence (together with their derivatives, in the sense that the covariant derivatives of gj with respect to gs are very small when j and s are both large). One checks in the usual way that such a Cauchy sequence converges. The origins pj are identified with each other, and hence with an origin p∞ in B∞ . Now it is the inverses of the maps identifying B(pj , rj ) with open subsets of B∞ that provide the diffeomorphisms of (relatively compact) open sets in B∞ into the geodesic balls B(pj , sj ) ⊂ Mj such that the pull-backs of the metrics gj converge to g∞ . This completes the proof of Step 3. Now it remains to prove both Claim 1 and Claim 2 in Step 1. Proof of Claim 1. It suffices to show the following general result: There exists a constant c > 0 depending only on the dimension, and constants Cq depending only on the dimension and q and bounds Bj on the curvature and its derivatives for j ≤ q where |Dj Rm| √ ≤ Bj , so that for any metric gkℓ in geodesic coordinates in the ball |x| ≤ r ≤ c/ B0 , we have 1 Ikℓ ≤ gkℓ ≤ 2Ikℓ 2 and ∂ ∂ j1 · · · jq gkℓ ≤ Cq , ∂x ∂x
where Ikℓ is the Euclidean metric.
Suppose√we are given a metric gij (x)dxi dxj in geodesic coordinates in the ball |x| ≤ r ≤ c/ B0 as in Claim 1. Then by definition every line through the origin is a geodesic (parametrized proportional to arc length) and gij = Iij at the origin. Also, the Gauss Lemma says that the metric gij is in geodesic coordinates if and only if gij xi = Iij xi . Note in particular that in geodesic coordinates |x|2 = gij xi xj = Iij xi xj is unambiguously defined. Also, in geodesic coordinates we have Γkij (0) = 0, and all the first derivatives for gjk vanish at the origin. Introduce the symmetric tensor Aij =
1 k ∂ x gij . 2 ∂xk
Since we have gjk xk = Ijk xk , we get xk
∂ ∂ gjk = Iij − gij = xk j gik i ∂x ∂x
276
H.-D. CAO AND X.-P. ZHU
and hence from the formula for Γijk xj Γijk = g iℓ Akℓ . Hence Akℓ xk = 0. Let Di be the covariant derivative with respect to the metric gij . Then Di xk = Iik + Γkij xj = Iik + g kℓ Aiℓ . Introduce the potential function P = |x|2 /2 =
1 gij xi xj . 2
We can use the formulas above to compute Di P = gij xj . Also we get Di Dj P = gij + Aij . The defining equation for P gives g ij Di P Dj P = 2P. If we take the covariant derivative of this equation we get g kℓ Dj Dk P Dℓ P = Dj P which is equivalent to Ajk xk = 0. But if we take the covariant derivative again we get g kℓ Di Dj Dk P Dℓ P + g kℓ Dj Dk P Di Dℓ P = Di Dj P. Now switching derivatives Di Dj Dk P = Di Dk Dj P = Dk Di Dj P + Rikjℓ g ℓm Dm P and if we use this and Di Dj P = gij + Aij and g kℓ Dℓ P = xk we find that xk Dk Aij + Aij + g kℓ Aik Ajℓ + Rikjℓ xk xℓ = 0. From our assumed curvature bounds we can take |Rijkℓ | ≤ B0 . Then we get the following estimate: |xk Dk Aij + Aij | ≤ C|Aij |2 + CB0 r2 on the ball |x| ≤ r for some constant C depending only on the dimension. We now show how to use the maximum principle on such equations. First of all, by a maximum principle argument, it is easy to show that if f is a function on a ball |x| ≤ r and λ > 0 is a constant, then ∂f λ sup |f | ≤ sup xk k + λf . ∂x
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
277
For any tensor T = {Ti···j } and any constant λ > 0, setting f = |T |2 in the above inequality, we have (4.1.8)
λ sup |T | ≤ sup |xk Dk T + λT |.
Applying this to the tensor Aij we get sup |Aij | ≤ C sup |Aij |2 + CB0 r2
|x|≤r
|x|≤r
for some constant depending only on the dimension. It is fairly elementary to see that there exist constants c > 0 and C0 < ∞ such that if√ the metric gij is in geodesic coordinates with |Rijkℓ | ≤ B0 in the ball of radius r ≤ c/ B0 then |Aij | ≤ C0 B0 r2 . Indeed, since the derivatives of gij vanish at the origin, so does Aij . Hence the estimate holds near the origin. But the inequality sup |Aij | ≤ C sup |Aij |2 + CB0 r2
|x|≤r
|x|≤r
says that |Aij | avoids an interval when c is chosen small. In fact the inequality X ≤ CX 2 + D is equivalent to |2CX − 1| ≥
√ 1 − 4CD
which makes X avoid an interval if 4CD < 1. (Hence in our case we need to choose c with 4C 2 c2 < 1.) Then if X is on the side containing 0 we get √ 1 − 1 − 4CD X≤ ≤ 2D. 2C This gives |Aij | ≤ C0 B0 r2 with C0 = 2C. We can also derive bounds on all the covariant derivatives of P in terms of bounds on the covariant derivatives of the curvature. To simplify the notation, we let Dq P = {Dj1 Dj2 · · · Djq P } denote the q th covariant derivative, and in estimating Dq P we will lump all the lower order terms into a general slush term Φq which will be a polynomial in D1 P, D2 P, . . . , Dq−1 P and Rm, D1 Rm, . . . , Dq−2 Rm. We already have estimates on a ball of radius r P ≤ r2 /2 |D1 P | ≤ r |Aij | ≤ C0 B0 r2
278
H.-D. CAO AND X.-P. ZHU
√ and since Di Dj P = gij + Aij and r ≤ c/ B0 if we choose c small we can make |Aij | ≤ 1/2, and we get |D2 P | ≤ C2 for some constant C2 depending only on the dimension. Start with the equation g ij Di P Dj P = 2P and apply repeated covariant derivatives. Observe that we get an equation which starts out g ij Di P Dq Dj P + · · · = 0 where the omitted terms only contain derivatives Dq P and lower. If we switch two derivatives in a term Dq+1 P or lower, we get a term which is a product of a covariant derivative of Rm of order at most q − 2 (since the two closest to P commute) and a covariant derivative of P of order at most q − 1; such a term can be lumped in with the slush term Φq . Therefore up to terms in Φq we can regard the derivatives as commuting. Then paying attention to the derivatives in D1 P we get an equation g ij Di P Dj Dk1 · · · Dkq P + g ij Di Dk1 P Dj Dk2 · · · Dkq P +g ij Di Dk2 P Dj Dk1 Dk3 · · · Dkq P + · · · + g ij Di Dkq P Dj Dk1 · · · Dkq−1 P = Dk1 · · · Dkq P + Φq . Recalling that Di Dj P = gij + Aij we can rewrite this as Φq = g ij Di P Dj Dk1 · · · Dkq + (q − 1)Dk1 · · · Dkq P
+ g ij Aik1 Dj Dk2 · · · Dkq P + · · · + g ij Aikq Dj Dk1 · · · Dkq−1 P.
Estimating the product of tensors in the usual way gives |xi Di Dq P + (q − 1)Dq P | ≤ q|A||Dq P | + |Φq |. Applying the inequality λ sup |T | ≤ sup |xk Dk T + λT | with T = Dq P gives (q − 1) sup |Dq P | ≤ sup(q|A||Dq P | + |Φq |). √ Now we can make |A| ≤ 1/2 by making r ≤ c/ B0 with c small; it is important here that c is independent of q! Then we get (q − 2) sup |Dq P | ≤ 2 sup |Φq | which is a good estimate for q ≥ 3. The term Φq is estimated inductively from the terms Dq−1 P and Dq−2 Rm and lower. This proves that there exist constants Cq for q ≥ 3 depending only on q and the dimension and on |Dj Rm| for j ≤ q − 2 such that |Dq P | ≤ Cq
√ on the ball r ≤ c/ B0 . Now we turn our attention to estimating the Euclidean metric Ijk and its covariant derivatives with respect to gjk . We will need the following elementary fact: suppose that f is a function on a ball |x| ≤ r with f (0) = 0 and ∂f i ≤ C|x|2 x i ∂x
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
279
for some constant C. Then |f | ≤ C|x|2
(4.1.9)
for the same constant C. As a consequence, if T = {Tj···k } is a tensor which vanishes at the origin and if |xi Di T | ≤ C|x|2 on a ball |x| ≤ r then |T | ≤ C|x|2 with the same constant C. (Simply apply the inequality (4.1.9) to the function f = |T |. In case this is not smooth, we can use p f = |T |2 + ǫ2 − ǫ and then let ǫ → 0.) Our application will be to the tensor Ijk which gives the Euclidean metric as a tensor in geodesic coordinates. We have Di Ijk = −Γpij Ipk − Γpik Ipj and since xi Γpij = g pq Ajq we get the equation xi Di Ijk = −g pq Ajp Ikq − g pq Akp Ijq . √ We already have |Ajk | ≤ C0 B0 |x|2 for |x| ≤ r ≤ c/ B0 . The tensor Ijk doesn’t vanish at the origin, but the tensor hjk = Ijk − gjk does. We can then use xi Di hjk = −g pq Ajp hkq − g pq Akq hjq − 2Ajk . Suppose M (s) = sup|x|≤s |hjk |. Then |xi Di hjk | ≤ 2[1 + M (s)]C0 B0 |x|2 and we get |hjk | ≤ 2[1 + M (s)]C0 B0 |x|2 on |x| ≤ s. This makes M (s) ≤ 2[1 + M (s)]C0 B0 s2 .
√ Then for s ≤ r ≤ c/ B0 with c small compared to C0 we get 2C0 B0 s2 ≤ 1/2 and M (s) ≤ 4C0 B0 s2 . Thus |Ijk − gjk | = |hjk | ≤ 4C0 B0 |x|2
√ for |x| ≤ r ≤ c/ B0 , and hence for c small enough
1 gjk ≤ Ijk ≤ 2gjk . 2
280
H.-D. CAO AND X.-P. ZHU
Thus the metrics are comparable. Note that this estimate only needs r small compared to B0 and does not need any bounds on the derivatives of the curvature. Now to obtain bounds on the covariant derivative of the Eucliden metric Ikℓ with respect to the Riemannian metric gkℓ we want to start with the equation xi Di Ikℓ + g mn Akm Iℓn + g mn Aℓm Ikn = 0 and apply q covariant derivatives Dj1 · · · Djq . Each time we do this we must interchange Dj and xi Di , and since this produces a term which helps we should look at it closely. If we write Rji = [Dj , Di ] for the commutator, this operator on tensors involves the curvature but no derivatives. Since Dj xi = Iji + g im Ajm we can compute [Dj , xi Di ] = Dj + g im Ajm Di + xi Rji and the term Dj in the commutator helps, while Ajm can be kept small and Rji is zero order. It follows that we get an equation of the form 0 = xi Di Dj1 · · · Djq Ikℓ + qDj1 · · · Djq Ikℓ +
q X
h=1
g im Ajh m Dj1 · · · Djh−1 Di Djh+1 · · · Djq Ikℓ
+ g mn Akm Dj1 · · · Djq Iℓn + g mn Aℓm Dj1 · · · Djq Ikn + Ψq , where the slush term Ψq is a polynomial in derivatives of Ikℓ of degree no more than q − 1 and derivatives of P of degree no more than q + 2 (remember xi = g ij Dj P and Aij = Di Dj P − gij ) and derivatives of the curvature Rm of degree no more than q − 1. We now estimate Dq Ikℓ = {Dj1 · · · Djq Ikℓ } by induction on q using (4.1.8) with λ = q. Noticing a total of q + 2 terms contracting Aij with a derivative of Ikℓ of degree q, we get the estimate q sup |Dq Ikℓ | ≤ (q + 2) sup |A| sup |Dq Ikℓ | + sup |Ψq |. and everything works. This proves that there exists a constant c > 0 depending only on the dimension, and constants Cq depending only on the dimension and q and bounds Bj on the curvature and its derivatives for j ≤ q where |Dj√Rm| ≤ Bj , so that for any metric gkℓ in geodesic coordinates in the ball |x| ≤ r ≤ c/ B0 the Euclidean metric Ikℓ satisfies 1 gkℓ ≤ Ikℓ ≤ 2gkℓ 2 and the covariant derivatives of Ikℓ with respect to gkℓ satisfy |Dj1 · · · Djq Ikℓ | ≤ Cq . The difference between a covariant derivative and an ordinary derivative is given by the connection −Γpij Ipk − Γpik Ipj
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
281
to get Γkij =
1 kℓ I (Dℓ Iij − Di Ijℓ − Dj Iiℓ ). 2
This gives us bounds on Γkij . We then obtain bounds on the first derivatives of gij from ∂ gjk = gkℓ Γℓij + gjℓ Γℓik . ∂xi Always proceeding inductively on the order of the derivative, we now get bounds on covariant derivatives of Γkij from the covariant derivatives of Ipk and bounds of the ordinary derivatives of Γkij by relating the to the covariant derivatives using the Γkij , and bounds on the ordinary derivatives of the gjk from bounds on the ordinary derivatives of the Γℓij . Consequently, we have estimates 1 Ikℓ ≤ gkℓ ≤ 2Ikℓ 2 and ∂ ∂ j1 · · · jq gkℓ ≤ C˜q ∂x ∂x
for similar constants C˜q . Therefore we have finished the proof of Claim 1.
Proof of Claim 2. We need to show how to estimate the derivatives of an isometry. We will prove that if y = F (x) is an isometry from a ball in Euclidean space with a metric gij dxi dxj to a ball in Euclidean space with a metric hkl dy k dy l . Then we can bound all of the derivatives of y with respect to x in terms of bounds on gij and its derivatives with respect to x and bound on hkl and its derivatives with respect to y. This would imply Claim 2. Since y = F (x) is an isometry we have the equation hpq
∂y p ∂y q = gjk . ∂xj ∂xk
Using bounds gjk ≤ CIjk and hpq ≥ cIpq comparing to the Euclidean metric, we easily get estimates ∂y p j ≤ C. ∂x
Now if we differentiate the equation with respect to xi we get hpq
∂ 2 y p ∂y q ∂y p ∂ 2 y q ∂gjk ∂hpq ∂y r ∂y p ∂y q + hpq j i k = − . i j k i ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂y r ∂xi ∂xj ∂xk
Now let Tijk = hpq
∂y p ∂ 2 y q ∂xi ∂xj ∂xk
282
H.-D. CAO AND X.-P. ZHU
and let Uijk =
∂hpq ∂y r ∂y p ∂y q ∂gjk − . i ∂x ∂y r ∂xi ∂xj ∂xk
Then the above equation says Tkij + Tjik = Uijk . Using the obvious symmetries Tijk = Tikj and Uijk = Uikj we can solve this in the usual way to obtain Tijk =
1 (Ujik + Ukij − Uijk ). 2
We can recover the second derivatives of y with respect to x from the formula ∂ 2yp ∂y p kℓ = g T . kij ∂xi ∂xj ∂xℓ Combining these gives an explicit formula giving ∂ 2 y p /∂xi ∂xj as a function of g ij , hpq , ∂gjk /∂xi , ∂hpq /∂y r , and ∂y p /∂y i . This gives bounds ∂ 2yp i j ≤ C ∂y ∂y
and bounds on all higher derivatives follow by differentiating the formula and using induction. This completes the proof of Claim 2 and hence the proof of Theorem 4.1.2. We now want to show how to use this convergence result on solutions to the Ricci flow. Let us first state the definition for the convergence of evolving manifolds. Definition 4.1.3. Let (Mk , gk (t), pk ) be a sequence of evolving marked complete Riemannian manifolds, with the evolving metrics gk (t) over a fixed time interval t ∈ (A, Ω], A < 0 ≤ Ω, and with the marked points pk ∈ Mk . We say a sequence of evolving marked (B0 (pk , sk ), gk (t), pk ) over t ∈ (A, Ω], where B0 (pk , sk ) are geodesic balls of (Mk , gk (0)) centered at pk with the radii sk → s∞ (≤ +∞), con∞ verges in the Cloc topology to an evolving marked (maybe noncomplete) manifold (B∞ , g∞ (t), p∞ ) over t ∈ (A, Ω], where, at the time t = 0, B∞ is a geodesic open ball centered at p∞ ∈ B∞ with the radius s∞ , if we can find a sequence of exhausting open sets Uk in B∞ containing p∞ and a sequence of diffeomorphisms fk of the sets Uk in B∞ to open sets Vk in B(pk , sk ) ⊂ Mk mapping p∞ to pk such that the pull-back metrics g˜k (t) = (fk )∗ gk (t) converge in C ∞ topology to g∞ (t) on every compact subset of B∞ × (A, Ω]. Now we fix a time interval A < t ≤ Ω with −∞ < A < 0 and 0 ≤ Ω < +∞. Consider a sequence of marked evolving complete manifolds (Mk , gk (t), pk ), t ∈ (A, Ω], with each gk (t), k = 1, 2, . . . , being a solution of the Ricci flow ∂ gk (t) = −2Ric k (t) ∂t on B0 (pk , sk ) × (A, Ω], where Rick is the Ricci curvature tensor of gk , and B0 (pk , sk ) is the geodesic ball of (Mk , gk (0)) centered at pk with the radii sk → s∞ (≤ +∞).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
283
Assume that for each r < s∞ there are positive constants C(r) and k(r) such that the curvatures of gk (t) satisfy the bound |Rm(gk )| ≤ C(r) on B0 (pk , r)×(A, Ω] for all k ≥ k(r). We also assume that (Mk , gk (t), pk ), k = 1, 2, . . . , have a uniform injectivity radius bound at the origins pk at t = 0. By Shi’s derivatives estimate (Theorem 1.4.1), the above assumption of uniform bound of the curvatures on the geodesic balls B0 (pk , r) (r < s∞ ) implies the uniform bounds on all the derivatives of the curvatures at t = 0 on the geodesic balls B0 (pk , r) (r < s∞ ). Then by Theorem 4.1.2 we can find a subsequence of marked evolving manifolds, still denoted by (Mk , gk (t), pk ) with t ∈ (A, Ω], so that the geodesic balls (B0 (pk , sk ), gk (0), pk ) ∞ converge in the Cloc topology to a geodesic ball (B∞ (p∞ , s∞ ), g∞ (0), p∞ ). From now on, we consider this subsequence of marked evolving manifolds. By Definition 4.1.1, we have a sequence of (relatively compact) exhausting covering {Uk } of B∞ (p∞ , s∞ ) containing p∞ and a sequence of diffeomorphisms fk of the sets Uk in B∞ (p∞ , s∞ ) to open sets Vk in B0 (pk , sk ) mapping p∞ to pk such that the pull-back metrics at t = 0 C∞
loc g˜k (0) = (fk )∗ gk (0) −→ g∞ (0),
as k → +∞, on B∞ (p∞ , s∞ ).
However, the pull-back metrics g˜k (t) = (fk )∗ gk (t) are also defined at all times A < t ≤ Ω (although g∞ (t) is not yet). We also have uniform bounds on the curvature of the pull-back metrics g˜k (t) and all their derivatives, by Shi’s derivative estimates (Theorem 1.4.1), on every compact subset of B∞ (p∞ , s∞ ) × (A, Ω]. What we claim next is that we can find uniform bounds on all the covariant derivatives of the g˜k taken with respect to the fixed metric g∞ (0). Lemma 4.1.4. Let (M, g) be a Riemannian manifold, K a compact subset of M , and g˜k (t) a collection of solutions to Ricci flow defined on neighborhoods of K × [α, β] with [α, β] containing 0. Suppose that for each l ≥ 0, (a) C0−1 g ≤ g˜k (0) ≤ C0 g, on K, f or all k, (b) |∇l g˜k (0)| ≤ Cl , on K, f or all k, ˜ l Rm(˜ (c) |∇ gk )|k ≤ Cl′ , on K × [α, β], f or all k, k for some positive constants Cl , Cl′ , l = 0, 1, . . . , independent of k, where Rm(˜ gk ) are ˜ k denote covariant derivative with respect the curvature tensors of the metrics g˜k (t), ∇ to g˜k (t), | · |k are the length of a tensor with respect to g˜k (t), and | · | is the length with respect to g. Then the metrics g˜k (t) satisfy −1 C˜0 g ≤ g˜k (t) ≤ C˜0 g, on K × [α, β]
and |∇l g˜k | ≤ C˜l , on K × [α, β], l = 1, 2, . . . , for all k, where C˜l , l = 0, 1, . . . , are positive constants independent of k. Proof. First by using the equation ∂ ˜ k g˜k = −2Ric ∂t and the assumption (c) we immediately get (4.1.10)
−1 C˜0 g ≤ g˜k (t) ≤ C˜0 g, on K × [α, β]
284
H.-D. CAO AND X.-P. ZHU
for some positive constant C˜0 independent of k. ˜ k of g˜k and the Next we want to bound ∇˜ gk . The difference of the connection Γ connection Γ of g is a tensor. Taking Γ to be fixed in time, we get ∂ ˜ ∂ 1 ∂ ∂ ∂ γδ (Γk − Γ) = (˜ gk ) (˜ gk )δβ + (˜ gk )δα − δ (˜ gk )αβ ∂t ∂t 2 ∂xα ∂xβ ∂x h 1 ˜ k )α (−2(Ric ˜ k )βδ ) + (∇ ˜ k )β (−2(Ric ˜ k )αδ ) gk )γδ (∇ = (˜ 2 i ˜ k )δ (−2(Ric ˜ k )αβ ) − (∇ and then by the assumption (c) and (4.1.10), ∂ (Γ ˜ k − Γ) ≤ C, ∂t
for all k.
Note also that at a normal coordinate of the metric g at a fixed point and at the time t = 0, ∂ ∂ ∂ γδ ˜ k )γ − Γγ = 1 (˜ (4.1.11) g ) (˜ g ) + (˜ g ) − (˜ g ) (Γ k k δβ k δα k αβ αβ αβ 2 ∂xα ∂xβ ∂xδ 1 = (˜ gk )γδ (∇α (˜ gk )δβ + ∇β (˜ gk )δα − ∇δ (˜ gk )αβ ), 2 thus by the assumption (b) and (4.1.10), ˜ k (0) − Γ| ≤ C, for all k. |Γ Integrating over time we deduce that ˜ k − Γ| ≤ C, on K × [α, β], for all k. (4.1.12) |Γ By using the assumption (c) and (4.1.10) again, we have ∂ ˜ (∇˜ g ) k = | − 2∇Ric k | ∂t
˜ k Ric ˜ k + (Γ ˜ k − Γ) ∗ Ric ˜ k| = | − 2∇ ≤ C, for all k.
Hence by combining with the assumption (b) we get bounds (4.1.13) |∇˜ gk | ≤ C˜1 , on K × [α, β],
for some positive constant C˜1 independent of k. Further we want to bound ∇2 g˜k . Again regarding ∇ as fixed in time, we see ∂ ˜ k ). (∇2 g˜k ) = −2∇2 (Ric ∂t
Write ˜ k = (∇ − ∇ ˜ k )(∇Ric ˜ k) + ∇ ˜ k (∇ − ∇ ˜ k )Ric ˜ k +∇ ˜ 2 Ric ˜ k ∇2 Ric k ˜ k ) ∗ ∇Ric ˜ k+∇ ˜ k ((Γ − Γ ˜ k ) ∗ Ric ˜ k) + ∇ ˜ 2 Ric ˜ k = (Γ − Γ k ˜ k ) ∗ [(∇ − ∇ ˜ k )Ric ˜ k+∇ ˜ k Ric ˜ k] = (Γ − Γ ˜ k (˜ ˜ k) + ∇ ˜ 2 Ric ˜ k +∇ gk−1 ∗ ∇˜ gk ∗ Ric k ˜ k ) ∗ [(Γ − Γ ˜ k ) ∗ Ric ˜ k+∇ ˜ k Ric ˜ k] = (Γ − Γ ˜ k ˜ k (˜ ˜ k) + ∇ ˜ 2 Ric +∇ gk−1 ∗ ∇˜ gk ∗ Ric k
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
285
where we have used (4.1.11). Then by the assumption (c), (4.1.10), (4.1.12) and (4.1.13) we have |
∂ 2 ˜ k ∇˜ ∇ g˜k | ≤ C + C · |∇ gk | ∂t ˜ k − Γ) ∗ ∇˜ = C + C · |∇2 g˜k + (Γ gk | ≤ C + C|∇2 g˜k |.
Hence by combining with the assumption (b) we get |∇2 g˜k | ≤ C˜2 , on K × [α, β],
for some positive constant C˜2 independent of k. The bounds on the higher derivatives can be derived by the same argument. Therefore we have completed the proof of the lemma. We now apply the lemma to the pull-back metrics g˜k (t) = (fk )∗ gk (t) on B∞ (p∞ , s∞ ) × (A, Ω]. Since the metrics g˜k (0) have uniform bounds on their curvature and all derivatives of their curvature on every compact set of B∞ (p∞ , s∞ ) ∞ and converge to the metric g∞ (0) in Cloc topology, the assumptions (a) and (b) are certainly held for every compact subset K ⊂ B∞ (p∞ , s∞ ) with g = g∞ (0). For every compact subinterval [α, β] ⊂ (A, Ω], we have already seen from Shi’s derivative estimates (Theorem 1.4.1) that the assumption (c) is also held on K × [α, β]. Then all of the ∇l g˜k are uniformly bounded with respect to the fixed metric g = g∞ (0) on every compact set of B∞ (p∞ , s∞ ) × (A, Ω]. By using the classical Arzela-Ascoli theorem, we can find a subsequence which converges uniformly together with all its derivatives on every compact subset of B∞ (p∞ , s∞ ) × (A, Ω]. The limit metric will agree with that obtained previously at t = 0, where we know its convergence already. The limit g∞ (t), t ∈ (A, Ω], is now clearly itself a solution of the Ricci flow. Thus we obtain the following Cheeger type compactness theorem to the Ricci flow, which is essentially obtained by Hamilton in [62] and is called Hamilton’s compactness theorem. Theorem 4.1.5 (Hamilton’s compactness theorem). Let (Mk , gk (t), pk ), t ∈ (A, Ω] with A < 0 ≤ Ω, be a sequence of evolving marked complete Riemannian manifolds. Consider a sequence of geodesic balls B0 (pk , sk ) ⊂ Mk of radii sk (0 < sk ≤ +∞), with sk → s∞ (≤ +∞), around the base points pk in the metrics gk (0). Suppose each gk (t) is a solution to the Ricci flow on B0 (pk , sk ) × (A, Ω]. Suppose also (i) for every radius r < s∞ there exist positive constants C(r) and k(r) independent of k such that the curvature tensors Rm(gk ) of the evolving metrics gk (t) satisfy the bound |Rm(gk )| ≤ C(r), on B0 (pk , r) × (A, Ω] for all k ≥ k(r), and (ii) there exists a constant δ > 0 such that the injectivity radii of Mk at pk in the metric gk (0) satisfy the bound inj (Mk , pk , gk (0)) ≥ δ > 0, for all k = 1, 2, . . .. Then there exists a subsequence of evolving marked (B0 (pk , sk ), gk (t), pk ) over t ∈ ∞ (A, Ω] which converge in Cloc topology to a solution (B∞ , g∞ (t), p∞ ) over t ∈ (A, Ω] to the Ricci flow, where, at the time t = 0, B∞ is a geodesic open ball centered at p∞ ∈ B∞ with the radius s∞ . Moreover the limiting solution is complete if s∞ = +∞.
286
H.-D. CAO AND X.-P. ZHU
4.2. Injectivity Radius Estimates. We will use rescaling arguments to understand the formation of singularities and long-time behaviors of the Ricci flow. In view of the compactness property obtained in the previous section, on one hand one needs to control the bounds on the curvature, and on the other hand one needs to control the lower bounds of the injectivity radius. In applications we usually rescale the solution so that the (rescaled) curvatures become uniformly bounded on compact subsets and leave the injectivity radii of the (rescaled) solutions to be estimated in terms of curvatures. In this section we will review a number of such injectivity radius estimates in Riemannian geometry. In the end we will combine these injectivity estimates with Perelman’s no local collapsing theorem I′ to give the well-known little loop lemma to the Ricci flow which was conjectured by Hamilton in [63]. Let M be a Riemannian manifold. Recall that the injectivity radius at a point p ∈ M is defined by inj (M, p) = sup{r > 0 | expp : B(O, r)(⊂ Tp M ) → M is injective}, and the injectivity radius of M is inj (M ) = inf{inj (M, p) | p ∈ M }. We begin with a basic lemma due to Klingenberg (see for example, Corollary 5.7 in Cheeger & Ebin [22]). Klingenberg’s Lemma. Let M be a complete Riemannian manifold and let p ∈ M . Let lM (p) denote the minimal length of a nontrivial geodesic loop starting and ending at p (maybe not smooth at p). Then the injectivity radius of M at p satisfies the inequality 1 π √ , lM (p) inj (M, p) ≥ min Kmax 2 where √ Kmax denotes the supermum of the sectional curvature on M and we understand π/ Kmax to be positive infinity if Kmax ≤ 0.
Based on this lemma and a second variation argument, Klingenberg proved that the injectivity radius of an even-dimensional, compact, simply connected √ Riemannian manifold of positive sectional curvature is bounded from below by π/ Kmax . For odddimensional, compact, simply connected Riemannian manifold of positive sectional curvature, the same injectivity radius estimates was also proved by Klingenberg under an additional assumption that the sectional curvature is strictly 41 -pinched (see for example Theorem 5.9 and 5.10 in Cheeger & Ebin [22]). We also remark that in dimension 7, there exists a sequence of simply connected, homogeneous Einstein spaces whose sectional curvatures are positive and uniformly bounded from above but their injectivity radii converge to zero. (See [2].) The next result due to Gromoll and Meyer [52] shows that for complete, noncompact Riemannian manifold with positive sectional curvature, the above injectivity radius estimate actually holds without any restriction on dimension. Since the result and proof were not explicitly given in [52], we include a proof here. Theorem 4.2.1 (The Gromoll-Meyer injectivity radius estimate). Let M be a complete, noncompact Riemannian manifold with positive sectional curvature. Then the injectivity radius of M satisfies the following estimate π . inj (M ) ≥ √ Kmax
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
287
Proof. Let O be an arbitrary fixed √ point in M . We need to show that the injectivity radius at O is not less than π/ Kmax . We argue by contradiction. Suppose not, then by Klingenberg’s lemma there exists a closed geodesic loop γ on M starting and ending at O (may be not smooth at O). Since M has positive sectional curvature, we know from the work of GromollMeyer [52] (see also Proposition 8.5 in Cheeger & Ebin [22]) that there exists a compact totally convex subset C of M containing the geodesic loop γ. Among all geodesic loops starting and ending at the same point and lying entirely in the compact totally convex set C there will be a shortest one. Call it γ0 , and suppose γ0 starts and ends at a point we call p0 . First we claim that γ0 must be also smooth at the point p0 . Indeed by the curvature bound and implicit function theorem, there will be a geodesic loop γ˜ close to γ0 starting and ending at any point p˜ close to p0 . Let p˜ be along γ0 . Then by total convexity of the set C, γ˜ also lies entirely in C. If γ0 makes an angle different from π at p0 , the first variation formula will imply that γ˜ is shorter than γ0 . This contradicts with the choice of the geodesic loop γ0 being the shortest. Now let L : [0, +∞) → M be a ray emanating from p0 . Choose r > 0 large enough and set q = L(r). Consider the distance between q and the geodesic loop γ0 . It is clear that the distance can be realized by a geodesic β connecting the point q to a point p on γ0 . Let X be the unit tangent vector of the geodesic loop γ0 at p. Clearly X is orthogonal to the tangent vector of β at p. We then translate the vector X along the geodesic β to get a parallel vector field X(t), 0 ≤ t ≤ r. By using this vector field we can form a variation fixing one endpoint q and the other on γ0 such that the variational vector field is (1 − rt )X(t). The second variation of the arclength of this family of curves is given by t t X(t), 1 − X(t) I 1− r r 2 Z r t ∇ ∂ 1 − = X(t) ∂t r 0 t ∂ t ∂ , 1− X(t), , 1 − X(t) dt −R ∂t r ∂t r 2 Z r 1 t ∂ ∂ = − 1− R , X(t), , X(t) dt r r ∂t ∂t 0 <0 when r is sufficiently large, since the sectional curvature of M is strictly positive everywhere. This contradicts with the fact that β is the shortest geodesic connecting the point q to the shortest geodesic loop γ0 . Thus we have proved the injectivity radius estimate. In contrast to the above injectivity radius estimates, the following well-known injectivity radius estimate of Cheeger (see for example, Theorem 5.8 in Cheeger & Ebin [22]) does not impose the restriction on the sign of the sectional curvature. Cheeger’s Lemma. Let M be an n-dimensional compact Riemannian manifold with the sectional curvature |KM | ≤ λ, the diameter d(M ) ≤ D, and the volume Vol (M ) ≥ v > 0. Then, we have inj (M ) ≥ Cn (λ, D, v)
288
H.-D. CAO AND X.-P. ZHU
for some positive constant Cn (λ, D, v) depending only on λ, D, v and the dimension n. For general complete manifolds, it is possible to relate a lower injectivity radius bound to some lower volume bound provided one localizes the relevant geometric quantities appropriately. The following injectivity radius estimate, which was first obtained by Cheng-Li-Yau [35] for heat kernel estimates and later by Cheeger-Gromov-Taylor [27] with a wave equation argument, is a localized version of the above Cheeger’s Lemma. We now present an argument adapted from Abresch and Meyer [1]. Theorem 4.2.2 (Cheng-Li-Yau [35]). Let B(x0 , 4r0 ), 0 < r0 < ∞, be a geodesic ball in an n-dimensional complete Riemannian manifold (M, g) such that the sectional curvature K of the metric g on B(x0 , 4r0 ) satisfies the bounds λ≤K≤Λ for some constants √ λ and Λ. Then for any positive constant r ≤ r0 (we will also require r ≤ π/(4 Λ) if Λ > 0) the injectivity radius of M at x0 can be bounded from below by inj(M, x0 ) ≥ r ·
Vol (B(x0 , r)) , Vol (B(x0 , r)) + Vλn (2r)
where Vλn (2r) denotes the volume of a geodesic ball of radius 2r in the n-dimensional simply connected space form Mλ with constant sectional curvature λ. Proof. It is well known (cf. Lemma 5.6 in Cheeger and Ebin [22]) that 1 inj(M, x0 ) = min conjugate radius of x0 , lM (x0 ) 2 where lM (x0 ) denotes the length of the shortest (nontrivial) closed geodesic starting √ and ending at x0 . Since by assumption r ≤ π/(4 Λ) if Λ > 0, the conjugate radius of x0 is at least 4r. Thus it suffices to show (4.2.1)
lM (x0 ) ≥ 2r ·
Vol (B(x0 , r)) . Vol (B(x0 , r)) + Vλn (2r)
Now we follow the argument presented in [1]. The idea for proving this inequality, as indicated in [1], is to compare the geometry of the ball B(x0 , 4r) ⊆ B(x0 , 4r0 ) ⊂ ˜4r ⊂ Tx0 (M ), via the exponential map expx , M with the geometry of its lifting B 0 ˜4r → B(x0 , 4r) is a equipped with the pull-back metric g˜ = exp∗x0 g. Thus expx0 : B length-preserving local diffeomorphism. ˜r ⊂ B ˜4r with x Let x ˜0 , x ˜1 , . . . , x ˜N be the preimages of x0 in B ˜0 = 0. Clearly they one-to-one correspond to the geodesic loops γ0 , γ1 , . . . , γN at x0 of length less than r, where γ0 is the trivial loop. Now for each point x ˜i there exists exactly one isometric ˜r → B ˜4r mapping 0 to x˜i and such that exp ϕi = exp . immersion ϕi : B x0 x0 Without loss of generality, we may assume γ1 is the shortest nontrivial geodesic ˜r loop at x0 . By analyzing short homotopies, one finds that ϕi (˜ x) 6= ϕj (˜ x) for all x˜ ∈ B and 0 ≤ i < j ≤ N . This fact has two consequences: (a) N ≥ 2m, where m = [r/lM (x0 )]. To see this, we first observe that the points ˜r because ϕ1 is an isometric immersion ϕk1 (0), −m ≤ k ≤ m, are preimages of x0 in B
289
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
satisfying expx0 ϕ1 = expx0 . Moreover we claim they are distinct. For otherwise ϕ1 would act as a permutation on the set {ϕk1 (0) | − m ≤ k ≤ m}. Since the induced ˜r has the injectivity radius at least 2r, it follows from metric g˜ at each point in B ˜r is geodesically convex. Then the Whitehead theorem (see for example [22]) that B there would exist the unique center of mass y˜ ∈ B˜r . But then y˜ = ϕ0 (˜ y ) = ϕ1 (˜ y ), a contradiction. ˜2r . (b) Each point in B(x0 , r) has at least N +1 preimages in Ω = ∪N xi , r) ⊂ B i=0 B(˜ Hence by the Bishop volume comparison, ˜2r ) ≤ V n (2r). (N + 1)Vol (B(x0 , r)) ≤ Volg˜ (Ω) ≤ Volg˜ (B λ Now the inequality (4.2.1) follows by combining the fact N ≥ 2[r/lM (x0 )] with the above volume estimate. For our purpose of application, we now consider in a complete Riemannian manifold M a geodesic ball B(p0 , s0 ) (0 < s0 ≤ ∞) with the property that there exists a positive increasing function Λ : [0, s0 ) → [0, ∞) such that for any 0 < s < s0 the sectional curvature K on the ball B(p0 , s) of radius s around p0 satisfies the bound |K| ≤ Λ(s). Using Theorem 4.2.2, we can control the injectivity radius at any point p ∈ B(p0 , s0 ) in terms a positive constant that depends only on the dimension n, the injectivity radius at the base point p0 , the function Λ and the distance d(p0 , p) from p to p0 . We now proceed to derive such an estimate. The geometric insight of the following argument belongs to Yau [128] where he obtained a lower bound estimate for volume by comparing various geodesic balls. Indeed, it is a finite version of Yau’s Busemann function argument which gives the information on comparing geodesic balls with centers far apart. For any point p ∈ B(p0 , s0 ) with d(p0 , p) = s, set r0 = (s0 − s)/4 (we define r0 =1 if s0 = ∞). Define the set S to be the union of minimal geodesic segments that connect p to each point in B(p0 , r0 ). Now any point q ∈ S has distance at most r0 + r0 + s = s + 2r0
p from p0 and hence S ⊆ B(p0 , s + 2r0 ). For any 0 < r ≤ min{π/4 Λ(s + 2r0 ), r0 }, we denote by α(p, r) the sector S ∩B(p, r) of radius r and by α(p, s+r0 ) = S ∩B(p, s+r0 ). Let α−Λ(s+2r0 ) (r0 ) (resp. α−Λ(s+2r0 ) (s + r0 )) be a corresponding sector of the same “angles” with radius r0 (resp. s + r0 ) in the n-dimensional simply connected space form with constant sectional curvature −Λ(s+2r0 ). Since B(p0 , r0 ) ⊂ S ⊂ α(p, s+r0 ) and α(p, r) ⊂ B(p, r), the Bishop-Gromov volume comparison theorem implies that Vol (α(p, s + r0 )) Vol (B(p0 , r0 )) ≤ Vol (B(p, r)) Vol (α(p, r)) ≤
n V−Λ(s+2r (s + r0 ) Vol (α−Λ(s+2r0 ) (s + r0 )) 0) = . n Vol (α−Λ(s+2r0 ) (r)) V−Λ(s+2r0 ) (r)
Combining this inequality with the local injectivity radius estimate in Theorem 4.2.2, we get inj (M, p) ≥r
n V−Λ(s+2r (r) · Vol (B(p0 , r0 )) 0)
n n n V−Λ(s+2r (r)Vol (B(p0 , r0 )) + V−Λ(s+2r (2r)V−Λ(s+2r (s + 2r0 ) 0) 0) 0)
.
290
H.-D. CAO AND X.-P. ZHU
Thus, we have proved the following Corollary 4.2.3. Suppose B(p0 , s0 ) (0 < s0 ≤ ∞) is a geodesic ball in an n-dimensional complete Riemannian manifold M having the property that for any 0 < s < s0 the sectional curvature K on B(p0 , s) satisfies the bound |K| ≤ Λ(s) for some positive increasing function Λ defined on [0, s0 ). Then for any point p p ∈ B(p0 , s0 ) with d(p0 , p) = s and any positive number r ≤ min{π/4 Λ(s + 2r0 ), r0 } with r0 = (s0 − s)/4, the injectivity radius of M at p is bounded below by inj (M, p) ≥r
n V−Λ(s+2r (r) · Vol (B(p0 , r0 )) 0) n n n V−Λ(s+2r0 ) (r)Vol (B(p0 , r0 )) + V−Λ(s+2r (2r)V−Λ(s+2r (s 0) 0)
+ 2r0 )
.
In particular, we have inj (M, p) ≥ ρn,δ,Λ (s)
(4.2.2)
where δ > 0 is a lower bound of the injectivity radius inj (M, p0 ) at the origin p0 and ρn,δ,Λ : [0, s0 ) → R+ is a positive decreasing function that depends only on the dimension n, the lower bound δ of the injectivity radius inj (M, p0 ), and the function Λ. We remark that in the above discussion if s0 = ∞ then we can apply the standard Bishop relative volume comparison theorem to geodesic balls directly. Indeed, for any ∆ p ∈ M and any positive constants r and r0 , we have B(p0 , r0 ) ⊆ B(p, rˆ) with rˆ = max{r, r0 +d(p0 , p)}. Suppose in addition the curvature K on M is uniformly bounded by λ ≤ K ≤ Λ for some constants λ and Λ, then the Bishop volume comparison theorem implies that Vol (B(p, rˆ)) Vλ (ˆ r) Vol (B(p0 , r0 )) ≤ ≤ . Vol (B(p, r)) Vol (B(p, r)) Vλ (r) Hence (4.2.3)
inj (M, p) ≥ r
Vλn (r) · Vol (B(p0 , r0 )) . Vλn (r)Vol (B(p0 , r0 )) + Vλn (2r)Vλn (ˆ r)
So we see that the injectivity radius inj(M, p) at p falls off at worst exponentially as the distance d(p0 , p) goes to infinity. In other words, (4.2.4)
√ √ c inj (M, p) ≥ √ (δ B)n e−C Bd(p,p0 ) B
where B is an upper bound on the absolute value of the √ sectional curvature, δ is a lower bound on the injectivity radius at p0 with δ < c/ B, and c > 0 and C < +∞ are positive constants depending only on the dimension n. Finally, by combining Theorem 4.2.2 with Perelman’s no local collapsing Theorem I′ (Theorem 3.3.3) we immediately obtain the following important Little Loop Lemma conjectured by Hamilton [63].
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
291
Theorem 4.2.4 (Little Loop Lemma). Let gij (t), 0 ≤ t < T < +∞, be a solution of the Ricci flow on a compact manifold M . Then there exists a constant ρ > 0 having the following property: if at a point x0 ∈ M and a time t0 ∈ [0, T ), |Rm|(·, t0 ) ≤ r−2
√
on Bt0 (x0 , r)
for some r ≤ T , then the injectivity radius of M with respect to the metric gij (t0 ) at x0 is bounded from below by inj (M, x0 , gij (t0 )) ≥ ρr. 4.3. Limiting Singularity Models. Consider a solution gij (x, t) of the Ricci flow on M × [0, T ), T ≤ +∞, where either M is compact or at each time t the metric gij (·, t) is complete and has bounded curvature. We say that gij (x, t) is a maximal solution if either T = +∞ or T < +∞ and |Rm| is unbounded as t → T . Denote by Kmax (t) = sup |Rm(x, t)|gij (t) . x∈M
Definition 4.3.1. We say that {(xk , tk ) ∈ M ×[0, T )}, k = 1, 2, . . ., is a sequence of (almost) maximum points if there exist positive constants c1 and α ∈ (0, 1] such that α |Rm(xk , tk )| ≥ c1 Kmax (t), t ∈ [tk − , tk ] Kmax (tk ) for all k. Definition 4.3.2. We say that the solution satisfies injectivity radius condition if for any sequence of (almost) maximum points {(xk , tk )}, there exists a constant c2 > 0 independent of k such that c2 inj (M, xk , gij (tk )) ≥ p Kmax (tk )
for all k.
Clearly, by the Little Loop Lemma, a maximal solution on a compact manifold with the maximal time T < +∞ always satisfies the injectivity radius condition. Also by the Gromoll-Meyer injectivity radius estimate, a solution on a complete noncompact manifold with positive sectional curvature also satisfies the injectivity radius condition. According to Hamilton [63], we classify maximal solutions into three types; every maximal solution is clearly of one and only one of the following three types: Type I:
T < +∞ and
sup (T − t)Kmax (t) < +∞;
t∈[0,T )
Type II: (a) T < +∞ but
sup (T − t)Kmax (t) = +∞;
t∈[0,T )
(b) T = +∞ but
sup tKmax (t) = +∞; t∈[0,T )
292
H.-D. CAO AND X.-P. ZHU
Type III:
(a) T = +∞,
sup tKmax (t) < +∞, and t∈[0,T )
lim sup tKmax (t) > 0; t→+∞
(b) T = +∞,
sup tKmax (t) < +∞, and t∈[0,T )
lim sup tKmax (t) = 0; t→+∞
It seems that Type III (b) is not compatible with the injectivity radius condition unless it is a trivial flat solution. Indeed under the Ricci flow the length of a curve γ connecting two points x0 , x1 ∈ M evolves by Z d Lt (γ) = −Ric (γ, ˙ γ)ds ˙ dt γ ≤ C(n)Kmax (t) · Lt (γ) ǫ ≤ Lt (γ), as t large enough, t
for arbitrarily fixed ǫ > 0. Thus when we are considering the Ricci flow on a compact manifold, the diameter of the evolving manifold grows at most as tǫ . But the curvature of the evolving manifold decays faster than t−1 . This says, as choosing ǫ > 0 small enough, diamt (M )2 · |Rm(·, t)| → 0, as t → +∞. Then it is well-known from Cheeger-Gromov [54] that the manifold is a nilmanifold and the injectivity radius condition can not be satisfied as t large enough. When we are considering the Ricci flow on a complete noncompact manifold with nonnegative curvature operator or on a complete noncompact K¨ahler manifold with nonnegative holomorphic bisectional curvature, Li-Yau-Hamilton inequalities imply that tR(x, t) is increasing in time t. Then Type III(b) occurs only when the solution is a trivial flat metric. For each type of solution we define a corresponding type of limiting singularity model. Definition 4.3.3. A solution gij (x, t) to the Ricci flow on the manifold M , where either M is compact or at each time t the metric gij (·, t) is complete and has bounded curvature, is called a singularity model if it is not flat and of one of the following three types: Type I: The solution exists for t ∈ (−∞, Ω) for some constant Ω with 0 < Ω < +∞ and |Rm| ≤ Ω/(Ω − t) everywhere with equality somewhere at t = 0;
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
293
Type II: The solution exists for t ∈ (−∞, +∞) and |Rm| ≤ 1 everywhere with equality somewhere at t = 0; Type III: The solution exists for t ∈ (−A, +∞) for some constant A with 0 < A < +∞ and |Rm| ≤ A/(A + t) everywhere with equality somewhere at t = 0. Theorem 4.3.4. For any maximal solution to the Ricci flow which satisfies the injectivity radius condition and is of Type I, II(a), (b), or III(a), there exists a sequence of dilations of the solution along (almost) maximum points which converges ∞ in the Cloc topology to a singularity model of the corresponding type. Proof. and
Type I: We consider a maximal solution gij (x, t) on M × [0, T ) with T < +∞ ∆
Ω = lim sup(T − t)Kmax (t) < +∞. t→T
First we note that Ω > 0. Indeed by the evolution equation of curvature, d 2 Kmax (t) ≤ Const · Kmax (t). dt This implies that Kmax (t) · (T − t) ≥ Const > 0, because lim sup Kmax (t) = +∞. t→T
Thus Ω must be positive. Choose a sequence of points xk and times tk such that tk → T and lim (T − tk )|Rm(xk , tk )| = Ω.
k→∞
Denote by ǫk = p
1 . |Rm(xk , tk )|
We translate in time so that tk becomes 0, dilate in space by the factor ǫk and dilate in time by ǫ2k to get (k)
2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t),
t˜ ∈ [−tk /ǫ2k , (T − tk )/ǫ2k ).
294
H.-D. CAO AND X.-P. ZHU
Then ∂ (k) ˜ ∂ g˜ (·, t) = ǫ−2 gij (·, t) · ǫ2k k ∂t ∂ t˜ ij = −2Rij (·, tk + ǫ2k t˜) ˜ (k) (·, t˜), = −2R ij
˜ (k) is the Ricci curvature of the metric g˜(k) . So g˜(k) (·, t˜) is still a solution to where R ij ij ij the Ricci flow which exists on the time interval [−tk /ǫ2k , (T − tk )/ǫ2k ), where tk /ǫ2k = tk |Rm(xk , tk )| → +∞ and (T − tk )/ǫ2k = (T − tk )|Rm(xk , tk )| → Ω. For any ǫ > 0 we can find a time τ < T such that for t ∈ [τ, T ), |Rm| ≤ (Ω + ǫ)/(T − t) (k) by the assumption. Then for t˜ ∈ [(τ − tk )/ǫ2k , (T − tk )/ǫ2k ), the curvature of g˜ij (·, t˜) is bounded by
˜ (k) | = ǫ2 |Rm| |Rm k
≤ (Ω + ǫ)/((T − t)|Rm(xk , tk )|) = (Ω + ǫ)/((T − tk )|Rm(xk , tk )| + (tk − t)|Rm(xk , tk )|) → (Ω + ǫ)/(Ω − t˜), as k → +∞.
This implies that {(xk , tk )} is a sequence of (almost) maximum points. And then by the injectivity radius condition and Hamilton’s compactness theorem 4.1.5, there (k) ∞ exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc topology to a (∞) ˜ (∞) ˜ limit metric g˜ij (t) on a limiting manifold M with t˜ ∈ (−∞, Ω) such that g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies the bound ˜ (∞) | ≤ Ω/(Ω − t˜) |Rm ˜ × (−∞, Ω) with the equality somewhere at t˜ = 0. everywhere on M Type II(a): We consider a maximal solution gij (x, t) on M × [0, T ) with T < +∞ and
lim sup(T − t)Kmax (t) = +∞. t→T
Let Tk < T < +∞ with Tk → T , and γk ր 1, as k → +∞. Pick points xk and times tk such that, as k → +∞, (Tk − tk )|Rm(xk , tk )| ≥ γk
sup x∈M,t≤Tk
(Tk − t)|Rm(x, t)| → +∞.
Again denote by ǫk = p
1 |Rm(xk , tk )|
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
295
and dilate the solution as before to get (k) 2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t),
t˜ ∈ [−tk /ǫ2k , (Tk − tk )/ǫ2k ),
which is still a solution to the Ricci flow and satisfies the curvature bound ˜ (k) | = ǫ2k |Rm| |Rm
1 (Tk − tk ) · γk (Tk − t) tk (Tk − tk ) 1 (Tk − tk )|Rm(xk , tk )| ˜ for t ∈ − 2 , = , γk [(Tk − tk )|Rm(xk , tk )| − t˜] ǫk ǫ2k p since t = tk +ǫ2k t˜ and ǫk = 1/ |Rm(xk , tk )|. Hence {(xk , tk )} is a sequence of (almost) maximum points. And then as before, by applying Hamilton’s compactness theorem (k) ∞ 4.1.5, there exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc (∞) ˜ and t˜ ∈ (−∞, +∞) such that topology to a limit g˜ij (t˜) on a limiting manifold M ≤
(∞)
g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies ˜ (∞) | ≤ 1 |Rm
˜ × (−∞, +∞) and the equality holds somewhere at t˜ = 0. everywhere on M
Type II(b): We consider a maximal solution gij (x, t) on M × [0, T ) with T = +∞ and
lim sup tKmax (t) = +∞. t→T
Again let Tk → T = +∞, and γk ր 1, as k → +∞. Pick xk and tk such that tk (Tk − tk )|Rm(xk , tk )| ≥ γk
sup x∈M,t≤Tk
t(Tk − t)|Rm(x, t)|.
Define (k) 2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t), p where ǫk = 1/ |Rm(xk , tk )|. Since
tk (Tk − tk )|Rm(xk , tk )| ≥ γk ≥ γk ≥ we have
sup x∈M,t≤Tk
t(Tk − t)|Rm(x, t)|
sup x∈M,t≤Tk /2
t(Tk − t)|Rm(x, t)|
Tk γk sup t|Rm(x, t)|, 2 x∈M,t≤Tk /2
Tk T k − tk
γk (Tk − tk ) = (Tk − tk )|Rm(xk , tk )| ≥ ǫ2k 2
tk γk = tk |Rm(xk , tk )| ≥ 2 ǫk 2
t˜ ∈ [−tk /ǫ2k , (Tk − tk )/ǫ2k ),
and
sup x∈M,t≤Tk /2
Tk tk
t|Rm(x, t)| → +∞,
sup x∈M,t≤Tk /2
t|Rm(x, t)| → +∞,
296
H.-D. CAO AND X.-P. ZHU
as k → +∞. As before, we also have ∂ (k) ˜ ˜ (k) (·, t˜) g˜ij (·, t) = −2R ij ˜ ∂t and ˜ (k) | |Rm = ǫ2k |Rm| 1 tk (Tk − tk ) · γk t(Tk − t) 1 tk (Tk − tk )|Rm(xk , tk )| = · γk (tk + ǫ2k t˜)[(Tk − tk ) − ǫ2k t˜] · |Rm(xk , tk )| tk (Tk − tk )|Rm(xk , tk )| 1 · = γk (tk + ǫ2k t˜)[(Tk − tk )|Rm(xk , tk )| − t˜] tk (Tk − tk )|Rm(xk , tk )| = γk (1+ t˜/(tk |Rm(xk , tk )|))[tk (Tk −tk )|Rm(xk , tk )|](1− t˜/((Tk −tk )|Rm(xk , tk )|)) ≤
→ 1,
as k → +∞.
Hence {(xk , tk )} is again a sequence of (almost) maximum points. As before, there (k) ∞ exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc topology to (∞) ˜ (∞) ˜ ˜ a limit g˜ij (t) on a limiting manifold M and t ∈ (−∞, +∞) such that g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies ˜ (∞) | ≤ 1 |Rm ˜ × (−∞, +∞) with the equality somewhere at t˜ = 0. everywhere on M Type III(a): T = +∞ and
We consider a maximal solution gij (x, t) on M × [0, T ) with lim sup tKmax (t) = A ∈ (0, +∞). t→T
Choose a sequence of xk and tk such that tk → +∞ and lim tk |Rm(xk , tk )| = A.
k→∞
p Set ǫk = 1/ |Rm(xk , tk )| and dilate the solution as before to get (k) 2˜ g˜ij (·, t˜) = ǫ−2 k gij (·, tk + ǫk t),
t˜ ∈ [−tk /ǫ2k , +∞)
which is still a solution to the Ricci flow. Also, for arbitrarily fixed ǫ > 0, there exists a sufficiently large positive constant τ such that for t ∈ [τ, +∞), ˜ (k) | = ǫ2k |Rm| |Rm A+ǫ 2 ≤ ǫk t A+ǫ 2 = ǫk tk + ǫ2k t˜
= (A + ǫ)/(tk |Rm(xk , tk )| + t˜),
for t˜ ∈ [(τ − tk )/ǫ2k , +∞).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
297
Note that (A + ǫ)/(tk |Rm(xk , tk )| + t˜) → (A + ǫ)/(A + t˜),
as k → +∞
and (τ − tk )/ǫ2k → −A,
as k → +∞.
Hence {(xk , tk )} is a sequence of (almost) maximum points. And then as before, there (k) ∞ exists a subsequence of the metrics g˜ij (t˜) which converges in the Cloc topology to (∞) ˜ (∞) ˜ ˜ a limit g˜ij (t) on a limiting manifold M and t ∈ (−A, +∞) such that g˜ij (t˜) is a complete solution of the Ricci flow and its curvature satisfies ˜ (∞) | ≤ A/(A + t˜) |Rm ˜ × (−A, +∞) with the equality somewhere at t˜ = 0. everywhere on M In the case of manifolds with nonnegative curvature operator, or K¨ahler metrics with nonnegative holomorphic bisectional curvature, we can bound the Riemannian curvature by the scalar curvature R upto a constant factor depending only on the dimension. Then we can slightly modify the statements in the previous theorem as follows Corollary 4.3.5. For any complete maximal solution to the Ricci flow with bounded and nonnegative curvature operator on a Riemannian manifold, or on a K¨ ahler manifold with bounded and nonnegative holomorphic bisectional curvature, there exists a sequence of dilations of the solution along (almost ) maximum points which converges to a singular model. For Type I solutions: the limit model exists for t ∈ (−∞, Ω) with 0 < Ω < +∞ and has R ≤ Ω/(Ω − t) everywhere with equality somewhere at t = 0. For Type II solutions: the limit model exists for t ∈ (−∞, +∞) and has R≤1 everywhere with equality somewhere at t = 0. For Type III solutions: the limit model exists for t ∈ (−A, +∞) with 0 < A < +∞ and has R ≤ A/(A + t) everywhere with equality somewhere at t = 0. A natural and important question is to understand each of the three types of singularity models. The following results obtained by Hamilton [66] and Chen-Zhu [30] characterize the Type II and Type III singularity models with nonnegative curvature operator and positive Ricci curvature respectively. The corresponding results in the K¨ahler case with nonnegative holomorphic bisectional curvature were obtained by the first author [14].
298
H.-D. CAO AND X.-P. ZHU
Theorem 4.3.6. (i) (Hamilton [66]) Any Type II singularity model with nonnegative curvature operator and positive Ricci curvature to the Ricci flow on a manifold M must be a (steady) Ricci soliton. (ii) (Chen-Zhu [30]) Any Type III singularity model with nonnegative curvature operator and positive Ricci curvature on a manifold M must be a homothetically expanding Ricci soliton. Proof. We only give the proof of (ii), since the proof of (i) is similar and easier. After a shift of the time variable, we may assume the Type III singularity model is defined on 0 < t < +∞ and tR assumes its maximum in space-time. Recall from the Li-Yau-Hamilton inequality (Theorem 2.5.4) that for any vectors V i and W i , (4.3.1)
Mij W i W j + (Pkij + Pkji )V k W i W j + Rikjl W i W j V k V l ≥ 0,
where 1 1 Mij = ∆Rij − ∇i ∇j R + 2Ripjq Rpq − g pq Rip Rjq + Rij 2 2t and Pijk = ∇i Rjk − ∇j Rik . Take the trace on W to get (4.3.2)
∆
Q=
∂R R + + 2∇i R · V i + 2Rij V i V j ≥ 0 ∂t t
for any vector V i . Let us choose V to be the vector field minimizing Q, i.e., 1 V i = − (Ric −1 )ik ∇k R, 2
(4.3.3)
where (Ric−1 )ik is the inverse of the Ricci tensor Rij . Substitute this vector field ˜ By a direct computation from the evolution V into Q to get a smooth function Q. equations of curvatures (see [61] for details), ∂ ˜ ˜ − 2 Q. ˜ Q ≥ ∆Q ∂t t
(4.3.4)
Suppose tR assumes its maximum at (x0 , t0 ) with t0 > 0, then ∂R R + = 0, ∂t t
at (x0 , t0 ).
This implies that the quantity Q=
∂R R + + 2∇i R · V i + 2Rij V i V j ∂t t
vanishes in the direction V = 0 at (x0 , t0 ). We claim that for any earlier time t < t0 and any point x ∈ M , there is a vector V ∈ Tx M such that Q = 0.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
299
We argue by contradiction. Suppose not, then there is x¯ ∈ M and 0 < t¯ < t0 such ˜ is positive at x = x¯ and t = t¯. We can find a nonnegative smooth function ρ that Q on M with support in a neighborhood of x ¯ so that ρ(¯ x) > 0 and ˜ ≥ ρ, Q t¯2 at t = t¯. Let ρ evolve by the heat equation ∂ρ = ∆ρ. ∂t It then follows from the standard strong maximum principle that ρ > 0 everywhere for any t > t¯. From (4.3.4) we see that ρ ∂ ˜ ˜− ρ −2 Q ˜− ρ Q− 2 ≥∆ Q ∂t t t2 t t2 Then by the maximum principle as in Chapter 2, we get ˜ ≥ ρ > 0, Q for all t ≥ t¯. t2 This gives a contradiction with the fact Q = 0 for V = 0 at (x0 , t0 ). We thus prove the claim. Consider each time t < t0 . The null vector field of Q satisfies the equation (4.3.5)
∇i R + 2Rij V j = 0,
by the first variation of Q in V . Since Rij is positive, we see that such a null vector field is unique and varies smoothly in space-time. Substituting (4.3.5) into the expression of Q, we have (4.3.6)
∂R R + + ∇i R · V i = 0 ∂t t
Denote by Qij = Mij + (Pkij + Pkji )V k + Rikjl V k V l . From (4.3.1) we see that Qij is nonnegative definite with its trace Q = 0 for such a null vector V . It follows that Qij = Mij + (Pkij + Pkji )V k + Rikjl V k V l = 0. Again from the first variation of Qij in V , we see that (4.3.7)
(Pkij + Pkji ) + (Rikjl + Rjkil )V l = 0,
and hence (4.3.8)
Mij − Rikjl V k V l = 0.
Applying the heat operator to (4.3.5) and (4.3.6) we get ∂ (4.3.9) − ∆ (∇i R + 2Rij V j ) 0= ∂t ∂ ∂ j −∆ V + − ∆ (∇i R) = 2Rij ∂t ∂t ∂ + 2V j − ∆ Rij − 4∇k Rij ∇k V j , ∂t
300
H.-D. CAO AND X.-P. ZHU
and (4.3.10)
∂ ∂R R i 0= −∆ + + ∇i R · V ∂t ∂t t ∂ ∂ i i −∆ V +V − ∆ (∇i R) − 2∇k ∇i R · ∇k V i = ∇i R ∂t ∂t ∂ ∂R R + −∆ + . ∂t ∂t t
Multiplying (4.3.9) by V i , summing over i and adding (4.3.10), as well as using the evolution equations on curvature, we get (4.3.11)
0 = 2V i (2∇i (|Rc|2 ) − Ril ∇l R) + 2V i V j (2Rpiqj Rpq − 2g pq Rpi Rqj ) − 4∇k Rij · ∇k V j · V i − 2∇k ∇i R · ∇k V i + 4Rij ∇i ∇j R R + 4g kl g mn g pq Rkm Rnp Rql + 4Rijkl Rik V j V l − 2 . t
From (4.3.5), we have the following equalities i l i j pq −2V Ril ∇ R − 4V V g Rpi Rqj = 0, (4.3.12) −4∇k Rij · ∇k V j · V i − 2∇k ∇i R · ∇k V i = 4Rij ∇k V i · ∇k V j , ∇i ∇j R = −2∇i Rjl · V l − 2Rjl ∇i V l . Substituting (4.3.12) into (4.3.11), we obtain
8Rij (∇k Rij · V k + Rikjl V k V l − ∇i Rjl · V l − Rjl ∇i V l ) R + 4Rij ∇k V i · ∇k V j + 4g kl g mn g pq Rkm Rnp Rql − 2 = 0. t By using (4.3.7), we know Rij (∇k Rij · V k + Rikjl V k V l − ∇i Rjl · V l ) = 0. Then we have (4.3.13)
−8Rij Rjl ∇i V l + 4Rij ∇k V i · ∇k V j + 4g kl g mn g pq Rkm Rnp Rql −
R = 0. t2
By taking the trace in the last equality in (4.3.12) and using (4.3.6) and the evolution equation of the scalar curvature, we can get (4.3.14)
Rij (Rij +
gij − ∇i Vj ) = 0. 2t
Finally by combining (4.3.13) and (4.3.14), we deduce gjk gik − ∇k Vi Rjk + − ∇k Vj = 0. 4Rij g kl Rik + 2t 2t Since Rij is positive definite, we get (4.3.15)
∇i Vj = Rij +
gij , 2t
for all i, j.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
301
This means that gij (t) is a homothetically expanding Ricci soliton. Remark 4.3.7. Recall from Section 1.5 that any compact steady Ricci soliton or expanding Ricci soliton must be Einstein. If the manifold M in Theorem 4.3.6 is noncompact and simply connected, then the steady (or expanding) Ricci soliton must be a steady (or expanding) gradient Ricci soliton. For example, we know that ∇i Vj is symmetric from (4.3.15). Also, by the simply connectedness of M there exists a function F such that ∇i ∇j F = ∇i Vj ,
on M.
So Rij = ∇i ∇j F −
gij , 2t
on M
This means that gij is an expanding gradient Ricci soliton. In the K¨ahler case, we have the following results for Type II and Type III singularity models with nonnegative holomorphic bisectional curvature obtained by the first author in [14]. Theorem 4.3.8 (Cao [14]). (i) Any Type II singularity model on a K¨ ahler manifold with nonnegative holomorphic bisectional curvature and positive Ricci curvature must be a steady K¨ ahler-Ricci soliton. (ii) Any Type III singularity model on a K¨ ahler manifold with nonnegative holomorphic bisectional curvature and positive Ricci curvature must be an expanding K¨ ahler-Ricci soliton. To conclude this section, we state a result of Sesum [113] on compact Type I singularity models. Recall that Perelman’s functional W, introduced in Section 1.5, is given by Z n W(g, f, τ ) = (4πτ )− 2 [τ (|∇f |2 + R) + f − n]e−f dVg M
with the function f satisfying the constraint Z n (4πτ )− 2 e−f dVg = 1. M
And recall from Corollary 1.5.9 that Z n µ(g(t)) = inf W(g(t), f, T − t)| (4π(T − t))− 2 e−f dVg(t) = 1 M
is strictly increasing along the Ricci flow unless we are on a gradient shrinking soliton. If one can show that µ(g(t)) is uniformly bounded from above and the minimizing functions f = f (·, t) have a limit as t → T , then the rescaling limit model will be a shrinking gradient soliton. As shown by Natasa Sesum in [113], Type I assumption guarantees the boundedness of µ(g(t)), while the compactness assumption of the rescaling limit guarantees the existence of the limit for the minimizing functions f (·, t). Therefore we have
302
H.-D. CAO AND X.-P. ZHU
Theorem 4.3.9 (Sesum [113]). Let (M, gij (t)) be a Type I singularity model obtained as a rescaling limit of a Type I maximal solution. Suppose M is compact. Then (M, gij (t)) must be a gradient shrinking Ricci soliton. It seems that the assumption on the compactness of the rescaling limit is superfluous. We conjecture that any noncompact Type I limit is also a gradient shrinking soliton. 4.4. Ricci Solitons. We will now examine the structure of a steady Ricci soliton of the sort we get as a Type II limit. Lemma 4.4.1. Suppose we have a complete gradient steady Ricci soliton gij with bounded curvature so that Rij = ∇i ∇j F for some function F on M . Assume the Ricci curvature is positive and the scalar curvature R attains its maximum Rmax at a point x0 ∈ M . Then (4.4.1)
|∇F |2 + R = Rmax
everywhere on M , and furthermore F is convex and attains its minimum at x0 . Proof. Recall that, from (1.1.15) and noting our F here is −f there, the steady gradient Ricci soliton has the property |∇F |2 + R = C0 for some constant C0 . Clearly, C0 ≥ Rmax . If C0 = Rmax , then ∇F = 0 at the point x0 . Since ∇i ∇j F = Rij > 0, we see that F is convex and F attains its minimum at x0 . If C0 > Rmax , consider a gradient path of F in a local coordinate neighborhood through x0 = (x10 , . . . , xn0 ) : ( i x = xi (u), u ∈ (−ε, ε), i = 1, . . . , n xi0 = xi (0),
and dxi = g ij ∇j F, du
u ∈ (−ε, ε).
Now |∇F |2 = C0 − R ≥ C0 − Rmax > 0 everywhere, while |∇F |2 is smallest at x = x0 since R is largest there. But we compute d d |∇F |2 = 2g jl ∇j F ∇l F du du = 2g ik g jl ∇i ∇j F · ∇k F ∇l F = 2g ik g jl Rij ∇k F ∇l F >0
since Rij > 0 and |∇F |2 > 0. Then |∇F |2 is not smallest at x0 , and we have a contradiction.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
303
We remark that when we are considering a complete expanding gradient Ricci soliton on M with positive Ricci curvature and Rij + ρgij = ∇i ∇j F for some constant ρ > 0 and some function F , the above argument gives |∇F |2 + R − 2ρF = C for some positive constant C. Moreover the function F is an exhausting and convex function. In particular, such an expanding gradient Ricci soliton is diffeomorphic to the Euclidean space Rn . Let us introduce a geometric invariant as follows. Let O be a fixed point in a Riemannian manifold M , s the distance to the fixed point O, and R the scalar curvature. We define the asymptotic scalar curvature ratio A = lim sup Rs2 . s→+∞
Clearly the definition is independent of the choice of the fixed point O and invariant under dilation. This concept is particular useful on manifolds with positive sectional curvature. The first type of gap theorem was obtained by Mok-Siu-Yau [93] in understanding the hypothesis of the paper of Siu-Yau [120]. Yau (see [49]) suggested that this should be a general phenomenon. This was later conformed by Greene-Wu [49, 50], Eschenberg-Shrader-Strake [45] and Drees [44] where they show that any complete noncompact n-dimensional (except n = 4 or 8) Riemannian manifold of positive sectional curvature must have A > 0. Similar results on complete noncompact K¨ahler manifolds of positive holomorphic bisectional curvature were obtained by Chen-Zhu [31] and Ni-Tam [100]. Theorem 4.4.2 (Hamilton [63]). For a complete noncompact steady gradient Ricci soliton with bounded curvature and positive sectional curvature of dimension n ≥ 3 where the scalar curvature assume its maximum at a point O ∈ M , the asymptotic scalar curvature ratio is infinite, i.e., A = lim sup Rs2 = +∞ s→+∞
where s is the distance to the point O. Proof. The solution to the Ricci flow corresponding to the soliton exists for −∞ < t < +∞ and is obtained by flowing along the gradient of a potential function F of the soliton. We argue by contradiction. Suppose Rs2 ≤ C. We will show that the limit g¯ij (x) = lim gij (x, t) t→−∞
exists for x 6= O on the manifold M and is a complete flat metric on M \ {O}. Since the sectional curvature of M is positive everywhere, it follows from Cheeger-Gromoll [23] that M is diffeomorphic to Rn . Thus M \ {O} is diffeomorphic to Sn−1 × R. But for n ≥ 3 there is no flat metric on Sn−1 × R, and this will finish the proof.
304
H.-D. CAO AND X.-P. ZHU
To see the limit metric exists, we note that R → 0 as s → +∞, so |∇F |2 → Rmax as s → +∞ by (4.4.1). The function F itself can be taken to evolve with time, using the definition ∂F ∂xi = ∇i F · = −|∇F |2 = ∆F − Rmax ∂t ∂t which pulls F back by the flow along the gradient of F . Then we continue to have ∇i ∇j F = Rij for all time, and |∇F |2 → Rmax as s → +∞ for each time. When we go backward in time, this is√equivalent to flowing outwards along the gradient of F , and our speed approaches Rmax . So, starting outside of any neighborhood of O we have p s dt (·, O) = → Rmax , |t| |t|
and (4.4.2)
R(·, t) ≤
C , Rmax · |t|2
as t → −∞
as |t| large enough.
Hence for |t| sufficiently large, 0 ≥ −2Rij ∂ = gij ∂t ≥ −2Rgij 2C gij ≥− Rmax · |t|2 which implies that for any tangent vector V , 0≤
d 2C (log(gij (t)V i V j )) ≤ . d|t| Rmax · |t|2
These two inequalities show that gij (t)V i V j has a limit g¯ij V i V j as t → −∞. Since the metrics are all essentially the same, it always takes an infinite length to get out to the infinity. This shows the limit g¯ij is complete at the infinity. One the other hand, any point P other than O will eventually be arbitrarily far from O, so the limit metric g¯ij is also complete away from O in M \ {O}. Using Shi’s derivative ∞ estimates in Chapter 1, it follows that gij (·, t) converges in the Cloc topology to a complete smooth limit metric g¯ij as t → −∞, and the limit metric is flat by (4.4.2). The above argument actually shows that (4.4.3)
lim sup Rs1+ε = +∞ s→+∞
for arbitrarily small ε > 0 and for any complete gradient Ricci soliton with bounded and positive sectional curvature of dimension n ≥ 3 where the scalar curvature assumes its maximum at a fixed point O. Finally we conclude this section with the important uniqueness of complete Ricci soliton on two-dimensional Riemannian manifolds.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
305
Theorem 4.4.3 (Hamilton [60]). The only complete steady Ricci soliton on a two-dimensional manifold with bounded curvature which assumes its maximum 1 at an origin is the “cigar” soliton on the plane R2 with the metric ds2 =
dx2 + dy 2 . 1 + x2 + y 2
Proof. Recall that the scalar curvature evolves by ∂R = ∆R + R2 ∂t on a two-dimensional manifold M . Denote by Rmin (t) = inf{R(x, t) | x ∈ M }. We see from the maximum principle (see for example Chapter 2) that Rmin (t) is strictly increasing whenever Rmin (t) 6= 0, for −∞ < t < +∞. This shows that the curvature of a steady Ricci soliton on a two-dimensional manifold M must be nonnegative and Rmin (t) = 0 for all t ∈ (−∞, +∞). Further by the strong maximum principle we see that the curvature is actually positive everywhere. In particular, the manifold must be noncompact. So the manifold M is diffiomorphic to R2 and the Ricci soliton must be a gradient soliton. Let F be a potential function of the gradient Ricci soliton. Then, by definition, we have ∇i Vj + ∇j Vi = Rgij with Vi = ∇i F . This says that the vector field V must be conformal. In complex ∂ coordinate a conformal vector field is holomorphic. Hence V is locally given by V (z) ∂z for a holomorphic function V (z). At a zero of V there will be a power series expansion V (z) = az p + · · · , (a 6= 0) and if p > 1 the vector field will have closed orbits in any neighborhood of the zero. Now the vector field is gradient and a gradient flow cannot have a closed orbit. Hence V (z) has only simple zeros. By Lemma 4.4.1, we know that F is strictly convex with the only critical point being the minima, chosen to be the origin of R2 . So the holomorphic vector field V must be V (z)
∂ ∂ = cz , ∂z ∂z
for z ∈ C,
for some complex number c. We now claim that c is real. Let us write the metric as ds2 = g(x, y)(dx2 + dy 2 ) with z = x +
√ √ ∂ means that if c = a + −1b, then −1y. Then ∇F = cz ∂z ∂F = (ax − by)g, ∂x
Taking the mixed partial derivatives gives b = 0, so c is real.
∂2F ∂x∂y
∂F = (bx + ay)g. ∂y and equating them at the origin x = y = 0
306
H.-D. CAO AND X.-P. ZHU
Let (
x = eu cos v,
−∞ < u < +∞,
u
0 ≤ v ≤ 2π.
y = e sin v,
Write ds2 = g(x, y)(dx2 + dy 2 ) = g(eu cos v, eu sin v)e2u (du2 + dv 2 ) ∆
= g(u, v)(du2 + dv 2 ). Then we get the equations ∂F = ag, ∂u
∂F =0 ∂v
∂ since the gradient of F is just a ∂u for a real constant a. The second equation shows that F = F (u) is a function of u only, then the first equation shows that g = g(u) is also a function of u only. Then we can write the metric as
(4.4.4)
ds2 = g(u)(du2 + dv 2 ) = g(u)e−2u (dx2 + dy 2 ).
This implies that e−2u g(u) must be a smooth function of x2 + y 2 = e2u . So as u → −∞, (4.4.5)
g(u) = b1 e2u + b2 (e2u )2 + · · · ,
with b1 > 0. The curvature of the metric is given by R=−
1 g
g′ g
′
where (·)′ is the derivative with respect to u. Note that the soliton is by translation in u with velocity c. Hence g = g(u + ct) satisfies ∂g = −Rg ∂t which becomes cg = ′
g′ g
′
.
Thus by (4.4.5), g′ = cg + 2 g and then by integrating e2u
1 c = − e2u + b1 g 2
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
307
i.e., e2u . b1 − 2c e2u
g(u) =
In particular, we have c < 0 since the Ricci soliton is not flat. Therefore ds2 = g(u)e−2u (dx2 + dy 2 ) =
dx2 + dy 2 α1 + α2 (x2 + y 2 )
for some constants α1 , α2 > 0. By the normalization condition that the curvature attains its maximum 1 at the origin, we conclude that ds2 =
dx2 + dy 2 . 1 + (x2 + y 2 )
5. Long Time Behaviors. Let M be a complete manifold of dimension n. Consider a solution of the Ricci flow gij (x, t) on M and on a maximal time interval [0, T ). When M is compact, we usually consider the normalized Ricci flow 2 ∂gij = rgij − 2Rij , ∂t n
R R where r = M RdV / M dV is the average scalar curvature. The factor r serves to normalize p the Ricci flow so that the volume is constant. To see this we observe that dV = det gij dx and then p ∂ 1 ∂ log det gij = g ij gij = r − R, ∂t 2 ∂t d dt
Z
dV = M
Z
M
(r − R)dV = 0.
The Ricci flow and the normalized Ricci flow differ only by a change of scale in space and a change of parametrization in time. Indeed, we first assume that gij (t) evolves by the (unnormalized) R Ricci flow and choose the normalization factor R ψ = ψ(t) so that g˜ij = ψgij , and M d˜ µ = 1. Next we choose a new time scale t˜ = ψ(t)dt. Then for the normalized metric g˜ij we have ˜ ij = Rij , R ˜ = 1 R, r˜ = 1 r. R ψ ψ R R n Because M dV˜ = 1, we see that M dV = ψ − 2 . Then Z 2 d d log ψ = − log dV dt n dt M R ∂p det gij dx 2 M ∂tR = − n M dV 2 = r, n
308
H.-D. CAO AND X.-P. ZHU
since
∂ ∂t gij
= −2Rij for the Ricci flow. Hence it follows that ∂ ∂ d g˜ij = gij + log ψ gij ∂t dt ∂ t˜ 2 ˜ ij . = r˜g˜ij − 2R n
Thus studying the behavior of the Ricci flow near the maximal time is equivalent to studying the long-time behavior of the normalized Ricci flow. In this chapter we will obtain long-time behavior of the normalized Ricci flow for the following special cases: (1) compact two-manifolds; (2) compact three-manifolds with nonnegative Ricci curvature; (3) compact four-manifolds with nonnegative curvature operator; and (4) compact three-manifolds with uniformly bounded normalized curvature. 5.1. The Ricci Flow on Two-manifolds. Let M be a compact surface, we will discuss in this section the evolution of a Riemannian metric gij under the normalized Ricci flow. On a surface, the Ricci curvature is given by Rij =
1 Rgij 2
so the normalized Ricci flow equation becomes ∂ gij = (r − R)gij . ∂t
(5.1.1)
Recall the Gauss-Bonnet formula says Z RdV = 4πχ(M ), M
where χ(M ) is the REuler characteristic number of M . Thus the average scalar curvature r = 4πχ(M )/ M dV is constant in time. To obtain the evolution equation of the normalized curvature, we recall a simple principle in [58] for converting from the unnormalized to the normalized evolution equation on an n-dimensional manifold. Let P and Q be two expressions formed from ˜ be the corresponding expressions the metric and curvature tensors, and let P˜ and Q for the normalized Ricci flow. Since they differ by dilations, they differ by a power of the normalized factor ψ = ψ(t). We say P has degree k if P˜ = ψ k P . Thus gij has degree 1, Rij has degree 0, R has degree −1. Lemma 5.1.1. Suppose P satisfies
∂P = ∆P + Q ∂t for the unnormalized Ricci flow, and P has degree k. Then Q has degree k − 1, and for the normalized Ricci flow, ∂ P˜ ˜ P˜ + Q ˜ + 2 k˜ r P˜ . =∆ ˜ n ∂t ˜ Then Proof. We first see Q has degree k − 1 since ∂ t˜/∂t = ψ and ∆ = ψ ∆. ψ
∂ −k ˜ ˜ −k P˜ ) + ψ −k+1 Q ˜ (ψ P ) = ψ ∆(ψ ∂ t˜
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
309
which implies ∂ P˜ ˜ P˜ + Q ˜ + k ∂ψ P˜ =∆ ˜ ψ ∂ t˜ ∂t ˜ P˜ + Q ˜ + 2 k˜ rP˜ =∆ n since
∂ log ψ ∂ t˜
∂ = ( ∂t log ψ)ψ −1 =
2 ˜. nr
We now come back to the normalized Ricci flow (5.1.1) on a compact surface. By applying the above lemma to the evolution equation of unnormalized scalar curvature, we have (5.1.2)
∂R = ∆R + R2 − rR ∂t
for the normalized scalar curvature R. As a direct consequence, by using the maximum principle, both nonnegative scalar curvature and nonpositive scalar curvature are preserved for the normalized Ricci flow on surfaces. Let us introduce a potential function ϕ as in the K¨ahler-Ricci flow (see for example [11]). Since R − r has mean value zero on a compact surface, there exists a unique function ϕ, with mean value zero, such that ∆ϕ = R − r.
(5.1.3)
Differentiating (5.1.3) in time, we have ∂ ∂ R = (∆ϕ) ∂t ∂t = (R − r)∆ϕ + g ij = (R − r)∆ϕ + ∆
∂ ∂t
∂ϕ ∂t
∂2ϕ ∂ϕ − Γkij k i j ∂x ∂x ∂x
.
Combining with the equation (5.1.2), we get ∂ϕ ∆ = ∆(∆ϕ) + r∆ϕ ∂t which implies that ∂ϕ = ∆ϕ + rϕ − b(t) ∂t R for some function b(t) of time only. Since M ϕdV = 0 for all t, we have Z Z Z d 0= ϕdµ = (∆ϕ + rϕ − b(t))dµ + ϕ(r − R)dµ dt M M M Z Z = −b(t) dµ + |∇ϕ|2 dµ. (5.1.4)
M
M
Thus the function b(t) is given by b(t) =
R
|∇ϕ|2 dµ . dµ M
MR
310
H.-D. CAO AND X.-P. ZHU
Define a function h by h = ∆ϕ + |∇ϕ|2 = (R − r) + |∇ϕ|2 , and set 1 Mij = ∇i ∇j ϕ − ∆ ϕgij 2 to be the traceless part of ∇i ∇j ϕ. Lemma 5.1.2. The function h satisfies the evolution equation (5.1.5)
∂h = ∆h − 2|Mij |2 + rh. ∂t
Proof. Under the normalized Ricci flow, ∂ ∂ ij ∂ |∇ϕ|2 = g ∇i ϕ∇j ϕ + 2g ij ∇i ϕ (∇j ϕ) ∂t ∂t ∂t
= (R − r)|∇ϕ|2 + 2g ij ∇i (∆ϕ + rϕ − b(t))∇j ϕ = (R + r)|∇ϕ|2 + 2g ij (∆∇i ϕ − Rik ∇k ϕ)∇j ϕ
= (R + r)|∇ϕ|2 + ∆|∇ϕ|2 − 2|∇2 ϕ|2 − 2g ij Rik ∇k ϕ∇j ϕ
= ∆|∇ϕ|2 − 2|∇2 ϕ|2 + r|∇ϕ|2 ,
where Rik = 12 Rgik on a surface. On the other hand we may rewrite the evolution equation (5.1.2) as ∂ (R − r) = ∆(R − r) + (∆ϕ)2 + r(R − r). ∂t Then the combination of above two equations yields ∂ 1 h = ∆h − 2(|∇2 ϕ|2 − (∆ϕ)2 ) + rh ∂t 2 = ∆h − 2|Mij |2 + rh as desired. As a direct consequence of the evolution equation (5.1.5) and the maximum principle, we have R ≤ C1 ert + r
(5.1.6)
for some positive constant C1 depending only on the initial metric. On the other hand, it follows from (5.1.2) that Rmin (t) = minx∈M R(x, t) satisfies d Rmin ≥ Rmin (Rmin − r) ≥ 0 dt whenever Rmin ≤ 0. This says that (5.1.7)
Rmin (t) ≥ −C2 ,
for all t > 0
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
311
for some positive constant C2 depending only on the initial metric. Thus the combination of (5.1.6) and (5.1.7) implies the following long time existence result. Proposition 5.1.3. For any initial metric on a compact surface, the normalized Ricci flow (5.1.1) has a solution for all time. To investigate the long-time behavior of the solution, let us now divide the discussion into three cases: χ(M ) < 0; χ(M ) = 0; and χ(M ) > 0. Case (1): χ(M ) < 0 (i.e., r < 0). From the evolution equation (5.1.2), we have d Rmin ≥ Rmin (Rmin − r) dt ≥ r(Rmin − r), on M × [0, +∞) which implies that R − r ≥ −C˜1 ert , on M × [0, +∞) for some positive constant C˜1 depending only on the initial metric. Thus by combining with (5.1.6) we have (5.1.8)
−C˜1 ert ≤ R − r ≤ C1 ert , on M × [0, +∞).
Theorem 5.1.4 (Hamilton [60]). On a compact surface with χ(M ) < 0, for any initial metric the solution of the normalized Ricci flow (5.1.1) exists for all time and converges in the C ∞ topology to a metric with negative constant curvature. Proof. The estimate (5.1.8) shows that the scalar curvature R(x, t) converges exponentially to the negative constant r as t → +∞. Fix a tangent vector v ∈ Tx M at a point x ∈ M and let |v|2t = gij (x, t)v i v j . Then we have d 2 ∂ |v|t = gij (x, t) v i v j dt ∂t = (r − R)|v|2t
which implies d log |v|2t ≤ Cert , for all t > 0 dt
for some positive constant C depending only on the initial metric (by using (5.1.8)). Thus |v|2t converges uniformly to a continuous function |v|2∞ as t → +∞ and |v|2∞ 6= 0 if v 6= 0. Since the parallelogram law continues to hold to the limit, the limiting norm |v|2∞ comes from an inner product gij (∞). This says, the metrics gij (t) are all equivalent and as t → +∞, the metric gij (t) converges uniformly to a positive-definite metric tensor gij (∞) which is continuous and equivalent to the initial metric. By the virtue of Shi’s derivative estimates of the unnormalized Ricci flow in Section 1.4, we see that all derivatives and higher order derivatives of the curvature of the solution gij of the normalized flow are uniformly bounded on M × [0, +∞).
312
H.-D. CAO AND X.-P. ZHU
This shows that the limiting metric gij (∞) is a smooth metric with negative constant curvature and the solution gij (t) converges to the limiting metric gij (∞) in the C ∞ topology as t → +∞. Case (2): χ(M ) = 0, i.e., r = 0. From (5.1.6) and (5.1.7) we know that the curvature remains bounded above and below. To get the convergence, we consider the potential function ϕ of (5.1.3) again. The evolution of ϕ is given by (5.1.4). We renormalize the function ϕ by Z ϕ(x, ˜ t) = ϕ(x, t) + b(t)dt, on M × [0, +∞). Then, since r = 0, ϕ˜ evolves by ∂ ϕ˜ = ∆ϕ, ˜ ∂t
(5.1.9)
on M × [0, +∞).
From the proof of Lemma 5.1.2, we get ∂ |∇ϕ| ˜ 2 = ∆|∇ϕ| ˜ 2 − 2|∇2 ϕ| ˜ 2. ∂t
(5.1.10) Clearly, we have
∂ 2 ϕ˜ = ∆ϕ˜2 − 2|∇ϕ| ˜ 2. ∂t
(5.1.11) Thus it follows that
∂ (t|∇ϕ| ˜ 2 + ϕ˜2 ) ≤ ∆(t|∇ϕ| ˜ 2 + ϕ˜2 ). ∂t Hence by applying the maximum principle, there exists a positive constant C3 depending only on the initial metric such that (5.1.12)
|∇ϕ| ˜ 2 (x, t) ≤
C3 , 1+t
on M × [0, +∞).
In the following we will use this decay estimate to obtain a decay estimate for the scalar curvature. By the evolution equations (5.1.2) and (5.1.10), we have ∂ (R + 2|∇ϕ| ˜ 2 ) = ∆(R + 2|∇ϕ| ˜ 2 ) + R2 − 4|∇2 ϕ| ˜2 ∂t ≤ ∆(R + 2|∇ϕ| ˜ 2 ) − R2 since R2 = (∆ϕ) ˜ 2 ≤ 2|∇2 ϕ| ˜ 2 . Thus by using (5.1.12), we have ∂ [t(R + 2|∇ϕ| ˜ 2 )] ∂t ≤ ∆[t(R + 2|∇ϕ| ˜ 2 )] − tR2 + R + 2|∇ϕ| ˜2
≤ ∆[t(R + 2|∇ϕ| ˜ 2 )] − t(R + 2|∇ϕ| ˜ 2 )2 + (1 + 4t|∇ϕ| ˜ 2 )(R + 2|∇ϕ| ˜ 2) ≤ ∆[t(R + 2|∇ϕ| ˜ 2 )] − [t(R + 2|∇ϕ| ˜ 2 ) − (1 + 4C3 )](R + 2|∇ϕ| ˜ 2) ≤ ∆[t(R + 2|∇ϕ| ˜ 2 )]
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
313
wherever t(R + 2|∇ϕ| ˜ 2 ) ≥ (1 + 4C3 ). Hence by the maximum principle, there holds R + 2|∇ϕ| ˜2≤
(5.1.13)
C4 , 1+t
on M × [0, +∞)
for some positive constant C4 depending only on the initial metric. On the other hand, the scalar curvature satisfies ∂R = ∆R + R2 , ∂t
on M × [0, +∞).
It is not hard to see that R≥
(5.1.14)
Rmin (0) , 1 − Rmin (0)t
on M × [0, +∞),
by using the maximum principle. So we obtain the decay estimate for the scalar curvature |R(x, t)| ≤
(5.1.15)
C5 , 1+t
on M × [0, +∞),
for some positive constant C5 depending only on the initial metric. Theorem 5.1.5 (Hamilton [60]). On a compact surface with χ(M ) = 0, for any initial metric the solution of the normalized Ricci flow (5.1.1) exists for all time and converges in C ∞ topology to a flat metric. Proof. Since
∂ϕ ˜ ∂t
= ∆ϕ, ˜ it follows from the maximum principle that |ϕ(x, ˜ t)| ≤ C6 ,
on M × [0, +∞)
for some positive constant C6 depending only on the initial metric. Recall that ∆ϕ˜ = R. We thus obtain for any tangent vector v ∈ Tx M at a point x ∈ M , d 2 ∂ |v|t = gij (x, t) v i v j dt ∂t = −R(x, t)|v|2t
and then Z t 2 d log |v|t = log vt |2 dt| |v|20 0 dt Z t = R(x, t)dt 0
= |ϕ(x, ˜ t) − ϕ(x, ˜ 0)| ≤ 2C6 ,
for all x ∈ M and t ∈ [0, +∞). This shows that the solution gij (t) of the normalized Ricci flow are all equivalent. This gives us control of the diameter and injectivity radius. As before, by Shi’s derivative estimates of the unnormalized Ricci flow, all derivatives and higher order derivatives of the curvature of the solution gij of the normalized Ricci flow (5.1.1) are uniformly bounded on M × [0, +∞). By the virtue of Hamilton’s
314
H.-D. CAO AND X.-P. ZHU
compactness theorem (Theorem 4.1.5) we see that the solution gij (t) subsequentially converges in C ∞ topology. The decay estimate (5.1.15) implies that each limit must be a flat metric on M . Clearly, we will finish the proof if we can show that limit is unique. Note that the solution gij (t) is changing conformally under the Ricci flow (5.1.1) on surfaces. Thus each limit must be conformal to the initial metric, denoted by g¯ij . Let us denote gij (∞) = eu g¯ij to be a limiting metric. Since gij (∞) is flat, it is easy to compute ¯ − ∆u), ¯ 0 = e−u (R
on M,
¯ is the curvature of g¯ij and ∆ ¯ is the Laplacian in the metric g¯ij . The solution where R of Poission equation ¯ = R, ¯ ∆u
on M
is unique up to constant. Moreover the constant must be also uniquely determined since the area of the solution of the normalized Ricci flow (5.1.1) is constant in time. So the limit is unique and we complete the proof of Theorem 5.1.5. Case (3): χ(M ) > 0, i.e., r > 0. This is the most difficult case. There exist several proofs by now but, in contrast to the previous two cases, none of them depend only on the maximum principle type of argument. In fact, all the proofs rely on some combination of the maximum principle argument and certain integral estimate of the curvature. In the pioneer work [60], Hamilton introduced an integral quantity Z E= R log R dV, M
which he calls entropy, for the (normalized) Ricci flow on a surface M with positive curvature, and showed that the entropy is monotone decreasing under the flow. By combining this entropy estimate with the Harnack inequality for the curvature (Corollary 2.5.3), Hamilton obtained the uniform bound on the curvature of the normalized Ricci flow on M with positive curvature. Furthermore, he showed that the evolving metric converges to a shrinking Ricci soliton on M and that the shrinking Ricci soliton must be a round metric on the 2-sphere S2 . Subsequently, Chow [36] extended Hamilton’s work to the general case when the curvature may change signs. More precisely, he proved that given any initial metric on a compact surface M with χ(M ) > 0, the evolving metric under the (normalized) Ricci flow will have positive curvature after a finite time. Hence, when combined with Hamilton’s result, we know the evolving metric on M converges to the round metric on S2 . In the following we present a new argument by combining the Li-Yau-Hamilton inequality of the curvature with Perelman’s no local collapsing theorem I′ , as was done in the recent joint work of Bing-Long Chen and the authors [15] where they considered the K¨ahler-Ricci flow on higher dimensional K¨ahler manifolds of nonnegative holomorphic bisectional curvature (see [15] for more details). (There are also other proofs for Case (3) by Bartz-Struwe-Ye [6] and Struwe [121].) Given any initial metric on M with χ(M ) > 0, we consider the solution gij (t) of the normalized Ricci flow (5.1.1). Recall that the (scalar) curvature R satisfies the evolution equation ∂ R = ∆R + R2 − rR. ∂t
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
315
The corresponding ODE is ds = s2 − rs. dt
(5.1.16)
Let us choose c > 1 and close to 1 so that r/(1 − c) < minx∈M R(x, 0). It is clear that the function s(t) = r/(1 − cert ) < 0 is a solution of the ODE (5.1.16) with s(0) < min R(x, 0). Then the difference of R and s evolves by x∈M
∂ (R − s) = ∆(R − s) + (R − r + s)(R − s). ∂t
(5.1.17)
Since minx∈M R(x, 0) − s(0) > 0, the maximum principle implies that R − s > 0 for all times. We first extend the Li-Yau-Hamilton inequality (Theorem 2.5.2) to the normalized Ricci flow whose curvature may change signs. As in the proof of Theorem 2.5.2, we consider the quantity L = log(R − s). It is easy to compute ∂L = ∆L + |∇L|2 + R − r + s. ∂t Then we set Q=
∂L − |∇L|2 − s = ∆L + R − r. ∂t
By a direct computation and using the estimate (5.1.8), we have ∂ Q=∆ ∂t
∂L ∂t
+ (R − r)∆L +
∂R ∂t
= ∆Q + 2|∇2 L|2 + 2h∇L, ∇(∆L)i + R|∇L|2 + (R − r)∆L + ∆R + R(R − r)
= ∆Q + 2|∇2 L|2 + 2h∇L, ∇Qi + 2(R − r)∆L + (R − r)2 + (r − s)∆L + s|∇L|2 + r(R − r)
= ∆Q + 2h∇L, ∇Qi + 2|∇2 L|2 + 2(R − r)∆L + (R − r)2 + (r − s)Q + s|∇L|2 + s(R − r)
≥ ∆Q + 2h∇L, ∇Qi + Q2 + (r − s)Q + s|∇L|2 − C. Here and below C is denoted by various positive constants depending only on the initial metric. In order to control the bad term s|∇L|2 , we consider ∂ (sL) = ∆(sL) + s|∇L|2 + s(R − r + s) + s(s − r)L ∂t ≥ ∆(sL) + 2h∇L, ∇(sL)i − s|∇L|2 − C
316
H.-D. CAO AND X.-P. ZHU
by using the estimate (5.1.8) again. Thus ∂ (Q + sL) ≥ ∆(Q + sL) + 2h∇L, ∇(Q + sL)i + Q2 + (r − s)Q − C ∂t 1 ≥ ∆(Q + sL) + 2h∇L, ∇(Q + sL)i + [(Q + sL)2 − C 2 ], 2 since sL is bounded by (5.1.8). This, by the maximum principle, implies that Q ≥ −C,
for all t ∈ [0, +∞).
Then for any two points x1 , x2 ∈ M and two times t2 > t1 ≥ 0, and a path γ : [t1 , t2 ] → M connecting x1 to x2 , we have Z
t2
d L(γ(t), t)dt dt t1 Z t2 ∂L + h∇L, γi ˙ dt = ∂t t1 1 ≥ − ∆ − C(t2 − t1 ) 4
L(x2 , t2 ) − L(x1 , t1 ) =
where ∆ = ∆(x1 , t1 ; x2 , t2 ) Z t2 2 = inf |γ(t)| ˙ gij (t) dt | γ : [t1 , t2 ] → M with γ(t1 ) = x1 , γ(t2 ) = x2 . t1
Thus we have proved the following Harnack inequality. Lemma 5.1.6 (Chow [36]). There exists a positive constant C depending only on the initial metric such that for any x1 , x2 ∈ M and t2 > t1 ≥ 0, ∆
R(x1 , t1 ) − s(t1 ) ≤ e 4 +C(t2 −t1 ) (R(x2 , t2 ) − s(t2 )) where ∆ = inf
Z
t2
t1
2 |γ(t)| ˙ t dt
| γ : [t1 , t2 ] → M with γ(t1 ) = x1 , γ(t2 ) = x2 .
We now state and prove the uniform bound estimate for the curvature. Proposition 5.1.7. Let (M, gij (t)) be a solution of the normalized Ricci flow on a compact surface with χ(M ) > 0. Then there exist a time t0 > 0 and a positive constant C such that the estimate C −1 ≤ R(x, t) ≤ C holds for all x ∈ M and t ∈ [t0 , +∞). Proof. Recall that R(x, t) ≥ s(t) =
r , 1 − cert
on M × [0, +∞).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
317
For any ε ∈ (0, r), there exists a large enough t0 > 0 such that R(x, t) ≥ −ε2 ,
(5.1.18)
on M × [t0 , +∞).
Let t be any fixed time with t ≥ t0 + 1. Obviously there is some point x0 ∈ M such that R(x0 , t + 1) = r. Consider the geodesic ball Bt (x0 , 1), centered at x0 and radius 1 with respect to the metric at the fixed time t. For any point x ∈ Bt (x0 , 1), we choose a geodesic γ: [t, t + 1] → M connecting x and x0 with respect to the metric at the fixed time t. Since ∂ gij = (r − R)gij ≤ 2rgij ∂t we have
Z
t+1 t
|γ(τ ˙ )|2τ dτ
≤e
2r
Z
on M × [t0 , +∞), t+1
t
|γ(τ ˙ )|2t dτ ≤ e2r .
Then by Lemma 5.1.6, we have (5.1.19)
R(x, t) ≤ s(t) + exp ≤ C1 ,
1 2r e +C 4
· (R(x0 , t + 1) − s(t + 1))
as x ∈ Bt (x0 , 1),
for some positive constant C1 depending only on the initial metric. Note the the corresponding unnormalized Ricci flow in this case has finite maximal time since its volume decreases at a fixed rate −4πχ(M ) < 0. Hence the no local collapsing theorem I′ (Theorem 3.3.3) implies that the volume of Bt (x0 , 1) with respect to the metric at the fixed time t is bounded from below by Vol t (Bt (x0 , 1)) ≥ C2
(5.1.20)
for some positive constant C2 depending only on the initial metric. We now want to bound the diameter of (M, gij (t)) from above. The following argument is analogous to Yau in [128] where he got a lower bound for the volume of geodesic balls of a complete Riemannian manifold with nonnegative Ricci curvature. Without loss of generality, we may assume that the diameter of (M, gij (t)) is at least 3. Choose a point x1 ∈ M such that the distance dt (x0 , x1 ) between x1 and x0 with respect to the metric at the fixed time t is at least a half of the diameter of (M, gij (t)). By (5.1.18), the standard Laplacian comparison theorem (c.f. [112]) implies ∆ρ2 = 2ρ∆ρ + 2 ≤ 2(1 + ερ) + 2 in the sense of distribution, where ρ is the distance function from x1 (with respect to the metric gij (t)). That is, for any ϕ ∈ C0∞ (M ), ϕ ≥ 0, we have Z Z (5.1.21) − ∇ρ2 · ∇ϕ ≤ [2(1 + ερ) + 2]ϕ. M
C0∞ (M )
M
Since functions can be approximated by Lipschitz functions in the above inequality, we can set ϕ(x) = ψ(ρ(x)), x ∈ M , where ψ(s) is given by 0 ≤ s ≤ dt (x0 , x1 ) − 1, 1, 1 ′ ψ(s) = ψ (s) = − 2 , dt (x0 , x1 ) − 1 ≤ s ≤ dt (x0 , x1 ) + 1, 0, s ≥ dt (x0 , x1 ) + 1.
318
H.-D. CAO AND X.-P. ZHU
Thus, by using (5.1.20), the left hand side of (5.1.21) is Z − ∇ρ2 · ∇ϕ M Z = ρ Bt (x1 ,dt (x0 ,x1 )+1)\Bt (x1 ,dt (x0 ,x1 )−1)
≥ (dt (x0 , x1 ) − 1)Vol t (Bt (x1 , dt (x0 , x1 ) + 1) \ Bt (x1 , dt (x0 , x1 ) − 1))
≥ (dt (x0 , x1 ) − 1)Vol t (Bt (x0 , 1)) ≥ (dt (x0 , x1 ) − 1)C2 , and the right hand side of (5.1.21) is Z Z [2(1 + ερ) + 2]ϕ ≤ M
[2(1 + ερ) + 2]
Bt (x1 ,dt (x0 ,x1 )+1)
≤ [2(1 + εdt (x0 , x1 )) + 4]Vol t (Bt (x1 , dt (x0 , x1 ) + 1)) ≤ [2(1 + εdt (x0 , x1 )) + 4]A
where A is the area of M with respect to the initial metric. Here we have used the fact that the area of solution of the normalized Ricci flow is constant in time. Hence C2 (dt (x0 , x1 ) − 1) ≤ [2(1 + εdt (x0 , x1 )) + 4]A, which implies, by choosing ε > 0 small enough, dt (x0 , x1 ) ≤ C3 for some positive constant C3 depending only on the initial metric. Therefore, the diameter of (M, gij (t)) is uniformly bounded above by (5.1.22)
diam (M, gij (t)) ≤ 2C3
for all t ∈ [t0 , +∞). We then argue, as in deriving (5.1.19), by applying Lemma 5.1.6 again to obtain R(x, t) ≤ C4 ,
on M × [t0 , +∞)
for some positive constant C4 depending only on the initial metric. It remains to prove a positive lower bound estimate of the curvature. First, we note that the function s(t) → 0 as t → +∞, and the average scalar curvature of the solution equals to r, a positive constant. Thus the Harnack inequality in Lemma 5.1.6 and the diameter estimate (5.1.22) imply a positive lower bound for the curvature. Therefore we have completed the proof of Proposition 5.1.7. Next we consider long-time convergence of the normalized flow. Recall that the trace-free part of the Hessian of the potential ϕ of the curvature is the tensor Mij defined by 1 Mij = ∇i ∇j ϕ − ∆ϕ · gij , 2 where by (5.1.3), ∆ϕ = R − r.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
Lemma 5.1.8. We have ∂ |Mij |2 = ∆|Mij |2 − 2|∇k Mij |2 − 2R|Mij |2 , on M × [0, +∞). ∂t
(5.1.23)
Proof. First we note the time-derivative of the Levi-Civita connection is ∂ 1 ∂ ∂ ∂ k Γij = g kl ∇i gjl + ∇j gil − ∇l gij ∂t 2 ∂t ∂t ∂t 1 −∇i R · δjk − ∇j R · δik + ∇k R · gij . = 2
By using this and (5.1.4), we have ∂ ∂ϕ ∂ k 1 ∂ Mij = ∇i ∇j − Γij ∇k ϕ − [(R − r)gij ] ∂t ∂t ∂t 2 ∂t 1 = ∇i ∇j ∆ϕ + (∇i R · ∇j ϕ + ∇j R · ∇i ϕ − h∇R, ∇ϕigij ) 2 1 − ∆R · gij + rMij . 2 Since on a surface, Rijkl =
1 R(gil gjk − gik gjl ), 2
we have ∇i ∇j ∆ϕ
= ∇i ∇k ∇j ∇k ϕ − ∇i (Rjl ∇l ϕ)
l = ∇k ∇i ∇j ∇k ϕ − Rikj ∇l ∇k ϕ − Ril ∇j ∇l ϕ − Rjl ∇i ∇l ϕ − ∇i Rjl ∇l ϕ
l l = ∆∇i ∇j ϕ − ∇k (Rikj ∇l ϕ) − Rikj ∇l ∇k ϕ
− Ril ∇j ∇l ϕ − Rjl ∇i ∇l ϕ − ∇i Rjl ∇l ϕ 1 = ∆∇i ∇j ϕ − (∇i R · ∇j ϕ + ∇j R · ∇i ϕ − h∇R, ∇ϕigij ) 2 1 − 2R ∇i ∇j ϕ − ∆ϕ · gij . 2
Combining these identities, we get 1 ∂ Mij = ∆∇i ∇j ϕ − ∆R · gij + (r − 2R)Mij ∂t 2 1 = ∆ ∇i ∇j ϕ − (R − r)gij + (r − 2R)Mij . 2 Thus the evolution Mij is given by (5.1.24)
∂Mij = ∆Mij + (r − 2R)Mij . ∂t
Now the lemma follows from (5.1.24) and a straightforward computation.
319
320
H.-D. CAO AND X.-P. ZHU
Proposition 5.1.7 tells us that the curvature R of the solution to the normalized Ricci flow is uniformly bounded from below by a positive constant for t large. Thus we can apply the maximum principle to the equation (5.1.23) in Lemma 5.1.8 to obtain the following estimate. Proposition 5.1.9. Let (M, gij (t)) be a solution of the normalized Ricci flow on a compact surface with χ(M ) > 0. Then there exist positive constants c and C depending only on the initial metric such that |Mij |2 ≤ Ce−ct ,
on M × [0, +∞).
Now we consider a modification of the normalized Ricci flow. Consider the equation ∂ gij = 2Mij = (r − R)gij + 2∇i ∇j ϕ. ∂t
(5.1.25)
As we saw in Section 1.3, the solution of this modified flow differs from that of the normalized Ricci flow only by a one parameter family of diffeomorphisms generated by the gradient vector field of the potential function ϕ. Since the quantity |Mij |2 is invariant under diffeomorphisms, the estimate |Mij |2 ≤ Ce−ct also holds for the solution of the modified flow (5.1.25). This exponential decay estimate then implies the solution gij (x, t) of the modified flow (5.1.25) converges exponentially to a continuous metric gij (∞) as t → +∞. Furthermore, by the virtue of Hamilton’s compactness theorem (Theorem 4.1.5) we see that the solution gij (x, t) of the modified flow actually converges exponentially in C ∞ topology to gij (∞). Moreover the limiting metric gij (∞) satisfies Mij = (r − R)gij + 2∇i ∇j ϕ = 0,
on M.
That is, the limiting metric is a shrinking gradient Ricci soliton on the surface M . The next result was first obtained by Hamilton in [60]. The following simplified proof by using the Kazdan-Warner identity was widely known to experts in the field. Proposition 5.1.10. On a compact surface there are no shrinking Ricci solitons other than constant curvature. Proof. By definition, a shrinking Ricci soliton on a compact surface M is given by ∇i Xj + ∇j Xi = (R − r)gij
(5.1.26)
for some vector field X = Xj . By contracting the above equation by Rg −1 , we have 2R(R − r) = 2R div X, and hence
Z
M
(R − r)2 dV =
Z
M
R(R − r)dV =
Z
R div XdV.
M
Since X is a conformal vector field (by the Ricci soliton equation (5.1.26)), by integrating by parts and applying the Kazdan-Warner identity [77], we obtain Z Z (R − r)2 dV = − h∇R, XidV = 0. M
M
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
321
Hence R ≡ r, and the lemma is proved. Now back to the solution of the modified flow (5.1.25). We have seen the curvature converges exponentially to its limiting value in the C ∞ topology. But since there are no nontrivial soliton on M , we must have R converging exponentially to the constant value r in the C ∞ topology. This then implies that the unmodified flow (5.1.1) will converge to a metric of positive constant curvature in the C ∞ topology. In conclusion, we have proved the following main theorem of this section. Theorem 5.1.11 (Hamilton [60], Chow [36]). On a compact surface with χ(M ) > 0, for any initial metric, the solution of the normalized Ricci flow (5.1.1) exists for all time, and converges in the C ∞ topology to a metric with positive constant curvature. 5.2. Differentiable Sphere Theorems in 3-D and 4-D. An important problem in Riemannian geometry is to understand the influence of curvatures, in particular the sign of curvatures, on the topology of underlying manifolds. Classical results of this type include sphere theorem and its refinements stated below (see, for example, Theorem 6.1 and Theorem 7.16 in Cheeger-Ebin [22], and Theorem 6.6 in CheegerEbin [22]). In this section we shall use the long-time behavior of the Ricci flow on positively curved manifolds to establish Hamilton’s differentiable sphere theorems in dimensions three and four. Let us first recall the classical sphere theorems. Given a Remannian manifold M , we denote by KM the sectional curvature of M . Classical Sphere Theorems. Let M be a complete, simply connected ndimensional manifold. (i) If 14 < KM ≤ 1, then M is homeomorphic to the n-sphere Sn . (ii) There exists a positive constant δ ∈ ( 14 , 1) such that if δ < KM ≤ 1, then M is diffeomorphic to the n-sphere Sn . Result (ii) is called the differentiable sphere theorem. If we relax the assumptions on the strict lower bound in (i), then we have the following rigidity result. Berger’s Rigidity Theorem. Let M be a complete, simply connected ndimensional manifold with 41 ≤ KM ≤ 1. Then either M is homeomorphic to Sn or M is isometric to a symmetric space. We remark that it follows from the classification of symmetric spaces (see for example [68]) thatn the only simply connected symmetric spaces with positive curvature n are Sn , CP 2 , QP 4 , and the Cayley plane. In early and mid 80’s respectively, Hamilton [58], [59] used the Ricci flow to prove the following differential sphere theorems. Theorem 5.2.1 (Hamilton [58]). A compact three-manifold with positive Ricci curvature must be diffeomorphic to the three-sphere S3 or a quotient of it by a finite group of fixed point free isometries in the standard metric. Theorem 5.2.2 (Hamilton [59]). A compact four-manifold with positive curvature operator is diffeomorphic to the four-sphere S4 or the real projective space RP4 . Note that in above two theorems, we only assume curvatures to be strictly positive, but not any strong pinching conditions as in the classical sphere theorems. In fact, one of the important special features discovered by Hamilton is that if the initial metric has positive curvature, then the metric will get rounder and rounder as it
322
H.-D. CAO AND X.-P. ZHU
evolves under the Ricci flow, at least in dimension three and four, so any small initial pinching will get improved. Indeed, the pinching estimate is a key step in proving both Theorem 5.2.1 and 5.2.2. The following results are concerned with compact three-manifolds or fourmanifolds with weakly positive curvatures. Theorem 5.2.3 (Hamilton [59]). (i) A compact three-manifold with nonnegative Ricci curvature is diffeomorphic to S3 , or a quotient of one of the spaces S3 or S2 × R1 or R3 by a group of fixed point free isometries in the standard metrics. (ii) A compact four-manifold with nonnegative curvature operator is diffeomorphic to S4 or CP2 or S2 × S2 , or a quotient of one of the spaces S4 or CP2 or S3 × R1 or S2 × S2 or S2 × R2 or R4 by a group of fixed point free isometries in the standard metrics. The rest of the section will be devoted to prove Theorems 5.2.1-5.2.3. Recall that the curvature operator Mαβ evolves by (5.2.1)
∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ . ∂t
2 where (see Section 1.3 and Section 2.4) Mαβ is the operator square 2 Mαβ = Mαγ Mβγ # and Mαβ is the Lie algebra so(n) square # Mαβ = Cαγζ Cβηθ Mγη Mζθ .
We begin with the curvature pinching estimates of the Ricci flow in three dimen# sions. In dimension n = 3, we know that Mαβ is the adjoint matrix of Mαβ . If we diagonalize Mαβ with eigenvalues λ ≥ µ ≥ ν so that λ , µ (Mαβ ) = ν # 2 then Mαβ and Mαβ are also diagonal, with 2 λ µν 2 and (M # ) = µ2 (Mαβ )= αβ ν2
λν λµ
,
and the ODE corresponding to PDE (5.2.1) is then given by the system d 2 dt λ = λ + µν, d 2 (5.2.2) dt µ = µ + λν, d 2 dt ν = ν + λµ. Lemma 5.2.4. For any ε ∈ [0, 13 ], the pinching condition Rij ≥ 0
and
Rij ≥ εRgij
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
323
is preserved by the Ricci flow. Proof. If we diagonalize the 3 × 3 curvature operator matrix Mαβ with eigenvalues λ ≥ µ ≥ ν, then nonnegative sectional curvature corresponds to ν ≥ 0 and nonnegative Ricci curvature corresponds to the inequality µ + ν ≥ 0. Also, the scalar curvature R = λ + µ + ν. So we need to show µ+ν ≥ 0
and µ + ν ≥ δλ, with δ = 2ε/(1 − 2ε),
are preserved by the Ricci flow. By Hamilton’s advanced maximum principle (Theorem 2.3.1), it suffices to show that the closed convex set K = {Mαβ | µ + ν ≥ 0 and µ + ν ≥ δλ} is preserved by the ODE system (5.2.2). Now suppose we have diagonalized Mαβ with eigenvalues λ ≥ µ ≥ ν at t = 0, # 2 then both Mαβ and Mαβ are diagonal, so the matrix Mαβ remains diagonal for t > 0. Moreover, since d (µ − ν) = (µ − ν)(µ + ν − λ), dt it is clear that µ ≥ ν for t > 0 also. Similarly, we have λ ≥ µ for t > 0. Hence the inequalities λ ≥ µ ≥ ν persist. This says that the solutions of the ODE system (5.2.2) agree with the original choice for the eigenvalues of the curvature operator. The condition µ + ν ≥ 0 is clearly preserved by the ODE, because d (µ + ν) = µ2 + ν 2 + λ(µ + ν) ≥ 0. dt It remains to check d d (µ + ν) ≥ δ λ dt dt or µ2 + λν + ν 2 + λµ ≥ δ(λ2 + µν) on the boundary where µ + ν = δλ ≥ 0. In fact, since (λ − ν)µ2 + (λ − µ)ν 2 ≥ 0, we have λ(µ2 + ν 2 ) ≥ (µ + ν)µν. Hence 2
2
µ + µλ + ν + νλ ≥
µ+ν λ
(λ2 + µν)
= δ(λ2 + µν).
324
H.-D. CAO AND X.-P. ZHU
So we get the desired pinching estimate. Proposition 5.2.5. Suppose that the initial metric of the solution to the Ricci flow on M 3 × [0, T ) has positive Ricci curvature. Then for any ε > 0 we can find Cε < +∞ such that Rij − 1 Rgij ≤ εR + Cε 3 for all subsequent t ∈ [0, T ).
Proof. Again we consider the ODE system (5.2.2). Let Mαβ be diagonalized with eigenvalues λ ≥ µ ≥ ν at t = 0. We saw in the proof of Lemma 5.2.4 the inequalities λ ≥ µ ≥ ν persist for t > 0. We only need to show that there are positive constants δ and C such that the closed convex set K = { Mαβ | λ − ν ≤ C(λ + µ + ν)1−δ } is preserved by the ODE. We compute d (λ − ν) = (λ − ν)(λ + ν − µ) dt and d (λ + µ + ν) = (λ + µ + ν)(λ + ν − µ) + µ2 dt + µ(µ + ν) + λ(µ − ν) ≥ (λ + µ + ν)(λ + ν − µ) + µ2 . Thus, without loss of generality, we may assume λ − ν > 0 and get d log(λ − ν) = λ + ν − µ dt and µ2 d log(λ + µ + ν) ≥ λ + ν − µ + . dt λ+µ+ν By Lemma 5.2.4, there exists a positive constant C depending only on the initial metric such that λ ≤ λ + µ ≤ C(µ + ν) ≤ 2Cµ, λ + ν − µ ≤ λ + µ + ν ≤ 6Cµ, and hence with ǫ = 1/36C 2, d log(λ + µ + ν) ≥ (1 + ǫ)(λ + ν − µ). dt
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
325
Therefore with (1 − δ) = 1/(1 + ǫ), d log((λ − ν)/(λ + µ + ν)1−δ ) ≤ 0. dt This proves the proposition. We now are ready to prove Theorem 5.2.1. Proof of Theorem 5.2.1. Let M be a compact three-manifold with positive Ricci curvature and let the metric evolve by the Ricci flow. By Lemma 5.2.4 we know that there exists a positive constant β > 0 such that Rij ≥ βRgij for all t ≥ 0 as long as the solution exists. The scalar curvature evolves by ∂R = ∆R + 2|Rij |2 ∂t 2 ≥ ∆R + R2 , 3 which implies, by the maximum principle, that the scalar curvature remains positive and tends to +∞ in finite time. We now use a blow up argument as in Section 4.3 to get the following gradient estimate. Claim. For any ε > 0, there exists a positive constant Cε < +∞ such that for any time τ ≥ 0, we have 3
max max |∇Rm(x, t)| ≤ ε max max |Rm(x, t)| 2 + Cε . t≤τ x∈M
t≤τ x∈M
We argue by contradiction. Suppose the above gradient estimate fails for some fixed ε0 > 0. Pick a sequence Cj → +∞, and pick points xj ∈ M and times τj such that 3
|∇Rm(xj , τj )| ≥ ε0 max max |Rm(x, t)| 2 + Cj , t≤τj x∈M
j = 1, 2, . . . .
Choose xj to be the origin, and pull the metric back to a small ball on the tangent space Txj M of radius rj proportional to the reciprocal of the square root of the maximum curvature up to time τj (i.e., maxt≤τj maxx∈M |Rm(x, t)|). Clearly the maximum curvatures go to infinity by Shi’s derivative estimate of curvature (Theorem 1.4.1). Dilate the metrics so that the maximum curvature max max |Rm(x, t)| t≤τj x∈M
becomes 1 and translate time so that τj becomes the time 0. By Theorem 4.1.5, we can take a (local) limit. The limit metric satisfies |∇Rm(0, 0)| ≥ ε0 > 0. However the pinching estimate in Proposition 5.2.5 tells us the limit metric has 1 Rij − Rgij ≡ 0. 3
326
H.-D. CAO AND X.-P. ZHU
By using the contracted second Bianchi identity 1 1 1 ∇i R = ∇j Rij = ∇j Rij − Rgij + ∇i R, 2 3 3 we get ∇i R ≡ 0
and then
∇i Rjk ≡ 0.
For a three-manifold, this in turn implies ∇Rm = 0 which is a contradiction. Hence we have proved the gradient estimate claimed. We can now show that the solution to the Ricci flow becomes round as the time t tends to the maximal time T . We have seen that the scalar curvature goes to infinity in finite time. Pick a sequence of points xj ∈ M and times τj where the curvature at xj is as large as it has been anywhere for 0 ≤ t ≤ τj and τj tends to the maximal time. Since |∇Rm| is very small compared to |Rm(xj , τj )| by the above gradient estimate and |Rij − 31 Rgij | is also very small compared to |Rm(xj , τj )| by Proposition 5.2.5, the curvature is nearly constant and positive in a large ball around xj at the time τj . But then the Bonnet-Myers’ theorem tells us this is the whole manifold. For j large enough, the sectional curvature of the solution at the time τj is sufficiently pinched. Then it follows from the Klingenberg injectivity radius estimate (see Section p 4.2) that the injectivity radius of the metric at time τj is bounded from below by c/ |Rm(xj , τj )| for some positive constant c independent of j. Dilate the metrics so that the maximum curvature |Rm(xj , τj )| = maxt≤τj maxx∈M |Rm(x, t)| becomes 1 and shift the time τj to the new time 0. Then we can apply Hamilton’s compactness theorem (Theorem 4.1.5) to take a limit. By the pinching estimate in Proposition 5.2.5, we know that the limit has positive constant curvature which is either the round S3 or a metric quotient of the round S3 . Consequently, the compact three-manifold M is diffeomorphic to the round S3 or a metric quotient of the round S3 . Next we consider the pinching estimates of the Ricci flow on a compact fourmanifold M with positive curvature operator. In dimension 4, we saw in Section 1.3 when we decompose orthogonally Λ2 = 2 Λ+ ⊕ Λ2− into the eigenspaces of Hodge star with eigenvalue ±1, we have a block decomposition of Mαβ as A B Mαβ = t B C and then # Mαβ =2
A# t # B
B# C#
where A# , B # , C # are the adjoints of 3 × 3 submatrices as before. Thus the ODE d # 2 Mαβ = Mαβ + Mαβ dt
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
327
corresponding to the PDE (5.2.1) breaks up into the system of three equations d 2 t # dt A = A + B B + 2A , d # (5.2.3) dt B = AB + BC + 2B , d 2 t # dt C = C + BB + 2C .
As shown in Section 1.3, by the Bianchi identity, we know that tr A = tr C. For the symmetric matrices A and C, we can choose an orthonormal basis x1 , x2 , x3 of Λ2+ such that a1 0 0 A = 0 a2 0 , 0
0
a3
and an orthonormal basis z1 , z2 , z3 of Λ2− such that
c1
C= 0 0
0 c2 0
0
0 .
c3
For matrix B, we can choose orthonormal basis y1+ , y2+ , y3+ of Λ2+ and y1− , y2− , y3− of Λ2− such that
b1
B= 0 0
0 b2 0
0
0 .
b3
with 0 ≤ b1 ≤ b2 ≤ b3 . We may also arrange the eigenvalues of A and C as a1 ≤ a2 ≤ a3 and c1 ≤ c2 ≤ c3 . In view of the advanced maximum principle Theorem 2.3.1, we only need to establish the pinching estimates for the ODE (5.2.3). Note that a1 = inf{A(x, x) | x ∈ Λ2+ and |x| = 1},
a3 = sup{A(x, x) | x ∈ Λ2+ and |x| = 1}, c1 = inf{C(z, z) | z ∈ Λ2− and |z| = 1},
c3 = sup{C(z, z) | z ∈ Λ2− and |z| = 1}.
We can compute their derivatives by Lemma 2.3.3 as follows: d 2 2 dt a1 ≥ a1 + b1 + 2a2 a3 , d dt a3 ≤ a23 + b23 + 2a1 a2 , (5.2.4) d 2 2 dt c1 ≥ c1 + b1 + 2c2 c3 , d 2 2 dt c3 ≤ c3 + b3 + 2c1 c2 .
328
H.-D. CAO AND X.-P. ZHU
We shall make the pinching estimates by using the functions b2 + b3 and a − 2b + c, where a = a1 + a2 + a3 = c = c1 + c2 + c3 and b = b1 + b2 + b3 . Since b2 + b3 = B(y2+ , y2− ) + B(y3+ , y3− ) = sup{B(y + , y − ) + B(˜ y + , y˜− ) | y + , y˜+ ∈ Λ2+ with |y + | = |˜ y + | = 1, y + ⊥˜ y + , and y − , y˜− ∈ Λ2− with |y − | = |˜ y − | = 1, y − ⊥˜ y − },
We compute by Lemma 2.3.3, (5.2.5)
d d d (b2 + b3 ) ≤ B(y2+ , y2− ) + B(y3+ , y3− ) dt dt dt = AB(y2+ , y2− ) + BC(y2+ , y2− ) + 2B # (y2+ , y2− ) + AB(y3+ , y3− ) + BC(y3+ , y3− ) + 2B # (y3+ , y3− ) = b2 A(y2+ , y2+ ) + b2 C(y2− , y2− ) + 2b1 b3 + b3 A(y3+ , y3+ ) + b3 C(y3− , y3− ) + 2b1 b2 ≤ a2 b2 + a3 b3 + b2 c2 + b3 c3 + 2b1 b2 + 2b1 b3 ,
where we used the facts that A(y2+ , y2+ ) + A(y3+ , y3+ ) ≤ a2 + a3 and C(y2− , y2− ) + C(y3− , y3− ) ≤ c2 + c3 . Note also that the function a = trA = c = trC is linear, and the function b is given by
Indeed,
b = B(y1+ , y1− ) + B(y2+ , y2− ) + B(y3+ , y3− ) n = sup B(T y1+ , T˜y1− ) + B(T y2+ , T˜y2− ) + B(T y3+ , T˜y3− ) | T, T˜ are o othogonal transformations of Λ2+ and Λ2− respectively . B(T y1+ , T˜y1− ) + B(T y2+ , T˜y2− ) + B(T y3+ , T˜y3− ) = B(y + , T −1 T˜(y − )) + B(y + , T −1 T˜(y − )) + B(y + , T −1 T˜(y − )) 1
1
2
2
3
3
= b1 t11 + b2 t22 + b3 t33 where t11 , t22 , t33 are diagonal elements of the orthogonal matrix T −1 T˜ with t11 , t22 , t33 ≤ 1. Thus by using Lemma 2.3.3 again, we compute d d d d (a − 2b + c) ≥ tr A−2 B+ C dt dt dt dt = tr ((A − B)2 + (C − B)2 + 2(A# − 2B # + C # ))
evaluated in those coordinates where B is diagonal as above. Recalling the definition of Lie algebra product P #Q =
1 εαβγ εζηθ Pβη Qγθ 2
with εαβγ being the permutation tensor, we see that the Lie algebra product # gives
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
329
a symmetric bilinear operation on matrices, and then tr (2(A# − 2B # + C # ))
= tr ((A − C)# + (A + 2B + C)#(A − 2B + C)) 1 1 = − tr (A − C)2 + (tr (A − C))2 2 2 + tr ((A + 2B + C)#(A − 2B + C)) 1 = − tr (A − C)2 + tr ((A + 2B + C)#(A − 2B + C)) 2
by the Bianchi identity. It is easy to check that 1 1 tr (A − B)2 + tr (C − B)2 − tr (A − C)2 = tr (A − 2B + C)2 ≥ 0. 2 2 Thus we obtain d (a − 2b + c) ≥ tr ((A + 2B + C)#(A − 2B + C)) dt Since Mαβ ≥ 0 and
Mαβ =
A
B
t
C
B
,
we see that A + 2B + C ≥ 0 and A − 2B + C ≥ 0, by applying Mαβ to the vectors (x, x) and (x, −x). It is then not hard to see tr ((A + 2B + C)#(A − 2B + C)) ≥ (a1 + 2b1 + c1 )(a − 2b + c). Hence we obtain (5.2.6)
d (a − 2b + c) ≥ (a1 + 2b1 + c1 )(a − 2b + c). dt
We now state and prove the following pinching estimates of Hamilton for the associated ODE (5.2.3). Proposition 5.2.6 (Hamilton [59]). If we choose successively positive constants G large enough, H large enough, δ small enough, J large enough, ε small enough, K large enough, θ small enough, and L large enough, with each depending on those chosen before, then the closed convex subset X of {Mαβ ≥ 0} defined by the inequalities (1) (b2 + b3 )2 ≤ Ga1 c1 , (2) a3 ≤ Ha1 and c3 ≤ Hc1 , (3) (b2 + b3 )2+δ ≤ Ja1 c1 (a − 2b + c)δ , (4) (b2 + b3 )2+ε ≤ Ka1 c1 , (5) a3 ≤ a1 + La1−θ and c3 ≤ c1 + Lc1−θ , 1 1 is preserved by ODE (5.2.3). Moreover every compact subset of {Mαβ > 0} lies in some such set X. Proof. Clearly the subset X is closed and convex. We first note that we may assume b2 + b3 > 0 because if b2 + b3 = 0, then from (5.2.5), b2 + b3 will remain
330
H.-D. CAO AND X.-P. ZHU
zero and then the inequalities (1), (3) and (4) concerning b2 + b3 are automatically satisfied. Likewise we may assume a3 > 0 and c3 > 0 from (5.2.4). Let G be a fixed positive constant. To prove the inequality (1) we only need to check a 1 c1 d log ≥0 dt (b2 + b3 )2
(5.2.7)
whenever (b2 + b3 )2 = Ga1 c1 and b2 + b3 > 0. Indeed, it follows from (5.2.4) and (5.2.5) that d (a1 − b1 )2 a3 log a1 ≥ 2b1 + 2a3 + + 2 (a2 − a1 ), dt a1 a1 (c1 − b1 )2 c3 d log c1 ≥ 2b1 + 2c3 + + 2 (c2 − c1 ), dt c1 c1
(5.2.8) (5.2.9) and (5.2.10)
b2 d log(b2 + b3 ) ≤ 2b1 + a3 + c3 − [(a3 − a2 ) + (c3 − c2 )], dt b2 + b3
which immediately give the desired inequality (5.2.7). By (5.2.4), we have d b2 2a1 log a3 ≤ a3 + 2a1 + 3 − (a3 − a2 ). dt a3 a3
(5.2.11)
From the inequality (1) there holds b23 ≤ Ga1 c1 . Since tr A = tr C, c1 ≤ c1 + c2 + c3 = a1 + a2 + a3 ≤ 3a3 which shows b23 ≤ 3Ga1 . a3
Thus by (5.2.8) and (5.2.11), d a3 log ≤ (3G + 2)a1 − a3 . dt a1 So if H ≥ (3G + 2), then the inequalities a3 ≤ Ha1 and likewise c3 ≤ Hc1 are preserved. For the inequality (3), we compute from (5.2.8)-(5.2.10) a 1 c1 (a1 − b1 )2 (c1 − b1 )2 a3 c3 d log ≥ + + 2 (a2 − a1 ) + 2 (c2 − c1 ) 2 dt (b2 + b3 ) a1 c1 a1 c1 2b2 + [(a3 − a2 ) + (c3 − c2 )]. b2 + b3 If b1 ≤ a1 /2, then (a1 − b1 )2 a1 1 ≥ ≥ a3 , a1 4 4H and if b1 ≥ a1 /2, then 2b2 1 2b2 2b2 2b2 ≥ √ . ≥√ ≥ √ ≥√ b2 + b3 Ga1 c1 3Ga1 a3 3GH · a1 3GH
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
331
Thus by combining with 3a3 ≥ c3 , we have a 1 c1 d log ≥ δ(a3 − a1 ) + δ(c3 − c1 ) dt (b2 + b3 )2 1 1 provided δ ≤ min( 24H , √3GH ). On the other hand, it follows from (5.2.6) and (5.2.10) that
d b2 + b3 log ≤ (a3 − a1 ) + (c3 − c1 ). dt a − 2b + c Therefore the inequality (3) (b2 + b3 )2+δ ≤ Ja1 c1 (a − 2b + c)δ will be preserved by any positive constant J. To verify the inequality (4), we first note that there is a small η > 0 such that b ≤ (1 − η)a, on the set defined by the inequality (3). Indeed, if b ≤ a2 , this is trivial and if b ≥ a2 , then a 2+δ ≤ (b2 + b3 )2+δ ≤ 2δ Ja2 (a − b)δ 3 which makes b ≤ (1 − η)a for some η > 0 small enough. Consequently, ηa ≤ a − b ≤ 3(a3 − b1 ) which implies either a3 − a1 ≥
1 ηa, 6
a1 − b 1 ≥
1 ηa. 6
or
Thus as in the proof of the inequality (3), we have d a 1 c1 log ≥ δ(a3 − a1 ) dt (b2 + b3 )2 and a 1 c1 (a1 − b1 )2 d log ≥ , dt (b2 + b3 )2 a1 which in turn implies d a 1 c1 log ≥ dt (b2 + b3 )2
1 1 2 max ηδ, η · a. 6 36
On the other hand, it follows from (5.2.10) that d log(b2 + b3 ) ≤ 2b1 + a3 + c3 ≤ 4a dt
332
H.-D. CAO AND X.-P. ZHU
since Mαβ ≥ 0. Then if ε > 0 is small enough a 1 c1 d log ≥0 dt (b2 + b3 )2+ε and it follows that the inequality (4) is preserved by any positive K. Finally we consider the inequality (5). From (5.2.8) we have d log a1 ≥ a1 + 2a3 dt and then for θ ∈ (0, 1), a1 + (1 − θ)La1−θ d 1 log(a1 + La11−θ ) ≥ (a1 + 2a3 ). dt a1 + La1−θ 1 On the other hand, the inequality (4) tells us ˜ 1−θ a3 b23 ≤ Ka 1 ˜ large enough with θ to be fixed small enough. And then for some positive constant K d ˜ 1−θ , log a3 ≤ a3 + 2a1 + Ka 1 dt by combining with (5.2.11). Thus by choosing θ ≤
1 6H
˜ and L ≥ 2K,
a1 + La1−θ La1−θ d 1 1 ˜ 1−θ log ≥ (a3 − a1 ) − θ (a1 + 2a3 ) − Ka 1 dt a3 a1 + La1−θ 1 La1−θ 1 ˜ 1−θ · 3Ha1 − Ka 1 a1 + La1−θ 1 ˜ 1−θ ≥ (a3 − a1 ) − (3θHL + K)a ≥ (a3 − a1 ) − θ
1
˜ 1−θ = [L − (3θHL + K)]a 1 ≥0
whenever a1 + La11−θ = a3 . Consequently the set {a1 + La1−θ ≥ a3 } is preserved. A 1 similar argument works for the inequality in C. This completes the proof of Proposition 5.2.6. The combination of the advanced maximum principle Theorem 2.3.1 and the pinching estimates of the ODE (5.2.3) in Proposition 5.2.6 immediately gives the following pinching estimate for the Ricci flow on a compact four-manifold. Corollary 5.2.7. Suppose that the initial metric of the solution to the Ricci flow on a compact four-manifold has positive curvature operator. Then for any ε > 0 we can find positive constant Cε < +∞ such that ◦
|Rm| ≤ εR + Cε ◦
for all t ≥ 0 as long as the solution exists, where Rm is the traceless part of the curvature operator.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
333
Proof of Theorem 5.2.2. Let M be a compact four-manifold with positive curvature operator and let us evolve the metric by the Ricci flow. Again the evolution equation of the scalar curvature tells us that the scalar curvature remains positive and becomes unbounded in finite time. Pick a sequence of points xj ∈ M and times τj where the curvature at xj is as large as it has been anywhere for 0 ≤ t ≤ τj . Dilate the metrics so that the maximum curvature |Rm(xj , τj )| = maxt≤τj maxx∈M |Rm(x, t)| becomes 1 and shift the time so that the time τj becomes the new time 0. The Klingenberg injectivity radius estimate in Section 4.2 tells us that the injectivity radii of the rescaled metrics at the origins xj and at the new time 0 are uniformly bounded from below. Then we can apply the Hamilton’s compactness theorem (Theorem 4.1.5) to take a limit. By the pinching estimate in Corollary 5.2.7, we know that the limit metric has positive constant curvature which is either S4 or RP4 . Therefore the compact four-manifold M is diffeomorphic to the sphere S4 or the real projective space RP4 . Remark 5.2.8. The proofs of Theorem 5.2.1 and Theorem 5.2.2 also show that the Ricci flow on a compact three-manifold with positive Ricci curvature or a compact four-manifold with positive curvature operator is subsequentially converging (up to scalings) in the C ∞ topology to the same underlying compact manifold with a metric of positive constant curvature. Of course, this subsequential convergence is in the sense of Hamilton’s compactness theorem (Theorem 4.1.5) which is also up to the pullbacks of diffeomorphisms. Actually in [58] and [59], Hamilton obtained the convergence in the stronger sense that the (rescaled) metrics converge (in the C ∞ topology) to a constant (positive) curvature metric. In the following we use the Hamilton’s strong maximum principle in Section 2.4 to prove Theorem 5.2.3. Proof of Theorem 5.2.3. In views of Theorem 5.2.1 and Theorem 5.2.2, we may assume the Ricci curvature (in dimension 3) and the curvature operator (in dimension 4) always have nontrivial kernels somewhere along the Ricci flow. (i) In the case of dimension 3, we consider the evolution equation (1.3.5) of the Ricci curvature ∂Rab = △Rab + 2Racbd Rcd ∂t in an orthonormal frame coordinate. At each point, we diagonalize Rab with eigenvectors e1 , e2 , e3 and eigenvalues λ1 ≤ λ2 ≤ λ3 . Since R1c1d Rcd = R1212 R22 + R1313 R33 1 = ((λ3 − λ2 )2 + λ1 (λ2 + λ3 )), 2 we know that if Rab ≥ 0, then Racbd Rcd ≥ 0. By Hamilton’s strong maximum principle (Theorem 2.2.1), there exists an interval 0 < t < δ on which the rank of Rab is constant and the null space of Rab is invariant under parallel translation and invariant in time and also lies in the null space of Racbd Rcd . If the null space of Rab has rank one, then λ1 = 0 and λ2 = λ3 > 0. In this case, by De Rham decomposition ˜ of the compact M splits isometrically as R × Σ2 theorem, the universal cover M 2 and the curvature of Σ has a positive lower bound. Hence Σ2 is diffeomorphic to S2 . Assume M = R × Σ2 /Γ, for some isometric subgroup Γ of R × Σ2 . Note that Γ
334
H.-D. CAO AND X.-P. ZHU
remains to be an isometric subgroup of R× Σ2 during the Ricci flow by the uniqueness (Theorem 1.2.4). Since the Ricci flow on R × Σ2 /Γ converges to the standard metric by Theorem 5.1.11, Γ must be an isometric subgroup of R× S2 in the standard metric. If the null space of Rab has rank greater than one, then Rab = 0 and the manifold is flat. This proves Theorem 5.2.3 part (i). (ii) In the case of dimension 4, we classify the manifolds according to the (restricted) holonomy algebra G. Note that the curvature operator has nontrivial kernel and G is the image of the the curvature operator, we see that G is a proper subalgebra of so(4). We divide the argument into two cases. Case 1. G is reducible.
˜ splits isometrically as M ˜1 × M ˜2 . By the above In this case the universal cover M results on two and three dimensional Ricci flow, we see that M is diffeomorphic to a quotient of one of the spaces R4 , R × S3 , R2 × S2 , S2 × S2 by a group of fixed point free isometries. As before by running the Ricci flow until it converges and using the uniqueness (Theorem 1.2.4), we see that this group is actually a subgroup of the isometries in the standard metrics. Case 2. G is not reducible (i.e., irreducible).
If the manifold is not Einstein, then by Berger’s classification theorem for holonomy groups [7], G = so(4) or u(2). Since the curvature operator is not strictly ˜ of M is K¨ positive, G = u(2), and the universal cover M ahler and has positive bisec˜ tional curvature. In this case M is biholomorphic to CP2 by the result of AndreottiFrankel [47] (also the resolution of the Frankel conjecture by Mori [96] and Siu-Yau [120]). If the manifold is Einstein, then by the block decomposition of the curvature operator matrix in four-manifolds (see the third section of Chapter 1), Rm(Λ2+ , Λ2− ) = 0. Let ϕ 6= 0, and ϕ = ϕ+ + ϕ− ∈ Λ2+ ⊕ Λ2− , lies in the kernel of the curvature operator, then 0 = Rm(ϕ+ , ϕ+ ) + Rm(ϕ− , ϕ− ). It follows that Rm(ϕ+ , ϕ+ ) = 0, and Rm(ϕ− , ϕ− ) = 0. We may assume ϕ+ 6= 0 (the argument for the other case is similar). We consider the restriction of Rm to Λ2+ , since Λ2+ is an invariant subspace of Rm and the intersection of Λ2+ with the null space of Rm is nontrivial. By considering the null space of Rm and its orthogonal complement in Λ2+ , we obtain a parallel distribution of rank one in Λ2+ . This parallel distribution gives a parallel translation invariant two-form ˜ of M . This two-form is nondegenerate, so it ω ∈ Λ2+ on the universal cover M ˜ . Since the K¨ induces a K¨ ahler structure of M ahler metric is parallel with respect to the original metric and the manifold is irreducible, the K¨ ahler metric is proportional ˜ is K¨ to the original metric. Hence the manifold M ahler-Einstein with nonnegative
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
335
˜ is curvature operator. Taking into account the irreducibility of G, it follows that M biholomorphic to CP2 . Therefore the proof of Theorem 5.2.3 is completed. To end this section, we mention the generalizations of Hamilton’s differential sphere theorem (Theorem 5.2.1 and Theorem 5.2.2) to higher dimensions. Using minimal surface theory, Micallef and Moore [88] proved that any compact simply connected n-dimensional manifold with positive curvature operator is homeomorphic to the sphere Sn . But it is still an open question whether a compact simply connected n-dimensional manifold with positive curvature operator is diffeomorphic to the sphere Sn . It is well-known that the curvature tensor Rm = {Rijkl } of a Riemannian manifold can be decomposed into three orthogonal components which have the same symmetries as Rm: Rm = W + V + U. Here W = {Wijkl } is the Weyl conformal curvature tensor, whereas V = {Vijkl } and U = {Uijkl } denote the traceless Ricci part and the scalar curvature part respectively. The following pointwisely pinching sphere theorem under the additional assumption that the manifold is compact was first obtained by Huisken [71], Margerin [83], [84] and Nishikawa [101] by using the Ricci flow. The compactness assumption was later removed by Chen and the second author in [30]. Theorem 5.2.9. Let n ≥ 4. Suppose M is a complete n-dimensional manifold with positive and bounded scalar curvature and satisfies the pointwisely pinching condition |W |2 + |V |2 ≤ δn (1 − ε)2 |U |2 , where ε > 0, δ4 = 51 , δ5 =
1 10 ,
and δn =
2 , n ≥ 6. (n − 2)(n + 1)
Then M is diffeomorphic to the sphere Sn or a quotient of it by a finite group of fixed point free isometries in the standard metric. In [30], Chen and the second author also used the Ricci flow to obtain the following flatness theorem for noncompact three-manifolds. Theorem 5.2.10. Let M be a three-dimensional complete noncompact Riemannian manifold with bounded and nonnegative sectional curvature. Suppose M satisfies the following Ricci pinching condition Rij ≥ εRgij ,
on M,
for some ε > 0. Then M is flat. The basic idea of proofs of these two theorems is to analyze the asymptotic behavior of the solution to the Ricci flow. For the details, one can consult the above cited literatures.
336
H.-D. CAO AND X.-P. ZHU
5.3. Nonsingular Solutions on Three-manifolds. We have seen in the previous section that a good understanding of the long time behaviors for solutions to the Ricci flow could lead to remarkable topological or geometric consequences for the underlying manifolds. Since one of the central themes of the Ricci flow is to study the geometry and topology of three-manifolds, we will start to analyze the long time behavior of the Ricci flow on a compact three-manifold by first considering a special class of solutions (i.e., the nonsingular solutions defined below) in this section. Our presentation follows closely the paper of Hamilton [65]. Let M be a compact three-manifold. We will consider the (unnormalized) Ricci flow ∂ gij = −2Rij , ∂t and the normalized Ricci flow ∂ 2 gij = rgij − 2Rij ∂t 3 where r = r(t) is the function of the average of the scalar curvature. Recall that the normalized flow differs from theRunnormalized flow only by rescaling in space and time so that the total volume V = M dµ remains constant. In this section we only consider a special class of solutions that we now define. Definition 5.3.1. A nonsingular solution of the Ricci flow is one where the solution of the normalized flow exists for all time 0 ≤ t < ∞, and the curvature remains bounded |Rm| ≤ C < +∞ for all time with some constant C independent of t. Clearly any solution to the Ricci flow on a compact three-manifold with nonnegative Ricci curvature is nonsingular. Currently there are few conditions which guarantee a solution will remain nonsingular. Nevertheless, the ideas and arguments of this section is extremely important. One will see in Chapter 7 that these arguments will be modified to analyze the long-time behavior of arbitrary solutions, or even the solutions with surgery, to the Ricci flow on three-manifolds. We begin with an improvement of Hamilton-Ivey pinching result (Theorem 2.4.1). Theorem 5.3.2 (Hamilton [65]). Suppose we have a solution to the (unnormalized ) Ricci flow on a three–manifold which is complete with bounded curvature for each t ≥ 0. Assume at t = 0 the eigenvalues λ ≥ µ ≥ ν of the curvature operator at each point are bounded below by ν ≥ −1. Then at all points and all times t ≥ 0 we have the pinching estimate R ≥ (−ν)[log(−ν) + log(1 + t) − 3] whenever ν < 0. Proof. As before, we study the ODE system dλ = λ2 + µν, dt dµ = µ2 + λν, dt dν = ν 2 + λµ. dt
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
337
Consider again the function y = f (x) = x(log x − 3) for e2 ≤ x < +∞, which is increasing and convex with range −e2 ≤ y < +∞. Its inverse function x = f −1 (y) is increasing and concave on −e2 ≤ y < +∞. For each t ≥ 0, we consider the set K(t) of 3×3 symmetric matrices defined by the inequalities: λ+µ+ν ≥−
(5.3.1)
3 , 1+t
and (5.3.2)
ν(1 + t) + f −1 ((λ + µ + ν)(1 + t)) ≥ 0,
which is closed and convex (as we saw in the proof of Theorem 2.4.1). By the assumptions at t = 0 and the advanced maximum principle Theorem 2.3.5, we only need to check that the set K(t) is preserved by the ODE system. Since R = λ + µ + ν, we get from the ODE that dR 2 1 ≥ R2 ≥ R2 dt 3 3 which implies that R≥−
3 , 1+t
for all t ≥ 0.
Thus the first inequality (5.3.1) is preserved. Note that the second inequality (5.3.2) is automatically satisfied when (−ν) ≤ 3/(1 + t). Now we compute from the ODE system, d R 1 dR d(−ν) ( − log(−ν)) = (−ν) · − (R + (−ν)) dt (−ν) (−ν)2 dt dt =
1 [(−ν)3 + (−ν)µ2 + λ2 ((−ν) + µ) − λµ(ν − µ)] (−ν)2
≥ (−ν) ≥
3 (1 + t)
≥
d [log(1 + t) − 3] dt
whenever R = (−ν)[log(−ν) + log(1 + t) − 3] and (−ν) ≥ 3/(1 + t). Thus the second inequality (5.3.2) is also preserved under the system of ODE. Therefore we have proved the theorem. Denote by ρˆ(t) = max{inj (x, gij (t)) | x ∈ M } where inj (x, gij (t)) is the injectivity radius of the manifold M at x with respect to the metric gij (t).
338
H.-D. CAO AND X.-P. ZHU
Definition 5.3.3. We say a solution to the normalized Ricci flow is collapsed if there is a sequence of times tk → +∞ such that ρˆ(tk ) → 0 as k → +∞. When a nonsingular solution of the Ricci flow on M is collapsed, it follows from the work of Cheeger-Gromov [24] [25] or Cheeger-Gromov-Fukaya [26] that the manifold M has an F -structure and then its topology is completely understood. In the following we always assume nonsingular solutions are not collapsed. Now suppose that we have a nonsingular solution which does not collapse. Then for arbitrary sequence of times tj → ∞, we can find a sequence of points xj and some δ > 0 so that the injectivity radius of M at xj in the metric at time tj is at least δ. Clearly the Hamilton’s compactness theorem (Theorem 4.1.5) also holds for the normalized Ricci flow. Then by taking the xj as origins and the tj as initial times, we can extract a convergent subsequence. We call such a limit a noncollapsing limit. Of course the limit has also finite volume. However the volume of the limit may be smaller than the original one if the diameter goes to infinity. The main result of this section is the following theorem of Hamilton [65]. Theorem 5.3.4 (Hamilton [65]). Let gij (t), 0 ≤ t < +∞, be a noncollapsing nonsingular solution of the normalized Ricci flow on a compact three-manifold M . Then either (i) there exist a sequence of times tk → +∞ and a sequence of diffeomorphisms ϕk : M → M so that the pull-back of the metric gij (tk ) by ϕk converges in the C ∞ topology to a metric on M with constant sectional curvature; or (ii) we can find a finite collection of complete noncompact hyperbolic threemanifolds H1 , . . . , Hm with finite volume, and for all t beyond some time T < +∞ we can find compact subsets K1 , . . . , Km of H1 , . . . , Hm respectively obtained by truncating each cusp of the hyperbolic manifolds along constant mean curvature torus of small area, and diffeomorphisms ϕl (t), 1 ≤ l ≤ m, of Kl into M so that as long as t sufficiently large, the pull-back of the solution metric gij (t) by ϕl (t) is as close as to the hyperbolic metric as we like on the compact sets K1 , . . . , Km ; and moreover if we call the exceptional part of M those points where they are not in the image of any ϕl , we can take the injectivity radii of the exceptional part at everywhere as small as we like and the boundary tori of each Kl are incompressible in the sense that each ϕl injects π1 (∂Kl ) into π1 (M ). Remark 5.3.5. The exceptional part has bounded curvature and arbitrarily small injectivity radii everywhere as t large enough. Moreover the boundary of the exceptional part consists of a finite disjoint union of tori with sufficiently small area and is convex. Then by the work of Cheeger-Gromov [24], [25] or Cheeger-GromovFukaya [26], there exists an F -structure on the exceptional part. In particular, the exceptional part is a graph manifold, which have been topologically classified. Hence any nonsingular solution to the normalized Ricci flow is geometrizable in the sense of Thurston (see the last section of Chapter 7 for details). The rest of this section is devoted to the proof of Theorem 5.3.4. We will divide the proof into three parts. Part I: Subsequence Convergence According to Lemma 5.1.1, the scalar curvature of the normalized flow evolves by
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
339
the equation ∂ 2 R = ∆R + 2|Ric|2 − rR ∂t 3 ◦ 2 = ∆R + 2| Ric | + R(R − r) 3
(5.3.3)
◦
where Ric is the traceless part of the Ricci tensor. As before, we denote by Rmin (t) = minx∈M R(x, t). It then follows from the maximum principle that (5.3.4)
d 2 Rmin ≥ Rmin (Rmin − r), dt 3
which implies that if Rmin ≤ 0 it must be nondecreasing, and if Rmin ≥ 0 it cannot go negative again. We can then divide the noncollapsing solutions of the normalized Ricci flow into three cases. Case (1): Rmin (t) > 0 for some t > 0; Case (2): Rmin (t) ≤ 0 for all t ∈ [0, +∞) and lim Rmin (t) = 0; t→+∞
Case (3): Rmin (t) ≤ 0 for all t ∈ [0, +∞) and lim Rmin (t) < 0. t→+∞
Let us first consider Case (1). In this case the maximal time interval [0, T ) of the corresponding solution of the unnormalized flow is finite, since the unnormalized ˜ satisfies scalar curvature R ∂ ˜ ˜ + 2|Ric| ˜ 2 R = ∆R ∂t ˜ + 2R ˜2 ≥ ∆R 3 which implies that the curvature of the unnormalized solution blows up in finite time. Without loss of generality, we may assume that for the initial metric at t = 0, the ˜ ≥µ eigenvalues λ ˜ ≥ ν˜ of the curvature operator are bounded below by ν˜ ≥ −1. It follows from Theorem 5.3.2 that the pinching estimate ˜ ≥ (−˜ R ν )[log(−˜ ν ) + log(1 + t) − 3] holds whenever ν˜ < 0. This shows that when the unnormalized curvature big, the negative ones are not nearly as large as the positive ones. Note that the unnormalized curvature becomes unbounded in finite time. Thus when we rescale the unnormalized flow to the normalized flow, the scaling factor must go to infinity. In the nonsingular case the rescaled positive curvature stay finite, so the rescaled negative curvature (if any) go to zero. Hence we can take a noncollapsing limit for the nonsingular solution of the normalized flow so that it has nonnegative sectional curvature. Since the volume of the limit is finite, it follows from a result of Calabi and Yau [112] that the limit must be compact and the limiting manifold is the original one. Then by the strong maximum principle as in the proof of Theorem 5.2.3 (i), either the limit is flat, or it is a compact metric quotient of the product of a positively curved surface Σ2 with R, or it has strictly positive curvature. By the work of Schoen-Yau [110], a flat three-manifold cannot have a metric of positive scalar curvature, but our manifold does in Case (1). This rules out the possibility of a flat limit. Clearly the limit is also a nonsingular solution to the normalized Ricci flow. Note that the
340
H.-D. CAO AND X.-P. ZHU
curvature of the surface Σ2 has a positive lower bound and is compact since it comes from the lifting of the compact limiting manifold. From Theorem 5.1.11, we see the metric of the two-dimensional factor Σ2 converges to the round two-sphere S2 in the normalized Ricci flow. Note also that the normalized factors in two-dimension and three-dimension are different. This implies that the compact quotient of the product Σ2 × R cannot be nonsingular, which is also ruled out for the limit. Thus the limit must have strictly positive sectional curvature. Since the convergence takes place everywhere for the compact limit, it follows that as t large enough the original nonsingular solution has strictly positive sectional curvature. This in turn shows that the corresponding unnormalized flow has strictly positive sectional curvature after some finite time. Then in views of the proof of Theorem 5.2.1, in particular the pinching estimate in Proposition 5.2.5, the limit has constant Ricci curvature and then constant sectional curvature for three-manifolds. This finishes the proof in Case (1). We next consider Case (2). In this case we only need to show that we can take a noncollapsing limit which has nonnegative sectional curvature. Indeed, if this is true, then as in the previous case, the limit is compact and either it is flat, or it splits as a product (or a quotient of a product) of a positively curved S 2 with a circle S 1 , or it has strictly positive curvature. But the assumption Rmin (t) ≤ 0 for all times t ≥ 0 in this case implies the limit must be flat. Let us consider the corresponding unnormalized flow g˜ij (t) associated to the noncollapsing nonsingular solution. The pinching estimate in Theorem 5.3.2 tells us that we may assume the unnormalized flow g˜ij (t) exists for all times 0 ≤ t < +∞, for otherwise, the scaling factor approaches infinity as in the previous case which implies the limit has nonnegative sectional curvature. The volume V˜ (t) of the unnormalized solution g˜ij (t) now changes. We divide the discussion into three subcases. Subcase (2.1): there is a sequence of times t˜k → +∞ such that V˜ (t˜k ) → +∞; Subcase (2.2): there is a sequence of times t˜k → +∞ such that V˜ (t˜k ) → 0; Subcase (2.3): there exist two positive constants C1 , C2 such that C1 ≤ V˜ (t) ≤ C2 for all 0 ≤ t < +∞. For Subcase (2.1), because dV˜ = −rV˜ dt we have Z t˜k V˜ (t˜k ) log =− r(t)dt → +∞, as k → +∞, V˜ (0) 0 which implies that there exists another sequence of times, still denoted by t˜k , such that t˜k → +∞ and r(t˜k ) ≤ 0. Let tk be the corresponding times for the normalized flow. Thus there holds for the normalized flow r(tk ) → 0,
as k → ∞,
since 0 ≥ r(tk ) ≥ Rmin (tk ) → 0 as k → +∞. Then Z (R − Rmin )dµ(tk ) = (r(tk ) − Rmin (tk ))V → 0, M
as k → ∞.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
341
As we take a noncollapsing limit along the time sequence tk , we get Z Rdµ∞ = 0 M∞
for the limit of the normalized solutions at the new time t = 0. But R ≥ 0 for the limit because lim Rmin (t) = 0 for the nonsingular solution. So R = 0 at t = 0 for t→+∞
the limit. Since the limit flow exists for −∞ < t < +∞ and the scalar curvature of the limit flow evolves by ∂ 2 R = ∆R + 2|Ric |2 − r∞ R, ∂t 3
t ∈ (−∞, +∞)
where r∞ is the limit of the function r(t) by translating the times tk as the new time t = 0. It follows from the strong maximum principle that R ≡ 0,
on M ∞ × (−∞, +∞).
This in turn implies, in view of the above evolution equation, that Ric ≡ 0,
on M ∞ × (−∞, +∞).
Hence this limit must be flat. Since the limit M ∞ is complete and has finite volume, the flat manifold M ∞ must be compact. Thus the underlying manifold M ∞ must agree with the original M (as a topological manifold). This says that the limit was taken on M . For Subcase (2.2), we may assume as before that for the initial metric at t = 0 ˜≥µ of the unnormalized flow g˜ij (t), the eigenvalues λ ˜ ≥ ν˜ of the curvature operator satisfy ν˜ ≥ −1. It then follows from Theorem 5.3.2 that ˜ ≥ (−˜ R ν )[log(−˜ ν ) + log(1 + t) − 3],
for all t ≥ 0
whenever ν˜ < 0. Let tk be the sequence of times in the normalized flow which corresponds to the sequence of times t˜k . Take a noncollapsing limit for the normalized flow along the times tk . Since V˜ (t˜k ) → 0, the normalized curvatures at the times tk are reduced by 2 multiplying the factor (V˜ (t˜k )) 3 . We claim the noncollapsing limit has nonnegative sectional curvature. Indeed if the maximum value of (−˜ ν ) at the time t˜k does not go to infinity, the normalized eigenvalue −ν at the corresponding time tk must get rescaled to tend to zero; while if the maximum value of (−˜ ν ) at the time t˜k does go to ˜ ˜ infinity, the maximum value of R at tk will go to infinity even faster from the pinching estimate, and when we normalize to keep the normalized scalar curvature R bounded at the time tk so the normalized (−ν) at the time tk will go to zero. Thus in either case the noncollapsing limit has nonnegative sectional curvature at the initial time t = 0 and then has nonnegative sectional curvature for all times t ≥ 0. For Subcase (2.3), normalizing the flow only changes quantities in a bounded way. As before we have the pinching estimate R ≥ (−ν)[log(−ν) + log(1 + t) − C] for the normalized Ricci flow, where C is a positive constant depending only on the constants C1 , C2 in the assumption of Subcase (2.3). If (−ν) ≤
A 1+t
342
H.-D. CAO AND X.-P. ZHU
for any fixed positive constant A, then (−ν) → 0 as t → +∞ and we can take a noncollapsing limit which has nonnegative sectional curvature. On the other hand if we can pick a sequence of times tk → ∞ and points xk where (−ν)(xk , tk ) = max(−ν)(x, tk ) satisfies x∈M
(−ν)(xk , tk )(1 + tk ) → +∞,
as k → +∞,
then from the pinching estimate, we have R(xk , tk ) → +∞, (−ν)(xk , tk )
as
k → +∞.
But R(xk , tk ) are uniformly bounded since normalizing the flow only changes quantities in bounded way. This shows sup(−ν)(·, tk ) → 0 as k → +∞. Thus we can take a noncollapsing limit along tk which has nonnegative sectional curvature. Hence we have completed the proof of Case (2). We now come to the most interesting Case (3) where Rmin increases monotonically to a limit strictly less than zero. By scaling we can assume Rmin (t) → −6 as t → +∞. Lemma 5.3.6. In Case (3) where Rmin → −6 as t → +∞, all noncollapsing limit are hyperbolic with constant sectional curvature −1. Proof. By (5.3.4) and the fact Rmin (t) ≤ −6, we have d Rmin (t) ≥ 4(r(t) − Rmin (t)) dt and Z
∞ 0
(r(t) − Rmin (t))dt < +∞.
Since r(t) − Rmin (t) ≥ 0 and Rmin (t) → −6 as t → +∞, it follows that the function r(t) has the limit r = −6, for any convergent subsequence. And since Z (R − Rmin (t))dµ = (r(t) − Rmin (t)) · V, M
it then follows that R ≡ −6 for the limit. The limit still has the following evolution equation for the limiting scalar curvature ◦ ∂ 2 R = ∆R + 2|Ric|2 + R(R − r). ∂t 3 ◦
Since R ≡ r ≡ −6 in space and time for the limit, it follows directly that |Ric| ≡ 0 for the limit. Thus the limit metric has λ = µ = ν = −2, so it has constant sectional curvature −1 as desired.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
343
If in the discussion above there exists a compact noncollapsing limit, then we know that the underlying manifold M is compact and we fall into the conclusion of Theorem 5.3.4(i) for the constant negative sectional curvature limit. Thus it remains to show when every noncollapsing limit is a complete noncompact hyperbolic manifold with finite volume, we have conclusion (ii) in Theorem 5.3.4. Now we first want to find a finite collection of persistent complete noncompact hyperbolic manifolds as stated in Theorem 5.3.4 (ii). Part II: Persistence of Hyperbolic Pieces We begin with the definition of the topology of C ∞ convergence on compact sets for maps F : M → N of one Riemannian manifold to another. For any compact set K ⊂⊂ M and any two maps F, G : M → N , we define dK (F, G) = sup d(F (x), G(x)) x∈K 0 where d(y, z) is the geodesic distance from y to z on N . This gives the Cloc topology k for maps between M and N . To define Cloc topology for any positive integer k ≥ 1, we consider the k-jet space J k M of a manifold M which is the collection of all
(x, J 1 , J 2 , . . . , J k ) where x is a point on M and J i is a tangent vector for 1 ≤ i ≤ k defined by the ith covariant derivative J i = ∇i∂ γ(0) for a path γ passing through the point x with ∂t
γ(0) = x. A smooth map F : M → N induces a map J kF :
J kM → J kN
defined by J k F (x, J 1 , . . . , J k ) = (F (x), ∇ ∂ (F (γ))(0), . . . , ∇k∂ (F (γ))(0)) ∂t
∂t
where γ is a path passing through the point x with J i = ∇i∂ γ(0), 1 ≤ i ≤ k. ∂t Define the k-jet distance between F and G on a compact set K ⊂⊂ M by dC k (K) (F, G) = dBJ k K (J k F, J k G) where BJ k K consists of all k-jets (x, J 1 , . . . , J k ) with x ∈ K and |J 1 |2 + |J 2 |2 + · · · + |J k |2 ≤ 1. Then the convergence in the metric dC k (K) for all positive integers k and all compact sets K defines the topology of C ∞ convergence on compact sets for the space of maps. We will need the following Mostow type rigidity result. Lemma 5.3.7. For any complete noncompact hyperbolic three-manifold H with finite volume with metric h, we can find a compact set K of H such that for every integer k and every ε > 0, there exist an integer q and a δ > 0 with the following property: if F is a diffeomorphism of K into another complete noncompact hyperbolic ˜ such that ˜ with no fewer cusps (than H), finite volume with metric h three-manifold H ˜ − hkC q (K) < δ kF ∗ h
344
H.-D. CAO AND X.-P. ZHU
˜ such that then there exists an isometry I of H to H dC k (K) (F, I) < ε. ˜ for an appropriate choice of Proof. First we claim that H is isometric to H compact set K, positive integer q and positive number δ. Let l : H → R be a function defined at each point by the length of the shortest non-contractible loop starting and ending at this point. Denote the Margulis constant by µ. Then by Margulis lemma (see for example [55] or [76]), for any 0 < ε0 < 21 µ, the set l−1 ([0, ε0 ]) ⊂ H consists of finitely many components and each of these components is isometric to a cusp or to a tube. Topologically, a tube is just a solid torus. Let ε0 be even smaller than one half of the minimum of the lengths of the all closed geodesics on the tubes. Then l−1 ([0, ε0 ]) consists of finite number of cusps. Set K0 = l−1 ([ε0 , ∞)). The boundary of K0 consists of flat tori with constant mean curvatures. Note that each embedded torus in a complete hyperbolic three-manifold with finite volume either bounds a solid torus or is isotopic to a standard torus in a cusp. The diffeomorphism F implies the boundary F (∂K0 ) are embedded tori. If one of components bounds a solid torus, then ˜ would have fewer cusps than H, which as δ sufficiently small and q sufficiently large, H o ˜ is diffeomorphic to F (K0 ). Here contradicts with our assumption. Consequently, H o
o
K0 is the interior of the set K0 . Since H is diffeomorphic to K0 , H is diffeomorphic ˜ Hence by Mostow’s rigidity theorem (see [97] and [107]), H is isometric to H. ˜ to H. ˜ So we can assume H = H. For K = K0 , we argue by contradiction. Suppose there is some k > 0 and ε > 0 so that there exist sequences of integers qj → ∞, δj → 0+ and diffeomorphisms Fj mapping K into H with kFj∗ h − hkC qj (K) < δj and dC k (K) (Fj , I) ≥ ε
for all isometries I of H to itself. We can extract a subsequence of Fj convergent to ∗ a map F∞ with F∞ h = h on K. We need to check that F∞ is still a diffeomorphism on K. Since F∞ is a local diffeomorphism and is the limit of diffeomorphisms, we can find an inverse of F∞ on o
o
F∞ (K). So F∞ is a diffeomorphism on K. We claim the image of the boundary can not touch the image of the interior. Indeed, if F∞ (x1 ) = F∞ (x2 ) with x1 ∈ ∂K and o
o
o
x2 ∈ K, then we can find x3 ∈ K near x1 and x4 ∈ K near x2 with F∞ (x3 ) = F∞ (x4 ), since F∞ is a local diffeomorphism. This contradicts with the fact that F∞ is a o
diffeomorphism on K. This proves our claim. Hence, the only possible overlap is at the boundary. But the image F∞ (∂K) is strictly concave, this prevents the boundary from touching itself. We conclude that the mapping F∞ is a diffeomorphism on K, hence an isometry. To extend F∞ to a global isometry, we argue as follows. For each truncated cusp end of K, the area of constant mean curvature flat torus is strictly decreasing. Since F∞ takes each such torus to another of the same area, we see that F∞ takes the foliation of an end by constant mean curvature flat tori to another such foliation. So F∞ takes cusps to cusps and preserves their foliations. Note that the isometric type
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
345
of a cusp is just the isometric type of the torus, more precisely, let (N, dr2 + e−2r gV ) be a cusp (where gV is the flat metric on the torus V), 0 < a < b are two constants, any isometry of N ∩ l−1 [a, b] to itself is just an isometry of V. Hence the isometry F∞ can be extended to the whole cusps. This gives a global isometry I contradicting our assumption when j large enough. The proof of the Lemma 5.3.7 is completed. In order to obtain the persistent hyperbolic pieces stated in Theorem 5.3.4 (ii), we will need to use a special parametrization given by harmonic maps. Lemma 5.3.8. Let (X, g) be a compact Riemannian manifold with strictly negative Ricci curvature and with strictly concave boundary. Then there are positive integer l0 and small number ε0 > 0 such that for each positive integer l ≥ l0 and positive number ε ≤ ε0 we can find positive integer q and positive number δ > 0 such that for every metric g˜ on X with ||˜ g − g||C q (X) ≤ δ we can find a unique diffeomorphism F of X to itself so that (a) F : (X, g) → (X, g˜) is harmonic, (b) F takes the boundary ∂X to itself and satisfies the free boundary condition that the normal derivative ∇N F of F at the boundary is normal to the boundary, (c) dC l (X) (F, Id) < ε, where Id is the identity map. Proof. Let Φ(X, ∂X) be the space of maps of X to itself which take ∂X to itself. Then Φ(X, ∂X) is a Banach manifold and the tangent space to Φ(X, ∂X) at the ∂ identity is the space of vector fields V = V i ∂x i tangent to the boundary. Consider the map sending F ∈ Φ(X, ∂X) to the pair {∆F, (∇N F )// } consisting of the harmonic map Laplacian and the tangential component (in the target) of the normal derivative of F at the boundary. By using the inverse function theorem, we only need to check that the derivative of this map is an isomorphism at the identity with g˜ = g. Let {xi }i=1,...,n be a local coordinates of (X, g) and {y α }α=1,...,n be a local coordinates of (X, g˜). The harmonic map Laplacian of F : (X, g) → (X, g˜) is given in local coordinates by ˜α ◦ F ) (∆F )α = ∆(F α ) + g ij (Γ βγ
∂F β ∂F γ ∂xi ∂xj
˜ α is the connection where ∆(F α ) is the Laplacian of the function F α on X and Γ βγ of g˜. Let F be a one-parameter family with F |s=0 = Id and dF ds |s=0 = V , a smooth vector field on X tangent to the boundary (with respect to g). At an arbitrary given point x ∈ X, we choose the coordinates {xi }i=1,...,n so that Γijk (x) = 0. We compute at the point x with g˜ = g, ∂ α d Γ V k. (∆F )α = ∆(V α ) + g ij ds s=0 ∂xk ij Since
β (∇i V )α = ∇i V α + (Γα iβ ◦ F )V ,
we have, at s = 0 and the point x, (∆V )α = ∆(V α ) + g ij
∂ α k Γ V . ∂xi jk
346
H.-D. CAO AND X.-P. ZHU
Thus we obtain (5.3.5)
d |s=0 (∆F )α = (∆V )α + g ij ds
∂ α ∂ α Γ − Γ Vk ∂xk ij ∂xi jk
= (∆V )α + g αi Rik V k . Since (∇N F )(F −1 (x)) = N i (F −1 (x))
∂F j −1 ∂ (F (x)) j (x) ∂xi ∂x
on ∂X,
we have (5.3.6)
d d |s=0 {(∇N F )// } = |s=0 (∇N F − h∇N F, N iN ) ds ds d d = |s=0 (∇N F ) − |s=0 ∇N F, N N ds ds
− h∇N F, ∇V N iN |s=0 − h∇N F, N i∇V N |s=0 d = |s=0 ∇N F − ∇V N ds // ∂ j i ∂ − II(V ) = −V (N ) i + N (V ) j ∂x ∂x //
= [N, V ]// − II(V )
= (∇N V )// − 2II(V )
where II is the second fundamental form of the boundary (as an automorphism of T (∂X)). Thus by (5.3.5) and (5.3.6), the kernel of the map sending F ∈ Φ(X, ∂X) to the pair {∆F, (∇N F )// } is the space of solutions of elliptic boundary value problem on X ∆V + Ric (V ) = 0 (5.3.7) V⊥ = 0, at ∂X, (∇N V )// − 2II(V ) = 0, at ∂X, where V⊥ is the normal component of V . Now using these equations and integrating by parts gives Z Z Z Z Z 2 |∇V | = Ric (V, V ) + 2 II(V, V ). X
X
∂X
Since Rc < 0 and II < 0 we conclude that the kernel is trivial. Clearly this elliptic boundary value is self-adjoint because of the free boundary condition. Thus the cokernel is trivial also. This proves the lemma. Now we can prove the persistence of hyperbolic pieces. Let gij (t), 0 ≤ t < +∞, be a noncollapsing nonsingular solution of the normalized Ricci flow on a compact three-manifold M . Assume that any noncollapsing limit of the nonsingular solution is a complete noncompact hyperbolic three-manifold with finite volume. Consider all the possible hyperbolic limits of the given nonsingular solution, and among them choose one such complete noncompact hyperbolic three-manifold H with the least possible number of cusps. In particular, we can find a sequence of times tk → +∞ and a
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
347
sequence of points Pk on M such that the marked three-manifolds (M, gij (tk ), Pk ) ∞ converge in the Cloc topology to H with hyperbolic metric hij and marked point P ∈ H. For any small enough a > 0 we can truncate each cusp of H along a constant mean curvature torus of area a which is uniquely determined; the remainder we denote by Ha . Clearly as a → 0 the Ha exhaust H. Pick a sufficiently small number a > 0 to truncate cusps so that Lemma 5.3.7 is applicable for the compact set K = Ha . Choose an integer l0 large enough and an ε0 sufficiently small to guarantee from Lemma 5.3.8 the uniqueness of the identity map Id among maps close to Id as a harmonic map F from Ha to itself with taking ∂Ha to itself, with the normal derivative of F at the boundary of the domain normal to the boundary of the target, and with dC l0 (Ha ) (F, Id) < ε0 . Then choose positive integer q0 and small number δ0 > 0 from Lemma 5.3.7 such that if F˜ is a diffeomorphism of Ha into another complete ˜ with no fewer cusps (than H), finite volume noncompact hyperbolic three-manifold H ˜ with metric hij satisfying ˜ ij − hij ||C q0 (H ) ≤ δ0 , ||F˜ ∗ h a ˜ such that then there exists an isometry I of H to H (5.3.8)
dC l0 (Ha ) (F˜ , I) < ε0 .
And we further require q0 and δ0 from Lemma 5.3.8 to guarantee the existence of harmonic diffeomorphism from (Ha , g˜ij ) to (Ha , hij ) for any metric g˜ij on Ha with ||˜ gij − hij ||C q0 (Ha ) ≤ δ0 . By definition, there exist a sequence of exhausting compact sets Uk of H (each Uk ⊃ Ha ) and a sequence of diffeomorphisms Fk from Uk into M such that Fk (P ) = Pk and ||Fk∗ gij (tk ) − hij ||C m (Uk ) → 0 as k → +∞ for all positive integers m. Note that ∂Ha is strictly concave and we can foliate a neighborhood of ∂Ha with constant mean curvature hypersurfaces where the area a has a nonzero gradient. As the approximating maps Fk : (Uk , hij ) → (M, gij (tk )) are close enough to isometries on this collar of ∂Ha , the metrics gij (tk ) on M will also admit a unique constant mean curvature hypersurface with the same area a near Fk (∂Ha )(⊂ M ) by the inverse function theorem. Thus we can change the map Fk by an amount which goes to zero as k → ∞ so that now Fk (∂Ha ) has constant mean curvature with the area a. Furthermore, by applying Lemma 5.3.8 we can again change Fk by an amount which goes to zero as k → ∞ so as to make Fk a harmonic diffeomorphism and take ∂Ha to the constant mean curvature hypersurface Fk (∂Ha ) and also satisfy the free boundary condition that the normal derivative of Fk at the boundary of the domain is normal to the boundary of the target. Hence for arbitrarily given positive integer q ≥ q0 and positive number δ < δ0 , there exists a positive integer k0 such that for the modified harmonic diffeomorphism Fk , when k ≥ k0 , ||Fk∗ gij (tk ) − hij ||C q (Ha ) < δ. For each fixed k ≥ k0 , by the implicit function theorem we can first find a constant mean curvature hypersurface near Fk (∂Ha ) in M with the metric gij (t) for t close to tk and with the same area for each component since ∂Ha is strictly concave and a neighborhood of ∂Ha is foliated by constant mean curvature hypersurfaces where the area a has a nonzero gradient and Fk : (Ha , hij ) → (M, gij (tk )) is close enough to an isometry and gij (t) varies smoothly. Then by applying Lemma 5.3.8 we can smoothly
348
H.-D. CAO AND X.-P. ZHU
continue the harmonic diffeomorphism Fk forward in time a little to a family of harmonic diffeomorphisms Fk (t) from Ha into M with the metric gij (t), with Fk (tk ) = Fk , where each Fk (t) takes ∂Ha into the constant mean curvature hypersurface we just found in (M, gij (t)) and satisfies the free boundary condition, and also satisfies ||Fk∗ (t)gij (t) − hij ||C q (Ha ) < δ. We claim that for all sufficiently large k, we can smoothly extend the harmonic diffeomorphism Fk to the family harmonic diffeomorphisms Fk (t) with ||Fk∗ (t)gij (t) − hij ||C q (Ha ) ≤ δ on a maximal time interval tk ≤ t ≤ ωk (or tk ≤ t < ωk when ωk = +∞); and if ωk < +∞, then (5.3.9)
||Fk∗ (ωk )gij (ωk ) − hij ||C q (Ha ) = δ.
Clearly the above argument shows that the set of t where we can extend the harmonic diffeomorphisms as desired is open. To verify claim (5.3.9), we thus only need to show that if we have a family of harmonic diffeomorphisms Fk (t) such as we desire for tk ≤ t < ω(< +∞), we can take the limit of Fk (t) as t → ω to get a harmonic diffeomorphism Fk (ω) satisfying ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) ≤ δ, and if ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) < δ, then we can extend Fk (ω) forward in time a little (i.e., we can find a constant mean curvature hypersurface near Fk (ω)(∂Ha ) in M with the metric gij (t) for each t close to ω and with the same area a for each component). Note that (5.3.10)
||Fk∗ (t)gij (t) − hij ||C q (Ha ) < δ.
for tk ≤ t < ω and the metrics gij (t) for tk ≤ t ≤ ω are uniformly equivalent. We can find a subsequence tn → ω for which Fk (tn ) converge to Fk (ω) in C q−1 (Ha ) and the limit map has ||Fk∗ (ω)gij (ω) − hij ||C q−1 (Ha ) ≤ δ. We need to check that Fk (ω) is still a diffeomorphism. We at least know Fk (ω) is a local diffeomorphism, and Fk (ω) is the limit of diffeomorphisms, so the only possibility of overlap is at the boundary. Hence we use the fact that Fk (ω)(∂Ha ) is still strictly concave since q is large and δ is small to prevent the boundary from touching itself. Thus Fk (ω) is a diffeomorphism. A limit of harmonic maps is harmonic, so Fk (ω) is a harmonic diffeomorphism from Ha into M with the metric gij (ω). Moreover Fk (ω) takes ∂Ha to the constant mean curvature hypersurface ∂(Fk (ω)(Ha )) of the area a in (M, gij (ω)) and continue to satisfy the free boundary condition. As a consequence of the standard regularity result of elliptic partial differential equations (see for example [48]), the map Fk (ω) ∈ C ∞ (Ha ) and then from (5.3.10) we have ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) ≤ δ. If ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) = δ, we then finish the proof of the claim. So we may assume that ||Fk∗ (ω)gij (ω) − hij ||C q (Ha ) < δ. We want to show that Fk (ω) can be extended forward in time a little.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
349
We argue by contradiction. Suppose not, then we consider the new sequence of the manifolds M with metric gij (ω) and the origins Fk (ω)(P ). Since Fk (ω) are close to isometries, the injectivity radii of the metrics gij (ω) at Fk (ω)(P ) do not go to zero, e with the and we can extract a subsequence which converges to a hyperbolic limit H e e e metric hij and the origin P and with finite volume. The new limit H has at least as many cusps as the old limit H, since we choose H with cusps as few as possible. By ek exhausting the definition of convergence, we can find a sequence of compact sets B e e e ek H and containing P , and a sequence of diffeomorphisms Fk of neighborhoods of B e e e e into M with Fk (P ) = Fk (ω)(P ) such that for each compact set B in H and each integer m ||Fek∗ (gij (ω)) − e hij ||C m (B) e →0
ek ) will contain all points out to any as k → +∞. For large enough k the set Fek (B fixed distance we need from the point Fk (ω)(P ), and then ek ) ⊃ Fk (ω)(Ha ) Fek (B
since the points of Ha have a bounded distance from P and Fk (ω) are reasonably close to preserving the metrics. Hence we can form the composition e Gk = Fek−1 ◦ Fk (ω) : Ha → H.
Arbitrarily fix δ ′ ∈ (δ, δ0 ). Since the Fek are as close to preserving the metric as we like, we have ||G∗k e hij − hij ||C q (Ha ) < δ ′
for all sufficiently large k. By Lemma 5.3.7, we deduce that there exists an isometry e and then (M, gij (ω), Fk (ω)(P )) (on compact subsets) is very close to I of H to H, (H, hij , P ) as long as δ small enough and k large enough. Since Fk (ω)(∂Ha ) is strictly concave and the foliation of a neighborhood of Fk (ω)(∂Ha ) by constant mean curvature hypersurfaces has the area as a function with nonzero gradient, by the implicit function theorem, there exists a unique constant mean curvature hypersurface with the same area a near Fk (ω)(∂Ha ) in M with the metric gij (t) for t close to ω. Hence, when k sufficiently large, Fk (ω) can be extended forward in time a little. This is a contradiction and we have proved claim (5.3.9). We further claim that there must be some k such that ωk = +∞ (i.e., we can smoothly continue the family of harmonic diffeomorphisms Fk (t) for all tk ≤ t < +∞, in other words, there must be at least one hyperbolic piece persisting). We argue by contradiction. Suppose for each k large enough, we can continue the family Fk (t) for tk ≤ t ≤ ωk < +∞ with ||Fk∗ (ωk )gij (ωk ) − hij ||C q (Ha ) = δ. Then as before, we consider the new sequence of the manifolds M with metrics gij (ωk ) and origins Fk (ωk )(P ). For sufficiently large k, we can obtain diffeomorphisms Fek of ek into M with Fek (Pe ) = Fk (ωk )(P ) which are as close to preserving neighborhoods of B ek is a sequence of compact sets, exhausting some the metric as we like, where B ˜ of finite volume and with no fewer cusps (than H), and hyperbolic three-manifold H,
350
H.-D. CAO AND X.-P. ZHU
ek ) will contain all the points out to any fixed containing P˜ ; moreover, the set Fek (B distance we need from the point Fk (ωk )(P ); and hence ek ) ⊇ Fk (ωk )(Ha ) Fek (B
since Ha is at bounded distance from P and Fk (ωk ) is reasonably close to preserving the metrics. Then we can form the composition Gk = Fek−1 ◦ Fk (ωk ) :
˜ Ha → H.
Since the Fek are as close to preserving the metric as we like, for any δe > δ we have hij − hij ||C q (Ha ) < δe ||G∗k e
for large enough k. Then a subsequence of Gk converges at least in C q−1 (Ha ) topology e By the same reason as in the argument of previous two to a map G∞ of Ha into H. paragraphs, the limit map G∞ is a smooth harmonic diffeomorphism from Ha into ˜ ij , and takes ∂Ha to a constant mean curvature hypersurface e with the metric h H e ˜ G∞ (∂Ha ) of (H, hij ) with the area a, and also satisfies the free boundary condition. Moreover we still have (5.3.11)
||G∗∞ e hij − hij ||C q (Ha ) = δ.
e with Now by Lemma 5.3.7 we deduce that there exists an isometry I of H to H dC l0 (Ha ) (G∞ , I) < ε0 .
ea and Ha , we see that the map I −1 ◦ G∞ is a harmonic By using I to identify H diffeomorphism of Ha to itself which satisfies the free boundary condition and dC l0 (Ha ) (I −1 ◦ G∞ , Id) < ε0 . From the uniqueness in Lemma 5.3.8 we conclude that I −1 ◦ G∞ = Id which contradicts with (5.3.11). This shows at least one hyperbolic piece persists. Moreover the pull-back of the solution metric gij (t) by Fk (t), for tk ≤ t < +∞, is as close to the hyperbolic metric hij as we like. We can continue to form other persistent hyperbolic pieces in the same way as long as there are any points Pk outside of the chosen pieces where the injectivity radius at times tk → ∞ are all at least some fixed positive number ρ > 0. The only modification in the proof is to take the new limit H to have the least possible number of cusps out of all remaining possible limits. Note that the volume of the normalized Ricci flow is constant in time. Therefore by combining with Margulis lemma (see for example [55] [76]), we have proved that there exists a finite collection of complete noncompact hyperbolic three-manifolds H1 , . . . , Hm with finite volume, a small number a > 0 and a time T < +∞ such that for all t beyond T we can find diffeomorphisms ϕl (t) of (Hl )a into M , 1 ≤ l ≤ m, so that the pull-back of the solution metric gij (t) by ϕl (t) is as close to the hyperbolic metrics as we like and the exceptional part of M where the points are not in the image of any ϕl has the injectivity radii everywhere as small as we like.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
351
Part III: Incompressibility We remain to show that the boundary tori of any persistent hyperbolic piece are incompressible, in the sense that the fundamental group of the torus injects into that of the whole manifold. The argument of this part is a parabolic version of Schoen and Yau’s minimal surface argument in [109, 110, 111]. Let B be a small positive number and assume the above positive number a is much smaller than B. Denote by Ma a persistent hyperbolic piece of the manifold M truncated by boundary tori of area a with constant mean curvature and denote by ◦
Mac = M \ Ma the part of M exterior to Ma . Thus there is a persistent hyperbolic piece MB ⊂ Ma of the manifold M truncated by boundary tori of area B with constant ◦
mean curvature. We also denote by MBc = M \ MB . By Van Kampen’s Theorem, if π1 (∂MB ) injects into π1 (MBc ) then it injects into π1 (M ) also. Thus we only need to show π1 (∂MB ) injects into π1 (MBc ). We will argue by contradiction. Let T be a torus in ∂MB . Suppose π1 (T ) does not inject into π1 (MBc ), then by Dehn’s Lemma the kernel is a cyclic subgroup of π1 (T ) generated by a primitive element. The work of Meeks-Yau [86] or Meeks-Simon-Yau [87] shows that among all disks in MBc whose boundary curve lies in T and generates the kernel, there is a smooth embedded disk normal to the boundary which has the least possible area. Let A = A(t) be the area of this disk. This is defined for all t sufficiently large. We will show that A(t) decreases at a rate bounded away from zero which will be a contradiction. Let us compute the rate at which A(t) changes under the Ricci flow. We need to show A(t) decrease at least at a certain rate, and since A(t) is the minimum area to bound any disk in the given homotopy class, it suffices to find some such disk whose area decreases at least that fast. We choose this disk as follows. Pick the minimal disk at time t0 , and extend it smoothly a little past the boundary torus since the minimal disk is normal to the boundary. For times t a little bigger than t0 , the boundary torus may need to move a little to stay constant mean curvature with area B as the metrics change, but we leave the surface alone and take the bounding disk to be the one cut ˜ of such disk comes from the off from it by the new torus. The change of the area A(t) change in the metric and the change in the boundary. For the change in the metric, we choose an orthonormal frame X, Y, Z at a point x in the disk so that X and Y are tangent to the disk while Z is normal and compute the rate of change of the area element dσ on the disk as T ∂ 1 2 − 2(Rij )T )dσ dσ = (g ij )T r(gij ∂t 2 3 2 = r − Ric (X, X) − Ric (Y, Y ) dσ, 3
since the metric evolves by the normalized Ricci flow. Here (·)T denotes the tangential projections on the disk. Notice the torus T may move in time to preserve constant mean curvature and constant area B. Suppose the boundary of the disk evolves with a normal velocity N . The change of the area at boundary along a piece of length ds ˜ is given by is given by N ds. Thus the total change of the area A(t) dA˜ = dt
Z Z
Z 2 r − Ric (X, X) − Ric (Y, Y ) dσ + N ds. 3 ∂
352
H.-D. CAO AND X.-P. ZHU
Note that Ric (X, X) + Ric (Y, Y ) = R(X, Y, X, Y ) + R(X, Z, X, Z) + R(Y, X, Y, X) + R(Y, Z, Y, Z) 1 = R + R(X, Y, X, Y ). 2 By the Gauss equation, the Gauss curvature K of the disk is given by K = R(X, Y, X, Y ) + det II where II is the second fundamental form of the disk in MBc . This gives at t = t0 , Z Z Z Z Z dA 2 1 ≤ r − R dσ − (K − det II)dσ + N ds dt 3 2 ∂ Since the bounding disk is a minimal surface, we have det II ≤ 0. The Gauss-Bonnet Theorem tells us that for a disk Z Z Z Kdσ + kds = 2π ∂
where k is the geodesic curvature of the boundary. Thus we obtain Z Z Z Z 2 1 dA ≤ r − R dσ + kds + N ds − 2π. (5.3.12) dt 3 2 ∂ ∂ Recall that we are assuming Rmin (t) increases monotonically to −6 as t → +∞. By the evolution equation of the scalar curvature, d Rmin (t) ≥ 4(r(t) − Rmin (t)) dt and then Z
∞ 0
(r(t) − Rmin (t))dt < +∞.
This implies that r(t) → −6 as t → +∞ by using the derivatives estimate for the curvatures. Thus for every ε > 0 we have 1 2 r − R ≤ −(1 − ε) 3 2 for t sufficiently large. And then the first term on RHS of (5.3.12) is bounded above by Z Z 1 2 r − R dσ ≤ −(1 − ε)A. 3 2 The geodesic curvature k of the boundary of the minimal disk is the acceleration of a curve moving with unit speed along the intersection of the disk with the torus;
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
353
since the disk and torus are normal, this is the same as the second fundamental form of the torus in the direction of the curve of intersection. Now if the metric were actually hyperbolic, the second fundamental form of the torus would be exactly 1 in all directions. Note that the persistent hyperbolic pieces are as close to the standard hyperbolic as we like. This makes that the second term of RHS of (5.3.12) is bounded above by Z kds ≤ (1 + ε0 )L ∂
for some sufficiently small positive number ε0 > 0, where L is the length of the boundary curve. Also since the metric on the persistent hyperbolic pieces are close to the standard hyperbolic as we like, its change under the normalized Ricci flow is as small as we like; So the motion of the constant mean curvature torus of fixed area B will have a normal velocity N as small as we like. This again makes the third term of RHS of (5.3.12) bounded above by Z N ds ≤ ε0 L. ∂
Combining these estimates, we obtain (5.3.13)
dA ≤ (1 + 2ε0 )L − (1 − ε0 )A − 2π dt
on the persistent hyperbolic piece, where ε0 is some sufficiently small positive number. We next need to bound the length L in terms of the area A. Since a is much smaller than B, for large t the metric is as close as we like to the standard hyperbolic one; not just on the persistent hyperbolic piece MB but as far beyond as we like. Thus for a long distance into MBc the metric will look nearly like a standard hyperbolic cusplike collar. Let us first recall a special coordinate system on the standard hyperbolic cusp projecting beyond torus T1 in ∂H1 as follows. The universal cover of the flat torus T1 can be mapped conformally to the x-y plane so that the deck transformation of T1 become translations in x and y, and so that the Euclidean area of the quotient is 1; then these coordinates are unique up to a translation. The hyperbolic cusp projecting beyond the torus T1 in ∂H1 can be parametrized by {(x, y, z) ∈ R3 | z > 0} with the hyperbolic metric (5.3.14)
ds2 =
dx2 + dy 2 + dz 2 . z2
Note that we can make the solution metric, in an arbitrarily large neighborhood of the torus T (of ∂MB ), as close to hyperbolic as we wish (in the sense that there exists a diffeomorphism from a large neighborhood of the torus TB (of ∂HB ) on the standard hyperbolic cusp to the above neighborhood of the torus T (of ∂MB ) such that the pull-back of the solution metric by the diffeomorphism is as close to the hyperbolic metric as we wish). Then by using this diffeomorphism (up to a slight modification) we can parametrize the cusplike tube of MBc projecting beyond the torus T in ∂MB by {(x, y, z) | z ≥ ζ} where the height ζ is chosen so that the torus in the hyperbolic cusp at height ζ has the area B. Now consider our minimal disk, and let L(z) be the length of the curve of the intersection of the disk with the torus at height z in the above coordinate system,
354
H.-D. CAO AND X.-P. ZHU
and also let A(z) be the area of the part of the disk between ζ and z. We now want to derive a monotonicity formula on the area A(z) for the minimal surface. For almost every z the intersection of the disk with the torus at height z is a smooth embedded curve or a finite union of them by the standard transversality theorem. If there is more than one curve, at least one of them is not homotopic to a point in T and represents the primitive generator in the kernel of π1 (T ) such that a part of the original disk beyond height z continues to a disk that bounds it. We extend this disk back to the initial height ζ by dropping the curve straight down. Let ˜ ˜ L(z) be the length of the curve we picked at height z; of course L(z) ≤ L(z) with ˜ equality if it is the only piece. Let L(w) denote the length of the same curve in the x-y plane dropped down to height w for ζ ≤ w ≤ z. In the hyperbolic space we would have z ˜ ˜ L(w) = L(z) w ˜ exactly. In our case there is a small error proportional to L(z) and we can also take it ˜ ˜ proportional to the distance z − w by which it drops since L(w)| w=z = L(z) and the ∞ solution metric is close to the hyperbolic in the Cloc topology. Thus, for arbitrarily given δ > 0 and ζ ∗ > ζ, as the solution metric is sufficiently close to hyperbolic, we have z ˜ ˜ ˜ ≤ δ(z − w)L(z) |L(w) − L(z)| w for all z and w in ζ ≤ w ≤ z ≤ ζ ∗ . Now given ε and ζ ∗ pick δ = 2ε/ζ ∗ . Then z˜ 2ε(z − w) ˜ (5.3.15) L(w) ≤ L(z) 1+ . w w When we drop the curve vertically for the construction of the new disk we get an area ˜ A(z) between ζ and z given by Z z ˜ L(w) ˜ dw. A(z) = (1 + o(1)) w ζ Here and in the following o(1) denotes various small error quantities as the solution metric close to hyperbolic. On the other hand if we do not drop vertically we pick up even more area, so the area A(z) of the original disk between ζ and z has Z z L(w) (5.3.16) A(z) ≥ (1 − o(1)) dw. w ζ Since the original disk minimized among all disks bounded a curve in the primitive generator of the kernel of π1 (T ), and the new disk beyond the height z is part of the original disk, we have ˜ A(z) ≤ A(z) and then by combining with (5.3.15), Z z Z z L(w) 1 − 2ε 2εz ˜ dw ≤ (1 + o(1))z L(z) + 3 dw w w2 w ζ ζ (z − ζ) z−ζ ≤ (1 + o(1))L(z) 1+ε . ζ ζ
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
355
˜ Here we used the fact that L(z) ≤ L(z). Since the solution metric is sufficiently close to hyperbolic, we have Z z d L(w) L(z) dw = dz w z ζ Z z ζ z−ζ L(w) 1−ε dw ≥ (1 − o(1)) z(z − ζ) ζ w ζ Z z L(w) 1 + 2ε 1 − dw, ≥ z−ζ z w ζ or equivalently (5.3.17)
d log dz
z 1+2ε (z − ζ)
Z
ζ
z
L(w) dw w
≥ 0.
This is the desired monotonicity formula for the area A(z). It follows directly from (5.3.16) and (5.3.17) that
or equivalently
z 1+2ε A(z) ≥ (1 − o(1))ζ 2ε L(ζ), (z − ζ) 2ε z z A(z) L(ζ) ≤ (1 + o(1)) ζ z−ζ
for all z ∈ [ζ, ζ ∗ ]. Since the solution metric, in an arbitrarily large neighborhood of the torus T (of ∂MB ), as √ close to hyperbolic as we wish, we may assume that√ζ ∗ is so √ ∗ ∗ ∗ large that ζ > ζ and √ζ ∗ζ−ζ is close to 1, and also ε > 0 is so small that ( ζζ )2ε is close to 1. Thus for arbitrarily small δ0 > 0, we have p ζ∗ . (5.3.18) L(ζ) ≤ (1 + δ0 )A
Now recall that (5.3.13) states
dA ≤ (1 + 2ε0 )L − (1 − ε0 )A − 2π. dt We now claim that if (1 + 2ε0 )L − (1 − ε0 )A ≥ 0 then L = L(ζ) is uniformly bounded from above. Indeed by the assumption we have A(ζ ∗ ) ≤
(1 + 2ε0 ) L(ζ) (1 − ε0 )
√ since A(ζ ∗ ) ≤ A. By combining with (5.3.16) we have some z0 ∈ ( ζ ∗ , ζ ∗ ) satisfying Z ζ∗ L(z0 ) ∗ p ∗ L(w) ζ − ζ ≤ √ dw z0 w ∗ ζ ≤ (1 + o(1))A(ζ ∗ ) 1 + 2ε0 ≤ (1 + o(1)) L(ζ). 1 − ε0
356
H.-D. CAO AND X.-P. ZHU
Thus for ζ ∗ suitably large, by noting that the solution metric on a large neighborhood of T (of ∂MB ) is sufficiently close to hyperbolic, we have (5.3.19)
L(z0 ) ≤ (1 + 4ε0 )
z0 L(ζ) ζ∗
√ for some z0 ∈ ( ζ ∗ , ζ ∗ ). It is clear that we may assume the intersection curve between the minimal disk with the torus at this height z0 is smooth and embedded. If the intersection curve at the height z0 has more than one piece, as before one of them will represent the primitive generator in the kernel of π1 (T ), and we can ignore the others. Let us move (the piece of) the intersection curve on the torus at height z0 through as small as possible area in the same homotopy class of π1 (T ) to a curve which is a geodesic circle in the flat torus coming from our special coordinates, and then drop this geodesic circle vertically in the special coordinates to obtain another new disk. We will compare the area of this new disk with the original minimal disk as follows. Denote by G the length of the geodesic circle in the standard hyperbolic cusp at height 1. Then the length of the geodesic circle at height z0 will be G/z0 . Observe that given an embedded curve of length l circling the cylinder S 1 × R of circumference w once, it is possible to deform the curve through an area not bigger than lw into a meridian circle. Note that (the piece of) the intersection curve represents the primitive generator in the kernel of π1 (T ). Note also that the solution metric is sufficiently close to the hyperbolic metric. Then the area of the deformation from (the piece of) the intersection curve on the torus at height z0 to the geodesic circle at height z0 is bounded by G · L(z0 ). (1 + o(1)) z0 The area to drop the geodesic circle from height z0 to height ζ is bounded by Z z0 G (1 + o(1)) dw. w2 ζ Hence comparing the area of the original minimal disk to that of this new disk gives 1 1 L(z0 ) + − . A(z0 ) ≤ (1 + o(1))G z0 ζ z0 √ By (5.3.18), (5.3.19) and the fact that z0 ∈ ( ζ ∗ , ζ ∗ ), this in turn gives L(ζ) ≤ (1 + δ0 )A(z0 ) L(ζ) 1 ≤ (1 + δ0 )G (1 + 4ε0 ) ∗ + . ζ ζ Since ζ ∗ is suitably large, we obtain L(ζ) ≤ 2G/ζ This gives the desired assertion since G is fixed from the geometry of the limit hyperbolic manifold H and ζ is very large as long as the area B of ∂MB small enough. Thus the combination of (5.3.13), (5.3.18) and the assertion implies that either d A ≤ −2π, dt
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
357
or d A ≤ (1 + 2ε0 )L − (1 − ε0 )A − 2π dt 2G ≤ (1 + 2ε0 ) − 2π ζ ≤ −π, since the solution metric on a very large neighborhood of the torus T (of ∂MB ) is sufficiently close to hyperbolic and ζ is very large as the area B of ∂MB small enough. This is impossible because A ≥ 0 and the persistent hyperbolic pieces go on forever. The contradiction shows that π1 (T ) in fact injects into π1 (MBc ). This proves that π1 (∂MB ) injects into π1 (M ). Therefore we have completed the proof of Theorem 5.3.4. 6. Ancient κ-solutions. Let us consider a solution of the Ricci flow on a compact manifold. If the solution blows up in finite time (i.e., the maximal solution exists only on a finite time interval), then as we saw in Chapter 4 a sequence of rescalings of the solution around the singularities converge to a solution which exists at least on the time interval (−∞, T ) for some finite number T . Furthermore, by Perelman’s no local collapsing theorem I (Theorem 3.3.2), we see that the limit is κ-noncollapsed on all scales for some positive constant κ. In addition, if the dimension n = 3 then the Hamilton-Ivey pinching estimate implies that the limiting solution must have nonnegative curvature operator. We call a solution to the Ricci flow an ancient κ-solution if it is complete (either compact or noncompact) and defined on an ancient time interval (−∞, T ) with T > 0, has nonnegative curvature operator and bounded curvature, and is κ-noncollapsed on all scales for some positive constant κ. In this chapter we study ancient κ-solutions of the Ricci flow. We will obtain crucial curvature estimates of such solutions and determine their structures in lower dimensional cases. 6.1. Preliminaries. We first present a useful geometric property, given by Chen and the second author in [34], for complete noncompact Riemannian manifolds with nonnegative sectional curvature. Let (M, gij ) be an n-dimensional complete Riemannian manifold and let ε be a positive constant. We call an open subset N ⊂ M an ε-neck of radius r if −1 (N, r−2 gij ) is ε-close, in the C [ε ] topology, to a standard neck Sn−1 × I, where Sn−1 is the round (n − 1)-sphere with scalar curvature 1 and I is an interval of length 2ε−1 . The following result is, to some extent, in similar spirit of Yau’s volume lower bound estimate [128]. Proposition 6.1.1 (Chen-Zhu [34]). There exists a positive constant ε0 = ε0 (n) such that every complete noncompact n-dimensional Riemannian manifold (M, gij ) of nonnegative sectional curvature has a positive constant r0 such that any ε-neck of radius r on (M, gij ) with ε ≤ ε0 must have r ≥ r0 . Proof. We argue by contradiction. Suppose there exist a sequence of positive constants εα → 0 and a sequence of n-dimensional complete noncompact Riemannian α manifolds (M α , gij ) such that for each fixed α, there exists a sequence of εα -necks Nk of radius at most 1/k in M α with centers Pk divergent to infinity.
358
H.-D. CAO AND X.-P. ZHU
Fix a point P on the manifold M α and connect each Pk to P by a minimizing geodesic γk . By passing to a subsequence we may assume the angle θkl between geodesic γk and γl at P is very small and tends to zero as k, l → +∞, and the length of γk+1 is much bigger than the length of γk . Let us connect Pk to Pl by a minimizing geodesic ηkl . For each fixed l > k, let P˜k be a point on the geodesic γl such that the geodesic segment from P to P˜k has the same length as γk and consider the triangle ∆P Pk P˜k in M α with vertices P , Pk and P˜k . By comparing with the corresponding triangle in the Euclidean plane R2 whose sides have the same corresponding lengths, Toponogov’s comparison theorem implies 1 ˜ d(Pk , Pk ) ≤ 2 sin θkl · d(Pk , P ). 2 Since θkl is very small, the distance from Pk to the geodesic γl can be realized by a geodesic ζkl which connects Pk to a point Pk′ on the interior of the geodesic γl and has length at most 2 sin( 12 θkl ) · d(Pk , P ). Clearly the angle between ζkl and γl at the intersection point Pk′ is π2 . Consider α to be fixed and sufficiently large. We claim that for large enough k, each minimizing geodesic γl with l > k, connecting P to Pl , goes through the neck Nk . Suppose not; then the angle between γk and ζkl at Pk is close to either zero or π since Pk is in the center of an εα -neck and α is sufficiently large. If the angle between γk and ζkl at Pk is close to zero, we consider the triangle ∆P Pk Pk′ in M α with vertices P , Pk , and Pk′ . Note that the length between Pk and Pk′ is much smaller than the lengths from Pk or Pk′ to P . By comparing the angles of this triangle with those of the corresponding triangle in the Euclidean plane with the same corresponding lengths and using Toponogov’s comparison theorem, we find that it is impossible. Thus the angle between γk and ζkl at Pk is close to π. We now consider the triangle ∆Pk Pk′ Pl in M α with the three sides ζkl , ηkl and the geodesic segment from Pk′ to Pl on γl . We have seen that the angle of ∆Pk Pk′ Pl at Pk is close to zero and the angle at Pk′ is π2 . ¯ P¯k P¯′ P¯l in the Euclidean plane R2 whose By comparing with corresponding triangle ∆ k sides have the same corresponding lengths, Toponogov’s comparison theorem implies 3 ∠P¯l P¯k P¯k′ + ∠P¯l P¯k′ P¯k ≤ ∠Pl Pk Pk′ + ∠Pl Pk′ Pk < π. 4 This is impossible since the length between P¯k and P¯k′ is much smaller than the length from P¯l to either P¯k or P¯k′ . So we have proved each γl with l > k passes through the neck Nk . Hence by taking a limit, we get a geodesic ray γ emanating from P which passes through all the necks Nk , k = 1, 2, . . . , except a finite number of them. Throwing these finite number of necks away, we may assume γ passes through all necks Nk , k = 1, 2, . . . . Denote the center sphere of Nk by Sk , and their intersection points with γ by pk ∈ Sk ∩ γ, k = 1, 2, . . . . Take a sequence of points γ(m) with m = 1, 2, . . . . For each fixed neck Nk , arbitrarily choose a point qk ∈ Nk near the center sphere Sk and draw a geodesic segment γ km from qk to γ(m). Now we claim that for any neck Nl with l > k, γ km will pass through Nl for all sufficiently large m. We argue by contradiction. Let us place all the necks Ni horizontally so that the geodesic γ passes through each Ni from the left to the right. We observe that the geodesic segment γ km must pass through the right half of Nk ; otherwise γ km cannot be minimal. Then for large enough m, the distance from pl to the geodesic segment
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
359
γ km must be achieved by the distance from pl to some interior point pk ′ of γ km . Let us draw a minimal geodesic η from pl to the interior point pk ′ with the angle at the intersection point pk ′ ∈ η ∩ γ km to be π2 . Suppose the claim is false. Then the angle between η and γ at pl is close to 0 or π since εα is small. If the angle between η and γ at pl is close to 0, we consider the triangle ∆pl pk ′ γ(m) ¯ pl p¯k ′ γ¯ (m) in the plane with the same correand construct a comparison triangle ∆¯ sponding length. Then by Toponogov’s comparison theorem, we see the sum of the ¯ pl p¯k ′ γ¯ (m) is less than 3π/4, which is imposinner angles of the comparison triangle ∆¯ sible. If the angle between η and γ at pl is close to π, by drawing a minimal geodesic ξ from qk to pl , we see that ξ must pass through the right half of Nk and the left half of Nl ; otherwise ξ cannot be minimal. Thus the three inner angles of the triangle ∆pl pk ′ qk are almost 0, π/2, and 0 respectively. This is also impossible by the Toponogov comparison theorem. Hence we have proved that the geodesic segment γ km passes through Nl for m large enough. Consider the triangle ∆pk qk γ(m) with two long sides pk γ(m)(⊂ γ) and qk γ(m)(= γ km ). For any s > 0, choose points p˜k on pk γ(m) and q˜k on qk γ(m) with d(pk , p˜k ) = d(qk , q˜k ) = s. By Toponogov’s comparison theorem, we have
d(˜p , q˜ ) 2 k
k
d(pk , qk ) =
¯ pk γ(m)˜ d(˜ pk , γ(m))2 + d(˜ qk , γ(m))2 − 2d(˜ pk , γ(m))d(˜ qk , γ(m)) cos ∡(˜ qk ) 2 2 ¯ k γ(m)qk ) d(pk , γ(m)) + d(qk , γ(m)) − 2d(pk , γ(m))d(qk , γ(m)) cos ∡(p
≥
¯ pk γ(m)˜ d(˜ pk , γ(m))2 + d(˜ qk , γ(m))2 − 2d(˜ pk , γ(m))d(˜ qk , γ(m)) cos ∡(˜ qk ) ¯ pk γ(m)˜ d(pk , γ(m))2 + d(qk , γ(m))2 − 2d(pk , γ(m))d(qk , γ(m)) cos ∡(˜ qk )
=
¯ pk γ(m)˜ (d(˜ pk , γ(m)) − d(˜ qk , γ(m)))2 + 2d(˜ pk , γ(m))d(˜ qk , γ(m))(1 − cos ∡(˜ qk )) 2 ¯ (d(˜ pk , γ(m)) − d(˜ qk , γ(m))) + 2d(pk , γ(m))d(qk , γ(m))(1 − cos ∡(˜ pk γ(m)˜ qk ))
≥
d(˜ pk , γ(m))d(˜ qk , γ(m)) d(pk , γ(m))d(qk , γ(m))
→1
¯ k γ(m)qk ) and ∡(˜ ¯ pk γ(m)˜ as m → ∞, where ∡(p qk ) are the corresponding angles of the comparison triangles. Letting m → ∞, we see that γ km has a convergent subsequence whose limit γ k is a geodesic ray passing through all Nl with l > k. Let us denote by pj = γ(tj ), j = 1, 2, . . .. From the above computation, we deduce that d(pk , qk ) ≤ d(γ(tk + s), γ k (s)) for all s > 0. Let ϕ(x) = limt→+∞ (t − d(x, γ(t))) be the Busemann function constructed from the ray γ. Note that the level set ϕ−1 (ϕ(pj )) ∩ Nj is close to the center sphere Sj for any j = 1, 2, . . .. Now let qk be any fixed point in ϕ−1 (ϕ(pk )) ∩ Nk . By the definition of Busemann function ϕ associated to the ray γ, we see that ϕ(γ k (s1 )) − ϕ(γ k (s2 )) = s1 − s2 for any s1 , s2 ≥ 0. Consequently, for each l > k, by choosing s = tl − tk , we see γ k (tl − tk ) ∈ ϕ−1 (ϕ(pl )) ∩ Nl . Since γ(tk + tl − tk ) = pl , it follows that d(pk , qk ) ≤ d(pl , γ k (s)).
360
H.-D. CAO AND X.-P. ZHU
with s = tl − tk > 0. This implies that the diameter of ϕ−1 (ϕ(pk )) ∩ Nk is not greater than the diameter of ϕ−1 (ϕ(pl )) ∩ Nl for any l > k, which is a contradiction for l much larger than k. Therefore we have proved the proposition. In [63], Hamilton discovered an important repulsion principle (cf. Theorem 21.4 of [63]) about the influence of a bump of strictly positive curvature in a complete noncompact manifold of nonnegative sectional curvature. Namely minimal geodesic paths that go past the bump have to avoid it. As a consequence he obtained a finite bump theorem (cf. Theorem 21.5 of [63]) that gives a bound on the number of bumps of curvature. Let M be a complete noncompact Riemannian manifold with nonnegative sectional curvature K ≥ 0. A geodesic ball B(p, r) of radius r centered at a point p ∈ M is called a curvature β-bump if sectional curvature K ≥ β/r2 at all points in the ball. The ball B(p, r) is called λ-remote from an origin O if d(p, O) ≥ λr. Finite Bump Theorem (Hamilton [63]). For every β > 0 there exists λ < ∞ such that in any complete manifold of nonnegative sectional curvature there are at most a finite number of disjoint balls which are λ-remote curvature β-bumps. This finite bump theorem played an important role in Hamilton’s study of the behavior of singularity models at infinity and in the dimension reduction argument he developed for the Ricci flow (cf. Section 22 of [63], see also [29] for application to the K¨ahler-Ricci flow and uniformization problem in complex dimension two). A special consequence of the finite bump theorem is that if we have a complete noncompact solution to the Ricci flow on an ancient time interval −∞ < t < T with T > 0 satisfying certain local injectivity radius bound, with curvature bounded at each time and with asymptotic scalar curvature ratio A = lim sup Rs2 = ∞, then we can find a sequence of points pj going to ∞ (as in the following Lemma 6.1.3) such that a cover of the limit of dilations around these points at time t = 0 splits as a product with a flat factor. The following result, obtained by Chen and the second author in [34], is in similar spirit as Hamilton’s finite bumps theorem and its consequence. The advantage is that we will get in the limit of dilations a product of the line with a lower dimensional manifold, instead of a quotient of such a product. Proposition 6.1.2 (Chen-Zhu [34]). Suppose (M, gij ) is a complete ndimensional Riemannian manifold with nonnegative sectional curvature. Let P ∈ M be fixed, and Pk ∈ M a sequence of points and λk a sequence of positive numbers with d(P, Pk ) → +∞ and λk d(P, Pk ) → +∞. Suppose also that the marked manifolds ∞ f. Then the (M, λ2k gij , Pk ) converge in the Cloc topology to a Riemannian manifold M f splits isometrically as the metric product of the form R × N , where N is a limit M Riemannian manifold with nonnegative sectional curvature. Proof. Let us denote by |OQ| = d(O, Q) the distance between two points O, Q ∈ M . Without loss of generality, we may assume that for each k, (6.1.1)
1 + 2|P Pk | ≤ |P Pk+1 |.
Draw a minimal geodesic γk from P to Pk and a minimal geodesic σk from Pk to Pk+1 , both parametrized by arclength. We may further assume (6.1.2)
θk = |∡(γ˙ k (0), γ˙ k+1 (0))| <
1 . k
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
361
∞ By assumption, the sequence (M, λ2k gij , Pk ) converges (in the Cloc topology) to a f e Riemannian manifold (M , geij , P ) with nonnegative sectional curvature. By a further choice of subsequences, we may also assume γk and σk converge to geodesic rays γ e f, and and σ e starting at Pe respectively. We will prove that γ˜ ∪ σ ˜ forms a line in M f must be splitted as R × N . then by the Toponogov splitting theorem [89] the limit M We argue by contradiction. Suppose γ e∪σ e is not a line; then for each k, there exist two points Ak ∈ γk and Bk ∈ σk such that as k → +∞, λ d(Pk , Ak ) → A > 0, k λk d(Pk , Bk ) → B > 0, (6.1.3) λk d(Ak , Bk ) → C > 0, but A + B > C.
Pk σk (((( ( ( B ( k ( δk ((((P ((( k+1 ( ( γk Ak ( (( ( ( ( ((( (((( ( ( (((( (((( ( ( P Now draw a minimal geodesic δk from Ak to Bk . Consider comparison triangles ¯ P¯k P¯ P¯k+1 and △ ¯ P¯k A¯k B ¯k in R2 with △ |P¯k P¯ | = |Pk P |, |P¯k P¯k+1 | = |Pk Pk+1 |, |P¯ P¯k+1 | = |P Pk+1 |, ¯k | = |Pk Bk |, |A¯k B ¯k | = |Ak Bk |. and |P¯k A¯k | = |Pk Ak |, |P¯k B By Toponogov’s comparison theorem [8], we have (6.1.4)
¯k ≥ ∡P¯ P¯k P¯k+1 . ∡A¯k P¯k B
On the other hand, by (6.1.2) and using Toponogov’s comparison theorem again, we have (6.1.5)
1 ∡P¯k P¯ P¯k+1 ≤ ∡Pk P Pk+1 < , k
and since |P¯k P¯k+1 | > |P¯ P¯k | by (6.1.1), we further have (6.1.6)
1 ∡P¯k P¯k+1 P¯ ≤ ∡P¯k P¯ P¯k+1 < . k
Thus the above inequalities (6.1.4)-(6.1.6) imply that ¯k > π − 2 . ∡A¯k P¯k B k
362
H.-D. CAO AND X.-P. ZHU
Hence (6.1.7)
¯k |2 ≥ |A¯k P¯k |2 + |P¯k B ¯k |2 − 2|A¯k P¯k | · |P¯k B ¯k | cos π − 2 . |A¯k B k
Multiplying the above inequality by λ2k and letting k → +∞, we get C ≥A+B which contradicts (6.1.3). Therefore we have proved the proposition. Let M be an n-dimensional complete noncompact Riemannian manifold. Pick an origin O ∈ M . Let s be the geodesic distance to the origin O of M , and R the scalar curvature. Recall that in Chapter 4 we have defined the asymptotic scalar curvature ratio A = lim sup Rs2 . s→+∞
We now state a useful lemma of Hamilton (Lemma 22.2 in [63]) about picking local (almost) maximum curvature points at infinity. Lemma 6.1.3. Given a complete noncompact Riemannian manifold with bounded curvature and with asymptotic scalar curvature ratio A = lim sup Rs2 = +∞, s→+∞
we can find a sequence of points xj divergent to infinity, a sequence of radii rj and a sequence of positive numbers δj → 0 such that (i) R(x) ≤ (1 + δj )R(xj ) for all x in the ball B(xj , rj ) of radius rj around xj , (ii) rj2 R(xj ) → +∞, (iii) λj = d(xj , O)/rj → +∞, (iv) the balls B(xj , rj ) are disjoint, where d(xj , O) is the distance of xj from the origin O. Proof. Pick a sequence of positive numbers ǫj → 0, then choose Aj → +∞ so that Aj ǫ2j → +∞. Let σj be the largest number such that sup{R(x)d(x, O)2 | d(x, O) ≤ σj } ≤ Aj . Then there exists some yj ∈ M such that R(yj )d(yj , O)2 = Aj
and d(yj , O) = σj .
Now pick xj ∈ M so that d(xj , O) ≥ σj and R(xj ) ≥
1 sup{R(x) | d(x, O) ≥ σj }. 1 + ǫj
Finally pick rj = ǫj σj . We check the properties (i)-(iv) as follows. (i) If x ∈ B(xj , rj ) ∩ {d(·, O) ≥ σj }, we have R(x) ≤ (1 + ǫj )R(xj )
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
363
by the choice of the point xj ; while if x ∈ B(xj , rj ) ∩ {d(·, O) ≤ σj }, we have R(x) ≤ Aj /d(x, O)2 ≤
1 (Aj /σj2 ) (1 − ǫj )2
=
1 R(yj ) (1 − ǫj )2
≤
(1 + ǫj ) R(xj ), (1 − ǫj )2
since d(x, O) ≥ d(xj , O) − d(x, xj ) ≥ σj − rj = (1 − ǫj )σj . Thus we have obtained R(x) ≤ (1 + δj )R(xj ), where δj = (ii)
(1+ǫj ) (1−ǫj )2
∀ x ∈ B(xj , rj ),
− 1 → 0 as j → +∞.
By the choices of rj , xj and yj , we have rj2 R(xj ) = ǫ2j σj2 R(xj ) 1 ≥ ǫ2j σj2 R(yj ) 1 + ǫj =
ǫ2j Aj → +∞, 1 + ǫj
as j → +∞.
(iii) Since d(xj , O) ≥ σj = rj /ǫj , it follows that λj = d(xj , O)/rj → +∞ as j → +∞. (iv) For any x ∈ B(xj , rj ), the distance from the origin d(x, O) ≥ d(xj , O) − d(x, xj ) ≥ σj − rj = (1 − ǫj )σj → +∞, as j → +∞. Thus any fixed compact set does not meet the balls B(xj , rj ) for large enough j. If we pass to a subsequence, the balls will all avoid each other. The above point picking lemma of Hamilton, as written down in Lemma 22.2 of [63], requires the curvature of the manifold to be bounded. When the manifold has unbounded curvature, we will appeal to the following simple lemma. Lemma 6.1.4. Given a complete noncompact Riemannian manifold with unbounded curvature, we can find a sequence of points xj divergent to infinity such that for each positive integer j, we have |Rm(xj )| ≥ j, and |Rm(x)| ≤ 4|Rm(xj )| for x ∈ B(xj , √
j ). |Rm(xj )|
Proof. Each xj can be constructed as a limit of a finite sequence {yi }, defined as follows. Let y0 be any fixed point with |Rm(y0 )| ≥ j. Inductively, if yi cannot be
364
H.-D. CAO AND X.-P. ZHU
taken as xj , then there is a yi+1 such that |Rm(yi+1 )| > 4|Rm(yi )|, j . d(yi , yi+1 ) 6 p |Rm(yi )| Thus we have
|Rm(yi )| > 4i |Rm(y0 )| ≥ 4i j, d(yi , y0 ) ≤ j
i X
k=1
p 1 p < 2 j. k−1 4 j
Since the manifold is smooth, the sequence {yi } must be finite. The last element fits. 6.2. Asymptotic Shrinking Solitons. We begin with the study of the asymptotic behavior of an ancient κ-solution gij (x, t), on M × (−∞, T ) with T > 0, to the Ricci flow as t → −∞. Pick an arbitrary point (p, t0 ) ∈ M × (−∞, 0] and recall from Chapter 3 that τ = t0 − t, 1 l(q, τ ) = √ inf 2 τ
Z
τ 0
for t < t0 ,
√ s R(γ(s), t0 − s) +
and V˜ (τ ) =
Z
2 |γ(s)| ˙ gij (t0 −s)
γ : [0, τ ] → M with ds γ(0) = p, γ(τ ) = q
n
(4πτ )− 2 exp(−l(q, τ ))dVt0 −τ (q). M
We first observe that Corollary 3.2.6 also holds for the general complete manifold ¯ τ ) = 4τ l(·, τ ) M . Indeed, since the scalar curvature is nonnegative, the function L(·, achieves its minimum on M for each fixed τ > 0. Thus the same argument in the proof of Corollary 3.2.6 shows there exists q = q(τ ) such that (6.2.1)
l(q(τ ), τ ) ≤
n 2
for each τ > 0. Recall from (3.2.11)-(3.2.13), the Li-Yau-Perelman distance l satisfies the following (6.2.2) (6.2.3) (6.2.4)
l 1 ∂ l = − + R + 3/2 K, ∂τ τ 2τ 1 l |∇l|2 = −R + − 3/2 K, τ τ n 1 ∆l ≤ −R + − 3/2 K, 2τ 2τ
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
365
and the equality in (6.2.4) holds R τ everywhere if and only if we are on a gradient shrinking soliton. Here K = 0 s3/2 Q(X)ds, Q(X) is the trace Li-Yau-Hamilton quadratic given by Q(X) = −Rτ −
R − 2h∇R, Xi + 2Ric (X, X) τ
and X is the tangential (velocity) vector field of an L-shortest curve γ : [0, τ ] → M connecting p to q. By applying the trace Li-Yau-Hamilton inequality (Corollary 2.5.5) to the ancient κ-solution, we have Q(X) = −Rτ − ≥−
R − 2h∇R, Xi + 2Ric (X, X) τ
R τ
and hence K=
Z
τ
s3/2 Q(X)ds 0
≥−
Z
τ
√ sRds
0
≥ −L(q, τ ). Thus by (6.2.3) we get |∇l|2 + R ≤
(6.2.5)
3l . τ
We now state and prove a result of Perelman [103] about the asymptotic shapes of ancient κ-solutions as the time t → −∞. Theorem 6.2.1 (Perelman [103]). Let gij (·, t), −∞ < t < T with some T > 0, be a nonflat ancient κ-solution for some κ > 0. Then there exist a sequence of points qk and a sequence of times tk → −∞ such that the scalings of gij (·, t) around qk with factor |tk |−1 and with the times tk shifting to the new time zero converge to a nonflat ∞ gradient shrinking soliton in Cloc topology. Proof. Clearly, we may assume that the nonflat ancient κ-solution is not a gradient shrinking soliton. For the arbitrarily fixed (p, t0 ), let q(τ )(τ = t0 − t) be chosen as in (6.2.1) with l(q(τ ), τ ) ≤ n2 . We only need to show that the scalings of gij (·, t) around q(τ ) with factor τ −1 converge along a subsequence of τ → +∞ to a nonflat gradient ∞ shrinking soliton in the Cloc topology. We first claim that for any A ≥ 1, one can find B = B(A) < +∞ such that for every τ¯ > 1 there holds (6.2.6)
l(q, τ ) ≤ B and τ R(q, t0 − τ ) ≤ B,
τ and d2t0 − τ¯ (q, q( τ2¯ )) ≤ A¯ τ. whenever 12 τ¯ ≤ τ ≤ A¯ 2
366
H.-D. CAO AND X.-P. ZHU
Indeed, by using (6.2.5) at τ = τ2¯ , we have r r τ¯ √ τ¯ n l(q, ) ≤ (6.2.7) + sup{|∇ l|} · dt0 − τ2¯ q, q 2 2 2 r r n 3 √ τ ≤ + · A¯ 2 2¯ τ r r n 3A = + , 2 2 and
(6.2.8)
τ¯ 3l(q, τ2¯ ) ≤ R q, t0 − 2 ( τ2¯ ) r !2 r 6 n 3A ≤ + , τ¯ 2 2
√ for q ∈ Bt0 − τ2¯ (q( τ2¯ ), A¯ τ ). Recall that the Li-Yau-Hamilton inequality implies that the scalar curvature of the ancient solution is pointwise nondecreasing in time. Thus we know from (6.2.8) that r !2 r n 3A + (6.2.9) τ R(q, t0 − τ ) ≤ 6A 2 2 τ and d2t0 − τ¯ (q, q( τ2¯ )) ≤ A¯ τ. whenever 21 τ¯ ≤ τ ≤ A¯ 2 By (6.2.2) and (6.2.3) we have ∂l 1 l R + |∇l|2 = − + . ∂τ 2 2τ 2 This together with (6.2.9) implies that r !2 r n 3A l 3A ∂l ≤− + + ∂τ 2τ τ 2 2 i.e., r !2 n 3A + 2 2 q, q τ2¯ ≤ A¯ τ . Hence by integrating this differ-
∂ √ 3A ( τ l) ≤ √ ∂τ τ
r
whenever 12 τ¯ ≤ τ ≤ A¯ τ and d2t0 − τ¯ 2 ential inequality, we obtain r !2 r r √ √ τ¯ τ¯ n 3A l q, ≤ 6A + τ l(q, τ ) − τ 2 2 2 2 and then by (6.2.7), (6.2.10)
r !2 r τ¯ n 3A l(q, τ ) ≤ l q, + 6A + 2 2 2 ! r 2 r n 3A ≤ 7A + 2 2
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
367
τ and d2t0 − τ¯ (q, q( τ2¯ )) ≤ A¯ τ . So we have proved claim (6.2.6). whenever 12 τ¯ ≤ τ ≤ A¯ 2 Recall that gij (τ ) = gij (·, t0 − τ ) satisfies (gij )τ = 2Rij . Let us take the scaling of the ancient κ-solution around q( τ2¯ ) with factor ( τ2¯ )−1 , i.e., g˜ij (s) =
2 τ¯ gij ·, t0 − s τ¯ 2
where s ∈ [0, +∞). Claim (6.2.6) says that for all s ∈ [1, 2A] and all q such that ˜ s) = τ¯ R(q, t0 − s ¯τ ) ≤ B. Now taking into dist2g˜ij (1) (q, q( τ2¯ )) ≤ A, we have R(q, 2 2 account the κ-noncollapsing assumption and Theorem 4.2.2, we can use Hamilton’s compactness theorem (Theorem 4.1.5) to obtain a sequence τ¯k → +∞ such that the (k) (k) marked evolving manifolds (M, g˜ij (s), q( τ¯2k )), with g˜ij (s) = τ¯2k gij (·, t0 − s τ¯2k ) and ¯ , g¯ij (s), q¯) with s ∈ [1, +∞), where g¯ij (s) is s ∈ [1, +∞), converge to a manifold (M ¯ also a solution to the Ricci flow on M . (k) Denote by ˜lk the corresponding Li-Yau-Perelman distance of g˜ij (s). It is easy to see that ˜lk (q, s) = l(q, τ¯2k s), for s ∈ [1, +∞). From (6.2.5), we also have (6.2.11)
˜ (k) ≤ 6˜lk , |∇˜lk |2g˜(k) + R ij
˜ (k) is the scalar curvature of the metric g˜(k) . Claim (6.2.6) says that ˜lk are where R ij uniformly bounded on compact subsets of M ×[1, +∞) (with the corresponding origins q( τ¯2k )). Thus the above gradient estimate (6.2.11) implies that the functions ˜lk tend ¯. (up to a subsequence) to a function ¯l which is a locally Lipschitz function on M ˜ We know from (6.2.2)-(6.2.4) that the Li-Yau-Perelman distance lk satisfies the following inequalities: (6.2.12)
˜ (k) + n ≥ 0, (˜lk )s − ∆˜lk + |∇˜lk |2 − R 2s
(6.2.13)
˜ ˜ (k) + lk − n ≤ 0. 2∆˜lk − |∇˜lk |2 + R s
We next show that the limit ¯l also satisfies the above two inequalities in the sense of distributions. Indeed the above two inequalities can be rewritten as n ∂ e (k) (4πs)− 2 exp(−e −△+R lk ) ≤ 0, (6.2.14) ∂s (6.2.15)
e lk
e(k) )e− 2 + −(4△ − R
˜lk − n lek e− 2 ≤ 0, s
in the sense of distributions. Note that the estimate (6.2.11) implies that ˜lk → ¯l in 0,α the Cloc norm for any 0 < α < 1. Thus the inequalities (6.2.14) and (6.2.15) imply that the limit l satisfies n ∂ − △ + R (4πs)− 2 exp(−l) ≤ 0, (6.2.16) ∂s (6.2.17)
l
−(4△ − R)e− 2 +
¯l − n l e− 2 ≤ 0, s
368
H.-D. CAO AND X.-P. ZHU
in the sense of distributions. (k) Denote by V˜ (k) (s) Perelman’s reduced volume of the scaled metric g˜ij (s). Since ˜lk (q, s) = l(q, τ¯2k s), we see that V˜ (k) (s) = V˜ ( τ¯2k s) where V˜ is Perelman’s reduced volume of the ancient κ-solution. The monotonicity of Perelman’s reduced volume (Theorem 3.2.8) then implies that (6.2.18)
lim V˜ (k) (s) = V¯ , for s ∈ [1, 2],
k→∞
for some nonnegative constant V¯ . (We remark that by the Jacobian comparison theorem (Theorem 3.2.7), (3.2.18) and (3.2.19), the integrand of V˜ (k) (s) is bounded by n n (4πs)− 2 exp(−˜lk (X, s))J˜(k) (s) ≤ (4π)− 2 exp(−|X|2 )
on Tp M , where J˜(k) (s) is the L-Jacobian of the L-exponential map of the metric (k) g˜ij (s) at Tp M . Thus we can apply the dominant convergence theorem to get the convergence in (6.2.18). But we are not sure whether the limiting V¯ is exactly Perel¯ , g¯ij (s)), because the points q( τ¯k ) man’s reduced volume of the limiting manifold (M 2 may diverge to infinity. Nevertheless, we can ensure that V¯ is not less than Perelman’s reduced volume of the limit.) Note by (6.2.5) that (6.2.19)
V˜ (k) (2) − V˜ (k) (1) Z 2 d ˜ (k) (V (s))ds = ds 1 Z 2 Z ∂ n (k) ˜ = ds −∆+R (4πs)− 2 exp(−˜lk ) dVg˜(k) (s) . ij ∂s 1 M
Thus we deduce that in the sense of distributions, ∂ ¯ (4πs)− n2 exp(−¯l) = 0, −∆+R (6.2.20) ∂s and
l¯
¯ −2 = (4∆ − R)e
¯l − n ¯l e− 2 s
or equivalently, (6.2.21)
¯ ¯ + l − n = 0, 2∆¯l − |∇¯l|2 + R s
¯ × [1, 2]. Thus by applying standard parabolic equation theory to (6.2.20) we on M find that ¯l is actually smooth. Here we used (6.2.2)-(6.2.4) to show that the equality in (6.2.16) implies the equality in (6.2.17). Set ¯ + ¯l − n] · (4πs)− n2 e−¯l . v = [s(2∆¯l − |∇¯l|2 + R) Then by applying Lemma 2.6.1, we have ∂ ¯ v = −2s|R ¯ ij + ∇i ∇j ¯l − 1 g¯ij |2 · (4πs)− n2 e−¯l . (6.2.22) −∆+R ∂s 2s
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
369
We see from (6.2.21) that the LHS of the equation (6.2.22) is identically zero. Thus the limit metric g¯ij satisfies (6.2.23)
¯ ij + ∇i ∇j ¯l − 1 g¯ij = 0, R 2s
so we have shown the limit is a gradient shrinking soliton. To show that the limiting gradient shrinking soliton is nonflat, we first show that the constant function V¯ (s) is strictly less than 1. Consider Perelman’s reduced volume V˜ (τ ) of the ancient κ-solution. By using Perelman’s Jacobian comparison theorem (Theorem 3.2.7), (3.2.18) and (3.2.19) as before, we have Z n V˜ (τ ) = (4πτ )− 2 e−l(X,τ ) J (τ )dX Z 2 n ≤ (4π)− 2 e−|X| dX Tp M
= 1.
Recall that we have assumed the nonflat ancient κ-solution is not a gradient shrinking soliton. Thus for τ > 0, we must have V˜ (τ ) < 1. Since the limiting function V¯ (s) is the limit of V˜ ( τ¯2k s) with τ¯k → +∞, we deduce that the constant V¯ (s) is strictly less than 1, for s ∈ [1, 2]. We now argue by contradiction. Suppose the limiting gradient shrinking soliton g¯ij (s) is flat. Then by (6.2.23), 1 n ∇i ∇j ¯l = g¯ij and ∆¯l = . 2s 2s Putting these into the identity (6.2.21), we get ¯l |∇¯l|2 = s √ Since the function ¯l is strictly convex, it follows that 4s¯l is a distance function (from ¯ . From the smoothness of the function some point) on the complete flat manifold M ¯l, we conclude that the flat manifold M ¯ must be Rn . In this case we would have its reduced distance to be ¯l and its reduced volume to be 1. Since V¯ is not less than the reduced volume of the limit, this is a contradiction. Therefore the limiting gradient shrinking soliton g¯ij is not flat. To conclude this section, we use the above theorem to derive the classification of all two-dimensional ancient κ-solutions which was obtained earlier by Hamilton in Section 26 of [63]. Theorem 6.2.2. The only nonflat ancient κ-solutions to Ricci flow on twodimensional manifolds are the round sphere S2 and the round real projective plane RP2 . Proof. Let gij (x, t) be a nonflat ancient κ-solution defined on M × (−∞, T ) (for some T > 0), where M is a two-dimensional manifold. Note that the ancient κ-solution satisfies the Li-Yau-Hamilton inequality (Corollary 2.5.5). In particular by Corollary 2.5.8, the scalar curvature of the ancient κ-solution is pointwise nondecreasing in
370
H.-D. CAO AND X.-P. ZHU
time. Moreover by the strong maximum principle, the ancient κ-solution has strictly positive curvature everywhere. By the above Theorem 6.2.1, we know that the scalings of the ancient κ-solution along a sequence of points qk in M and a sequence of times tk → −∞ converge to a ¯ , g¯ij (x, t)) with −∞ < t ≤ 0. nonflat gradient shrinking soliton (M ¯ , g¯ij (x, t)) has uniWe first show that the limiting gradient shrinking soliton (M formly bounded curvature. Clearly, the limiting soliton has nonnegative curvature and is κ-noncollapsed on all scales, and its scalar curvature is still pointwise nondecreasing in time. Thus we only need to show that the limiting soliton has bounded curvature at t = 0. We argue by contradiction. Suppose the curvature of the limiting soliton ¯ is noncompact. is unbounded at t = 0. Of course in this case the limiting soliton M Then by applying Lemma 6.1.4, we can choose a sequence of points xj , j = 1, 2, . . . , ¯ of the limit satisfies divergent to infinity such that the scalar curvature R ¯ j , 0) ≥ j and R(x, ¯ 0) ≤ 4R(x ¯ j , 0) R(x p ¯ j , 0)). And then by the nondecreasing (in for all j = 1, 2, . . . , and x ∈ B0 (xj , j/ R(x time) of the scalar curvature, we have ¯ t) ≤ 4R(x ¯ j , 0), R(x,
p ¯ j , 0)) and t ≤ 0. By combining with Hamilfor all j = 1, 2, . . ., x ∈ B0 (xj , j/ R(x ton’s compactness theorem (Theorem 4.1.5) and the κ-noncollapsing, we know that a subsequence of the rescaling solutions ¯ , R(x ¯ j , 0)¯ ¯ j , 0)), xj ), (M gij (x, t/R(x
j = 1, 2, . . . ,
∞ converges in the Cloc topology to a nonflat smooth solution of the Ricci flow. Then Proposition 6.1.2 implies that the new (two-dimensional) limit must be flat. This arrives at a contradiction. So we have proved that the limiting gradient shrinking soliton has uniformly bounded curvature. We next show that the limiting soliton is compact. Suppose the limiting soliton is (complete and) noncompact. By the strong maximum principle we know that the limiting soliton also has strictly positive curvature everywhere. After a shift of the time, we may assume that the limiting soliton satisfies the following equation
(6.2.24)
¯ ij + 1 g¯ij = 0, ∇i ∇j f + R 2t
on − ∞ < t < 0,
everywhere for some function f . Differentiating the equation (6.2.24) and switching the order of differentiations, as in the derivation of (1.1.14), we get (6.2.25)
¯ = 2R ¯ ij ∇j f. ∇i R
Fix some t < 0, say t = −1, and consider a long shortest geodesic γ(s), 0 ≤ s ≤ s. Let x0 = γ(0) and X(s) = γ(s). ˙ Let V (0) be any unit vector orthogonal to γ(0) ˙ and translate V (0) along γ(s) to get a parallel vector field V (s), 0 ≤ s ≤ s on γ. Set for 0 ≤ s ≤ 1, sV (s), b V (s) = V (s), for 1 ≤ s ≤ s − 1, (s − s)V (s), for s − 1 ≤ s ≤ s.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
371
It follows from the second variation formula of arclength that Z
0
Thus we clearly have
s
˙ ¯ (|Vb (s)|2 − R(X, Vb , X, Vb ))ds ≥ 0. Z
s
¯ R(X, Vb , X, Vb )ds ≤ const.,
0
and then (6.2.26)
Z
s
0
¯ Ric(X, X)ds ≤ const..
By integrating the equation (6.2.24) we get X(f (γ(s))) − X(f (γ(0))) +
Z
s 0
1 ¯ Ric(X, X)ds − s = 0 2
and then by (6.2.26), we deduce d s (f ◦ γ(s)) ≥ − const., ds 2
and f ◦ γ(s) ≥
s2 − const. · s − const. 4
for s > 0 large enough. Thus at large distances from the fixed point x0 the function f has no critical points and is proper. It then follows from the Morse theory that any two high level sets of f are diffeomorphic via the gradient curves of f . Since by (6.2.25), d ¯ ¯ η(s)i R(η(s), −1) = h∇R, ˙ ds ¯ ij ∇i f ∇j f = 2R ≥0
¯ −1) has for any integral curve η(s) of ∇f , we conclude that the scalar curvature R(x, ¯ , which contradicts the Bonnet-Myers Theorem. So we a positive lower bound on M have proved that the limiting gradient shrinking soliton is compact. By Proposition 5.1.10, the compact limiting gradient shrinking soliton has constant curvature. This says that the scalings of the ancient κ-solution (M, gij (x, t)) along a sequence of points qk ∈ M and a sequence of times tk → −∞ converge in the C ∞ topology to the round S2 or the round RP2 . In particular, by looking at the time derivative of the volume and the Gauss-Bonnet theorem, we know that the ancient κ-solution (M, gij (x, t)) exists on a maximal time interval (−∞, T ) with T < +∞. Consider the scaled entropy of Hamilton [60] Z E(t) = R log[R(T − t)]dVt . M
372
H.-D. CAO AND X.-P. ZHU
We compute (6.2.27)
d E(t) = dt
Z
Z
log[R(T − t)]∆RdVt + M |∇R|2 + R2 − rR dVt = − R M Z |∇R|2 2 + (R − r) dVt = − R M Z
M
R ∆R + R − dVt (T − t) 2
R R where r = M RdVt /V olt (M ) and we have used Vol t (M ) = ( M RdVt ) · (T − t) (by the Gauss-Bonnet theorem). For a smooth function f on the surface M , one can readily check 2 Z Z Z 1 2 R|∇f |2 , (∆f ) = 2 ∇i ∇j f − 2 (∆f )gij + M M M Z Z Z Z |∇R|2 |∇R + R∇f |2 = −2 R(∆f ) + R|∇f |2 , R R M M M M and then Z
Z |∇R|2 + (∆f )(∆f − 2R) R M M 2 Z Z 1 =2 ∇i ∇j f − 2 (∆f )gij + M
By choosing ∆f = R − r, we get Z Z |∇R|2 − (R − r)2 R M M 2 Z Z ∇i ∇j f − 1 (∆f )gij + =2 2 M
M
M
|∇R + R∇f |2 . R
|∇R + R∇f |2 ≥ 0. R
If the equality holds, then we have
1 ∇i ∇j f − (∆f )gij = 0 2 i.e., ∇f is conformal. By the Kazdan-Warner identity [77], it follows that Z ∇R · ∇f = 0, M
so
0=− =−
Z
R∆f
M
Z
M
(R − r)2 .
Hence we have proved the following inequality due to Chow [37] Z Z |∇R|2 (6.2.28) ≥ (R − r)2 , R M M
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
373
and the equality holds if and only if R ≡ r. The combination (6.2.27) and (6.2.28) shows that the scaled entropy E(t) is strictly decreasing along the Ricci flow unless we are on the round sphere S2 or its quotient RP2 . Moreover the convergence result in Theorem 5.1.11 shows that the scaled entropy E has its minimum value at the constant curvature metric (round S2 or round RP2 ). We had shown that the scalings of the nonflat ancient κ-solution along a sequence of times tk → −∞ converge to the constant curvature metric. Then E(t) has its minimal value at t = −∞, so it was constant all along, hence the ancient κ-solution must have constant curvature for each t ∈ (−∞, T ). This proves the theorem. 6.3. Curvature Estimates via Volume Growth. For solutions to the Ricci flow, Perelman’s no local collapsing theorems tell us that the local curvature upper bounds imply the local volume lower bounds. Conversely, one would expect to get local curvature upper bounds from local volume lower bounds. If this is the case, one will be able to establish an elliptic type estimate for the curvatures of solutions to the Ricci flow. This will provide the key estimate for the canonical neighborhood structure and thick-thin decomposition of the Ricci flow on three-manifolds. In this section we derive such curvature estimates for nonnegatively curved solutions. In the next chapter we will derive similar estimates for all smooth solutions, as well as surgically modified solutions, of the Ricci flow on three-manifolds. Let M be an n-dimensional complete noncompact Riemannian manifold with nonnegative Ricci curvature. Pick an origin O ∈ M . The well-known Bishop-Gromov volume comparison theorem tells us the ratio V ol(B(O, r))/rn is monotone nonincreasing in r ∈ [0, +∞). Thus there exists a limit νM = lim
r→+∞
Vol (B(O, r)) . rn
Clearly the number νM is invariant under dilation and is independent of the choice of the origin. νM is called the asymptotic volume ratio of the Riemannian manifold M. The following result obtained by Perelman in [103] shows that any ancient κsolution must have zero asymptotic volume ratio. This result for the Ricci flow on K¨ahler manifolds was obtained by Chen and the second author in [32] independently. Moreover in the K¨ahler case, as shown by Chen, Tang and the second author in [29] (for complex two dimension) and by Ni in [98] (for all dimensions), the condition of nonnegative curvature operator can be replaced by the weaker condition of nonnegative bisectional curvature. Lemma 6.3.1. Let M be an n-dimensional complete noncompact Riemannian manifold. Suppose gij (x, t), x ∈ M and t ∈ (−∞, T ) with T > 0, is a nonflat ancient solution of the Ricci flow with nonnegative curvature operator and bounded curvature. Then the asymptotic volume ratio of the solution metric satisfies νM (t) = lim
r→+∞
Vol t (Bt (O, r)) =0 rn
for each t ∈ (−∞, T ). Proof. The proof is by induction on the dimension. When the dimension is two, the lemma is valid by Theorem 6.2.2. For dimension ≥ 3, we argue by contradiction.
374
H.-D. CAO AND X.-P. ZHU
Suppose the lemma is valid for dimensions ≤ n − 1 and suppose νM (t0 ) > 0 for some n-dimensional nonflat ancient solution with nonnegative curvature operator and bounded curvature at some time t0 ≤ 0. Fixing a point x0 ∈ M , we consider the asymptotic scalar curvature ratio A=
lim sup dt0 (x,x0 )→+∞
R(x, t0 )d2t0 (x, x0 ).
We divide the proof into three cases. Case 1: A = +∞. By Lemma 6.1.3, there exist sequences of points xk ∈ M divergent to infinity, of radii rk → +∞, and of positive constants δk → 0 such that (i) R(x, t0 ) ≤ (1+δk )R(xk , t0 ) for all x in the ball Bt0 (xk , rk ) of radius rk around xk , (ii) rk2 R(xk , t0 ) → +∞ as k → +∞, (iii) dt0 (xk , x0 )/rk → +∞. By scaling the solution around the points xk with factor R(xk , t0 ), and shifting the time t0 to the new time zero, we get a sequence of rescaled solutions s gk (s) = R(xk , t0 )g ·, t0 + R(xk , t0 ) to the Ricci flow. Since the ancient solution has nonnegative curvature operator and bounded curvature, there holds the Li-Yau-Hamilton inequality (Corollary 2.5.5). Thus the rescaled solutions satisfy Rk (x, s) ≤ (1 + δk ) p for all s ≤ 0 and x ∈ Bgk (0) (xk , rk R(xk , t0 )). Since νM (t0 ) > 0, it follows from the standard volume comparison and Theorem 4.2.2 that the injectivity radii of the rescaled solutions gk at the points xk and the new time zero is uniformly bounded below by a positive number. Then by Hamilton’s compactness theorem (Theorem 4.1.5), ˜ , g˜(s), O) after passing to a subsequence, (M, gk (s), xk ) will converge to a solution (M to the Ricci flow with ˜ s) ≤ 1, for all s ≤ 0 and y ∈ M ˜, R(y, and ˜ R(O, 0) = 1. Since the metric is shrinking, by (ii) and (iii), we get R(xk , t0 )d2g(·,t0 +
s R(xk ,t0 ) )
(x0 , xk ) ≥ R(xk , t0 )d2g(·,t0 ) (x0 , xk )
which tends to +∞, as k → +∞, for all s ≤ 0. Thus by Proposition 6.1.2, for ˜ , g˜(s)) splits off a line. We now consider the lifting of the solution each s ≤ 0, (M ˜˜ g˜˜(s)), s ≤ 0. Clearly we ˜ , g˜(s)), s ≤ 0, to its universal cover and denote it by (M, (M still have νM˜ (0) > 0 and νM˜˜ (0) > 0.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
375
By applying Hamilton’s strong maximum principle and the de Rham decomposi˜˜ splits isometrically as X × R for some (n − 1)tion theorem, the universal cover M dimensional nonflat (complete) ancient solution X with nonnegative curvature operator and bounded curvature. These imply that νX (0) > 0, which contradicts the induction hypothesis. Case 2: 0 < A < +∞. Take a sequence of points xk divergent to infinity such that R(xk , t0 )d2t0 (xk , x0 ) → A, as k → +∞. Consider the rescaled solutions (M, gk (s)) (around the fixed point x0 ), where s gk (s) = R(xk , t0 )g ·, t0 + , s ∈ (−∞, 0]. R(xk , t0 ) Then there is a constant C > 0 such that Rk (x, 0) ≤ C/d2k (x, x0 , 0), Rk (xk , 0) = 1, (6.3.1) √ dk (xk , x0 , 0) → A > 0,
where dk (·, x0 , 0) is the distance function from the point x0 in the metric gk (0). Since νM (t0 ) > 0, it is a basic result in Alexandrov space theory (see for example Theorem 7.6 of [20]) that a subsequence of (M, gk (s), x0 ) converges in the Gromov˜ , g˜(0), x0 ) with vertex x0 . Hausdorff sense to an n-dimensional metric cone (M By (6.3.1), the standard volume comparison and Theorem 4.2.2, we know that the injectivity radius of (M, gk (0)) at xk is uniformly bounded from below by a positive number ρ0 . After taking a subsequence, we may assume xk converges to a point x∞ ˜ \ {x0 }. Then by Hamilton’s compactness theorem (Theorem 4.1.5), we can take in M a subsequence such that the metrics gk (s) on the metric balls B0 (xk , 21 ρ0 )(⊂ M with ∞ respect to the metric gk (0)) converge in the Cloc topology to a solution of the Ricci 1 ∞ flow on a ball B0 (x∞ , 2 ρ0 ). Clearly the Cloc limit has nonnegative curvature operator and it is a piece of the metric cone at the time s = 0. By (6.3.1), we have (6.3.2)
˜ ∞ , 0) = 1. R(x
Let x be any point in the limiting ball B0 (x∞ , 21 ρ0 ) and e1 be any radial direction ˜ 1 , e1 ) = 0. Recall that the evolution equation of the Ricci tensor at x. Clearly Ric(e in frame coordinates is ∂ ˜ ˜R ˜ ab + 2R ˜ acbd R ˜ cd . Rab = △ ∂t Since the curvature operator is nonnegative, by applying Hamilton’s strong maximum ˜ principle (Theorem 2.2.1) to the above equation, we deduce that the null space of Ric is invariant under parallel translation. In particular, all radial directions split off locally and isometrically. While by (6.3.2), the piece of the metric cone is nonflat. This gives a contradiction. Case 3: A = 0.
376
H.-D. CAO AND X.-P. ZHU
The gap theorem as was initiated by Mok-Siu-Yau [93] and established by GreeneWu [49, 50], Eschenberg-Shrader-Strake [45], and Drees [44] shows that a complete noncompact n-dimensional (except n = 4 or 8) Riemannian manifold with nonnegative sectional curvature and the asymptotic scalar curvature ratio A = 0 must be flat. So the present case is ruled out except in dimension n = 4 or 8. Since in our situation the asymptotic volume ratio is positive and the manifold is the solution of the Ricci flow, we can give an alternative proof for all dimensions as follows. We claim the sectional curvature of (M, gij (x, t0 )) is positive everywhere. Indeed, by Theorem 2.2.2, the image of the curvature operator is just the restricted holonomy algebra G of the manifold. If the sectional curvature vanishes for some two-plane, then the holonomy algebra G cannot be so(n). We observe the manifold is not Einstein since it is noncompact, nonflat and has nonnegative curvature operator. If G is irreducible, then by Berger’s Theorem [7], G = u( n2 ). Thus the manifold is K¨ahler with bounded and nonnegative bisectional curvature and with curvature decay faster than quadratic. Then by the gap theorem obtained by Chen and the second author in [31], this K¨ahler manifold must be flat. This contradicts the assumption. Hence the holonomy algebra ˜ 1 ×M ˜ 2 nontrivially. G is reducible and the universal cover of M splits isometrically as M ˜ 1 and Clearly the universal cover of M has positive asymptotic volume ratio. So M ˜ M2 still have positive asymptotic volume ratio and at least one of them is nonflat. By the induction hypothesis, this is also impossible. Thus our claim is proved. Now we know that the sectional curvature of (M, gij (x, t0 )) is positive everywhere. Choose a sequence of points xk divergent to infinity such that R(xk , t0 )d2t0 (xk , x0 ) = sup{R(x, t0 )d2t0 (x, x0 ) | dt0 (x, x0 ) ≥ dt0 (xk , x0 )}, dt0 (xk , x0 ) ≥ k, R(xk , t0 )d2t0 (xk , x0 ) = εk → 0. Consider the rescaled metric
gk (0) = R(xk , t0 )g(·, t0 ) as before. Then ( (6.3.3)
Rk (x, 0) ≤ εk /d2k (x, x0 , 0), √ dk (xk , x0 , 0) = εk → 0.
for dk (x, x0 , 0) ≥
√ εk ,
As in Case 2, the rescaled marked solutions (M, gk (0), x0 ) will converge in the Gromov˜ , g˜(0), x0 ). And by the virtue of Hamilton’s Hausdorff sense to a metric cone (M compactness theorem (Theorem 4.1.5), up to a subsequence, the convergence is in the ∞ ˜ \ {x0 }. We next claim the metric cone (M ˜ , g˜(0), x0 ) is isometric Cloc topology in M to Rn . ˜ as a warped product R+ ×r X n−1 for Indeed, let us write the metric cone M n−1 some (n − 1)-dimensional manifold X . By (6.3.3), the metric cone must be flat and X n−1 is isometric to a quotient of the round sphere Sn−1 by fixed point free ˜ is isometric to Rn , we only need to isometries in the standard metric. To show M n−1 verify that X is simply connected. Let ϕ be the Busemann function of (M, gij (·, t0 )) with respect to the point x0 . Since (M, gij (·, t0 )) has nonnegative sectional curvature, it is easy to see that for any small ε > 0, there is a r0 > 0 such that (1 − ε)dt0 (x, x0 ) ≤ ϕ(x) ≤ dt0 (x, x0 )
377
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
for all x ∈ M \ Bt0 (x0 , r0 ). The strict positivity of the sectional curvature of the manifold (M, gij (·, t0 )) implies that the square of the Busemann function is strictly convex (and exhausting). Thus every level set ϕ−1 (a), with a > inf{ϕ(x) | x ∈ M }, of the Busemann function ϕ is diffeomorphic to the (n − 1)-sphere Sn−1 . In particular, ϕ−1 ([a, 32 a]) is simply connected for a > inf{ϕ(x) | x ∈ M } since n ≥ 3. ˜ = R+ ×r X n−1 . Consider an annulus portion [1, 2] × X n−1 of the metric cone M It is the limit of (Mk , gk (0)), where ) ( 2 1 ≤ dt0 (x, x0 ) ≤ p . Mk = x ∈ M p R(xk , t0 ) R(xk , t0 ) It is clear that " ϕ
−1
2(1 − ε) p ,p R(xk , t0 ) R(xk , t0 ) 1
#!
⊂ Mk ⊂ ϕ
−1
"
p
1−ε
R(xk , t0 )
,p
2 R(xk , t0 )
#!
for k large enough. Thus any closed loop in { 23 } × X n−1 can be shrunk to a point by a homotopy in [1, 2] × X n−1 . This shows that X n−1 is simply connected. Hence we ˜ , g˜(0), x0 ) is isometric to Rn . Consequently, have proven that the metric cone (M \ Bg(t0 ) x0 , √ σr Vol g(t0 ) Bg(t0 ) x0 , √ r R(xk ,t0 ) R(xk ,t0 ) n = αn (1−σ n ) lim k→+∞ r √ R(xk ,t0 )
for any r > 0 and 0 < σ < 1, where αn is the volume of the unit ball in the Euclidean space Rn . Finally, by combining with the monotonicity of the Bishop-Gromov volume comparison, we conclude that the manifold (M, gij (·, t0 )) is flat and isometric to Rn . This contradicts the assumption. Therefore, we have proved the lemma. Finally we would like to include an alternative simpler argument, inspired by Ni [98], for the above Case 2 and Case 3 to avoid the use of the gap theorem, holonomy groups, and asymptotic cone structure. Alternative Proof for Case 2 and Case 3. Let us consider the situation of 0 ≤ A < +∞ in the above proof. Observe that νM (t) is nonincreasing in time t by using Lemma 3.4.1(ii) and the fact that the metric is shrinking in t. Suppose νM (t0 ) > 0, then the solution gij (·, t) is κ-noncollapsed for t ≤ t0 for some uniform κ > 0. By combining with Theorem 6.2.1, there exist a sequence of points qk and a sequence of times tk → −∞ such that the scalings of gij (·, t) around qk with factor |tk |−1 and with the times tk shifting to the new time zero converge to a nonflat gradient shrinking ¯ in the C ∞ topology. This gradient soliton also has maximal volume growth soliton M loc (i.e. νM¯ (t) > 0) and satisfies the Li-Yau-Hamilton estimate (Corollary 2.5.5). If the ¯ at the time −1 is bounded, then we see from curvature of the shrinking soliton M the proof of Theorem 6.2.2 that by using the equations (6.2.24)-(6.2.26), the scalar ¯ at the time −1. In particular, curvature has a positive lower bound everywhere on M this implies the asymptotic scalar curvature ratio A = ∞ for the soliton at the time −1, which reduces to Case 1 and arrives at a contradiction by the dimension reduction argument. On the other hand, if the scalar curvature is unbounded, then by Lemma
378
H.-D. CAO AND X.-P. ZHU
6.1.4, the Li-Yau-Hamilton estimate (Corollary 2.5.5) and Lemma 6.1.2, we can do the same dimension reduction as in Case 1 to arrive at a contradiction also. The following lemma is a local and space-time version of Lemma 6.1.4 on picking local (almost) maximum curvature points. We formulate it from Perelman’s arguments in the section 10 of [103]. Lemma 6.3.2. For any positive constants B, C with B > 4 and C > 1000, there 1 exists 1 ≤ A < min{ 41 B, 1000 C} which tends to infinity as B and C tend to infinity and satisfies the following property. Suppose we have a (not necessarily complete) solution gij (t) to the Ricci flow, defined on M × [−t0 , 0], so that at each time t ∈ [−t0 , 0] the metric ball Bt (x0 , 1) is compactly contained in M . Suppose there exists a point (x′ , t′ ) ∈ M × (−t0 , 0] such that dt′ (x′ , x0 ) ≤
1 and |Rm(x′ , t′ )| > C + B(t′ + t0 )−1 . 4
Then we can find a point (¯ x, t¯) ∈ M × (−t0 , 0] such that dt¯(¯ x, x0 ) <
1 with Q = |Rm(¯ x, t¯)| > C + B(t¯ + t0 )−1 , 3
and |Rm(x, t)| ≤ 4Q for all (−t0 <) t¯ − AQ−1 ≤ t ≤ t¯ and dt (x, x¯) ≤
1 1 − 12 2 . 10 A Q
Proof. We first claim that there exists a point (¯ x, t¯) with −t0 < t¯ ≤ 0 and dt¯(¯ x, x0 ) < 31 such that Q = |Rm(¯ x, t¯)| > C + B(t¯ + t0 )−1 , and |Rm(x, t)| ≤ 4Q
(6.3.4)
1
wherever t¯ − AQ−1 ≤ t ≤ t¯, dt (x, x0 ) ≤ dt¯(¯ x, x0 ) + (AQ−1 ) 2 . We will construct such (¯ x, t¯) as a limit of a finite sequence of points. Take an arbitrary (x1 , t1 ) such that dt1 (x1 , x0 ) ≤
1 , 4
−t0 < t1 ≤ 0 and |Rm(x1 , t1 )| > C + B(t1 + t0 )−1 .
Such a point clearly exists by our assumption. Assume we have already constructed (xk , tk ). If we cannot take the point (xk , tk ) to be the desired point (¯ x, t¯), then there exists a point (xk+1 , tk+1 ) such that tk − A|Rm(xk , tk )|−1 ≤ tk+1 ≤ tk , and 1
dtk+1 (xk+1 , x0 ) ≤ dtk (xk , x0 ) + (A|Rm(xk , tk )|−1 ) 2 , but |Rm(xk+1 , tk+1 )| > 4|Rm(xk , tk )|.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
379
It then follows that dtk+1 (xk+1 , x0 ) ≤ dt1 (x1 , x0 ) + A 1 1 ≤ + A2 4
k X
2
1 2
k X i=1
−(i−1)
i=1
− 12
|Rm(xi , ti )|
− 12
|Rm(x1 , t1 )|
!
!
1 1 + 2(AC −1 ) 2 4 1 < , 3
≤
tk+1 − (−t0 ) =
k X i=1
≥−
(ti+1 − ti ) + (t1 − (−t0 ))
k X i=1
≥ −A ≥− ≥
A|Rm(xi , ti )|−1 + (t1 − (−t0 ))
k X i=1
4−(i−1) |Rm(x1 , t1 )|−1 + (t1 − (−t0 ))
2A (t1 + t0 ) + (t1 + t0 ) B
1 (t1 + t0 ), 2
and |Rm(xk+1 , tk+1 )| > 4k |Rm(x1 , t1 )|
≥ 4k C → +∞ as k → +∞.
Since the solution is smooth, the sequence {(xk , tk )} is finite and its last element fits. Thus we have proved assertion (6.3.4). From the above construction we also see that the chosen point (¯ x, t¯) satisfies dt¯(¯ x, x0 ) <
1 3
and Q = |Rm(¯ x, t¯)| > C + B(t¯ + t0 )−1 . Clearly, up to some adjustment of the constant A, we only need to show that |Rm(x, t)| ≤ 4Q
(6.3.4)′ 1
1
1
1 1 wherever t¯ − 200n A 2 Q−1 ≤ t ≤ t¯ and dt (x, x¯) ≤ 10 A 2 Q− 2 . 1 1 1 − ¯ For any point (x, t) with dt¯(x, x¯) ≤ 10 A 2 Q 2 , we have
dt¯(x, x0 ) ≤ dt¯(¯ x, x0 ) + dt¯(x, x¯)
1
≤ dt¯(¯ x, x0 ) + (AQ−1 ) 2
380
H.-D. CAO AND X.-P. ZHU
and then by (6.3.4) |Rm(x, t¯)| ≤ 4Q. 1 1 A 2 Q−1 , t¯] such that Thus by continuity, there is a minimal t¯′ ∈ [t¯ − 200n 1 1 −1 ′ ¯ ¯ 2 2 ≤ 5Q. (6.3.5) sup |Rm(x, t)| | t ≤ t ≤ t, dt (x, x¯) ≤ A Q 10
For any point (x, t) with t¯′ ≤ t ≤ t¯ and dt (x, x¯) ≤ discussion into two cases.
1 −1 12 ) , 10 (AQ
we divide the
1
3 Case (1): dt (¯ x, x0 ) ≤ 10 (AQ−1 ) 2 . From assertion (6.3.4) we see that
(6.3.5)′
1 sup{|Rm(x, t)| | t¯′ ≤ t ≤ t¯, dt (x, x0 ) ≤ (AQ−1 ) 2 } ≤ 4Q.
Since dt (¯ x, x0 ) ≤
3 −1 21 ) , 10 (AQ
we have
dt (x, x0 ) ≤ dt (x, x¯) + dt (¯ x, x0 ) 1 1 1 3 ≤ (AQ−1 ) 2 + (AQ−1 ) 2 10 10 1 ≤ (AQ−1 ) 2 which implies the estimate |Rm(x, t)| ≤ 4Q from (6.3.5)′ . 1
3 Case (2): dt (¯ x, x0 ) > 10 (AQ−1 ) 2 . From the curvature bounds in (6.3.5) and (6.3.5)′ , we can apply Lemma 3.4.1 (ii) 1 1 Q− 2 to get with r0 = 10
d 1 (dt (¯ x, x0 )) ≥ −40(n − 1)Q 2 dt and then 1
dt (¯ x, x0 ) ≤ dtˆ(¯ x, x0 ) + 40n(Q 2 )
1 1 A 2 Q−1 200n
1 1 ≤ dtˆ(¯ x, x0 ) + (AQ−1 ) 2 5
where tˆ ∈ (t, t¯] satisfies the property that ds (¯ x, x0 ) ≥ So we have either
3 −1 12 ) 10 (AQ
whenever s ∈ [t, tˆ].
dt (x, x0 ) ≤ dt (x, x¯) + dt (¯ x, x0 ) 1 1 1 3 1 1 ≤ (AQ−1 ) 2 + (AQ−1 ) 2 + (AQ−1 ) 2 10 10 5 1 ≤ (AQ−1 ) 2 , or dt (x, x0 ) ≤ dt (x, x¯) + dt (¯ x, x0 ) 1 1 1 1 x, x0 ) + (AQ−1 ) 2 ≤ (AQ−1 ) 2 + dt¯(¯ 10 5 1 ≤ dt¯(¯ x, x0 ) + (AQ−1 ) 2 .
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
381
It then follows from (6.3.4) that |Rm(x, t)| ≤ 4Q. Hence we have proved |Rm(x, t)| ≤ 4Q 1 1 for any point (x, t) with t¯′ ≤ t ≤ t¯ and dt (x, x ¯) ≤ 10 (AQ−1 ) 2 . By combining with 1 1 the choice of t¯′ in (6.3.5), we must have t¯′ = t¯ − 200n A 2 Q−1 . This proves assertion ′ (6.3.4) . Therefore we have completed the proof of the lemma.
We now use the volume lower bound assumption to establish the crucial curvature upper bound estimate of Perelman [103] for the Ricci flow. For the Ricci flow on K¨ahler manifolds, a global version of this estimate (i.e., curvature decaying linear in time and quadratic in space) was independently obtained in [29] and [32]. Note that the volume estimate conclusion in the following Theorem 6.3.3 (ii) was not stated in Corollary 11.6 (b) of Perelman [103]. The estimate will be used later in the proof of Theorem 7.2.2 and Theorem 7.5.2. Theorem 6.3.3 (Perelman [103]). For every w > 0 there exist B = B(w) < +∞, C = C(w) < +∞, τ0 = τ0 (w) > 0, and ξ = ξ(w) > 0 (depending also on the dimension) with the following properties. Suppose we have a (not necessarily complete) solution gij (t) to the Ricci flow, defined on M × [−t0 r02 , 0], so that at each time t ∈ [−t0 r02 , 0] the metric ball Bt (x0 , r0 ) is compactly contained in M. (i) If at each time t ∈ [−t0 r02 , 0], Rm(., t) ≥ −r0−2 on Bt (x0 , r0 ) and Vol t (Bt (x0 , r0 )) ≥ wr0n , then we have the estimate |Rm(x, t)| ≤ Cr0−2 + B(t + t0 r02 )−1 whenever −t0 r02 < t ≤ 0 and dt (x, x0 ) ≤ 41 r0 . (ii) If for some 0 < τ¯ ≤ t0 , Rm(x, t) ≥ −r0−2 for t ∈ [−¯ τ r02 , 0], x ∈ Bt (x0 , r0 ), and Vol 0 (B0 (x0 , r0 )) ≥ wr0n , then we have the estimates Vol t (Bt (x0 , r0 )) ≥ ξr0n for all max{−¯ τ r02 , −τ0 r02 } ≤ t ≤ 0, and |Rm(x, t)| ≤ Cr0−2 + B(t − max{−¯ τ r02 , −τ0 r02 })−1 whenever max{−¯ τ r02 , −τ0 r02 } < t ≤ 0 and dt (x, x0 ) ≤ 14 r0 .
382
H.-D. CAO AND X.-P. ZHU
Proof. By scaling we may assume r0 = 1. (i) By the standard (relative) volume comparison, we know that there exists some w′ > 0, with w′ ≤ w, depending only on w, such that for each point (x, t) with −t0 ≤ t ≤ 0 and dt (x, x0 ) ≤ 13 , and for each r ≤ 13 , there holds Vol t (Bt (x, r)) ≥ w′ rn .
(6.3.6)
We argue by contradiction. Suppose there are sequences B, C → +∞, of solutions gij (t) and points (x′ , t′ ) such that dt′ (x′ , x0 ) ≤
1 , 4
−t0 < t′ ≤ 0 and |Rm(x′ , t′ )| > C + B(t′ + t0 )−1 .
Then by Lemma 6.3.2, we can find a sequence of points (¯ x, t¯) such that dt¯(¯ x, x0 ) <
1 , 3
Q = |Rm(¯ x, t¯)| > C + B(t¯ + t0 )−1 , and |Rm(x, t)| ≤ 4Q 1 1 1 A 2 Q− 2 , where A tends to infinity wherever (−t0 <) t¯− AQ−1 ≤ t ≤ t¯, dt (x, x¯) ≤ 10 with B, C. Thus we may take a blow-up limit along the points (¯ x, t¯) with factors Q (∞) and get a non-flat ancient solution (M∞ , gij (t)) with nonnegative curvature operator and with the asymptotic volume ratio νM∞ (t) ≥ w′ > 0 for each t ∈ (−∞, 0] (by (6.3.6)). This contradicts Lemma 6.3.1.
(ii) Let B(w), C(w) be good for the first part of the theorem. By the volume assumption at t = 0 and the standard (relative) volume comparison, we still have the estimate (6.3.6)′
Vol 0 (B0 (x, r)) ≥ w′ rn
for each x ∈ M with d0 (x, x0 ) ≤ 31 and r ≤ 13 . We will show that ξ = 5−n w′ , B = B(5−n w′ ) and C = C(5−n w′ ) are good for the second part of the theorem. By continuity and the volume assumption at t = 0, there is a maximal subinterval [−τ, 0] of the time interval [−¯ τ , 0] such that Vol t (Bt (x0 , 1)) ≥ w ≥ 5−n w′
for all t ∈ [−τ, 0].
This says that the assumptions of (i) hold with 5−n w′ in place of w and with τ in place of t0 . Thus the conclusion of the part (i) gives us the estimate (6.3.7)
|Rm(x, t)| ≤ C + B(t + τ )−1
whenever t ∈ (−τ, 0] and dt (x, x0 ) ≤ 14 . We need to show that one can choose a positive τ0 depending only on w and the dimension such that the maximal τ ≥ min{¯ τ , τ0 }.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
For t ∈ (−τ, 0] and
1 8
383
≤ dt (x, x0 ) ≤ 14 , we use (6.3.7) and Lemma 3.4.1(ii) to get
√ √ √ d dt (x, x0 ) ≥ −10(n − 1)( C + ( B/ t + τ )) dt which further gives √ √ d0 (x, x0 ) ≥ d−τ (x, x0 ) − 10(n − 1)(τ C + 2 Bτ ). This means (6.3.8)
√ √ 1 1 B(−τ ) (x0 , ) ⊃ B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) . 4 4
Note that the scalar curvature R ≥ −C(n) for some constant C(n) depending only on the dimension since Rm ≥ −1. We have √ √ d 1 Vol t B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) dt 4 Z (−R)dVt = √ √ B0 (x0 , 41 −10(n−1)(τ
C+2 Bτ ))
√ √ 1 ≤ C(n)Vol t B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4
and then (6.3.9)
√ √ 1 Vol t B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4 √ √ 1 . ≤ eC(n)τ Vol (−τ ) B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4
Thus by (6.3.6)′ , (6.3.8) and (6.3.9), Vol (−τ ) (B(−τ ) )(x0 , 1) 1 ≥ Vol (−τ ) (B(−τ ) ) x0 , 4 √ √ 1 ≥ Vol (−τ ) B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4 √ √ 1 −C(n)τ ≥e Vol 0 B0 x0 , − 10(n − 1)(τ C + 2 Bτ ) 4 n √ √ 1 −C(n)τ ′ ≥e w − 10(n − 1)(τ C + 2 Bτ ) . 4 So it suffices to choose τ0 = τ0 (w) small enough so that e−C(n)τ0
n n p √ 1 1 ≥ − 10(n − 1)(τ0 C + 2 Bτ0 ) . 4 5
Therefore we have proved the theorem.
384
H.-D. CAO AND X.-P. ZHU
6.4. Ancient κ-solutions on Three-manifolds. In this section we will determine the structures of ancient κ-solutions on three-manifolds. First of all, we consider a special class of ancient solutions — gradient shrinking Ricci solitons. Recall that a solution gij (t) to the Ricci flow is said to be a gradient shrinking Ricci soliton if there exists a smooth function f such that 1 gij = 0 for − ∞ < t < 0. 2t A gradient shrinking Ricci soliton moves by the one parameter group of diffeomorphisms generated by ∇f and shrinks by a factor at the same time. The following result of Perelman [103] gives a complete classification for all threedimensional complete κ-noncollapsed gradient shrinking solitons with bounded and nonnegative sectional curvature. (6.4.1)
∇i ∇j f + Rij +
Lemma 6.4.1 (Classification of three-dimensional shrinking solitons). Let (M, gij (t)) be a nonflat gradient shrinking soliton on a three-manifold. Suppose (M, gij (t)) has bounded and nonnegative sectional curvature and is κ-noncollapsed on all scales for some κ > 0. Then (M, gij (t)) is one of the following: (i) the round three-sphere S3 , or a metric quotient of S3 ; (ii) the round infinite cylinder S2 × R, or one of its Z2 quotients. Proof. We first consider the case that the sectional curvature of the nonflat gradient shrinking soliton is not strictly positive. Let us pull back the soliton to its universal cover. Then the pull-back metric is again a nonflat ancient κ-solution. By Hamilton’s strong maximum principle (Theorem 2.2.1), we know that the pull-back solution splits as the metric product of a two-dimensional nonflat ancient κ-solution and R. Since the two-dimensional nonflat ancient κ-solution is simply connected, it follows from Theorem 6.2.2 that it must be the round sphere S2 . Thus, the gradient shrinking soliton must be S2 × R/Γ, a metric quotient of the round cylinder. For each σ ∈ Γ and (x, s) ∈ S2 × R, we write σ(x, s) = (σ1 (x, s), σ2 (x, s)) ∈ S2 × R. Since σ sends lines to lines, and sends cross spheres to cross spheres, we have σ2 (x, s) = σ2 (y, s), for all x, y ∈ S2 . This says that σ2 reduces to a function of s alone on R. Moreover, for any (x, s), (x′ , s′ ) ∈ S2 × R, since σ preserves the distances between cross spheres S2 × {s} and S2 × {s′ }, we have |σ2 (x, s) − σ2 (x′ , s′ )| = |s − s′ |. So the projection Γ2 of Γ to the second factor R is an isometry subgroup of R. If the metric quotient S2 × R/Γ were compact, it would not be κ-noncollapsed on sufficiently large scales as t → −∞. Thus the metric quotient S2 × R/Γ is noncompact. It follows that Γ2 = {1} or Z2 . In particular, there is a Γ-invariant cross sphere S2 in the round cylinder S2 × R. Denote it by S2 × {0}. Then Γ acts on the round two-sphere S2 × {0} isometrically without fixed points. This implies Γ is either {1} or Z2 . Hence we conclude that the gradient shrinking soliton is either the round cylinder S2 × R, or ˜ where Z2 flips both S2 and R. RP2 × R, or the twisted product S2 ×R We next consider the case that the gradient shrinking soliton is compact and has strictly positive sectional curvature everywhere. By the proof of Theorem 5.2.1 (see also Remark 5.2.8) we see that the compact gradient shrinking soliton is getting round and tends to a space form (with positive constant curvature) as the time approaches the maximal time t = 0. Since the shape of a gradient shrinking Ricci soliton does not change up to reparametrizations and homothetical scalings, the gradient shrinking soliton has to be the round three-sphere S3 or a metric quotient of S3 . Finally we want to exclude the case that the gradient shrinking soliton is noncompact and has strictly positive sectional curvature everywhere.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
385
Suppose there is a (complete three-dimensional) noncompact κ-noncollapsed gradient shrinking soliton gij (t), −∞ < t < 0, with bounded and positive sectional curvature at each t ∈ (−∞, 0) and satisfying the equation (6.4.1). Then as in (6.2.25), we have ∇i R = 2Rij ∇j f.
(6.4.2)
Fix some t < 0, say t = −1, and consider a long shortest geodesic γ(s), 0 ≤ s ≤ s¯. Let x0 = γ(0) and X(s) = γ(s). ˙ Let U (0) be any unit vector orthogonal to γ(0) ˙ and translate U (0) along γ(s) to get a parallel vector field U (s), 0 ≤ s ≤ s¯, on γ. Set sU (s), for 0 ≤ s ≤ 1, e U (s), for 1 ≤ s ≤ s¯ − 1 U (s) = (¯ s − s)U (s), for s¯ − 1 ≤ s ≤ s¯. It follows from the second variation formula of arclength that Z s¯ e˙ (s)|2 − R(X, U e , X, U e ))ds ≥ 0. (|U 0
Since the curvature of the metric gij (−1) is bounded, we clearly have Z s¯ R(X, U, X, U )ds ≤ const. 0
and then Z
(6.4.3)
0
s¯
Ric (X, X)ds ≤ const..
Moreover, since the curvature of the metric gij (−1) is positive, it follows from the Cauchy-Schwarz inequality that for any unit vector field Y along γ and orthogonal to X(= γ(s)), ˙ we have Z s¯ Z s¯ Ric (X, X)Ric (Y, Y )ds |Ric (X, Y )|2 ds ≤ 0 0 Z s¯ ≤ const. · Ric (X, X)ds 0
≤ const.
and then Z
(6.4.4)
s¯ 0
√ |Ric (X, Y )|ds ≤ const. · ( s¯ + 1).
From (6.4.1) we know ∇X ∇X f + Ric (X, X) −
1 =0 2
and by integrating this equation we get X(f (γ(¯ s))) − X(f (γ(0))) +
Z
s¯ 0
1 Ric (X, X)ds − s¯ = 0. 2
386
H.-D. CAO AND X.-P. ZHU
Thus by (6.4.3) we deduce (6.4.5)
s¯ s¯ − const. ≤ hX, ∇f (γ(¯ s))i ≤ + const.. 2 2
Similarly by integrating (6.4.1) and using (6.4.4) we can deduce √ (6.4.6) |hY, ∇f (γ(¯ s))i| ≤ const. · ( s¯ + 1). These two inequalities tell us that at large distances from the fixed point x0 the function f has no critical point, and its gradient makes a small angle with the gradient of the distance function from x0 . Now from (6.4.2) we see that at large distances from x0 , R is strictly increasing along the gradient curves of f , in particular ¯= R
lim sup d(−1) (x,x0 )→+∞
R(x, −1) > 0.
¯ By the noncolLet us choose a sequence of points (xk , −1) where R(xk , −1) → R. lapsing assumption we can take a limit along this sequence of points of the gradient soliton and get an ancient κ-solution defined on −∞ < t < 0. By Proposition 6.1.2, we deduce that the limiting ancient κ-solution splits off a line. Since the soliton has positive sectional curvature, we know from Gromoll-Meyer [52] that it is orientable. Then it follows from Theorem 6.2.2 that the limiting solution is the shrinking round ¯ at time t = −1. Since the limiting solution infinite cylinder with scalar curvature R ¯ exists on (−∞, 0), we conclude that R ≤ 1. Hence R(x, −1) < 1 when the distance from x to the fixed x0 is large enough on the gradient shrinking soliton. Let us consider the level surface {f = a} of f . The second fundamental form of the level surface is given by ∇f hij = ∇i , ej |∇f | = ∇i ∇j f /|∇f |,
i, j = 1, 2,
where {e1 , e2 } is an orthonormal basis of the level surface. By (6.4.1), we have ∇ei ∇ei f =
1 1 R − Ric (ei , ei ) ≥ − > 0, 2 2 2
i = 1, 2,
since for a three-manifold the positivity of sectional curvature is equivalent to R ≥ 2Ric . It then follows from the first variation formula that Z ∇f d (6.4.7) Area {f = a} = div da |∇f | {f =a} Z 1 ≥ (1 − R) |∇f | {f =a} Z 1 ¯ (1 − R) > |∇f | {f =a} ≥0
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
387
for a large enough. We conclude that Area {f = a} strictly increases as a increases. From (6.4.5) we see that for s large enough df − s ≤ const., ds 2 and then
Thus we get from (6.4.7)
2 f − s ≤ const. · (s + 1). 4
¯ 1−R d Area {f = a} > √ Area {f = a} da 2 a for a large enough. This implies that √ ¯ a − const. log Area {f = a} > (1 − R) for a large enough. But it is clear from (6.4.7) that Area {f = a} is uniformly bounded ¯ for all large a. Thus from above by the area of the round sphere of scalar curvature R ¯ we deduce that R = 1. So Area {f = a} < 8π
(6.4.8)
for a large enough. Denote by X the unit normal vector to the level surface {f = a}. By using the Gauss equation and (6.4.1), the intrinsic curvature of the level surface {f = a} can be computed as (6.4.9)
intrinsic curvature = R1212 + det(hij ) = ≤ = = <
1 det(Hess (f )) (R − 2Ric (X, X)) + 2 |∇f |2 1 1 (R − 2Ric (X, X)) + (tr (Hess (f )))2 2 4|∇f |2 1 1 (R − 2Ric (X, X)) + (1 − (R − Ric (X, X)))2 2 4|∇f |2 1 (1 − R + Ric (X, X))2 1 − Ric (X, X) − (1 − R + Ric (X, X)) + 2 2|∇f |2 1 2
for sufficiently large a, since (1 − R + Ric(X, X)) > 0 and |∇f | is large when a is large. Thus the combination of (6.4.8) and (6.4.9) gives a contradiction to the Gauss-Bonnet formula. Therefore we have proved the lemma. As a direct consequence, there is a universal positive constant κ0 such that any nonflat three-dimensional gradient shrinking soliton, which is also an ancient
388
H.-D. CAO AND X.-P. ZHU
κ-solution, to the Ricci flow must be κ0 -noncollapsed on all scales unless it is a metric quotient of round three-sphere. The following result, claimed by Perelman in the section 1.5 of [104], shows that this property actually holds for all nonflat threedimensional ancient κ-solutions. Proposition 6.4.2 (Universal noncollapsing). There exists a positive constant κ0 with the following property. Suppose we have a nonflat three-dimensional ancient κ-solution for some κ > 0. Then either the solution is κ0 -noncollapsed on all scales, or it is a metric quotient of the round three-sphere. Proof. Let gij (x, t), x ∈ M and t ∈ (−∞, 0], be a nonflat ancient κ-solution for some κ > 0. For an arbitrary point (p, t0 ) ∈ M × (−∞, 0], we define as in Chapter 3 that τ = t0 − t,
for t < t0 , Z τ √ 1 2 l(q, τ ) = √ inf s(R(γ(s), t0 − s) + |γ(s)| ˙ gij (t0 −s) )ds| 2 τ 0 γ : [0, τ ] → M with γ(0) = p, γ(τ ) = q , Z 3 and Ve (τ ) = (4πτ )− 2 exp(−l(q, τ ))dVt0 −τ (q). M
Recall from (6.2.1) that for each τ > 0 we can find q = q(τ ) such that l(q, τ ) ≤ 32 . In view of Lemma 6.4.1, we may assume that the ancient κ-solution is not a gradient shrinking Ricci soliton. Thus by (the proof of) Theorem 6.2.1, the scalings of gij (t0 −τ ) at q(τ ) with factor τ −1 converge along a subsequence of τ → +∞ to a nonflat gradient shrinking soliton with nonnegative curvature operator which is κ-noncollapsed on all scales. We now show that the limit has bounded curvature. ¯ , g¯ij (x, t)) with −∞ < Denote the limiting nonflat gradient shrinking soliton by (M t ≤ 0. Note that there holds the Li-Yau-Hamilton inequality (Theorem 2.5.4) on any ancient κ-solution and in particular, the scalar curvature of the ancient κ-solution is pointwise nondecreasing in time. This implies that the scalar curvature of the limiting ¯ , g¯ij (x, t)) is still pointwise nondecreasing in time. Thus we only need to soliton (M show that the limiting soliton has bounded curvature at t = 0. We argue by contradiction. By lifting to its orientable cover, we may assume ¯ is orientable. Suppose the curvature of the limiting soliton is unbounded at that M ¯ is noncompact. Then by applying t = 0. Of course in this case the limiting soliton M Lemma 6.1.4, we can choose a sequence of points xj , j = 1, 2, . . . , divergent to infinity ¯ of the limit satisfies such that the scalar curvature R ¯ j , 0) ≥ j and R(x, ¯ 0) ≤ 4R(x ¯ j , 0) R(x p ¯ j , 0)) and j = 1, 2, . . .. Since the scalar curvature is nondefor all x ∈ B0 (xj , j/ R(x creasing in time, we have (6.4.10) p
¯ t) ≤ 4R(x ¯ j , 0), R(x,
¯ j , 0)), all t ≤ 0 and j = 1, 2, . . .. By combining with for all x ∈ B0 (xj , j/ R(x Hamilton’s compactness theorem (Theorem 4.1.5) and the κ-noncollapsing, we know that a subsequence of the rescaled solutions ¯ , R(x ¯ j , 0)¯ ¯ j , 0)), xj ), j = 1, 2, . . . , (M gij (x, t/R(x
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
389
∞ converges in the Cloc topology to a nonflat smooth solution of the Ricci flow. Then Proposition 6.1.2 implies that the new limit at the new time {t = 0} must split off a line. By pulling back the new limit to its universal cover and applying Hamilton’s strong maximum principle, we deduce that the pull-back of the new limit on the universal cover splits off a line for all time t ≤ 0. Thus by combining with Theorem 6.2.2 and the argument in the proof of Lemma 6.4.1, we further deduce that the new ¯ is orientable, limit is either the round cylinder S2 × R or the round RP2 × R. Since M ¯ , g¯ij (x, 0)) has nonnegative curvature operator the new limit must be S2 × R. Since (M ¯ j , 0) → +∞, this gives a contradiction and the points {xj } going to infinity and R(x to Proposition 6.1.1. So we have proved that the limiting gradient shrinking soliton has bounded curvature at each time. Hence by Lemma 6.4.1, the limiting gradient shrinking soliton is either the round three-sphere S3 or its metric quotients, or the infinite cylinder S2 × R or one of its Z2 quotients. If the asymptotic gradient shrinking soliton is the round three-sphere S3 or its metric quotients, it follows from Lemma 5.2.4 and Proposition 5.2.5 that the ancient κ-solution must be round. Thus in the following we may assume the asymptotic gradient shrinking soliton is the infinite cylinder S2 × R or a Z2 quotient of S2 × R. We now come back to consider the original ancient κ-solution (M, gij (x, t)). By rescaling, we can assume that R(x, t) ≤ 1 for all (x, t) satisfying dt0 (x, p) ≤ 2 and t ∈ [t0 − 1, t0 ]. We will argue as in the proof of Theorem 3.3.2 (Perelman’s no local collapsing theorem I) to obtain a positive lower bound for Vol t0 (Bt0 (p, 1)). 1
3 Denote by ξ = Vol t0 (Bt0 (p, 1)) √ . For any v ∈ Tp M we can find an L-geodesic γ(τ ), starting at p, with limτ →0+ τ γ(τ ˙ ) = v. It follows from the L-geodesic equation (3.2.1) that
√ 1√ d √ ˙ − τ ∇R + 2Ric ( τ γ, ˙ ·) = 0. ( τ γ) dτ 2 By integrating as before we see that for τ ≤ ξ with the property γ(σ) ∈ Bt0 (p, 1) as long as σ < τ , there holds √ | τ γ(τ ˙ ) − v| ≤ Cξ(|v| + 1) where C is some positive constant depending only on the dimension. Without loss of 1 1 . Then for v ∈ Tp M with |v| ≤ 14 ξ − 2 generality, we may assume Cξ ≤ 14 and ξ ≤ 100 and for τ ≤ ξ with the property γ(σ) ∈ Bt0 (p, 1) as long as σ < τ , we have dt0 (p, γ(τ )) ≤
Z
τ
|γ(σ)|dσ ˙ Z 1 − 1 τ dσ 2 √ < ξ 2 σ 0 = 1. 0
This shows (6.4.11)
1 1 L exp |v| ≤ ξ − 2 (ξ) ⊂ Bt0 (p, 1). 4
390
H.-D. CAO AND X.-P. ZHU
We decompose Perelman’s reduced volume Ve (ξ) as
(6.4.12)
Ve (ξ) =
Z
n
1
L exp |v|≤ 41 ξ − 2
+
Z
o
(ξ)
n
M\L exp
1 |v|≤ 41 ξ − 2
3
o
(ξ)
(4πξ)− 2 exp(−l(q, ξ))dVt0 −ξ (q).
By using (6.4.11) and the metric evolution equation of the Ricci flow, the first term on the RHS of (6.4.12) can be estimated by Z
3
−1 L exp{|v|≤ 41 ξ 2
Z
≤
(4πξ)− 2 exp(−l(q, ξ))dVt0 −ξ (q) }(ξ) 3
(4πξ)− 2 e3ξ dVt0 (q) Bt0 (p,1) 3
3
= (4π)− 2 e3ξ ξ 2 3
< ξ2, while by using Theorem 3.2.7 (Perelman’s Jacobian comparison theorem), the second term on the RHS of (6.4.12) can be estimated by (6.4.13)
Z
n
M\L exp
≤
Z
1 |v|≤ 14 ξ − 2
3
(4πξ)− 2 exp(−l(q, ξ))dVt0 −ξ (q)
o
(ξ) 3
1 {|v|> 41 ξ − 2
− 32
= (4π)
Z
(4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dv }
1
exp(−|v|2 )dv
{|v|> 41 ξ − 2 }
3
< ξ2 3
since limτ →0+ τ − 2 J (τ ) = 1 and limτ →0+ l(τ ) = |v|2 by (3.2.18) and (3.2.19) respectively. Thus we obtain (6.4.14)
3 Ve (ξ) < 2ξ 2 .
On the other hand, we recall that there exist a sequence τk → +∞ and a sequence of points q(τk ) ∈ M with l(q(τk ), τk ) ≤ 32 so that the scalings of the ancient κ-solution at q(τk ) with factor τk−1 converge to either round S2 × R or one of its Z2 quotients. For sufficiently large k, we construct a path γ : [0, 2τk ] → M , connecting p to any given point q ∈ M , as follows: the first half path γ|[0,τk ] connects p to q(τk ) such that 1 l(q(τk ), τk ) = √ 2 τk
Z
0
τk
√
τ (R + |γ(τ ˙ )|2 )dτ ≤ 2,
and the second half path γ|[τk ,2τk ] is a shortest geodesic connecting q(τk ) to q with respect to the metric gij (t0 − τk ). Note that the rescaled metric τk−1 gij (t0 − τ ) over √ the domain Bt0 −τk (q(τk ), τk ) × [t0 − 2τk , t0 − τk ] is sufficiently close to the round
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
391
S2 × R or its Z2 quotients. Then there is a universal positive constant β such that Z τk Z 2τk √ 1 + τ (R + |γ(τ ˙ )|2 )dτ l(q, 2τk ) ≤ √ 2 2τk 0 τk Z 2τk √ √ 1 ≤ 2+ √ τ (R + |γ(τ ˙ )|2 )dτ 2 2τk τk ≤β √ for all q ∈ Bt0 −τk (q(τk ), τk ). Thus Z 3 Ve (2τk ) = (4π(2τk ))− 2 exp(−l(q, 2τk ))dVt0 −2τk (q) M Z 3 −β (4π(2τk ))− 2 dVt0 −2τk (q) ≥e √ Bt0 −τk (q(τk ), τk )
≥ β˜
˜ Here we have used the curvature estimate for some universal positive constant β. (6.2.6). By combining with the monotonicity of Perelman’s reduced volume (Theorem 3.2.8) and (6.4.14), we deduce that
This proves
3 β˜ ≤ Ve (2τk ) ≤ Ve (ξ) < 2ξ 2 .
Vol t0 (Bt0 (p, 1)) ≥ κ0 > 0 for some universal positive constant κ0 . So we have proved that the ancient κ-solution is also an ancient κ0 -solution. The important Li-Yau-Hamilton inequality gives rise to a parabolic Harnack estimate (Corollary 2.5.7) for solutions of the Ricci flow with bounded and nonnegative curvature operator. As explained in the previous section, the no local collapsing theorem of Perelman implies a volume lower bound from a curvature upper bound, while the estimate in the previous section implies a curvature upper bound from a volume lower bound. The combination of these two estimates as well as the Li-Yau-Hamilton inequality will give an important elliptic type property for three-dimensional ancient κ-solutions. This elliptic type property was first implicitly given by Perelman in [103] and it will play a crucial role in the analysis of singularities. Theorem 6.4.3 (Elliptic type estimate). There exist a positive constant η and a positive increasing function ω : [0, +∞) → (0, +∞) with the following properties. Suppose we have a three-dimensional ancient κ-solution (M, gij (t)), −∞ < t ≤ 0, for some κ > 0. Then (i) for every x, y ∈ M and t ∈ (−∞, 0], there holds R(x, t) ≤ R(y, t) · ω(R(y, t)d2t (x, y)); (ii) for all x ∈ M and t ∈ (−∞, 0], there hold ∂R 3 (x, t) ≤ ηR2 (x, t). 2 |∇R|(x, t) ≤ ηR (x, t) and ∂t
392
H.-D. CAO AND X.-P. ZHU
Proof. (i) Consider a three-dimensional nonflat ancient κ-solution gij (x, t) on M × (−∞, 0]. In view of Proposition 6.4.2, we may assume that the ancient solution is universal κ0 -noncollapsed. Obviously we only need to establish the estimate at t = 0. Let y be an arbitrarily fixed point in M . By rescaling, we can assume R(y, 0) = 1. Let us first consider the case that sup{R(x, 0)d20 (x, y) | x ∈ M } > 1. Define z to be the closest point to y (at time t = 0) satisfying R(z, 0)d20 (z, y) = 1. We want to 1 bound R(x, 0)/R(z, 0) from above for x ∈ B0 (z, 2R(z, 0)− 2 ). Connect y and z by a shortest geodesic and choose a point z˜ lying on the geo1 desic satisfying d0 (˜ z , z) = 14 R(z, 0)− 2 . Denote by B the ball centered at z˜ and with 1 radius 14 R(z, 0)− 2 (with respect to the metric at t = 0). Clearly the ball B lies in 1 1 B0 (y, R(z, 0)− 2 ) and lies outside B0 (y, 21 R(z, 0)− 2 ). Thus for x ∈ B, we have R(x, 0)d20 (x, y) ≤ 1 and d0 (x, y) ≥
1 1 R(z, 0)− 2 2
and hence R(x, 0) ≤
1 1 ( 12 R(z, 0)− 2 )2
for all x ∈ B.
Then by the Li-Yau-Hamilton inequality and the κ0 -noncollapsing, we have 3 1 − 21 , Vol 0 (B) ≥ κ0 R(z, 0) 4 and then 1
Vol 0 (B0 (z, 8R(z, 0)− 2 )) ≥
1 κ0 (8R(z, 0)− 2 )3 . 215
So by Theorem 6.3.3(ii), there exist positive constants B(κ0 ), C(κ0 ), and τ0 (κ0 ) such that (6.4.15)
R(x, 0) ≤ (C(κ0 ) +
B(κ0 ) )R(z, 0) τ0 (κ0 )
1
for all x ∈ B0 (z, 2R(z, 0)− 2 ). We now consider the remaining case. If R(x, 0)d20 (x, y) ≤ 1 everywhere, we choose a point z satisfying sup{R(x, 0) | x ∈ M } ≤ 2R(z, 0). Obviously we also have the estimate (6.4.15) in this case. We next want to bound R(z, 0) for the chosen z ∈ M . By (6.4.15) and the Li-Yau-Hamilton inequality, we have R(x, t) ≤ (C(κ0 ) +
B(κ0 ) )R(z, 0) τ0 (κ0 )
1
for all x ∈ B0 (z, 2R(z, 0)− 2 ) and all t ≤ 0. It then follows from the local derivative estimates of Shi that
which implies (6.4.16)
∂R e 0 )R(z, 0)2 , (z, t) ≤ C(κ ∂t
for all − R−1 (z, 0) ≤ t ≤ 0
R(z, −cR−1(z, 0)) ≥ cR(z, 0)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
393
for some small positive constant c depending only on κ0 . On the other hand, by using the Harnack estimate in Corollary 2.5.7, we have (6.4.17)
1 = R(y, 0) ≥ e cR(z, −cR−1 (z, 0))
1
for some small positive constant e c depending only on κ0 , since d0 (y, z) ≤ R(z, 0)− 2 and the metric gij (t) is equivalent on 1
B0 (z, 2R(z, 0)− 2 ) × [−cR−1 (z, 0), 0]
with c > 0 small enough. Thus we get from (6.4.16) and (6.4.17) that e R(z, 0) ≤ A
(6.4.18)
e depending only on κ0 . for some positive constant A 1 1 1 e − 12 , the combiSince B0 (z, 2R(z, 0)− 2 ) ⊃ B0 (y, R(z, 0)− 2 ) and R(z, 0)− 2 ≥ (A) nation of (6.4.15) and (6.4.18) gives R(x, 0) ≤ (C(κ0 ) +
(6.4.19)
B(κ0 ) e )A τ0 (κ0 )
e − 2 ). Then by the κ0 -noncollapsing there exists a positive whenever x ∈ B0 (y, (A) constant r0 depending only on κ0 such that 1
Vol 0 (B0 (y, r0 )) ≥ κ0 r03 .
For any fixed R0 ≥ r0 , we then have Vol 0 (B0 (y, R0 )) ≥ κ0 r03 = κ0 (
r0 3 ) · R03 . R0
By applying Theorem 6.3.3 (ii) again and noting that the constant κ0 is universal, there exists a positive constant ω(R0 ) depending only on R0 such that R(x, 0) ≤ ω(R02 )
1 for all x ∈ B0 (y, R0 ). 4
This gives the desired estimate. (ii) This follows immediately from conclusion (i), the Li-Yau-Hamilton inequality and the local derivative estimate of Shi. As a consequence, we have the following compactness result due to Perelman [103]. Corollary 6.4.4 (Compactness of ancient κ0 -solutions). The set of nonflat three-dimensional ancient κ0 -solutions is compact modulo scaling in the sense that for any sequence of such solutions and marking points (xk , 0) with R(xk , 0) = 1, we ∞ can extract a Cloc converging subsequence whose limit is also an ancient κ0 -solution. Proof. Consider any sequence of three-dimensional ancient κ0 -solutions and marking points (xk , 0) with R(xk , 0) = 1. By Theorem 6.4.3(i), the Li-Yau-Hamilton inequality and Hamilton’s compactness theorem (Theorem 4.1.5), we can extract a ∞ ¯ , g¯ij (x, t)), with −∞ < t ≤ 0, Cloc converging subsequence such that the limit (M is an ancient solution to the Ricci flow with nonnegative curvature operator and κ0 noncollapsed on all scales. Since any ancient κ0 -solution satisfies the Li-Yau-Hamilton
394
H.-D. CAO AND X.-P. ZHU
¯ t) of the limit (M ¯ , g¯ij (x, t)) is inequality, it implies that the scalar curvature R(x, pointwise nondecreasing in time. Thus it remains to show that the limit solution has bounded curvature at t = 0. ¯ is noncompact. By pulling Obviously we may assume the limiting manifold M back the limiting solution to its orientable cover, we can assume that the limiting man¯ is orientable. We now argue by contradiction. Suppose the scalar curvature ifold M ¯ of the limit at t = 0 is unbounded. R ¯,j = By applying Lemma 6.1.4, we can choose a sequence of points xj ∈ M ¯ of the limit satisfies 1, 2, . . . , divergent to infinity such that the scalar curvature R ¯ j , 0) ≥ j and R(x, ¯ 0) ≤ 4R(x ¯ j , 0) R(x p ¯ j , 0)). Then from the fact that the limiting for all j = 1, 2, . . . , and x ∈ B0 (xj , j/ R(x ¯ scalar curvature R(x, t) is pointwise nondecreasing in time, we have ¯ t) ≤ 4R(x ¯ j , 0) R(x, p ¯ j , 0)) and t ≤ 0. By combining with Hamilton’s for all j = 1, 2, . . ., x ∈ B0 (xj , j/ R(x compactness theorem (Theorem 4.1.5) and the κ0 -noncollapsing, we know that a subsequence of the rescaled solutions
(6.4.20)
¯ , R(x ¯ j , 0)¯ ¯ j , 0)), xj ), j = 1, 2, . . . , (M gij (x, t/R(x ∞ converges in the Cloc topology to a nonflat smooth solution of the Ricci flow. Then Proposition 6.1.2 implies that the new limit at the new time {t = 0} must split off a line. By pulling back the new limit to its universal cover and applying Hamilton’s strong maximum principle, we deduce that the pull-back of the new limit on the universal cover splits off a line for all time t ≤ 0. Thus by combining with Theorem 6.2.2 and the argument in the proof of Lemma 6.4.1, we further deduce that the new ¯ is orientable, limit is either the round cylinder S2 × R or the round RP2 × R. Since M 2 ¯ the new limit must be S ×R. Moreover, since (M , g¯ij (x, 0)) has nonnegative curvature ¯ j , 0) → +∞, this gives a operator and the points {xj } are going to infinity and R(x ¯ , g¯ij (x, t)) has contradiction to Proposition 6.1.1. So we have proved that the limit (M uniformly bounded curvature.
Arbitrarily fix ε > 0. Let gij (x, t) be a nonflat ancient κ-solution on a threemanifold M for some κ > 0. We say that a point x0 ∈ M is the center of an evolving ε-neck at t = 0, if the solution gij (x, t) in the set {(x, t) | − ε−2 Q−1 < t ≤ 0, d2t (x, x0 ) < ε−2 Q−1 }, where Q = R(x0 , 0), is, after scaling with factor Q, ε-close (in −1 the C [ε ] topology) to the corresponding set of the evolving round cylinder having scalar curvature one at t = 0. As another consequence of the elliptic type estimate, we have the following global structure result obtained by Perelman in [103] for noncompact ancient κ-solutions. Corollary 6.4.5. For any ε > 0 there exists C = C(ε) > 0, such that if gij (t) is a nonflat ancient κ-solution on a noncompact three-manifold M for some κ > 0, and Mε denotes the set of points in M which are not centers of evolving ε-necks at t = 0, then at t = 0, either the whole manifold M is the round cylinder S2 × R or its Z2 metric quotients, or Mε satisfies the following (i) Mε is compact, 1 (ii) diam Mε ≤ CQ− 2 and C −1 Q ≤ R(x, 0) ≤ CQ, whenever x ∈ Mε , where Q = R(x0 , 0) for some x0 ∈ ∂Mε .
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
395
Proof. We first consider the easy case that the curvature operator of the ancient κ-solution has a nontrivial null vector somewhere at some time. Let us pull back the solution to its universal cover. By applying Hamilton’s strong maximum principle and Theorem 6.2.2, we see that the universal cover is the evolving round cylinder S2 × R. Thus in this case, by the argument in the proof of Lemma 6.4.1, we conclude that the ancient κ-solution is either isometric to the round cylinder S2 × R or one of its Z2 ˜ where Z2 flips both S2 , metric quotients (i.e., RP2 × R, or the twisted product S2 ×R or R). We then assume that the curvature operator of the nonflat ancient κ-solution is positive everywhere. Firstly we want to show Mε is compact. We argue by contradiction. Suppose there exists a sequence of points zk , k = 1, 2, . . ., going to infinity (with respect to the metric gij (0)) such that each zk is not the center of any evolving ε-neck. For an arbitrarily fixed point z0 ∈ M , it follows from Theorem 6.4.3(i) that 0 < R(z0 , 0) ≤ R(zk , 0) · ω(R(zk , 0)d20 (zk , z0 ))
which implies that lim R(zk , 0)d20 (zk , z0 ) = +∞.
k→∞
Since the sectional curvature of the ancient κ-solution is positive everywhere, the underlying manifold is diffeomorphic to R3 , and in particular, orientable. Then as before, by Proposition 6.1.2, Theorem 6.2.2 and Corollary 6.4.4, we conclude that zk is the center of an evolving ε-neck for k sufficiently large. This is a contradiction, so we have proved that Mε is compact. Again, we notice that M is diffeomorphic to R3 since the curvature operator is positive. According to the resolution of the Schoenflies conjecture in three-dimensions, every approximately round two-sphere cross-section through the center of an evolving ε-neck divides M into two parts such that one of them is diffeomorphic to the three-ball B3 . Let ϕ be the Busemann function on M , it is a standard fact that ϕ is convex and proper. Since Mε is compact, Mε is contained in a compact set K = ϕ−1 ((−∞, A]) for some large A. We note that each point x ∈ M \Mε is the center of an ε-neck. It is clear that there is an ε-neck N lying entirely outside K. Consider a point x on one of the boundary components of the ε-neck N . Since x ∈ M \ Mε , there is an ε-neck adjacent to the initial ε-neck, producing a longer neck. We then take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Mε or go on infinitely to produce a semi-infinite (topological) cylinder. The same procedure can be repeated for the other boundary component of the initial ˜ . If N ˜ never touches ε-neck. This procedure will give a maximal extended neck N Mε , the manifold will be diffeomorphic to the standard infinite cylinder, which is ˜ touch Mε , then there is a geodesic connecting a contradiction. If both ends of N two points of Mε and passing through N . This is impossible since the function ϕ ˜ will touch Mε and the other end will is convex. So we conclude that one end of N tend to infinity to produce a semi-infinite (topological) cylinder. Thus we can find an approximately round two-sphere cross-section which encloses the whole set Mε and 1 touches some point x0 ∈ ∂Mε . We next want to show that R(x0 , 0) 2 · diam(Mε ) is bounded from above by some positive constant C = C(ε) depending only on ε. Suppose not; then there exists a sequence of nonflat noncompact threedimensional ancient κ-solutions with positive curvature operator such that for the above chosen points x0 ∈ ∂Mε there would hold (6.4.21)
1
R(x0 , 0) 2 · diam (Mε ) → +∞.
396
H.-D. CAO AND X.-P. ZHU
By Proposition 6.4.2, we know that the ancient solutions are κ0 -noncollapsed on all scales for some universal positive constant κ0 . Let us dilate the ancient solutions around the points x0 with the factors R(x0 , 0). By Corollary 6.4.4, we can extract a convergent subsequence. From the choice of the points x0 and (6.4.21), the limit has at least two ends. Then by Toponogov’s splitting theorem the limit is isometric to X × R for some nonflat two-dimensional ancient κ0 -solution X. Since M is orientable, we conclude from Theorem 6.2.2 that limit must be the evolving round cylinder S2 ×R. This contradicts the fact that each chosen point x0 is not the center of any evolving ε-neck. Therefore we have proved 1
diam (Mε ) ≤ CQ− 2
for some positive constant C = C(ε) depending only on ε, where Q = R(x0 , 0). Finally by combining this diameter estimate with Theorem 6.4.3(i), we immediately deduce e −1 Q ≤ R(x, 0) ≤ CQ, e C whenever x ∈ Mε ,
e depending only on ε. for some positive constant C
We now can describe the canonical structures for three-dimensional nonflat (compact or noncompact) ancient κ-solutions. The following theorem was given by Perelman in the section 1.5 of [104]. Recently in [34], this canonical neighborhood result has been extended to four-dimensional ancient κ-solutions with isotropic curvature pinching. Theorem 6.4.6 (Canonical neighborhood theorem). For any ε > 0 one can find positive constants C1 = C1 (ε) and C2 = C2 (ε) with the following property. Suppose we have a three-dimensional nonflat (compact or noncompact) ancient κsolution (M, gij (x, t)). Then either the ancient solution is the round RP2 × R, or every point (x, t) has an open neighborhood B, with Bt (x, r) ⊂ B ⊂ Bt (x, 2r) for 1 some 0 < r < C1 R(x, t)− 2 , which falls into one of the following three categories: (a) B is an evolving ε-neck (in the sense that it is the slice at the time t of the parabolic region {(x′ , t′ ) | x′ ∈ B, t′ ∈ [t − ε−2 R(x, t)−1 , t]} which is, after scaling with factor R(x, t) and shifting the time t to zero, ε-close (in −1 the C [ε ] topology) to the subset (S2 × I) × [−ε−2 , 0] of the evolving standard round cylinder with scalar curvature 1 and length 2ε−1 to I at the time zero), or (b) B is an evolving ε-cap (in the sense that it is the time slice at the time t of an evolving metric on B3 or RP3 \ B¯3 such that the region outside some suitable compact subset of B3 or RP3 \ B¯3 is an evolving ε-neck), or (c) B is a compact manifold (without boundary) with positive sectional curvature (thus it is diffeomorphic to the round three-sphere S3 or a metric quotient of S3 ); furthermore, the scalar curvature of the ancient κ-solution on B at time t is between C2−1 R(x, t) and C2 R(x, t), and the volume of B in case (a) and case (b) satisfies 3
(C2 R(x, t))− 2 ≤ Vol t (B) ≤ εr3 . Proof. As before, we first consider the easy case that the curvature operator has a nontrivial null vector somewhere at some time. By pulling back the solution to
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
397
its universal cover and applying Hamilton’s strong maximum principle and Theorem 6.2.2, we deduce that the universal cover is the evolving round cylinder S2 × R. Then exactly as before, by the argument in the proof of Lemma 6.4.1, we conclude that the ancient κ-solution is isometric to the round S2 × R, RP2 × R, or the twisted product ˜ where Z2 flips both S2 and R. Clearly each point of the round cylinder S2 × R S2 ×R ˜ has a neighborhood falling into the category (a) or (b) or the twisted product S2 ×R (over RP3 \ B¯3 ). We now assume that the curvature operator of the nonflat ancient κ-solution is positive everywhere. Then the manifold is orientable by the Cheeger-Gromoll theorem [23] for the noncompact case or the Synge theorem [22] for the compact case. 1 Without loss of generality, we may assume ε is suitably small, say 0 < ε < 100 . If the nonflat ancient κ-solution is noncompact, the conclusions follow immediately from the combination of Corollary 6.4.5 and Theorem 6.4.3(i). Thus we may assume the nonflat ancient κ-solution is compact. By Proposition 6.4.2, either the compact ancient κ-solution is isometric to a metric quotient of the round S3 , or it is κ0 noncollapsed on all scales for the universal positive constant κ0 . Clearly each point of a metric quotient of the round S3 has a neighborhood falling into category (c). Thus we may further assume the ancient κ-solution is also κ0 -noncollapsing. 1 ), there exist a We argue by contradiction. Suppose that for some ε ∈ (0, 100 sequence of compact orientable ancient κ0 -solutions (Mk , gk ) with positive curvature operator, a sequence of points (xk , 0) with xk ∈ Mk and sequences of positive constants 2 C1k → ∞ and C2k = ω(4C1k ), with the function ω given in Theorem 6.4.3, such 1 that for every radius r, 0 < r < C1k R(xk , 0)− 2 , any open neighborhood B, with B0 (xk , r) ⊂ B ⊂ B0 (xk , 2r), does not fall into one of the three categories (a), (b) and (c), where in the case (a) and case (b), we require the neighborhood B to satisfy the volume estimate 3
(C2k R(xk , 0))− 2 ≤ Vol 0 (B) ≤ εr3 . By Theorem 6.4.3(i) and the choice of the constants C2k we see that the diameter 1 of each Mk at t = 0 is at least C1k R(xk , 0)− 2 ; otherwise we can choose suitable 1 r ∈ (0, C1k R(xk , 0)− 2 ) and B = Mk , which falls into the category (c) with the scalar −1 curvature between C2k R(x, 0) and C2k R(x, 0) on B. Now by scaling the ancient κ0 -solutions along the points (xk , 0) with factors R(xk , 0), it follows from Corollary ∞ 6.4.4 that a sequence of the ancient κ0 -solutions converge in the Cloc topology to a noncompact orientable ancient κ0 -solution. If the curvature operator of the noncompact limit has a nontrivial null vector somewhere at some time, it follows exactly as before by using the argument in the proof of Lemma 6.4.1 that the orientable limit is isometric to the round S2 × R, or ˜ where Z2 flips both S2 and R. Then for k large enough, the twisted product S2 ×R a suitable neighborhood B (for suitable r) of the point (xk , 0) would fall into the category (a) or (b) (over RP3 \ B¯3 ) with the desired volume estimate. This is a contradiction. If the noncompact limit has positive sectional curvature everywhere, then by using Corollary 6.4.5 and Theorem 6.4.3(i) for the noncompact limit we see that for k large enough, a suitable neighborhood B (for suitable r) of the point (xk , 0) would fall into category (a) or (b) (over B3 ) with the desired volume estimate. This is also a contradiction. Finally, the statement on the curvature estimate in the neighborhood B follows directly from Theorem 6.4.3(i).
398
H.-D. CAO AND X.-P. ZHU
7. Ricci Flow on Three-manifolds. We will use the Ricci flow to study the topology of compact orientable three-manifolds. Let M be a compact threedimensional orientable manifold. Arbitrarily given a Riemannian metric on the manifold, we evolve it by the Ricci flow. The basic idea is to understand the topology of the underlying manifold by studying long-time behavior of the solution of the Ricci flow. We have seen in Chapter 5 that for a compact three-manifold with positive Ricci curvature as initial data, the solution to the Ricci flow tends, up to scalings, to a metric of positive constant curvature. Consequently, a compact three-manifold with positive Ricci curvature is diffeomorphic to the round three-sphere or a metric quotient of it. However, for general initial metrics, the Ricci flow may develop singularities in some parts while it keeps smooth in other parts. Naturally one would like to cut off the singularities and continue to run the Ricci flow. If the Ricci flow still develops singularities after a while, one can do the surgeries and run the Ricci flow again. By repeating this procedure, one will get a kind of “weak” solution to the Ricci flow. Furthermore, if the “weak” solution has only a finite number of surgeries at any finite time interval and one can remember what had been cut during the surgeries, and if the “weak” solution has a well-understood long-time behavior, then one will also get the topology structure of the initial manifold. This theory of surgically modified Ricci flow was first developed by Hamilton [64] for compact four-manifolds and further developed more recently by Perelman [104] for compact orientable three-manifolds. The main purpose of this chapter is to give a complete and detailed discussion of the Ricci flow with surgery on three-manifolds. 7.1. Canonical Neighborhood Structures. Let us call a Riemannian metric on a compact orientable three-dimensional manifold normalized if the eigenvalues 1 1 ≥ λ ≥ µ ≥ ν ≥ − 10 , and of its curvature operator at every point are bounded by 10 every geodesic ball of radius one has volume at least one. By the evolution equation of the curvature and the maximum principle, it is easy to see that any solution to the Ricci flow with (compact and three-dimensional) normalized initial metric exists on a maximal time interval [0, tmax ) with tmax > 1. Consider a smooth solution gij (x, t) to the Ricci flow on M × [0, T ), where M is a compact orientable three-manifold and T < +∞. After rescaling, we may always assume the initial metric gij (·, 0) is normalized. By Theorem 5.3.2, the solution gij (·, t) then satisfies the pinching estimate R ≥ (−ν)[log(−ν) + log(1 + t) − 3]
(7.1.1)
whenever ν < 0 on M × [0, T ). Recall the function y = f (x) = x(log x − 3), for e2 ≤ x < +∞, is increasing and convex with range −e2 ≤ y < +∞, and its inverse function is also increasing and satisfies lim f −1 (y)/y = 0.
y→+∞
We can rewrite the pinching estimate (7.1.1) as (7.1.2)
Rm(x, t) ≥ −[f −1 (R(x, t)(1 + t))/(R(x, t)(1 + t))]R(x, t)
on M × [0, T ).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
399
Suppose that the solution gij (·, t) becomes singular as t → T . Let us take a sequence of times tk → T , and a sequence of points pk ∈ M such that for some positive constant C, |Rm|(x, t) ≤ CQk with Qk = |Rm(pk , tk )| for all x ∈ M and t ∈ [0, tk ]. Thus, (pk , tk ) is a sequence of (almost) maximum points. By applying Hamilton’s compactness theorem and Perelman’s no local collapsing theorem I as well as the pinching estimate (7.1.2), a sequence of the scalings of the solution gij (x, t) around the points pk with factors Qk converges to a nonflat complete three-dimensional orientable ancient κ-solution (for some κ > 0). For an arbitrarily given ε > 0, the canonical neighborhood theorem (Theorem 6.4.6) in the previous chapter implies that each point in the ancient κ-solution has a neighborhood which is either an evolving ε-neck, or an evolving ε-cap, or a compact (without boundary) positively curved manifold. This gives the structure of singularities coming from a sequence of (almost) maximum points. However the above argument does not work for singularities coming from a sequence of points (yk , τk ) with τk → T and |Rm(yk , τk )| → +∞ when |Rm(yk , τk )| is not comparable with the maximum of the curvature at the time τk , since we cannot take a limit directly. In [103], Perelman developed a refined rescaling argument to obtain the following singularity structure theorem. We remark that our statement of the singularity structure theorem below is slightly different from Perelman’s original statement (cf. Theorem 12.1 of [103]). While Perelman assumed the condition of κ-noncollapsing on scales less than r0 , we assume that the initial metric is normalized so that from the rescaling argument one can get the κ-noncollapsing on all scales for the limit solutions. Theorem 7.1.1 (Singularity structure theorem). Given ε > 0 and T0 > 1, one can find r0 > 0 with the following property. If gij (x, t), x ∈ M and t ∈ [0, T ) with 1 < T ≤ T0 , is a solution to the Ricci flow on a compact orientable three-manifold M with normalized initial metric, then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) | d2t0 (x, x0 ) < ε−2 Q−1 , t0 − ε−2 Q−1 ≤ t ≤ t0 } −1 is, after scaling by the factor Q, ε-close (in the C [ε ] -topology) to the corresponding subset of some orientable ancient κ-solution (for some κ > 0). Proof. Since the initial metric is normalized, it follows from the no local collapsing theorem I or I’ (and their proofs) that there is a positive constant κ, depending only on T0√ , such that the solution in Theorem 7.1.1 is κ-noncollapsed on all scales less than T0 . Let C(ε) be a positive constant larger than or equal to ε−2 . It suffices to prove that there exists r0 > 0 such that for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in the parabolic region {(x, t) ∈ M × [0, T ) | d2t0 (x, x0 ) < C(ε)Q−1 , t0 − C(ε)Q−1 ≤ t ≤ t0 } is, after scaling by the factor Q, ε-close to the corresponding subset of some orientable ancient κ-solution. The constant C(ε) will be determined later. We argue by contradiction. Suppose for some ε > 0, there exist a sequence of solutions (Mk , gk (·, t)) to the Ricci flow on compact orientable three-manifolds with normalized initial metrics, defined on the time intervals [0,Tk ) with 1 < Tk ≤ T0 , a sequence of positive numbers rk → 0, and a sequence of points xk ∈ Mk and times tk ≥ 1 with Qk = Rk (xk , tk ) ≥ rk−2 such that each solution (Mk , gk (·, t)) in the −1 parabolic region {(x, t) ∈ Mk × [0, Tk ) | d2tk (x, xk ) < C(ε)Q−1 k , tk − C(ε)Qk ≤ t ≤ tk } is not, after scaling by the factor Qk , ε-close to the corresponding subset of any orientable ancient κ-solution, where Rk denotes the scalar curvature of (Mk , gk ). For each solution (Mk , gk (·, t)), we may adjust the point (xk , tk ) with tk ≥ 21
400
H.-D. CAO AND X.-P. ZHU
and with Qk = Rk (xk , tk ) to be as large as possible so that the conclusion of the theorem fails at (xk , tk ), but holds for any (x, t) ∈ Mk × [tk − Hk Q−1 k , tk ] satisfying Rk (x, t) ≥ 2Qk , where Hk = 41 rk−2 → +∞ as k → +∞. Indeed, suppose not, by setting (xk1 , tk1 ) = (xk , tk ), we can choose a sequence of points (xkl , tkl ) ∈ Mk × [tk(l−1) − Hk Rk (xk(l−1) , tk(l−1) )−1 , tk(l−1) ] such that Rk (xkl , tkl ) ≥ 2Rk (xk(l−1) , tk(l−1) ) and the conclusion of the theorem fails at (xkl , tkl ) for each l = 2, 3, . . . . Since the solution is smooth, but Rk (xkl , tkl ) ≥ 2Rk (xk(l−1) , tk(l−1) ) ≥ · · · ≥ 2l−1 Rk (xk , tk ), and tkl ≥ tk(l−1) − Hk Rk (xk(l−1) , tk(l−1) )−1 ≥ t k − Hk ≥
1 , 2
l−1 X 1 Rk (xk , tk )−1 i−1 2 i=1
this process must terminate after a finite number of steps and the last element fits. Let (Mk , g˜k (·, t), xk ) be the rescaled solutions obtained by rescaling (Mk , gk (·, t)) around xk with the factors Qk = Rk (xk , tk ) and shifting the time tk ˜ k the rescaled scalar curvature. We will show that to the new time zero. Denote by R a subsequence of the orientable rescaled solutions (Mk , g˜k (·, t), xk ) converges in the ∞ Cloc topology to an orientable ancient κ-solution, which is a contradiction. In the following we divide the argument into four steps. Step 1. First of all, we need a local bound on curvatures. ¯ ¯ Lemma 7.1.2. For each (¯ x, t¯) with tk − 21 Hk Q−1 k ≤ t ≤ tk , we have Rk (x, t) ≤ 4Qk −1 −1 ¯ ≤ t ≤ t¯ and d2¯ (x, x ¯ , where Q ¯ k = Qk + Rk (¯ whenever t¯− cQ ¯ ) ≤ cQ x, t¯) and c > 0 k k t is a small universal constant. ¯ −1 , t¯] with c > 0 to be ¯ −1 ) 12 ) × [t¯− cQ Proof. Consider any point (x, t) ∈ Bt¯(¯ x, (cQ k k determined. If Rk (x, t) ≤ 2Qk , there is nothing to show. If Rk (x, t) > 2Qk , consider a space-time curve γ from (x, t) to (¯ x, t¯) that goes straight from (x, t) to (x, t¯) and goes from (x, t¯) to (¯ x, t¯) along a minimizing geodesic (with respect to the metric gk (·, t¯)). If there is a point on γ with the scalar curvature 2Qk , let y0 be the nearest such point to (x, t). If not, put y0 = (¯ x, t¯). On the segment of γ from (x, t) to y0 , the scalar curvature is at least 2Qk . According to the choice of the point (xk , tk ), the solution along the segment is ε-close to that of some ancient κ-solution. It follows from Theorem 6.4.3 (ii) that ∂ − 12 −1 |∇(Rk )| ≤ 2η and (Rk ) ≤ 2η ∂t
on the segment. (Here, without loss of generality, we may assume ε is suitably small). Then by choosing c > 0 (depending only on η) small enough we get the desired curvature bound by integrating the above derivative estimates along the segment. This proves the lemma. Step 2. Next we want to show that for each A < +∞, there exist a positive constant C(A) (independent of k) such that the curvatures of the rescaled solutions
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
401
g˜k (·, t) at the new time t = 0 (corresponding to the original times tk ) satisfy the estimate g k |(y, 0) ≤ C(A) |Rm
whenever dg˜k (·,0) (y, xk ) ≤ A and k ≥ 1. For each ρ ≥ 0, set
˜ k (x, 0) | k ≥ 1, x ∈ Mk with d0 (x, xk ) ≤ ρ} M (ρ) = sup{R and ρ0 = sup{ρ ≥ 0 | M (ρ) < +∞}. By the pinching estimate (7.1.1), it suffices to show ρ0 = +∞. Note that ρ0 > 0 by applying Lemma 7.1.2 with (¯ x, t¯) = (xk , tk ). We now argue by contradiction to show ρ0 = +∞. Suppose not, we may find (after passing to a subsequence if necessary) a sequence of points yk ∈ Mk with d0 (xk , yk ) → ρ0 < +∞ ˜ k (yk , 0) → +∞. Let γk (⊂ Mk ) be a minimizing geodesic segment from xk to and R ˜ k (zk , 0) = 2, and let βk be the yk . Let zk ∈ γk be the point on γk closest to yk with R subsegment of γk running from zk to yk . By Lemma 7.1.2 the length of βk is bounded away from zero independent of k. By the pinching estimate (7.1.1), for each ρ < ρ0 , we have a uniform bound on the curvatures on the open balls B0 (xk , ρ) ⊂ (Mk , g˜k ). The injectivity radii of the rescaled solutions g˜k at the points xk and the time t = 0 are also uniformly bounded from below by the κ-noncollapsing property. Therefore by Lemma 7.1.2 and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked sequence (B0 (xk , ρ0 ), g˜k (·, 0), xk ) ∞ converges in the Cloc topology to a marked (noncomplete) manifold (B∞ , g˜∞ , x∞ ), the segments γk converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ ¯∞ denote emanating from x∞ , and βk converges to a subsegment β∞ of γ∞ . Let B ¯∞ the limit point of γ∞ . the completion of (B∞ , g˜∞ ), and y∞ ∈ B ˜ ∞ the scalar curvature of (B∞ , g˜∞ ). Since the rescaled scalar curvaDenote by R ˜ k along βk are at least 2, it follows from the choice of the points (xk , 0) that for tures R ˜∞ (q0 ))−1 } any q0 ∈ β∞ , the manifold (B∞ , g˜∞ ) in {q ∈ B∞ | dist2g˜∞ (q, q0 ) < C(ε)(R is 2ε-close to the corresponding subset of (a time slice of) some orientable ancient κ-solution. Then by Theorem 6.4.6, we know that the orientable ancient κ-solution 1 at each point (x, t) has a radius r, 0 < r < C1 (2ε)R(x, t)− 2 , such that its canonical neighborhood B, with Bt (x, r) ⊂ B ⊂ Bt (x, 2r), is either an evolving 2ε-neck, or an evolving 2ε-cap, or a compact manifold (without boundary) diffeomorphic to a metric quotient of the round three-sphere S3 , and moreover the scalar curvature is between (C2 (2ε))−1 R(x, t) and C2 (2ε)R(x, t), where C1 (2ε) and C2 (2ε) are the positive constants in Theorem 6.4.6. We now choose C(ε) = max{2C12 (2ε), ε−2 }. By the local curvature estimate ˜ ∞ becomes unbounded when in Lemma 7.1.2, we see that the scalar curvature R approaching y∞ along γ∞ . This implies that the canonical neighborhood around q0 cannot be a compact manifold (without boundary) diffeomorphic to a metric quotient of the round three-sphere S3 . Note that γ∞ is shortest since it is the limit of a sequence of shortest geodesics. Without loss of generality, we may assume ε is suitably small 1 (say, ε ≤ 100 ). These imply that as q0 gets sufficiently close to y∞ , the canonical neighborhood around q0 cannot be an evolving 2ε-cap. Thus we conclude that each q0 ∈ γ∞ sufficiently close to y∞ is the center of an evolving 2ε-neck.
402
H.-D. CAO AND X.-P. ZHU
Let U=
[
˜ ∞ (q0 ))− 21 ) (⊂ (B∞ , g˜∞ )), B(q0 , 4π(R
q0 ∈γ∞
˜ ∞ (q0 ))− 21 ) is the ball centered at q0 ∈ B∞ with the radius where B(q0 , 4π(R ˜ ∞ (q0 ))− 12 . Clearly U has nonnegative sectional curvature by the pinching es4π(R timate (7.1.1). Since the metric g˜∞ is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ , we see that the metric space U = U ∪ {y∞ } by adding in the point y∞ , is locally complete and intrinsic near y∞ . Furthermore y∞ cannot be an interior point of any geodesic segment in U . This implies the curvature of U at y∞ is nonnegative in the Alexandrov sense. It is a basic result in Alexandrov space theory (see for example Theorem 10.9.3 and Corollary 10.9.5 of [9]) that there exists a three-dimensional tangent cone Cy∞ U at y∞ which is a metric cone. It is clear that its aperture is ≤ 10ε, thus the tangent cone is nonflat. Pick a point p ∈ Cy∞ U such that the distance from the vertex y∞ to p is one and it is nonflat around p. Then the ball B(p, 21 ) ⊂ Cy∞ U is the Gromov-Hausdorff limit of the scalings of a sequence of balls B0 (pk , sk ) ⊂ (Mk , g˜k (·, 0)) by some factors ak , where sk → 0+ . Since the tangent cone is three-dimensional and nonflat around ˜ k (pk , 0). By using the local curvature p, the factors ak must be comparable with R ∞ estimate in Lemma 7.1.2, we actually have the convergence in the Cloc topology for the solutions g˜k (·, t) on the balls B0 (pk , sk ) and over some time interval t ∈ [−δ, 0] for some sufficiently small δ > 0. The limiting ball B(p, 12 ) ⊂ Cy∞ U is a piece of the nonnegative curved and nonflat metric cone whose radial directions are all Ricci flat. On the other hand, by applying Hamilton’s strong maximum principle to the evolution equation of the Ricci curvature tensor as in the proof of Lemma 6.3.1, the limiting ball B(p, 21 ) would split off all radial directions isometrically (and locally). Since the limit is nonflat around p, this is impossible. Therefore we have proved that the curvatures of the rescaled solutions g˜k (·, t) at the new times t = 0 (corresponding to the original times tk ) stay uniformly bounded at bounded distances from xk for all k. We have proved that for each A < +∞, the curvature of the marked manifold (Mk , g˜k (·, 0), xk ) at each point y ∈ Mk with distance from xk at most A is bounded by C(A). Lemma 7.1.2 extends this curvature control to a backward parabolic neighborhood centered at y whose radius depends only on the distance from y to xk . Thus by Shi’s local derivative estimates (Theorem 1.4.2) we can control all derivatives of the curvature in such backward parabolic neighborhoods. Then by using the κ-noncollapsing and Hamilton’s compactness theorem (Theorem 4.1.5), we can take ∞ a Cloc subsequent limit to obtain (M∞ , g˜∞ (·, t), x∞ ), which is κ-noncollapsed on all scales and is defined on a space-time open subset of M∞ × (−∞, 0] containing the time slice M∞ × {0}. Clearly it follows from the pinching estimate (7.1.1) that the limit (M∞ , g˜∞ (·, 0), x∞ ) has nonnegative curvature operator (and hence nonnegative sectional curvature). Step 3. We further claim that the limit (M∞ , g˜∞ (·, 0), x∞ ) at the time slice {t = 0} has bounded curvature. We know that the sectional curvature of the limit (M∞ , g˜∞ (·, 0), x∞ ) is nonnegative everywhere. Argue by contradiction. Suppose the curvature of (M∞ , g˜∞ (·, 0), x∞ ) is not bounded, then by Lemma 6.1.4, there exists a sequence of points qj ∈ M∞ di˜ ∞ (qj , 0) → +∞ as j → +∞ verging to infinity such that their scalar curvatures R
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
403
and ˜ ∞ (x, 0) ≤ 4R ˜ ∞ (qj , 0) R q ˜ ∞ (qj , 0)) ⊂ (M∞ , g˜∞ (·, 0)). By combining with Lemma 7.1.2 for x ∈ B(qj , j/ R and the κ-noncollapsing, a subsequence of the rescaled and marked manifolds ∞ ˜ ∞ (qj , 0)˜ (M∞ , R g∞ (·, 0), qj ) converges in the Cloc topology to a smooth nonflat limit Y . By Proposition 6.1.2, the new limit Y is isometric to a metric product N × R for some two-dimensional manifold N . On the other hand, in view of the choice of the points (xk , tk ), the original limit (M∞ , g˜∞ (·, 0), x∞ ) at the point qj has a canonical neighborhood which is either a 2ε-neck, a 2ε-cap, or a compact manifold (without boundary) diffeomorphic to a metric quotient of the round S3 . It follows that for j ˜ ∞ (qj , 0))− 12 . Without loss of large enough, qj is the center of a 2ε-neck of radius (R generality, we may further assume that 2ε < ε0 , where ε0 is the positive constant ˜∞ (qj , 0))− 21 → 0 as j → +∞, this contradicts given in Proposition 6.1.1. Since (R Proposition 6.1.1. So the curvature of (M∞ , g˜∞ (·, 0)) is bounded. Step 4. Finally we want to extend the limit backwards in time to −∞. By Lemma 7.1.2 again, we now know that the limiting solution (M∞ , g˜∞ (·, t)) is defined on a backward time interval [−a, 0] for some a > 0. Denote by t′ = inf{t˜| we can take a smooth limit on (t˜, 0] (with bounded curvature at each time slice) from a subsequence of the convergent rescaled solutions g˜k }. We first claim that there is a subsequence of the rescaled solutions g˜k which converges ∞ in the Cloc topology to a smooth limit (M∞ , g˜∞ (·, t)) on the maximal time interval ′ (t , 0]. Indeed, let t′k be a sequence of negative numbers such that t′k → t′ and there exist k smooth limits (M∞ , g˜∞ (·, t)) defined on (t′k , 0]. For each k, the limit has nonnegative sectional curvature and has bounded curvature at each time slice. Moreover by Lemma 7.1.2, the limit has bounded curvature on each subinterval [−b, 0] ⊂ (t′k , 0]. Denote ˜ the scalar curvature upper bound of the limit at time zero (where Q ˜ is the same by Q for all k). Then we can apply the Li-Yau-Hamilton estimate (Corollary 2.5.7) to get −t′k k ˜ ˜ , R∞ (x, t) ≤ Q t − t′k k k ˜∞ where R (x, t) are the scalar curvatures of the limits (M∞ , g˜∞ (·, t)). Hence by the definition of convergence and the above curvature estimates, we can find a subsequence ∞ of the above convergent rescaled solutions g˜k which converges in the Cloc topology to ′ a smooth limit (M∞ , g˜∞ (·, t)) on the maximal time interval (t , 0]. We next claim that t′ = −∞. Suppose not, then by Lemma 7.1.2, the curvature of the limit (M∞ , g˜∞ (·, t)) becomes unbounded as t → t′ > −∞. By applying the maximum principle to the evolution equation of the scalar curvature, we see that the infimum of the scalar ˜ ∞ (x∞ , 0) = 1. Thus there exists curvature is nondecreasing in time. Note that R some point y∞ ∈ M∞ such that ˜ ∞ y ∞ , t′ + c < 3 R 10 2
404
H.-D. CAO AND X.-P. ZHU
where c > 0 is the universal constant in Lemma 7.1.2. By using Lemma 7.1.2 again c ) we see that the limit (M∞ , g˜∞ (·, t)) in a small neighborhood of the point (y∞ , t′ + 10 c c ′ ′ extends backwards to the time interval [t − 10 , t + 10 ]. We remark that the distances at time t and time 0 are roughly equivalent in the following sense (7.1.3)
dt (x, y) ≥ d0 (x, y) ≥ dt (x, y) − const.
for any x, y ∈ M∞ and t ∈ (t′ , 0]. Indeed from the Li-Yau-Hamilton inequality (Corollary 2.5.7) we have the estimate −t′ ˜ ˜ R∞ (x, t) ≤ Q , on M∞ × (t′ , 0]. t − t′ By applying Lemma 3.4.1 (ii), we have q ˜ dt (x, y) ≤ d0 (x, y) + 30(−t′ ) Q
for any x, y ∈ M∞ and t ∈ (t′ , 0]. On the other hand, since the curvature of the limit metric g˜∞ (·, t) is nonnegative, we have dt (x, y) ≥ d0 (x, y) for any x, y ∈ M∞ and t ∈ (t′ , 0]. Thus we obtain the estimate (7.1.3). Let us still denote by (Mk , g˜k (·, t)) the subsequence which converges on the maximal time interval (t′, 0]. Consider the rescaled sequence (Mk , g˜k (·, t)) with the marked points xk replaced by the associated sequence of points yk → y∞ and the (original −1 c c ′ unshifted) times tk replaced by any sk ∈ [tk + (t′ − 20 )Q−1 k , tk + (t + 20 )Qk ]. It follows from Lemma 7.1.2 that for k large enough, the rescaled solutions (Mk , g˜k (·, t)) at yk satisfy ˜ k (yk , t) ≤ 10 R c c , t′ + 10 ]. By applying the same arguments as in the above Step 2, for all t ∈ [t′ − 10 we conclude that for any A > 0, there is a positive constant C(A) < +∞ such that
˜ k (x, t) ≤ C(A) R c c for all (x, t) with dt (x, yk ) ≤ A and t ∈ [t′ − 20 , t′ + 20 ]. The estimate (7.1.3) implies c that there is a positive constant A0 such that for arbitrarily given small ǫ′ ∈ (0, 100 ), for k large enough, there hold
dt (xk , yk ) ≤ A0 for all t ∈ [t′ + ǫ′ , 0]. By combining with Lemma 7.1.2, we then conclude that for any ˜ A > 0, there is a positive constant C(A) such that for k large enough, the rescaled solutions (Mk , g˜k (·, t)) satisfy ˜ k (x, t) ≤ C(A) ˜ R ˜0 (xk , A) and t ∈ [t′ − c (C(A))−1 , 0]. for all x ∈ B 100 Now, by taking convergent subsequences from the (original) rescaled solutions (Mk , g˜k (·, t), xk ), we see that the limiting solution (M∞ , g˜∞ (·, t)) is defined on a spacetime open subset of M∞ ×(−∞, 0] containing M∞ ×[t′ , 0]. By repeating the argument
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
405
of Step 3 and using Lemma 7.1.2, we further conclude the limit (M∞ , g˜∞ (·, t)) has uniformly bounded curvature on M∞ × [t′ , 0]. This is a contradiction. Therefore we have proved a subsequence of the rescaled solutions (Mk , g˜k (·, t), xk ) converges to an orientable ancient κ-solution, which gives the desired contradiction. This completes the proof of the theorem. We remark that this singularity structure theorem has been extended by Chen and the second author in [34] to the Ricci flow on compact four-manifolds with positive isotropic curvature. 7.2. Curvature Estimates for Smooth Solutions. Let us consider solutions to the Ricci flow on compact orientable three-manifolds with normalized initial metrics. The above singularity structure theorem (Theorem 7.1.1) tells us that the solutions around high curvature points are sufficiently close to ancient κ-solutions. It is thus reasonable to expect that the elliptic type estimate (Theorem 6.4.3) and the curvature estimate via volume growth (Theorem 6.3.3) for ancient κ-solutions are heritable to general solutions of the Ricci flow on three-manifolds. The main purpose of this section is to establish such curvature estimates. In the fifth section of this chapter, we will further extend these estimates to surgically modified solutions. The first result of this section is an extension of the elliptic type estimate (Theorem 6.4.3). This result is reminiscent of the second step in the proof of Theorem 7.1.1. Theorem 7.2.1 (Perelman [103]). For any A < +∞, there exist K = K(A) < +∞ and α = α(A) > 0 with the following property. Suppose we have a solution to the Ricci flow on a three-dimensional, compact and orientable manifold M with normalized initial metric. Suppose that for some x0 ∈ M and some r0 > 0 with r0 < α, the solution is defined for 0 ≤ t ≤ r02 and satisfies |Rm|(x, t) ≤ r0−2 ,
for 0 ≤ t ≤ r02 , d0 (x, x0 ) ≤ r0 ,
and Vol 0 (B0 (x0 , r0 )) ≥ A−1 r03 . Then R(x, r02 ) ≤ Kr0−2 whenever dr02 (x, x0 ) < Ar0 . Proof. Given any large A > 0 and letting α > 0 be chosen later, by Perelman’s no local collapsing theorem II (Theorem 3.4.2), there exists a positive constant κ = κ(A) (independent of α) such that any complete solution satisfying the assumptions of the theorem is κ-noncollapsed on scales ≤ r0 over the region {(x, t) | 15 r02 ≤ t ≤ r02 , dt (x, x0 ) ≤ 5Ar0 }. Set 1 1 ε0 , , ε = min 4 100 where ε0 is the positive constant in Proposition 6.1.1. We first prove the following assertion. Claim. For the above fixed ε > 0, one can find K = K(A, ε) < +∞ such that if we have a three-dimensional complete orientable solution with normalized initial metric and satisfying |Rm|(x, t) ≤ r0−2
for 0 ≤ t ≤ r02 , d0 (x, x0 ) ≤ r0 ,
406
H.-D. CAO AND X.-P. ZHU
and Vol 0 (B0 (x0 , r0 )) ≥ A−1 r03 for some x0 ∈ M and some r0 > 0, then for any point x ∈ M with dr02 (x, x0 ) < 3Ar0 , either R(x, r02 ) < Kr0−2 or the subset {(y, t) | d2r2 (y, x) ≤ ε−2 R(x, r02 )−1 , r02 − ε−2 R(x, r02 )−1 ≤ t ≤ r02 } around 0 the point (x, r02 ) is ε-close to the corresponding subset of an orientable ancient κsolution. Notice that in this assertion we don’t impose the restriction of r0 < α, so we can consider for the moment r0 > 0 to be arbitrary in proving the above claim. Note that the assumption on the normalization of the initial metric is just to ensure the pinching estimate. By scaling, we may assume r0 = 1. The proof of the claim is essentially adapted from that of Theorem 7.1.1. But we will meet the difficulties of adjusting points and verifying a local curvature estimate. Suppose that the claim is not true. Then there exist a sequence of solutions (Mk , gk (·, t)) to the Ricci flow satisfying the assumptions of the claim with the origins x0k , and a sequence of positive numbers Kk → ∞, times tk = 1 and points xk ∈ Mk with dtk (xk , x0k ) < 3A such that Qk = Rk (xk , tk ) ≥ Kk and the solution in −1 2 {(x, t) | tk − C(ε)Q−1 k ≤ t ≤ tk , dtk (x, xk ) ≤ C(ε)Qk } is not, after scaling by the factor Qk , ε-close to the corresponding subset of any orientable ancient κ-solution, where Rk denotes the scalar curvature of (Mk , gk (·, t)) and C(ε)(≥ ε−2 ) is the constant defined in the proof of Theorem 7.1.1. As before we need to first adjust the point (xk , tk ) with tk ≥ 12 and dtk (xk , x0k ) < 4A so that Qk = Rk (xk , tk ) ≥ Kk and the conclusion of the claim fails at (xk , tk ), but holds for any (x, t) satisfying Rk (x, t) ≥ 1
−1
2Qk , tk − Hk Q−1 ≤ t ≤ tk and dt (x, x0k ) < dtk (xk , x0k ) + Hk2 Qk 2 , where Hk = k 1 4 Kk → ∞, as k → +∞. Indeed, by starting with (xk1 , tk1 ) = (xk , 1) we can choose (xk2 , tk2 ) ∈ Mk × (0, 1] with tk1 − Hk Rk (xk1 , tk1 )−1 ≤ tk2 ≤ tk1 , and dtk2 (xk2 , x0k ) < dtk1 (xk1 , x0k ) + 1
1
Hk2 Rk (xk1 , tk1 )− 2 such that Rk (xk2 , tk2 ) ≥ 2Rk (xk1 , tk1 ) and the conclusion of the claim fails at (xk2 , tk2 ); otherwise we have the desired point. Repeating this process, we can choose points (xki , tki ), i = 2, . . . , j, such that Rk (xki , tki ) ≥ 2Rk (xki−1 , tki−1 ), tki−1 − Hk Rk (xki−1 , tki−1 )−1 ≤ tki ≤ tki−1 , 1
1
dtki (xki , x0k ) < dtki−1 (xki−1 , x0k ) + Hk2 Rk (xki−1 , tki−1 )− 2 , and the conclusion of the claim fails at the points (xki , tki ), i = 2, . . . , j. These inequalities imply Rk (xkj , tkj ) ≥ 2j−1 Rk (xk1 , tk1 ) ≥ 2j−1 Kk , 1 ≥ tkj ≥ tk1 − Hk
j−2 X 1 1 Rk (xk1 , tk1 )−1 ≥ , 2i 2 i=0
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
407
and 1
dtkj (xkj , x0k ) < dtk1 (xk1 , x0k ) + Hk2
j−2 X i=0
1 1 √ Rk (xk1 , tk1 )− 2 < 4A. ( 2)i
Since the solutions are smooth, this process must terminate after a finite number of steps to give the desired point, still denoted by (xk , tk ). For each adjusted (xk , tk ), let [t′ , tk ] be the maximal subinterval of [tk − 1 −2 −1 ε Qk , tk ] so that the conclusion of the claim with K = 2Qk holds on 2 1 1 −1 P xk , tk , Hk2 Qk 2 , t′ − tk 10 1 1 −1 = (x, t) | x ∈ Bt xk , Hk2 Qk 2 , t ∈ [t′ , tk ] 10 for all sufficiently large k. We now want to show t′ = tk − 12 ε−2 Q−1 k . Consider the scalar curvature Rk at the point xk over the time interval [t′ , tk ]. If there is a time t˜ ∈ [t′ , tk ] satisfying Rk (xk , t˜) ≥ 2Qk , we let t˜ be the first of such time from tk . Then the solution (Mk , gk (·, t)) around the point xk over the time ˜ interval [t˜ − 12 ε−2 Q−1 k , t] is ε-close to some orientable ancient κ-solution. Note from the Li-Yau-Hamilton inequality that the scalar curvature of any ancient κ-solution is pointwise nondecreasing in time. Consequently, we have the following curvature estimate Rk (xk , t) ≤ 2(1 + ε)Qk ′ ˜ for t ∈ [t˜ − 21 ε−2 Q−1 k , tk ] (or t ∈ [t , tk ] if there is no such time t). By combining with the elliptic type estimate for ancient κ-solutions (Theorem 6.4.3) and the HamiltonIvey pinching estimate, we further have
|Rm(x, t)| ≤ 5ω(1)Qk
(7.2.1)
1 ′ for all x ∈ Bt (xk , (3Qk )− 2 ) and t ∈ [t˜− 21 ε−2 Q−1 k , tk ] (or t ∈ [t , tk ]) and all sufficiently large k, where ω is the positive function in Theorem 6.4.3. ′ For any point (x, t) with t˜ − 12 ε−2 Q−1 k ≤ t ≤ tk (or t ∈ [t , tk ]) and dt (x, xk ) ≤ 1 − 21 1 2 10 Hk Qk ,
we divide the discussion into two cases.
Case (1): dt (xk , x0k ) ≤
1 − 12 3 2 10 Hk Qk .
dt (x, x0k ) ≤ dt (x, xk ) + dt (xk , x0k ) 3 1 −1 1 12 − 12 Hk Qk + Hk2 Qk 2 ≤ 10 10 1 12 − 12 ≤ Hk Q k . 2
(7.2.2)
1
−1
3 Hk2 Qk 2 . Case (2): dt (xk , x0k ) > 10 From the curvature bound (7.2.1) and the assumption, we apply Lemma 3.4.1(ii) − 21
with r0 = Qk
to get 1 d (dt (xk , x0k )) ≥ −20(ω(1) + 1)Qk2 , dt
408
H.-D. CAO AND X.-P. ZHU
and then for k large enough, − 12
dt (xk , x0k ) ≤ dtˆ(xk , x0k ) + 20(ω(1) + 1)ε−2 Qk 1 1 −1 ≤ dtˆ(xk , x0k ) + Hk2 Qk 2 , 10 where tˆ ∈ (t, tk ] satisfies the property that ds (xk , x0k ) ≥ So we have (7.2.3)
1 − 12 3 2 10 Hk Qk
whenever s ∈ [t, tˆ].
dt (x, x0k ) ≤ dt (x, xk ) + dt (xk , x0k ) 1 12 − 12 1 1 −1 ≤ Hk Qk + dtˆ(xk , x0k ) + Hk2 Qk 2 10 10 1 12 − 12 ≤ dtk (xk , x0k ) + Hk Qk , 2
for all sufficiently large k. Then the combination of (7.2.2), (7.2.3) and the choice of the points (xk , tk ) implies t′ = tk − 21 ε−2 Q−1 k for all sufficiently large k. (Here we also used the maximality of the subinterval [t′ , tk ] in the case that there is no time in [t′ , tk ] with Rk (xk , ·) ≥ 2Qk .) Now we rescale the solutions (Mk , gk (·, t)) into (Mk , g˜k (·, t)) around the points xk by the factors Qk = Rk (xk , tk ) and shift the times tk to the new times zero. Then the same arguments from Step 1 to Step 3 in the proof of Theorem 7.1.1 prove that ∞ a subsequence of the rescaled solutions (Mk , g˜k (·, t)) converges in the Cloc topology to a limiting (complete) solution (M∞ , g˜∞ (·, t)), which is defined on a backward time interval [−a, 0] for some a > 0. (The only modification is in Lemma 7.1.2 of Step 1 ¯ by further requiring tk − 41 ε−2 Q−1 k ≤ t ≤ tk ). We next study how to adapt the argument of Step 4 in the proof of Theorem 7.1.1. As before, we have a maximal time interval (t∞ , 0] for which we can take a smooth limit (M∞ , g˜∞ (·, t), x∞ ) from a subsequence of the rescaled solutions (Mk , g˜k (·, t), xk ). We want to show t∞ = −∞. Suppose not; then t∞ > −∞. Let c > 0 be a positive constant much smaller than 1 −2 ε . Note that the infimum of the scalar curvature is nondecreasing in time. Then 10 we can find some point y∞ ∈ M∞ and some time t = t∞ + θ with 0 < θ < 3c such ˜ ∞ (y∞ , t∞ + θ) ≤ 3 . that R 2 Consider the (unrescaled) scalar curvature Rk of (Mk , gk (·, t)) at the point xk ˜ over the time interval [tk + (t∞ + 2θ )Q−1 k , tk ]. Since the scalar curvature R∞ of the θ limit on M∞ × [t∞ + 3 , 0] is uniformly bounded by some positive constant C, we have the curvature estimate Rk (xk , t) ≤ 2CQk for all t ∈ [tk + (t∞ + θ2 )Q−1 k , tk ] and all sufficiently large k. Then by repeating the same arguments as in deriving (7.2.1), (7.2.2) and (7.2.3), we deduce that the conclusion of the claim with K = 2Qk holds on the parabolic neighborhood 1 −1 1 Hk2 Qk 2 , (t∞ + θ2 )Q−1 P (xk , tk , 10 k ) for all sufficiently large k. Let (yk , tk + (t∞ + θk )Q−1 ) be a sequence of associated points and times in the k (unrescaled) solutions (Mk , gk (·, t)) so that after rescaling, the sequence converges to the (y∞ , t∞ + θ) in the limit. Clearly θ2 ≤ θk ≤ 2θ for all sufficiently large k. Then, by considering the scalar curvature Rk at the point yk over the time interval −1 [tk + (t∞ − 3c )Q−1 k , tk + (t∞ + θk )Qk ], the above argument (as in deriving the similar
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
409
estimates (7.2.1)-(7.2.3)) implies that the conclusion of the claim with K = 2Qk holds 1
−1
1 on the parabolic neighborhood P (yk , tk , 10 Hk2 Qk 2 , (t∞ − 3c )Q−1 k ) for all sufficiently large k. In particular, we have the curvature estimate
Rk (yk , t) ≤ 4(1 + ε)Qk −1 for t ∈ [tk + (t∞ − 3c )Q−1 k , tk + (t∞ + θk )Qk ] for all sufficiently large k. We now consider the rescaled sequence (Mk , g˜k (·, t)) with the marked points re−1 c placed by yk and the times replaced by sk ∈ [tk + (t∞ − 4c )Q−1 k , tk + (t∞ + 4 )Qk ]. By applying the same arguments from Step 1 to Step 3 in the proof of Theorem 7.1.1 and the Li-Yau-Hamilton inequality as in Step 4 of Theorem 7.1.1, we conclude that there is some small constant a′ > 0 such that the original limit (M∞ , g˜∞ (·, t)) is actually well defined on M∞ × [t∞ − a′ , 0] with uniformly bounded curvature. This is a contradiction. Therefore we have checked the claim. To finish the proof, we next argue by contradiction. Suppose there exist sequences of positive numbers Kk → +∞, αk → 0, as k → +∞, and a sequence of solutions (Mk , gk (·, t)) to the Ricci flow satisfying the assumptions of the theorem with origins x0k and with radii r0k satisfying r0k < αk such that for some points xk ∈ Mk with dr02 (xk , x0k ) < Ar0k we have k
R(xk , r02k ) > Kk r0−2 k
(7.2.4)
for all k. Let (Mk , gˆk (·, t), x0k ) be the rescaled solutions of (Mk , gk (·, t)) around the origins x0k by the factors r0−2 and shifting the times r02k to the new times zero. k The above claim tells us that for k large, any point (y, 0) ∈ (Mk , gˆk (·, 0), x0k ) with ˆ k (y, 0) > Kk has a dgˆk (·,0) (y, x0k ) < 3A and with the rescaled scalar curvature R canonical neighborhood which is either a 2ε-neck, or a 2ε-cap, or a compact manifold (without boundary) diffeomorphic to a metric quotient of the round three-sphere. Note that the pinching estimate (7.1.1) and the condition αk → 0 imply any subsequential limit of the rescaled solutions (Mk , gˆk (·, t), x0k ) must have nonnegative sectional curvature. Thus the same argument as in Step 2 of the proof of Theorem 7.1.1 shows that for all sufficiently large k, the curvatures of the rescaled solutions at the time zero stay uniformly bounded at those points whose distances from the origins x0k do not exceed 2A. This contradicts (7.2.4) for k large enough. Therefore we have completed the proof of the theorem. The next result is a generalization of the curvature estimate via volume growth in Theorem 6.3.3 (ii) where the condition on the curvature lower bound over a time interval is replaced by that at a time slice only. Theorem 7.2.2 (Perelman [103]). For any w > 0 there exist τ = τ (w) > 0, K = K(w) < +∞, α = α(w) > 0 with the following property. Suppose we have a three-dimensional, compact and orientable solution to the Ricci flow defined on M × [0, T ) with normalized initial metric. Suppose that for some radius r0 > 0 with r0 < α and a point (x0 , t0 ) ∈ M × [0, T ) with T > t0 ≥ 4τ r02 , the solution on the ball Bt0 (x0 , r0 ) satisfies Rm(x, t0 ) ≥ −r0−2 and
on Bt0 (x0 , r0 ),
Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 .
410
H.-D. CAO AND X.-P. ZHU
Then R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and dt (x, x0 ) ≤ 41 r0 . Proof. If we knew that Rm(x, t) ≥ −r0−2 for all t ∈ [0, t0 ] and dt (x, x0 ) ≤ r0 , then we could just apply Theorem 6.3.3 (ii) and take τ (w) = τ0 (w)/2, K(w) = C(w) + 2B(w)/τ0 (w). Now fix these values of τ and K. We argue by contradiction. Consider a three-dimensional, compact and orientable solution gij (t) to the Ricci flow with normalized initial metric, a point (x0 , t0 ) and some radius r0 > 0 with r0 < α, for α > 0 a sufficiently small constant to be determined later, such that the assumptions of the theorem do hold whereas the conclusion does not. We first claim that we may assume that any other point (x′ , t′ ) and radius r′ > 0 with the same property has either t′ > t0 or t′ < t0 − 2τ r02 , or 2r′ > r0 . Indeed, suppose otherwise. Then there exist (x′0 , t′0 ) and r0′ with t′0 ∈ [t0 −2τ r02 , t0 ] and r0′ ≤ 21 r0 , for which the assumptions of the theorem hold but the conclusion does not. Thus, there is a point (x, t) such that i h τ t ∈ [t′0 − τ (r0′ )2 , t′0 ] ⊂ t0 − 2τ r02 − r02 , t0 4 and R(x, t) > K(r0′ )−2 ≥ 4Kr0−2 . If the point (x′0 , t′0 ) and the radius r0′ satisfy the claim then we stop, and otherwise we iterate the procedure. Since t0 ≥ 4τ r02 and the solution is smooth, the iteration must terminate in a finite number of steps, which provides the desired point and the desired radius. Let τ ′ ≥ 0 be the largest number such that (7.2.5)
Rm(x, t) ≥ −r0−2
whenever t ∈ [t0 − τ ′ r02 , t0 ] and dt (x, x0 ) ≤ r0 . If τ ′ ≥ 2τ , we are done by Theorem 6.3.3 (ii). Thus we may assume τ ′ < 2τ . By applying Theorem 6.3.3(ii), we know that at time t′ = t0 − τ ′ r02 , the ball Bt′ (x0 , r0 ) has (7.2.6)
Vol t′ (Bt′ (x0 , r0 )) ≥ ξ(w)r03
for some positive constant ξ(w) depending only on w. We next claim that there exists a ball (at time t′ = t0 − τ ′ r02 ) Bt′ (x′ , r′ ) ⊂ Bt′ (x0 , r0 ) with (7.2.7)
Vol t′ (Bt′ (x′ , r′ )) ≥
1 α3 (r′ )3 2
and with (7.2.8)
r0 > r′ ≥ c(w)r0 2
for some small positive constant c(w) depending only on w, where α3 is the volume of the unit ball B3 in the Euclidean space R3 . Indeed, suppose that it is not true. Then after rescaling, there is a sequence of Riemannian manifolds Mi , i = 1, 2, . . . , with balls B(xi , 1) ⊂ Mi so that (7.2.5)′
Rm ≥ −1 on B(xi , 1)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
411
and (7.2.6)′
Vol (B(xi , 1)) ≥ ξ(w)
for all i, but all balls B(x′i , ri′ ) ⊂ B(xi , 1) with (7.2.9)
Vol (B(x′i , ri′ )) <
1 2
> ri′ ≥
1 i
satisfy
1 α3 (ri′ )3 . 2
It follows from basic results in Alexandrov space theory (see for example Theorem 10.7.2 and Theorem 10.10.10 of [9]) that, after taking a subsequence, the marked balls (B(xi , 1), xi ) converge in the Gromov-Hausdorff topology to a marked length space (B∞ , x∞ ) with curvature bounded from below by −1 in the Alexandrov space sense, and the associated Riemannian volume forms dVol Mi over (B(xi , 1), xi ) converge weakly to the Hausdorff measure µ of B∞ . It is well-known that the Hausdorff dimension of any Alexandrov space is either an integer or infinity (see for example Theorem 10.8.2 of [9]). Then by (7.2.6)′ , we know the limit (B∞ , x∞ ) is a threedimensional Alexandrov space of curvature ≥ −1. In the Alexandrov space theory, a point p ∈ B∞ is said to be regular if the tangent cone of B∞ at p is isometric to R3 . It is also a basic result in Alexandrov space theory (see for example Corollary 10.9.13 of [9]) that the set of regular points in B∞ is dense and for each regular point there is a small neighborhood which is almost isometric to an open set of the Euclidean ′ ′ space R3 . Thus for any ε > 0, there are balls B(x′∞ , r∞ ) ⊂ B∞ with 0 < r∞ < 13 and satisfying ′ ′ 3 µ(B(x′∞ , r∞ )) ≥ (1 − ε)α3 (r∞ ) .
This is a contradiction with (7.2.9). Without loss of generality, we may assume w ≤ 41 α3 . Since τ ′ < 2τ , it follows from the choice of the point (x0 , t0 ) and the radius r0 and (7.2.5), (7.2.7), (7.2.8) that the conclusion of the theorem holds for (x′ , t′ ) and r′ . Thus we have the estimate R(x, t) ≤ K(r′ )−2 whenever t ∈ [t′ − τ (r′ )2 , t′ ] and dt (x, x′ ) ≤ 14 r′ . For α > 0 small, by combining with the pinching estimate (7.1.1), we have |Rm(x, t)| ≤ K ′ (r′ )−2 whenever t ∈ [t′ − τ (r′ )2 , t′ ] and dt (x, x′ ) ≤ 14 r′ , where K ′ is some positive constant depending only on K. Note that this curvature estimate implies the evolving metrics are equivalent over a suitable subregion of {(x, t) | t ∈ [t′ − τ (r′ )2 , t′ ] and dt (x, x′ ) ≤ 1 ′ 4 r }. Now we can apply Theorem 7.2.1 to choose α = α(w) > 0 so small that (7.2.10)
′ −2 −2 −2 ˜ ˜ R(x, t) ≤ K(w)(r ) ≤ K(w)c(w) r0
whenever t ∈ [t′ − τ2 (r′ )2 , t′ ] and dt (x, x′ ) ≤ 10r0 . Then the combination of (7.2.10) with the pinching estimate (7.1.2) would imply Rm(x, t) ≥ −[f −1 (R(x, t)(1 + t))/(R(x, t)(1 + t))]R(x, t) 1 ≥ − r0−2 2
412
H.-D. CAO AND X.-P. ZHU
on the region {(x, t) | t ∈ [t′ − τ2 (r′ )2 , t′ ] and dt (x, x0 ) ≤ r0 } when α = α(w) > r0 small enough. This contradicts the choice of τ ′ . Therefore we have proved the theorem. The combination of the above two theorems immediately gives the following consequence. Corollary 7.2.3. For any w > 0 and A < +∞, there exist τ = τ (w, A) > 0, K = K(w, A) < +∞, and α = α(w, A) > 0 with the following property. Suppose we have a three-dimensional, compact and orientable solution to the Ricci flow defined on M × [0, T ) with normalized initial metric. Suppose that for some radius r0 > 0 with r0 < α and a point (x0 , t0 ) ∈ M × [0, T ) with T > t0 ≥ 4τ r02 , the solution on the ball Bt0 (x0 , r0 ) satisfies Rm(x, t0 ) ≥ −r0−2 and
on Bt0 (x0 , r0 ),
Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 .
Then R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and dt (x, x0 ) ≤ Ar0 . We can also state the previous corollary in the following version.
Corollary 7.2.4 (Perelman [103]). For any w > 0 one can find ρ = ρ(w) > 0 such that if gij (t) is a complete solution to the Ricci flow defined on M × [0, T ) with T > 1 and with normalized initial metric, where M is a three-dimensional, compact and orientable manifold, and if Bt0 (x0 , r0 ) is a metric ball at time t0 ≥ 1, with r0 < ρ, such that min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , then Vol t0 (Bt0 (x0 , r0 )) ≤ wr03 . Proof. We argue by contradiction. Suppose for any ρ > 0, there is a solution and a ball Bt0 (x0 , r0 ) satisfying the assumption of the corollary with r0 < ρ, t0 ≥ 1, and with min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , but Vol t0 (Bt0 (x0 , r0 )) > wr03 . We can apply Corollary 7.2.3 to get R(x, t) ≤ Kr0−2 whenever t ∈ [t0 −τ r02 , t0 ] and dt (x, x0 ) ≤ 2r0 , provided ρ > 0 is so small that 4τ ρ2 ≤ 1 and ρ < α, where τ, α and K are the positive constants obtained in Corollary 7.2.3. Then for r0 < ρ and ρ > 0 sufficiently small, it follows from the pinching estimate (7.1.2) that Rm(x, t) ≥ −[f −1 (R(x, t)(1 + t))/(R(x, t)(1 + t))]R(x, t) 1 ≥ − r0−2 2
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
413
in the region {(x, t) | t ∈ [t0 − τ (r0 )2 , t0 ] and dt (x, x0 ) ≤ 2r0 }. In particular, this would imply min{Rm(x, t0 )|x ∈ Bt0 (x0 , r0 )} > −r0−2 . This contradicts the assumption. 7.3. Ricci Flow with Surgery. One of the central themes of the Ricci flow theory is to give a classification of all compact orientable three-manifolds. As we mentioned before, the basic idea is to obtain long-time behavior of solutions to the Ricci flow. However the solutions will in general become singular in finite time. Fortunately, we now understand the precise structures of the solutions around the singularities, thanks to Theorem 7.1.1. When a solution develops singularities, one can perform geometric surgeries by cutting off the canonical neighborhoods around the singularities and gluing back some known pieces, and then continue running the Ricci flow. By repeating this procedure, one hopes to get a kind of “weak” solution. In this section we will give a detailed description of this surgery procedure and define a global “weak” solution to the Ricci flow. Given any ε > 0, based on the singularity structure theorem (Theorem 7.1.1), we can get a clear picture of the solution near the singular time as follows. Let (M, gij (·, t)) be a maximal solution to the Ricci flow on the maximal time interval [0, T ) with T < +∞, where M is a connected compact orientable threemanifold and the initial metric is normalized. For the given ε > 0 and the solution (M, gij (·, t)), we can find r0 > 0 such that each point (x, t) with R(x, t) ≥ r0−2 satisfies the derivative estimates ∂ 3 2 (7.3.1) |∇R(x, t)| < ηR (x, t) and R(x, t) < ηR2 (x, t), ∂t
where η > 0 is a universal constant, and has a canonical neighborhood which is either an evolving ε-neck, or an evolving ε-cap, or a compact positively curved manifold (without boundary). In the last case the solution becomes extinct at the maximal time T and the manifold M is diffeomorphic to the round three-sphere S3 or a metric quotient of S3 by Theorem 5.2.1. Let Ω denote the set of all points in M where the curvature stays bounded as t → T . The gradient estimates in (7.3.1) imply that Ω is open and that R(x, t) → ∞ as t → T for each x ∈ M \ Ω. If Ω is empty, then the solution becomes extinct at time T . In this case, either the manifold M is compact and positively curved, or it is entirely covered by evolving ε-necks and evolving ε-caps shortly before the maximal time T . So the manifold M is diffeomorphic to either S3 , or a metric quotient of the round S3 , or S2 × S1 , or RP3 #RP3 . The reason is as follows. Clearly, we only need to consider the situation that the manifold M is entirely covered by evolving ε-necks and evolving ε-caps shortly before the maximal time T . If M contains a cap C, then there is a cap or a neck adjacent to the neck-like end of C. The former case implies that M is diffeomorphic to S3 , RP3 , or RP3 #RP3 . In the latter case, we get a new longer cap and continue. Finally, we must end up with a cap, producing a S3 , RP3 , or RP3 #RP3 . If M contains no caps, we start with a neck N . By connecting with the necks that are adjacent to the boundary of N , we get a longer neck and continue. After a finite number of steps, the resulting neck must repeat itself. Since M is orientable, we conclude that M is diffeomorphic to S2 × S1 .
414
H.-D. CAO AND X.-P. ZHU
We can now assume that Ω is nonempty. By using the local derivative estimates of Shi (Theorem 1.4.2), we see that as t → T the solution gij (·, t) has a smooth limit ¯ g¯ij (·) on Ω. Let R(x) denote the scalar curvature of g¯ij . For any ρ < r0 , let us consider the set ¯ Ωρ = {x ∈ Ω | R(x) ≤ ρ−2 }. By the evolution equation of the Ricci flow, we see that the initial metric gij (·, 0) and the limit metric gij (·) are equivalent over any fixed region where the curvature remains uniformly bounded. Note that for any fixed x ∈ ∂Ω, and any sequence of points xj ∈ Ω with xj → x with respect to the initial metric gij (·, 0), we have R(xj ) → +∞. In fact, if there were a subsequence xjk so that limk→∞ R(xjk ) exists and is finite, then it would follow from the gradient estimates (7.3.1) that R is uniformly bounded in some small neighborhood of x ∈ ∂Ω (with respect to the induced topology of the initial metric gij (·, 0)); this is a contradiction. From this observation and the compactness of the initial manifold, we see that Ωρ is compact (with respect to the metric gij (·)). For further discussions, let us introduce the following terminologies. Denote by I an interval. Recall that an ε-neck (of radius r) is an open set with a Riemannian metric, which is, after scaling the metric with factor r−2 , ε-close to the standard neck S2 × I with the product metric, where S2 has constant scalar curvature one and I has length −1 2ε−1 and the ε-closeness refers to the C [ε ] topology. 2 A metric on S × I, such that each point is contained in some ε-neck, is called an ε-tube, or an ε-horn, or a double ε-horn, if the scalar curvature stays bounded on both ends, or stays bounded on one end and tends to infinity on the other, or tends to infinity on both ends, respectively. ¯ 3 is called a ε-cap if the region outside some suitable A metric on B3 or RP3 \ B ¯ 3 is called an capped ε-horn compact subset is an ε-neck. A metric on B3 or RP3 \ B if each point outside some compact subset is contained in an ε-neck and the scalar curvature tends to infinity on the end. Now take any ε-neck in (Ω, g¯ij ) and consider a point x on one of its boundary components. If x ∈ Ω \ Ωρ , then there is either an ε-cap or an ε-neck, adjacent to the initial ε-neck. In the latter case we can take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Ωρ or an ε-cap, or go on indefinitely, producing an ε-horn. The same procedure can be repeated for the other boundary component of the initial ε-neck. Therefore, taking into account that Ω has no compact components, we conclude that each ε-neck of (Ω, g¯ij ) is contained in a subset of Ω of one of the following types: (a) an ε-tube with boundary components in Ωρ , or
(7.3.2)
(b)
an ε-cap with boundary in Ωρ , or
(c)
an ε-horn with boundary in Ωρ , or
(d)
a capped ε-horn, or
(e)
a double ε-horn.
Similarly, each ε-cap of (Ω, g¯ij ) is contained in a subset of Ω of either type (b) or type (d). It is clear that there is a definite lower bound (depending on ρ) for the volume of subsets of types (a), (b) and (c), so there can be only a finite number of them. Thus
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
415
we conclude that there is only a finite number of components of Ω containing points of Ωρ , and every such component has a finite number of ends, each being an ε-horn. On the other hand, every component of Ω, containing no points of Ωρ , is either a capped ε-horn, or a double ε-horn. If we look at the solution at a slightly earlier time, the above argument shows each ε-neck or ε-cap of (M, gij (·, t)) is contained in a subset of types (a) or (b), while the ε-horns, capped ε-horns and double ε-horns (at the maximal time T) are connected together to form ε-tubes and ε-caps at the times t shortly before T .
Ωρ R @
Ωρ R @
@ I ε-horn
6 double ε-horn
@ I ε-tube
6 capped ε-horn
Hence, by looking at the solution at times shortly before T , we see that the topology of M can be reconstructed as follows: take the components Ωj , 1 ≤ j ≤ k, of Ω which contain points of Ωρ , truncate their ε-horns, and glue to the boundary ¯ 3. components of truncated Ωj a finite collection of tubes S2 × I and caps B3 or RP3 \ B ¯ j , 1 ≤ j ≤ k, with a finite number Thus, M is diffeomorphic to a connected sum of Ω of copies of S2 × S1 (which correspond to gluing a tube to two boundary components ¯ j denotes Ωj with each of the same Ωj ), and a finite number of copies of RP3 . Here Ω ¯ j in the following ε-horn one point compactified. More geometrically, one can get Ω way: in every ε-horn of Ωj one can find an ε-neck, cut it along the middle two-sphere, remove the horn-shaped end, and glue back a cap (or more precisely, a differentiable three-ball). Thus to understand the topology of M , one only needs to understand the ¯ j , 1 ≤ j ≤ k. topologies of the compact orientable three-manifolds Ω ¯ j by the Ricci flow again and, when singularities Naturally one can evolve each Ω develop again, perform the above surgery for each ε-horn to get new compact orientable three-manifolds. By repeating this procedure indefinitely, it will likely give a long-time “weak” solution to the Ricci flow. The following abstract definition for this kind of “weak” solution was introduced by Perelman in [104] . Definition 7.3.1. Suppose we are given a (finite or countably infinite) collection + k of three-dimensional smooth solutions gij (t) to the Ricci flow defined on Mk × [t− k , tk )
416
H.-D. CAO AND X.-P. ZHU
and go singular as t → t+ k , where each manifold Mk is compact and orientable, possibly k disconnected with only a finite number of connected components. Let (Ωk , g¯ij ) be the + k limits of the corresponding solutions gij (t) as t → tk , as above. Suppose also that for + k−1 k − each k we have t− ¯ij ) and (Mk , gij (tk )) contain compact k = tk−1 , and that (Ωk−1 , g (possibly disconnected) three-dimensional submanifolds with smooth boundary which are isometric. Then by identifying these isometric submanifolds, we say the collection k of solutions gij (t) is a solution to the Ricci flow with surgery (or a surgically modified solution to the Ricci flow) on the time interval which is the union of all + + [t− k , tk ), and say the times tk are surgery times. To get the topology of the initial manifold from the solution to the Ricci flow with surgery, one has to overcome the following two difficulties: (i) how to prevent the surgery times from accumulating? (ii) how to obtain the long time behavior of the solution to the Ricci flow with surgery? Thus it is natural to consider those solutions having “good” properties. For any arbitrarily fixed positive number ε, we will only consider those solutions to the Ricci flow with surgery which satisfy the following a priori assumptions (with accuracy ε). Pinching assumption. The eigenvalues λ ≥ µ ≥ ν of the curvature operator of the solution to the Ricci flow with surgery at each point and each time satisfy (7.3.3)
R ≥ (−ν)[log(−ν) + log(1 + t) − 3]
whenever ν < 0. Canonical neighborhood assumption (with accuracy ε). For any given ε > 0, there exist positive constants C1 and C2 depending only on ε, and a nonincreasing positive function r : [0, +∞) → (0, +∞) such that at each time t > 0, every point x where scalar curvature R(x, t) is at least r−2 (t) has a neighborhood B, with 1 Bt (x, σ) ⊂ B ⊂ Bt (x, 2σ) for some 0 < σ < C1 R− 2 (x, t), which falls into one of the following three categories: (a) B is a strong ε-neck (in the sense B is the slice at time t of the parabolic neighborhood {(x′ , t′ ) | x′ ∈ B, t′ ∈ [t − R(x, t)−1 , t]}, where the solution is well defined on the whole parabolic neighborhood and is, after scaling with −1 factor R(x, t) and shifting the time to zero, ε-close (in the C [ε ] topology) 2 to the subset (S × I) × [−1, 0] of the evolving standard round cylinder with scalar curvature 1 to S2 and length 2ε−1 to I at time zero), or (b) B is an ε-cap, or (c) B is a compact manifold (without boundary) of positive sectional curvature. Furthermore, the scalar curvature in B at time t is between C2−1 R(x, t) and C2 R(x, t), satisfies the gradient estimates ∂R 3 < ηR2 , (7.3.4) |∇R| < ηR 2 and ∂t and the volume of B in case (a) and case (b) satisfies 3
(C2 R(x, t))− 2 ≤ Vol t (B) ≤ εσ 3 . Here η is a universal positive constant.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
417
Without loss of generality, we always assume the above constants C1 and C2 are twice bigger than the corresponding constants C1 ( 2ε ) and C2 ( 2ε ) in Theorem 6.4.6 with the accuracy 2ε . We remark that the above definition of the canonical neighborhood assumption is slightly different from that of Perelman in [104] in two aspects: (1) it allows the parameter r to depend on time; (2) it also includes an volume upper bound for the canonical neighborhoods of types (a) and (b). Arbitrarily given a compact orientable three-manifold with a Riemannian metric, by scaling, we may assume the Riemannian metric is normalized. In the rest of this section and the next section, we will show the Ricci flow with surgery, with the normalized metric as initial data, has a long-time solution which satisfies the above a priori assumptions and has only a finite number of surgery times at each finite time interval. The construction of the long-time solution will be given by an induction argument. First, for the arbitrarily given compact orientable normalized three-dimensional Riemannian manifold (M, gij (x)), the Ricci flow with it as initial data has a maximal solution gij (x, t) on a maximal time interval [0, T ) with T > 1. It follows from Theorem 5.3.2 and Theorem 7.1.1 that the a priori assumptions (with accuracy ε) hold for the smooth solution on [0, T ). If T = +∞, we have the desired long time solution. Thus, without loss of generality, we may assume the maximal time T < +∞ so that the solution goes singular at time T . Suppose that we have a solution to the Ricci flow with surgery, with the normalized metric as initial data, satisfying the a priori assumptions (with accuracy ε), defined on [0, T ) with T < +∞, going singular at time T and having only a finite number of surgery times on [0, T ). Let Ω denote the set of all points where the curvature stays bounded as t → T . As we have seen before, the canonical neighborhood assumption implies that Ω is open and that R(x, t) → ∞ as t → T for all x lying outside Ω. Moreover, as t → T , the solution gij (x, t) has a smooth limit g¯ij (x) on Ω. For some δ > 0 to be chosen much smaller than ε, we let ρ = δr(T ) where r(t) is the positive nonincreasing function in the definition of the canonical neighborhood assumption. We consider the corresponding compact set ¯ Ωρ = {x ∈ Ω | R(x) ≤ ρ−2 }, ¯ where R(x) is the scalar curvature of g¯ij . If Ωρ is empty, the manifold (near the maximal time T ) is entirely covered by ε-tubes, ε-caps and compact components with positive curvature. Clearly, the number of the compact components is finite. Then in this case the manifold (near the maximal time T ) is diffeomorphic to the union of a finite number of copies of S3 , or metric quotients of the round S3 , or S2 × S1 , or a connected sum of them. Thus when Ωρ is empty, the procedure stops here, and we say the solution becomes extinct. We now assume Ωρ is not empty. Then we know that every point x ∈ Ω \ Ωρ lies in one of the subsets of Ω listed in (7.3.2), or in a compact component with positive curvature, or in a compact component which is contained in Ω \ Ωρ and is diffeomorphic to either S3 , or S2 × S1 or RP3 #RP3 . Note again that the number of the compact components is finite. Let us throw away all the compact components lying in Ω \ Ωρ and all the compact components with positive curvature, and then consider those components Ωj , 1 ≤ j ≤ k, of Ω which contain points of Ωρ . (We will consider those components of Ω \ Ωρ consisting of capped ε-horns and double ε-horns later). We will perform surgical procedures, as we roughly described before, by finding an ε-neck in every horn of Ωj , 1 ≤ j ≤ k,
418
H.-D. CAO AND X.-P. ZHU
and then cutting it along the middle two-sphere, removing the horn-shaped end, and gluing back a cap. In order to maintain the a priori assumptions with the same accuracy after the surgery, we will need to find sufficient “fine” necks in the ε-horns and to glue sufficient “fine” caps. Note that δ > 0 will be chosen much smaller than ε > 0. The following lemma due to Perelman [104] gives us the “fine” necks in the ε-horns. (At the first sight, we should also cut off all those ε-tubes and ε-caps in the surgery procedure. However, in general we are not able to find a “fine” neck in an ε-tube or in an ε-cap, and surgeries at “rough” ε-necks will certainly lose some accuracy. If we perform surgeries at the necks with some fixed accuracy ε in the high curvature region at each surgery time, then it is possible that the errors of surgeries may accumulate to a certain amount so that at some later time we cannot recognize the structure of very high curvature regions. This prevents us from carrying out the whole process in finite time with a finite number of steps. This is the reason why we will only perform the surgeries at the ε-horns.) 1 Lemma 7.3.2 (Perelman [104]). Given 0 < ε ≤ 100 , 0 < δ < ε and 0 < T < +∞, there exists a radius 0 < h < δρ, depending only on δ and r(T ), such that if we have a solution to the Ricci flow with surgery, with a normalized metric as initial data, satisfying the a priori assumptions (with accuracy ε), defined on [0, T ), going singular at time T and having only a finite number of surgery times on [0, T ), then for each ¯ − 12 (x) ≤ h in an ε-horn of (Ω, g¯ij ) with boundary in Ωρ , the point x with h(x) = R neighborhood BT (x, δ −1 h(x)) , {y ∈ Ω | dT (y, x) ≤ δ −1 h(x)} is a strong δ-neck (i.e., BT (x, δ −1 h(x)) × [T − h2 (x), T ] is, after scaling with factor h−2 (x), δ-close (in the −1 C [δ ] topology) to the corresponding subset of the evolving standard round cylinder 2 S × R over the time interval [−1, 0] with scalar curvature 1 at time zero).
strong δ-neck
Ωρ
Proof. We argue by contradiction. Suppose that there exists a sequence of soluk tions gij (·, t), k = 1, 2, . . ., to the Ricci flow with surgery, satisfying the a priori ask sumptions (with accuracy ε), defined on [0, T ) with limit metrics (Ωk , g¯ij ), k = 1, 2, . . ., k k k and points x , lying inside an ε-horn of Ω with boundary in Ωρ , and having h(xk ) → 0 such that the neighborhoods BT (xk , δ −1 h(xk )) = {y ∈ Ωk | dT (y, xk ) ≤ δ −1 h(xk )} are not strong δ-necks. k ¯ k ) = h−2 (xk ) Let geij (·, t) be the solutions obtained by rescaling by the factor R(x k around x and shifting the time T to the new time zero. We now want to show that a k subsequence of geij (·, t), k = 1, 2, . . ., converges to the evolving round cylinder, which will give a contradiction. k Note that e gij (·, t), k = 1, 2, . . . , are solutions modified by surgery. So, we cannot apply Hamilton’s compactness theorem directly since it is stated only for smooth k solutions. For each (unrescaled) surgical solution g¯ij (·, t), we pick a point z k , with k 2 −2 k k ¯ ) = 2C (ε)ρ , in the ε-horn of (Ω , g¯ ) with boundary in Ωk , where C2 (ε) is R(z 2 ρ ij the positive constant in the canonical neighborhood assumption. From the definition
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
419
of ε-horn and the canonical neighborhood assumption, we know that each point x k k k k lying inside the ε-horn of (Ωk , g¯ij ) with dg¯ij k (x, Ωρ ) ≥ dg k (z , Ωρ ) has a strong ε-neck ¯ij
as its canonical neighborhood. Since h(xk ) → 0, each xk lies deeply inside an ε-horn. k Thus for each positive A < +∞, the rescaled (surgical) solutions e gij (·, t) with the k k k marked origins x over the geodesic balls Begij k (·,0) (x , A), centered at x of radii A
k (with respect to the metrics geij (·, 0)), will be smooth on some uniform (size) small k time intervals for all sufficiently large k, if the curvatures of the rescaled solutions e gij k at t = 0 in Begij k (·,0) (x , A) are uniformly bounded. In such a situation, Hamilton’s compactness theorem is applicable. Then we can apply the same argument as in the second step of the proof of Theorem 7.1.1 to conclude that for each A < +∞, there exists a positive constant C(A) such that the curvatures of the rescaled solutions k geij (·, t) at the new time 0 satisfy the estimate
e k |(y, 0) ≤ C(A) |Rm
k whenever degij k (·,0) (y, x ) ≤ A and k ≥ 1; otherwise we would get a piece of a non-flat nonnegatively curved metric cone as a blow-up limit, which contradicts Hamilton’s strong maximum principle. Moreover, by Hamilton’s compactness theorem (Theorem k ∞ ∞ 4.1.5), a subsequence of the rescaled solutions geij (·, t) converges to a Cloc limit e gij (·, t), defined on a spacetime set which is relatively open in the half spacetime {t ≤ 0} and contains the time slice t = 0. By the pinching assumption, the limit is a complete manifold with nonnegative sectional curvature. Since xk was contained in an ε-horn with boundary in Ωkρ and h(xk )/ρ → 0, the limiting manifold has two ends. Thus, by Toponogov’s splitting theorem, the limiting manifold admits a metric splitting Σ2 × R, where Σ2 is diffeomorphic to the two-sphere S2 because xk was the center of a strong ε-neck. By combining with the canonical neighborhood assumption (with accuracy ε), we see that the limit is defined on the time interval [−1, 0] and is ε-close to the evolving standard round cylinder. In particular, the scalar curvature of the limit at time t = −1 is ε-close to 1/2. Since h(xk )/ρ → 0, each point in the limiting manifold at time t = −1 also has a strong ε-neck as its canonical neighborhood. Thus the limit is defined at least on the time interval [−2, 0] and the limiting manifold at time t = −2 is, after rescaling, ε-close to the standard round cylinder. By using the canonical neighborhood assumption again, every point in the limiting manifold at time t = −2 still has a strong ε-neck as its canonical neighborhood. Also note that the scalar curvature of the limit at t = −2 is not bigger than 1/2 + ε. Thus the limit is defined at least on the time interval [−3, 0] and the limiting manifold at time t = −3 is, after rescaling, ε-close to the standard round cylinder. By repeating this argument we prove that the limit exists on the ancient time interval (−∞, 0]. The above argument also shows that at every time, each point of the limit has a strong ε-neck as its canonical neighborhood. This implies that the limit is κnoncollaped on all scales for some κ > 0. Therefore, by Theorem 6.2.2, the limit is the evolving round cylinder S2 × R, which gives the desired contradiction.
In the above lemma, the property that the radius h depends only on δ and the time T but is independent of the surgical solution is crucial; otherwise we will not be able to cut off enough volume at each surgery to guarantee the number of surgeries being finite in each finite time interval. We also remark that the above proof actually
420
H.-D. CAO AND X.-P. ZHU
proves a stronger result: the parabolic region {(y, t) | y ∈ BT (x, δ −1 h(x)), t ∈ [T − −1 δ −2 h2 (x), T ]} is, after scaling with factor h−2 (x), δ-close (in the C [δ ] topology) to the corresponding subset of the evolving standard round cylinder S2 × R over the time interval [−δ −2 , 0] with scalar curvature 1 at the time zero. This fact will be used later in the proof of Proposition 7.4.1. We next want to construct “fine” caps. Take a rotationally symmetric metric on R3 with nonnegative sectional curvature and positive scalar curvature such that outside some compact set it is a semi-infinite standard round cylinder (i.e. the metric product of a ray with the round two-sphere of scalar curvature 1). We call such a metric on R3 a standard capped infinite cylinder. By the short-time existence theorem of Shi (Theorem 1.2.3), the Ricci flow with a standard capped infinite cylinder as initial data has a complete solution on a maximal time interval [0, T ) such that the curvature of the solution is bounded on R3 × [0, T ′] for each 0 < T ′ < T . Such a solution is called a standard solution by Perelman [104]. The following result proved by Chen and the second author in [34] gives the curvature estimate for standard solutions. Proposition 7.3.3. Let gij be a complete Riemannian metric on Rn (n ≥ 3) with nonnegative curvature operator and positive scalar curvature which is asymptotic to a round cylinder of scalar curvature 1 at infinity. Then there is a complete solution gij (·, t) to the Ricci flow, with gij as initial metric, which exists on the time interval n−1 [0, n−1 2 ), has bounded curvature at each time t ∈ [0, 2 ), and satisfies the estimate R(x, t) ≥
C −1 n−1 2 −t
for some C depending only on the initial metric gij . Proof. Since the initial metric has bounded curvature operator and a positive lower bound on its scalar curvature, the Ricci flow has a solution gij (·, t) defined on a maximal time interval [0, T ) with T < ∞ which has bounded curvature on Rn × [0, T ′] for each 0 < T ′ < T . By Proposition 2.1.4, the solution gij (·, t) has nonnegative curvature operator for all t ∈ [0, T ). Note that the injectivity radius of the initial metric has a positive lower bound. As we remarked at the beginning of Section 3.4, the same proof of Perelman’s no local√collapsing theorem I concludes that gij (·, t) is κ-noncollapsed on all scales less than T for some κ > 0 depending only on the initial metric. We will first prove the following assertion. Claim 1. There is a positive function ω : [0, ∞) → [0, ∞) depending only on the initial metric and κ such that R(x, t) ≤ R(y, t)ω(R(y, t)d2t (x, y)) for all x, y ∈ Rn , t ∈ [0, T ). The proof is similar to that of Theorem 6.4.3. Notice that the initial metric has nonnegative curvature operator and its scalar curvature satisfies the bounds (7.3.5)
C −1 6 R(x) 6 C
for some positive constant C. By the maximum principle, we know T ≥ 1 1 R(x, t) ≤ 2C for t ∈ [0, 4nC ]. The assertion is clearly true for t ∈ [0, 4nC ].
1 2nC
and
421
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
1 . Let z be the closest point to y with Now fix (y, t0 ) ∈ Rn × [0, T ) with t0 ≥ 4nC 2 the property R(z, t0 )dt0 (z, y) = 1 (at time t0 ). Draw a shortest geodesic from y to z 1 and choose a point z˜ on the geodesic satisfying dt0 (z, z˜) = 14 R(z, t0 )− 2 , then we have
R(x, t0 ) ≤
1 1
( 21 R(z, t0 )− 2 )2
on Bt0
− 12 1 z˜, R(z, t0 ). 4
Note that R(x, t) > C −1 everywhere by the evolution equation of the scalar curvature. Then by the Li-Yau-Hamilton inequality (Corollary 2.5.5), for all (x, t) ∈ 1 1 1 1 Bt0 (˜ z , 8nC R(z, t0 )− 2 ) × [t0 − ( 8nC R(z, t0 )− 2 )2 , t0 ], we have R(x, t) ≤
t0 −
t0
1 √ 8n C
2
1 1 R(z, t0 )− 2 ≤ 8nC
−2
1
1 − 12 2 R(z, t0 )
2
.
Combining this with the κ-noncollapsing property, we have n 1 1 1 1 ≥κ Vol Bt0 z˜, R(z, t0 )− 2 R(z, t0 )− 2 8nC 8nC and then
− 21 Vol Bt0 z, 8R(z, t0) ≥κ
1 64nC
n
1
8R(z, t0 )− 2
n
.
So by Theorem 6.3.3 (ii), we have 1 R(x, t0 ) ≤ C(κ)R(z, t0 ) for all x ∈ Bt0 z, 4R(z, t0)− 2 .
Here and in the following we denote by C(κ) various positive constants depending only on κ, n and the initial metric. Now by the Li-Yau-Hamilton inequality (Corollary 2.5.5) and local gradient estimate of Shi (Theorem 1.4.2), we obtain ∂ R(x, t) ≤ C(κ)R(z, t0 ) and R (x, t) ≤ C(κ)(R(z, t0 ))2 ∂t 1
1
1 for all (x, t) ∈ Bt0 (z, 2R(z, t0)− 2 )) × [t0 − ( 8nC R(z, t0 )− 2 )2 , t0 ]. Therefore by combining with the Harnack estimate (Corollary 2.5.7), we obtain
R(y, t0 ) ≥ C(κ)−1 R(z, t0 − C(κ)−1 R(z, t0 )−1 ) ≥ C(κ)−2 R(z, t0 )
and
Consequently, we have showed that there is a constant C(κ) such that n 1 1 Vol Bt0 y, R(y, t0 )− 2 ≥ C(κ)−1 R(y, t0 )− 2 1 R(x, t0 ) ≤ C(κ)R(y, t0 ) for all x ∈ Bt0 y, R(y, t0 )− 2 .
422
H.-D. CAO AND X.-P. ZHU 1
In general, for any r ≥ R(y, t0 )− 2 , we have Vol (Bt0 (y, r)) ≥ C(κ)−1 (r2 R(y, t0 ))− 2 rn . n
By applying Theorem 6.3.3(ii) again, there exists a positive ω(r2 R(y, t0 )) depending only on the constant r2 R(y, t0 ) and κ such that 1 2 R(x, t0 ) ≤ R(y, t0 )ω(r R(y, t0 )) for all x ∈ Bt0 y, r . 4
constant
This proves the desired Claim 1. Now we study the asymptotic behavior of the solution at infinity. For any 0 < t0 < T , we know that the metrics gij (x, t) with t ∈ [0, t0 ] has uniformly bounded curvature. Let xk be a sequence of points with d0 (x0 , xk ) → ∞. After passing to a subsequence, gij (x, t) around xk will converge to a solution to the Ricci flow on R × Sn−1 with round cylinder metric of scalar curvature 1 as initial data. Denote the limit by g˜ij . Then by the uniqueness theorem (Theorem 1.2.4), we have ˜ t) = R(x, It follows that T ≤ assertion.
n−1 2 .
n−1 2 n−1 2 −
t
for all t ∈ [0, t0 ].
In order to show T =
n−1 2 ,
it suffices to prove the following
n Claim 2. Suppose T < n−1 2 . Fix a point x0 ∈ R , then there is a δ > 0, such −1 that for any x ∈ M with d0 (x, x0 ) ≥ δ , we have
R(x, t) ≤ 2C +
n−1 −t
n−1 2
for all t ∈ [0, T ),
where C is the constant in (7.3.5). In view of Claim 1, if Claim 2 holds, then n−1 −2 2C + sup R(y, t) ≤ ω δ 2C + n−1 M n ×[0,T ) 2 −T <∞
n−1 n−1 2 −T
which will contradict the definition of T . To show Claim 2, we argue by contradiction. Suppose for each δ > 0, there is a point (xδ , tδ ) with 0 < tδ < T such that R(xδ , tδ ) > 2C +
n−1 and d0 (xδ , x0 ) ≥ δ −1 . − tδ
n−1 2
Let (
Since
lim
t¯δ = sup t
d0 (y,x0 )→∞
R(y, t) =
n−1 sup R(y, t) < 2C + n−1 n −1 M \B0 (x0 ,δ ) 2 −t
n−1 2 n−1 2 −t
)
.
1 and supM×[0, 4nC ] R(y, t) ≤ 2C, we know
tδ and there is a x ¯δ such that d0 (x0 , x ¯δ ) ≥ δ −1 and R(¯ xδ , t¯δ ) = 2C +
n−1 n−1 ¯ 2 −tδ
1 4nC
≤ t¯δ ≤
. By Claim
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
423
1 and Hamilton’s compactness theorem (Theorem 4.1.5), for δ → 0 and after taking −1 a subsequence, the metrics gij (x, t) on B0 (¯ xδ , δ 2 ) over the time interval [0, t¯δ ] will converge to a solution g˜ij on R × Sn−1 with the standard metric of scalar curvature 1 as initial data over the time interval [0, t¯∞ ], and its scalar curvature satisfies n−1 , − t¯∞ ˜ t) 6 2C + n − 1 , R(x, n−1 ¯ 2 − t∞
˜ x∞ , t¯∞ ) = 2C + R(¯
n−1 2
for all t ∈ [0, t¯∞ ],
where (¯ x∞ , t¯∞ ) is the limit of (¯ xδ , t¯δ ). On the other hand, by the uniqueness theorem (Theorem 1.2.4) again, we know ˜ x∞ , t¯∞ ) = R(¯
n−1 2 n−1 ¯ 2 − t∞
which is a contradiction. Hence we have proved Claim 2 and then have verified T = n−1 2 . Now we are ready to show (7.3.6)
R(x, t) ≥
C˜ −1 , −t
n−1 2
h n − 1 for all (x, t) ∈ Rn × 0, , 2
for some positive constant C˜ depending only on the initial metric. For any (x, t) ∈ Rn × [0, n−1 2 ), by Claim 1 and κ-noncollapsing, there is a constant C(κ) > 0 such that 1
1
Vol t (Bt (x, R(x, t)− 2 )) ≥ C(κ)−1 (R(x, t)− 2 )n . Then by the well-known volume estimate of Calabi-Yau (see for example [128] or [112]) for complete manifolds with Ric ≥ 0, for any a ≥ 1, we have Vol t (Bt (x, aR(x, t)
− 21
)) ≥ C(κ)−1
a −1 (R(x, t) 2 )n . 8n
On the other hand, since (Rn , gij (·, t)) is asymptotic to a cylinder of scalar curvature n−1 ( n−1 2 )/( 2 − t), for sufficiently large a > 0, we have !! r n2 n−1 n−1 −t ≤ C(n)a −t Vol t Bt x, a . 2 2 Combining the two inequalities, for all sufficiently large a, we have: q n2 n−1 1 n−1 2 −t − R(x, t) 2 C(n)a −t ≥ Vol t Bt x, a 1 2 R(x, t)− 2 q n−1 n − t a 2 − 12 ≥ C(κ)−1 R(x, t) , 8n R(x, t)− 12
which gives the desired estimate (7.3.6). Therefore the proof of the proposition is complete.
424
H.-D. CAO AND X.-P. ZHU
We now fix a standard capped infinite cylinder for dimension n = 3 as follows. Consider the semi-infinite standard round cylinder N0 = S2 × (−∞, 4) with the metric g0 of scalar curvature 1. Denote by z the coordinate of the second factor (−∞, 4). Let f be a smooth nondecreasing convex function on (−∞, 4) defined by f (z) = 0, z ≤ 0, f (z) = ce− Pz , z ∈ (0, 3], (7.3.7) f (z) is strictly convex on z ∈ [3, 3.9], f (z) = − 21 log(16 − z 2 ), z ∈ [3.9, 4),
where the (small) constant c > 0 and (big) constant P > 0 will be determined later. Let us replace the standard metric g0 on the portion S2 × [0, 4) of the semi-infinite cylinder by gˆ = e−2f g0 . Then the resulting metric gˆ will be smooth on R3 obtained by adding a point to S2 × (−∞, 4) at z = 4. We denote by C(c, P ) = (R3 , gˆ). Clearly, C(c, P ) is a standard capped infinite cylinder. We next use a compact portion of the standard capped infinite cylinder C(c, P ) and the δ-neck obtained in Lemma 7.3.2 to perform the following surgery procedure due to Hamilton [64]. Consider the metric g¯ at the maximal time T < +∞. Take an ε-horn with boundary in Ωρ . By Lemma 7.3.2, there exists a δ-neck N of radius 0 < h < δρ in the −1 ε-horn. By definition, (N, h−2 g¯) is δ-close (in the C [δ ] topology) to the standard round neck S2 × I of scalar curvature 1 with I = (−δ −1 , δ −1 ). Using the parameter z ∈ I, we see the above function f is defined on the δ-neck N . T Let us cut the δ-neck N along the middle (topological) two-sphere N {z = 0}.T Without loss of generality, we may assume that the right hand half portion N {z ≥ 0} is contained in the horn-shaped end. Let ϕ be a smooth bump function with ϕ = 1 for z ≤ 2, and ϕ = 0 for z ≥ 3. Construct a new metric g˜ on a (topological) three-ball B3 as follows g¯, z = 0, e−2f g¯, z ∈ [0, 2], (7.3.8) g˜ = −2f −2f 2 ϕe g¯ + (1 − ϕ)e h g0 , z ∈ [2, 3], h2 e−2f g , z ∈ [3, 4]. 0 The surgery is to replace the horn-shaped end by the cap (B3 , g˜). We call such surgery procedure a δ-cutoff surgery. The following lemma determines the constants c and P in the δ-cutoff surgery so that the pinching assumption is preserved under the surgery.
Lemma 7.3.4 (Justification of the pinching assumption). There are universal positive constants δ0 , c0 and P0 such that if we take a δ-cutoff surgery at a δ-neck of radius h at time T with δ ≤ δ0 and h−2 ≥ 2e2 log(1 + T ), then we can choose c = c0 and P = P0 in the definition of f (z) such that after the surgery, there still holds the pinching condition (7.3.9)
˜ ≥ (−˜ R ν )[log(−˜ ν ) + log(1 + T ) − 3]
˜ is the scalar curvature of the metric g˜ and ν˜ is the least whenever ν˜ < 0, where R eigenvalue of the curvature operator of g˜. Moreover, after the surgery, any metric ball
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
425
1
of radius δ − 2 h with center near the tip (i.e., the origin of the attached cap) is, after −1
1
scaling with factor h−2 , δ 2 -close (in the C [δ 2 ] topology) to the corresponding ball of the standard capped infinite cylinder C(c0 , P0 ). Proof. First, we consider the metric g˜ on the portion {0 ≤ z ≤ 2}. Under the ˜ ijkl is given by conformal change g˜ = e−2f g¯, the curvature tensor R h ˜ ijkl = e−2f R ¯ ijkl + |∇f ¯ |2 (¯ R gil g¯jk − g¯ik g¯jl ) + (fik + fi fk )¯ gjl i + (fjl + fj fl )¯ gik − (fil + fi fl )¯ gjk − (fjk + fj fk )¯ gil .
∂ ∂ If {F¯a = F¯ai ∂x ¯ij , then {F˜a = ef F¯a = F˜ai ∂x i } is an orthonormal frame for g i } is an orthonormal frame for g˜ij . Let
¯ abcd = R ¯ ijkl F¯ai F¯ j F¯ck F¯dl , R b ˜ abcd = R ˜ ijkl F˜ i F˜ j F˜ k F˜ l , R a b c d then (7.3.10)
h ˜ abcd = e2f R ¯ abcd + |∇f ¯ |2 (δad δbc − δac δbd ) + (fac + fa fc )δbd R
i + (fbd + fb fd )δac − (fad + fa fd )δbc − (fbc + fb fc )δad ,
and
¯ − 2|∇f ˜ = e2f (R ¯ + 4△f ¯ |2 ). R
(7.3.11) Since
df P P d2 f P = ce− z 2 , = ce− z dz z dz 2
P2 2P − 3 z4 z
,
then for any small θ > 0, we may choose c > 0 small and P > 0 large such that for z ∈ [0, 3], we have 2 df df d2 f d2 f 2f (7.3.12) |e − 1| + + < θ. < θ 2, dz dz dz dz 2 On the other hand, by the definition of δ-neck of radius h, we have |¯ g − h2 g0 |g0 < δh2 , o
|∇j g¯|g0 < δh2 , for 1 ≤ j ≤ [δ −1 ], where g0 is the standard metric of the round cylinder S2 × R. Note that in three dimensions, we can choose the orthonormal frame {F¯1 , F¯2 , F¯3 } for √ g¯ so √ the metric that√its curvature operator is diagonal in the orthonormal frame { 2F¯2 ∧ F¯3 , 2F¯3 ∧ ¯ and F¯1 , 2F¯1 ∧ F¯2 } with eigenvalues ν¯ ≤ µ ¯≤λ ¯ = 2R ¯ 2323 , µ ¯ 3131 , λ ¯ 1212 . ν¯ = 2R ¯ = 2R
426
H.-D. CAO AND X.-P. ZHU
Since h−2 g¯ is δ-close to the standard round cylinder metric g0 on the δ-neck, we have ¯3131 | + |R ¯ 2323 | < δ 78 h−2 , |R ¯1212 − 1 h−2 | < δ 78 h−2 , |R (7.3.13) 2 |F¯3 − h−1 ∂ |g0 < δ 78 h−1 , ∂z
¯ a z = ∇z( ¯ F¯a ) and ∇ ¯ a∇ ¯ bz = ∇ ¯ 2 z(F¯a , F¯b ), it follows for suitably small δ > 0. Since ∇ that 7
¯ 3 z − h−1 | < δ 8 h−1 , |∇ 7
¯ 1 z| + |∇ ¯ 2 z| < δ 8 h−1 , |∇ and 7
¯ a∇ ¯ b z| < δ 8 h−2 , for 1 ≤ a, b ≤ 3. |∇ By combining with 2 ¯ a z, ∇ ¯ a∇ ¯ b f = df ∇ ¯ a∇ ¯ bz + d f ∇ ¯ az∇ ¯ bz ¯ a f = df ∇ ∇ dz dz dz 2
and (7.3.12), we get ¯ f | < 2θh−1 d2 f2 , for 1 ≤ a ≤ 3, |∇ dz a 2 3 d f −2 ¯ ¯ (7.3.14) |∇a ∇b f | < δ 4 h dz2 , unless a = b = 3, 2 2 3 ¯ ¯ d f d f |∇3 ∇3 f − h−2 dz2 | < δ 4 h−2 dz2 .
By combining (7.3.10) and (7.3.14), we have ˜ 1212 ≥ R ¯ 1212 − (θ 21 + δ 58 )h−2 d2 f2 , R dz R ˜ 3131 ≥ R ¯ 3131 + (1 − θ 12 − δ 58 )h−2 d2 f2 , dz (7.3.15) 1 5 d2 f −2 ˜ ¯ 2 8 R2323 ≥ R2323 + (1 − θ − δ )h dz2 , 2 1 5 ˜ otherwise, |Rabcd | ≤ (θ 2 + δ 8 )h−2 d f2 , dz
where θ and δ are suitably small. Then it follows that
2
˜≥R ¯ + [4 − 6(θ 31 + δ 12 )]h−2 d f , R dz 2 1
1
−˜ ν ≤ −¯ ν − [2 − 2(θ 3 + δ 2 )]h−2
d2 f , dz 2
for suitably small θ and δ. If 0 < −˜ ν ≤ e2 , then by the assumption that h−2 ≥ 2e2 log(1 + T ), we have ˜≥R ¯ R 1 ≥ h−2 2 ≥ e2 log(1 + T ) ≥ (−˜ ν )[log(−˜ ν ) + log(1 + T ) − 3].
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
427
While if −˜ ν > e2 , then by the pinching estimate of g¯, we have ˜≥R ¯ R ≥ (−¯ ν )[log(−¯ ν ) + log(1 + T ) − 3]
≥ (−˜ ν )[log(−˜ ν ) + log(1 + T ) − 3].
So we have verified the pinching condition on the portion {0 ≤ z ≤ 2}. Next, we consider the metric g˜ on the portion {2 ≤ z ≤ 4}. Let θ be a fixed suitably small positive number. Then the constant c = c0 and P = P0 are fixed. 2 So ζ = minz∈[1,4] ddzf2 > 0 is also fixed. By the same argument as in the derivation of (7.3.15) from (7.3.10), we see that the curvature of the metric gˆ = e−2f g0 of the standard capped infinite cylinder C(c0 , P0 ) on the portion {1 ≤ z ≤ 4} is bounded from below by 32 ζ > 0. Since h−2 g¯ is δ-close to the standard round metric g0 , the 3 metric h−2 g˜ defined by (7.3.8) is clearly δ 4 -close to the metric gˆ = e−2f g0 of the standard capped infinite cylinder on the portion {1 ≤ z ≤ 4}. Thus as δ is sufficiently small, the curvature operator of g˜ on the portion {2 ≤ z ≤ 4} is positive. Hence the pinching condition (7.3.9) holds trivially on the portion {2 ≤ z ≤ 4}. The last statement in Lemma 7.3.4 is obvious from the definition (7.3.8). Recall from Lemma 7.3.2 that the δ-necks at a time t > 0, where we performed Hamilton’s surgeries, have their radii 0 < h < δρ = δ 2 r(t). Without loss of generality, we may assume the positive nonincreasing function r(t) in the definition of the canonical neighborhood assumption is less than 1 and the universal constant δ0 in Lemma ¯ by 7.3.4 is also less than 1. We define a positive function δ(t) (7.3.16)
¯ = min δ(t)
1 , δ0 2 2e log(1 + t)
for t ∈ [0, +∞).
¯ for any δ-cutoff surgery at time From now on, we always assume 0 < δ < δ(t) t > 0 and assume c = c0 and P = P0 . As a result, the standard capped infinite cylinder and the standard solution are also fixed. The following lemma, which will be used in the next section, gives the canonical neighborhood structure for the fixed standard solution. Lemma 7.3.5. Let gij (x, t) be the above fixed standard solution to the Ricci flow on R3 × [0, 1). Then for any ε > 0, there is a positive constant C(ε) such that each point (x, t) ∈ R3 × [0, 1) has an open neighborhood B, with Bt (x, r) ⊂ B ⊂ Bt (x, 2r) 1 for some 0 < r < C(ε)R(x, t)− 2 , which falls into one of the following two categories: either (a) B is an ε-cap, or (b) B is an ε-neck and it is the slice at the time t of the parabolic neighborhood 1 Bt (x, ε−1 R(x, t)− 2 ) × [t − min{R(x, t)−1 , t}, t], on which the standard solution is, after scaling with the factor R(x, t) and shifting the time t to zero, −1 ε-close (in the C [ε ] topology) to the corresponding subset of the evolving standard cylinder S2 × R over the time interval [− min{tR(x, t), 1}, 0] with scalar curvature 1 at the time zero. Proof. The proof of the lemma is reduced to two assertions. We now state and prove the first assertion which takes care of those points with times close to 1.
428
H.-D. CAO AND X.-P. ZHU
Assertion 1. For any ε > 0, there is a positive number θ = θ(ε) with 0 < θ < 1 such that for any (x0 , t0 ) ∈ R3 × [θ, 1), the standard solution on the parabolic neigh1 borhood Bt0 (x, ε−1 R(x0 , t0 )− 2 ) × [t0 − ε−2 R(x0 , t0 )−1 , t0 ] is well-defined and is, after −1 scaling with the factor R(x0 , t0 ), ε-close (in the C [ε ] topology) to the corresponding subset of some orientable ancient κ-solution. We argue by contradiction. Suppose Assertion 1 is not true, then there exist ε¯ > 0 and a sequence of points (xk , tk ) with tk → 1, such that for each k, the standard solution on the parabolic neighborhood 1
Btk (xk , ε¯−1 R(xk , tk )− 2 ) × [tk − ε¯−2 R(xk , tk )−1 , tk ] is not, after scaling by the factor R(xk , tk ), ε¯-close to the corresponding subset of any ancient κ-solution. Note that by Proposition 7.3.3, there is a constant C > 0 (depending only on the initial metric, hence it is universal ) such that R(x, t) ≥ C −1 /(1 − t). This implies ε¯−2 R(xk , tk )−1 ≤ C ε¯−2 (1 − tk ) < tk , and then the standard solution on the parabolic neighborhood Btk (xk , 1 ε¯−1 R(xk , tk )− 2 ) × [tk − ε¯−2 R(xk , tk )−1 , tk ] is well-defined for k large. By Claim 1 in the proof of Proposition 7.3.3, there is a positive function ω : [0, ∞) → [0, ∞) such that R(x, tk ) ≤ R(xk , tk )ω(R(xk , tk )d2tk (x, xk )) for all x ∈ R3 . Now by scaling the standard solution gij (·, t) around xk with the factor R(xk , tk ) and shifting the time tk to zero, we get a sequence of the rescaled k solutions g˜ij (x, t˜) = R(xk , tk )gij (x, tk + t˜/R(xk , tk )) to the Ricci flow defined on R3 with t˜ ∈ [−R(xk , tk )tk , 0]. We denote the scalar curvature and the distance of the k ˜ By combining with Claim 1 in the proof of Proposition ˜ k and d. rescaled metric g˜ij by R 7.3.3 and the Li-Yau-Hamilton inequality, we get ˜ k (x, 0) ≤ ω(d˜2 (x, xk )) R 0 ˜ k (x, t˜) ≤ R
R(xk , tk )tk ω(d˜20 (x, xk )) t˜ + R(xk , tk )tk
for any x ∈ R3 and t˜ ∈ (−R(xk , tk )tk , 0]. Note that R(xk , tk )tk → ∞ by Proposition 7.3.3. We have shown in the proof of Proposition 7.3.3 that the standard solution is κnoncollapsed on all scales less than 1 for some κ > 0. Then from the κ-noncollapsing property, the above curvature estimates and Hamilton’s compactness theorem, we k know g˜ij (x, t˜) has a convergent subsequence (as k → ∞) whose limit is an ancient, κnoncollapsed, complete and orientable solution with nonnegative curvature operator. This limit must have bounded curvature by the same proof of Step 3 in the proof of Theorem 7.1.1. This gives a contradiction. Hence Assertion 1 is proved. We now fix the constant θ(ε) obtained in Assertion 1. Let O be the tip of the standard capped infinite cylinder R3 (it is rotationally symmetric about O at time 0, and it remains so as t > 0 by the uniqueness Theorem 1.2.4). Assertion 2. There are constants B1 (ε) and B2 (ε) depending only on ε, such that if (x0 , t0 ) ∈ R3 × [0, θ) with dt0 (x0 , O) ≤ B1 (ε), then there is a 0 < r < B2 (ε)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
429
such that Bt0 (x0 , r) is an ε-cap; if (x0 , t0 ) ∈ R3 × [0, θ) with dt0 (x0 , O) ≥ B1 (ε), then 1 the parabolic neighborhood Bt0 (x0 , ε−1 R(x0 , t0 )− 2 ) ×[t0 − min{R(x0 , t0 )−1 , t0 }, t0 ] is after scaling with the factor R(x0 , t0 ) and shifting the time t0 to zero, ε-close (in the −1 C [ε ] topology) to the corresponding subset of the evolving standard cylinder S2 × R over the time interval [− min{t0 R(x0 , t0 ), 1}, 0] with scalar curvature 1 at time zero. Since the standard solution exists on the time interval [0, 1), there is a constant B0 (ε) such that the curvatures on [0, θ(ε)] are uniformly bounded by B0 (ε). This implies that the metrics in [0, θ(ε)] are equivalent. Note that the initial metric is asymptotic to the standard capped infinite cylinder. For any sequence of points xk with d0 (O, xk ) → ∞, after passing to a subsequence, gij (x, t) around xk will converge to a solution to the Ricci flow on R × S2 with round cylinder metric of scalar curvature 1 as initial data. By the uniqueness theorem (Theorem 1.2.4), the limit solution must be the standard evolving round cylinder. This implies that there is a constant B1 (ε) > 0 depending on ε such that for any (x0 , t0 ) with t0 ≤ θ(ε) and dt0 (x0 , O) ≥ B1 (ε), the standard solution on the parabolic neighbor1 hood Bt0 (x0 , ε−1 R(x0 , t0 )− 2 ) × [t0 − min{R(x0 , t0 )−1 , t0 }, t0 ] is, after scaling with the factor R(x0 , t0 ), ε-close to the corresponding subset of the evolving round cylinder. Since the solution is rotationally symmetric around O, the cap neighborhood structures of those points x0 with dt0 (x0 , O) ≤ B1 (ε) follow directly. Hence Assertion 2 is proved. The combination of these two assertions proves the lemma. Since there are only a finite number of horns with the other end connected to Ωρ , we perform only a finite number of such δ-cutoff surgeries at time T . Besides those horns, there could be capped horns and double horns which lie in Ω\Ωρ . As explained before, they are connected to form tubes or capped tubes at any time slightly before T . So we can regard the capped horns and double horns (of Ω \ Ωρ ) to be extinct and throw them away at time T . We only need to remember that the connected sums were broken there. Remember that we have thrown away all compact components, either lying in Ω \ Ωρ or with positive sectional curvature, each of which is diffeomorphic to either S3 , or a metric quotient of S3 , or S2 × S1 or RP3 #RP3 . So we have also removed a finite number of copies of S3 , or metric quotients of S3 , or S2 × S1 or RP3 #RP3 at the time T . Let us agree to declare extinct every compact component either with positive sectional curvature or lying in Ω \ Ωρ ; in particular, this allows us to exclude the components with positive sectional curvature from the list of canonical neighborhoods. In summary, our surgery at time T consists of the following four procedures: (1) perform δ-cutoff surgeries for all ε-horns, whose other ends are connected to Ωρ ; (2) declare extinct every compact component which has positive sectional curvature; (3) throw away all capped horns and double horns lying in Ω \ Ωρ ; (4) declare extinct all compact components lying in Ω \ Ωρ . (In Sections 7.6 and 7.7, we will add one more procedure by declaring extinct every compact component which has nonnegative scalar curvature.) By Lemma 7.3.4, after performing surgeries at time T , the pinching assumption (7.3.3) still holds for the surgically modified manifold. With this surgically modified manifold (possibly disconnected) as initial data, we now continue our solution under
430
H.-D. CAO AND X.-P. ZHU
the Ricci flow until it becomes singular again at some time T ′ (> T ). Therefore, we have extended the solution to the Ricci flow with surgery, originally defined on [0, T ) with T < +∞, to the new time interval [0, T ′ ) with T ′ > T . By the proof of Theorem 5.3.2, we see that the solution to the Ricci flow with surgery also satisfies the pinching assumption on [0, T ′ ). It remains to verify the canonical neighborhood assumption (with accuracy ε) for the solution on the time interval [T, T ′) and to prove that this extension procedure works indefinitely (unless it becomes extinct at some finite time) and that there exists at most a finite number of surgeries at every finite time interval. We leave these arguments to the next section. Before we end this section, we check the following two assertions of Perelman in [104] which will be used in the next section to estimate the Li-Yau-Perelman distance of space-time curves which stretch to surgery regions. Lemma 7.3.6 (Perelman [104]). For any 0 < ε ≤ 1/100, 1 < A < +∞ and ¯ 0 < θ < 1, one can find δ¯ = δ(A, θ, ε) with the following property. Suppose we have a solution to the Ricci flow with surgery which satisfies the a priori assumptions (with accuracy ε) on [0, T ] and is obtained from a compact orientable three-manifold by a ¯ Suppose we have a cutoff surgery finite number of δ-cutoff surgeries with each δ < δ. at time T0 ∈ (0, T ), let x0 be any fixed point on the gluing caps (i.e., the regions affected by the cutoff surgeries at time T0 ), and let T1 = min{T, T0 + θh2 }, where h is the cutoff radius around x0 obtained in Lemma 7.3.2. Then either (i) the solution is defined on P (x0 , T0 , Ah, T1 − T0 ) , {(x, t) | x ∈ Bt (x0 , Ah), t ∈ [T0 , T1 ]} and is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to a corresponding subset of the standard solution, or (ii) the assertion (i) holds with T1 replaced by some time t+ ∈ (T0 , T1 ), where t+ is a surgery time; moreover, for each point in BT0 (x0 , Ah), the solution is defined for t ∈ [T0 , t+ ) but is not defined past t+ (i.e., the whole ball BT0 (x0 , Ah) is cut off at the time t+ ). Proof. Let Q be the maximum of the scalar curvature of the standard solution in −T0 ) the time interval [0, θ] and choose a large positive integer N so that ∆t = (T1N < −1 −1 2 εη Q h , where the positive constant η is given in the canonical neighborhood assumption. Set tk = T0 + k∆t, k = 0, 1, . . . , N . 1 From Lemma 7.3.4, the geodesic ball BT0 (x0 , A0 h) at time T0 , with A0 = δ − 2 1 −2 is, after scaling with factor h , δ 2 -close to the corresponding ball in the standard capped infinite cylinder with the center near the tip. Assume first that for each point in BT0 (x0 , A0 h), the solution is defined on [T0 , t1 ]. By the gradient estimates (7.3.4) in the canonical neighborhood assumption and the choice of ∆t we have a uniform curvature bound on this set for h−2 -scaled metric. Then by the uniqueness theorem 1 1 (Theorem 1.2.4), if δ 2 → 0 (i.e. A0 = δ − 2 → +∞), the solution with h−2 -scaled ∞ metric will converge to the standard solution in the Cloc topology. Therefore we can define A1 , depending only on A0 and tending to infinity with A0 , such that the solution in the parabolic region P (x0 , T0 , A1 h, t1 − T0 ) , {(x, t) | x ∈ Bt (x0 , A1 h), t ∈ [T0 , T0 + (t1 − T0 )]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 1 -close to the corresponding subset in the standard solution. In particular, the scalar curvature on this subset does not exceed 2Qh−2 . Now if for each point in BT0 (x0 , A1 h) the solution is defined on [T0 , t2 ], then we can repeat the procedure, defining A2 , such that the solution in the parabolic region P (x0 , T0 , A2 h, t2 − T0 ) , {(x, t) | x ∈ Bt (x0 , A2 h), t ∈ [T0 , T0 + (t2 − T0 )]} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 2 -close to the corresponding subset in the standard
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
431
solution. Again, the scalar curvature on this subset still does not exceed 2Qh−2 . Continuing this way, we eventually define AN . Note that N is depends only on θ ¯ ¯ we have and ε. Thus there exists a positive δ¯ = δ(A, θ, ε) such that for δ < δ, A0 > A1 > · · · > AN > A, and assertion (i) holds when the solution is defined on BT0 (x0 , A(N −1) h) × [T0 , T1 ]. The above argument shows that either assertion (i) holds, or there exists some k (0 ≤ k ≤ N − 1) and a surgery time t+ ∈ (tk , tk+1 ] such that the solution on BT0 (x0 , Ak h) is defined on [T0 , t+ ), but for some point of this set it is not defined past t+ . Now we consider the latter case. Clearly the above argument also shows that the parabolic region P (x0 , T0 , Ak+1 h, t+ − T0 ) , {(x, t) | x ∈ Bt (x, Ak+1 h), t ∈ [T0 , t+ )} is, after scaling with factor h−2 and shifting time T0 to zero, A−1 k+1 -close to the corresponding subset in the standard solution. In particular, as time tends to t+ , the ball BT0 (x0 , Ak+1 h) keeps on looking like a cap. Since the scalar curvature on BT0 (x0 , Ak h) × [T0 , tk ] does not exceed 2Qh−2 , it follows from the pinching assumption, the gradient estimates in the canonical neighborhood assumption and the evolution equation of the metric that the diameter of the set BT0 (x0 , Ak h) at any 1 time t ∈ [T0 , t+ ) is bounded from above by 4δ − 2 h. These imply that no point of the ball BT0 (x0 , Ak h) at any time near t+ can be the center of a δ-neck for any 1 ¯ ¯ 0 < δ < δ(A, θ, ε) with δ(A, θ, ε) > 0 small enough, since 4δ − 2 h is much smaller than δ −1 h. However the solution disappears somewhere in the set BT0 (x0 , Ak h) at time t+ by a cutoff surgery and the surgery is always done along the middle two-sphere of a δ-neck. So the set BT0 (x0 , Ak h) at time t+ is a part of a capped horn. (Recall that we have declared extinct every compact component with positive curvature and every compact component lying in Ω \ Ωρ ). Hence for each point of BT0 (x0 , Ak h) the solution terminates at t+ . This proves assertion (ii). Corollary 7.3.7 (Perelman [104]). For any l < ∞ one can find A = A(l) < ∞ and θ = θ(l), 0 < θ < 1, with the following property. Suppose we are in the ¯ situation of the lemma above, with δ < δ(A, θ, ε). Consider smooth curves γ in the set BT0 (x0 , Ah), parametrized by t ∈ [T0 , Tγ ], such that γ(T0 ) ∈ BT0 (x0 , Ah 2 ) and either Tγ = T1 < T , or Tγ < T1 and γ(Tγ ) ∈ ∂BT0 (x0 , Ah), where x0 is any fixed point on a gluing cap at T0 and T1 = min{T, T0 + θh2 }. Then Z
Tγ
T0
2 (R(γ(t), t) + |γ(t)| ˙ )dt > l.
Proof. We know from Proposition 7.3.3 that on the standard solution, Z
0
θ
Rdt ≥ const.
Z
θ 0
(1 − t)−1 dt
= −const. · log(1 − θ).
By choosing θ = θ(l) sufficiently close to 1 we have the desired estimate for the standard solution. Let us consider the first case: Tγ = T1 < T . For θ = θ(l) fixed above, by Lemma 7.3.6, our solution in the subset BT0 (x0 , Ah) and in the time interval [T0 , Tγ ] is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding
432
H.-D. CAO AND X.-P. ZHU
subset in the standard solution for any sufficiently large A. So we have Z Tγ Z θ 2 (R(γ(t), t) + |γ(t)| ˙ )dt ≥ const. (1 − t)−1 dt 0
T0
= −const. · log(1 − θ).
Hence we have obtained the desired estimate in the first case. We now consider the second case: Tγ < T1 and γ(Tγ ) ∈ ∂BT0 (x0 , Ah). Let θ = θ(l) be chosen above and let Q = Q(l) be the maximum of the scalar curvature on the standard solution in the time interval [0, θ]. On the standard solution, we can choose A = A(l) so large that for each t ∈ [0, θ], distt (x0 , ∂B0 (x0 , A)) ≥ dist0 (x0 , ∂B0 (x0 , A)) − 4(Q + 1)t ≥ A − 4(Q + 1)θ 4 ≥ A 5
and A A ≤ , distt x0 , ∂B0 x0 , 2 2 where we have used Lemma 3.4.1(ii) in the first inequality. Now our solution in the subset BT0 (x0 , Ah) and in the time interval [T0 , Tγ ] is, after scaling with factor h−2 and shifting time T0 to zero, A−1 -close to the corresponding subset in the standard solution. This implies that for A = A(l) large enough 1 Ah ≤ 5
Z
Tγ
T0
|γ(t)|dt ˙ ≤
Z
Tγ
T0
2 |γ(t)| ˙ dt
! 12
1
· (Tγ − T0 ) 2 ,
Hence Z
Tγ
T0
2 (R(γ(t), t) + |γ(t)| ˙ )dt ≥
A2 > l. 25θ
This proves the desired estimate. 7.4. Justification of the Canonical Neighborhood Assumptions. We continue the induction argument for the construction of a long-time solution to the Ricci flow with surgery. Let us recall what we have done in the previous section. Let ε be an arbitrarily given positive constant satisfying 0 < ε ≤ 1/100. For an arbitrarily given compact orientable normalized three-manifold, we evolve it by the Ricci flow. We may assume that the solution goes singular at some time 0 < t+ 1 < +∞ and know that the solution satisfies the a priori assumptions (with accuracy ε) on [0, t+ 1 ) for a nonincreasing positive function r = r1 (t) (defined on [0, +∞)). Suppose that we have a solution + + + to the Ricci flow with surgery, defined on [0, t+ k ) with 0 < t1 < t2 < · · · < tk < +∞, satisfying the a priori assumptions (with accuracy ε) for some nonincreasing positive function r = rk (t) (defined on [0, +∞)), going singular at time t+ k and having δi -cutoff + ¯ surgeries at each time t+ , 1 ≤ i ≤ k − 1, where δ < δ(t ) for each 1 ≤ i ≤ k − 1. Then i i i ¯ + ), we can perform δk -cutoff surgeries at the time t+ and extend for any 0 < δk < δ(t k k + + ¯ the solution to the interval [0, t+ k+1 ) with tk+1 > tk . Here δ(t) is the positive function
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
433
defined in (7.3.16). We have already shown in Lemma 7.3.4 that the extended solution still satisfies the pinching assumption on [0, t+ k+1 ). In view of Theorem 7.1.1, there always is a nonincreasing positive function r = rk+1 (t), defined on [0, +∞), such that the canonical neighborhood assumption (with accuracy ε) holds on the extended time interval [0, t+ k+1 ) with the positive function r = rk+1 (t). Nevertheless, in order to prevent the surgery times from accumulating, the key is to choose the nonincreasing positive functions r = ri (t), i = 1, 2, . . ., uniformly. That is, to justify the canonical neighborhood assumption (with accuracy ε) for the indefinitely extending solution, we need to show that there exists a nonincreasing positive function re(t), defined on [0, +∞), which is independent of k, such that the above chosen nonincreasing positive functions satisfy ri (t) ≥ re(t),
on [0, +∞),
for all i = 1, 2, . . . , k + 1. ¯ By a further restriction on the positive function δ(t), we can verify this after proving the following assertion which was stated by Perelman in [104]. Proposition 7.4.1 (Justification of the canonical neighborhood assumption). Given any small ε > 0, there exist decreasing sequences 0 < rej < ε, κj > 0, and 0 < δej < ε2 , j = 1, 2, · · · , with the following property. Define the positive function e on [0, +∞) by δ(t) e = δej for t ∈ [(j − 1)ε2 , jε2 ). Suppose there is a surgically δ(t) modified solution, defined on [0, T ) with T < +∞, to the Ricci flow which satisfies the following: (1) it starts on a compact orientable three-manifold with normalized initial metric, and (2) it has only a finite number of surgeries such that each surgery at a time t ∈ (0, T ) is a δ(t)-cutoff surgery with e δ(t)}. ¯ 0 < δ(t) ≤ min{δ(t), T Then on each time interval [(j − 1)ε2 , jε2 ] [0, T ), j = 1, 2, · · · , the solution satisfies the κj -noncollapsing condition on all scales less than ε and the canonical neighborhood assumption (with accuracy ε) with r = rej .
Here and in the following, we call a (three-dimensional) surgically modified solution gij (t), 0 ≤ t < T , κ-noncollapsed at (x0 , t0 ) on the scales less than ρ (for some κ > 0, ρ > 0) if it satisfies the following property: whenever r < ρ and |Rm(x, t)| ≤ r−2 for all those (x, t) ∈ P (x0 , t0 , r, −r2 ) = {(x′ , t′ ) | x′ ∈ Bt′ (x0 , r), t′ ∈ [t0 − r2 , t0 ]}, for which the solution is defined, we have Vol t0 (Bt0 (x0 , r)) ≥ κr3 . Before we give the proof of the proposition, we need to verify a κ-noncollapsing estimate which was given by Perelman in [104]. Lemma 7.4.2. Given any 0 < ε ≤ ε¯0 (for some sufficiently small universal constant ε¯0 ), suppose we have constructed the sequences satisfying the proposition for 1 ≤ j ≤ m (for some positive integer m). Then there exists κ > 0, such that for any
434
H.-D. CAO AND X.-P. ZHU
e ε), 0 < δe < ε2 , which may also depend on the r, 0 < r < ε, one can find δe = δ(r, already constructed sequences, with the following property. Suppose we have a solution with a compact orientable normalized three-manifold as initial data, to the Ricci flow with finite number of surgeries on a time interval [0, T¯] with mε2 ≤ T¯ < (m + 1)ε2 , satisfying the assumptions and the conclusions of Proposition 7.4.1 on [0, mε2 ), and the canonical neighborhood assumption (with accuracy ε) with r on [mε2 , T¯], as well e δ(t)} ¯ as 0 < δ(t) ≤ min{δ, for any δ-cutoff surgery with δ = δ(t) at a time t ∈ 2 ¯ [(m − 1)ε , T ]. Then the solution is κ-noncollapsed on [0, T¯] for all scales less than ε. Proof. Consider a parabolic neighborhood P (x0 , t0 , r0 , −r02 ) , {(x, t) | x ∈ Bt (x0 , r0 ), t ∈ [t0 − r02 , t0 ]} with mε2 ≤ t0 ≤ T¯ and 0 < r0 < ε, where the solution satisfies |Rm| ≤ r0−2 , whenever it is defined. We will use an argument analogous to the proof of Theorem 3.3.2 (no local collapsing theorem I) to prove (7.4.1)
Vol t0 (Bt0 (x0 , r0 )) ≥ κr03 .
Let η be the universal positive constant in the definition of the canonical neighborhood assumption. Without loss of generality, we always assume η ≥ 10. Firstly, 1 we want to show that one may assume r0 ≥ 2η r. Obviously, the curvature satisfies the estimate |Rm(x, t)| ≤ 20r0−2 , 1 1 2 1 1 2 for those (x, t) ∈ P (x0 , t0 , 2η r0 , − 8η r0 ) = {(x, t) | x ∈ Bt (x0 , 2η r0 ), t ∈ [t0 − 8η r0 , t0 ]}, 1 ′ for which the solution is defined. When r0 < 2η r, we can enlarge r0 to some r0 ∈ [r0 , r] so that
|Rm| ≤ 20r0′−2 1 ′2 1 ′ r0 , − 8η r0 ) (whenever it is defined), and either the equality holds someon P (x0 , t0 , 2η
1 ′ 1 ′2 where in P (x0 , t0 , 2η r0 , −( 8η r 0 + ǫ′ )) for any arbitrarily small ǫ′ > 0 or r0′ = r. In the case that the equality holds somewhere, it follows from the pinching assumption that we have
R > 10r0′−2 1 ′2 1 ′ r0 , −( 8η r 0 + ǫ′ )) for any arbitrarily small ǫ′ > 0. Here, somewhere in P (x0 , t0 , 2η without loss of generality, we have assumed r is suitably small. Then by the gradient estimates in the definition of the canonical neighborhood assumption, we know
R(x0 , t0 ) > r0′−2 ≥ r−2 . Hence the desired noncollapsing estimate (7.4.1) in this case follows directly from the canonical neighborhood assumption. (Recall that we have excluded every compact component which has positive sectional curvature in the surgery procedure and then we have excluded them from the list of canonical neighborhoods. Here we also used the standard volume comparison when the canonical neighborhood is an ε-cap.)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
435
While in the case that r0′ = r, we have the curvature bound |Rm(x, t)| ≤
1 r 2η
−2
,
1 1 1 r, −( 2η r)2 ) = {(x, t) | x ∈ Bt (x0 , 2η r), t ∈ [t0 − for those (x, t) ∈ P (x0 , t0 , 2η 1 ( 2η r)2 , t0 ]}, for which the solution is defined. It follows from the standard volume 1 comparison that we only need to verify the noncollapsing estimate (7.4.1) for r0 = 2η r. 1 Thus we have reduced the proof to the case r0 ≥ 2η r. Recall from Theorem 3.3.2 that if a solution is smooth everywhere, we can get a lower bound for the volume of the ball Bt0 (x0 , r0 ) as follows: define τ (t) = t0 − t and consider Perelman’s reduced volume function and the Li-Yau-Perelman distance associated to the point x0 ; take a point x ¯ at the time t = ε2 so that the Li-YauPerelman distance l attains its minimum lmin (τ ) = l(¯ x, τ ) ≤ 23 for τ = t0 − ε2 ; use it to obtain an upper bound for the Li-Yau-Perelman distance from x0 to each point of B0 (¯ x, 1), thus getting a lower bound for Perelman’s reduced volume at τ = t0 ; apply the monotonicity of Perelman’s reduced volume to deduce a lower bound for Perelman’s reduced volume at τ near 0, and then get the desired estimate for the volume of the ball Bt0 (x0 , r0 ). Now since our solution has undergone surgeries, we need to localize this argument to the region which is unaffected by surgery. We call a space-time curve in the solution track admissible if it stays in the spacetime region unaffected by surgery, and we call a space-time curve in the solution track a barely admissible curve if it is on the boundary of the set of admissible curves. First of all, we want to estimate the L-length of a barely admissible curve.
¯ r, rem , ε) > 0 with the following Claim. For any L < ∞ one can find δ¯ = δ(L, property. Suppose that we have a curve γ, parametrized by t ∈ [T0 , t0 ], (m − 1)ε2 ≤ T0 < t0 , such that γ(t0 ) = x0 , T0 is a surgery time, and γ(T0 ) lies in the gluing cap. ¯ Then we Suppose also each δ-cutoff surgery at a time in [(m − 1)ε2 , T¯] has δ ≤ δ. have an estimate Z t0 √ 2 (7.4.2) t0 − t(R+ (γ(t), t) + |γ(t)| ˙ )dt ≥ L T0
where R+ = max{R, 0}. 1 r and |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ) (whenever it is defined), we Since r0 ≥ 2η can require δ¯ > 0, depending on r and rem , to be so small that γ(T0 ) does not lie in the region P (x0 , t0 , r0 , −r02 ). Let ∆t be the maximal number such that γ|[t0 −∆t,t0 ] ⊂ P (x0 , t0 , r0 , −∆t) (i.e., t0 − ∆t is the first time when γ escapes the parabolic region P (x0 , t0 , r0 , −r02 )). Obviously we only need to consider the case:
Z
t0
t0 −∆t
√ 2 t0 − t(R+ (γ(t), t) + |γ(t)| ˙ )dt < L.
We observe that ∆t can be bounded from below in terms of L and r0 . Indeed, if ∆t ≥ r02 , there is nothing to prove. Thus we may assume ∆t < r02 . By the curvature bound |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ) and the Ricci flow equation we see Z
t0
t0 −∆t
|γ(t)|dt ˙ ≥ cr0
436
H.-D. CAO AND X.-P. ZHU
for some universal positive constant c. On the other hand, by the Cauchy-Schwarz inequality, we have Z
t0
t0 −∆t
|γ(t)|dt ˙ ≤
Z
t0
t0 −∆t
√ t0 − t(R+ + |γ| ˙ 2 )dt
1
1
12 Z ·
t0
t0 −∆t
√
≤ (2L) 2 (∆t) 4 ,
1 t0 − t
dt
12
which implies c2 r02 . 2L
1
(∆t) 2 ≥
(7.4.3) Thus Z
t0
T0
√ t0 − t(R+ + |γ| ˙ 2 )dt ≥
Z
t0 −∆t √
T0 1
≥ (∆t) 2
Z
t0 − t(R+ + |γ| ˙ 2 )dt
t0 −∆t
T0
(R+ + |γ| ˙ 2 )dt
Z t0 −∆t 2 2 c r0 (R+ + |γ| ˙ 2 )dt, , r0 ≥ min 2L T0 ¯ r, rem , ε) > 0 so small that while by Corollary 7.3.7, we can find δ¯ = δ(L, Z
t0 −∆t
T0
2 2 −1 c r0 (R+ + |γ| ˙ )dt ≥ L min , r0 . 2L 2
Then we have proved the desired assertion (7.4.2). Recall that for a curve γ, parametrized by τ = t0 − t ∈ [0, τ¯], with γ(0) = x0 and R τ¯ √ ˙ 2 )dτ . We can also define L+ (γ) τ¯ ≤ t0 − (m − 1)ε2 , we have L(γ) = 0 τ (R + |γ| by replacing R with R+ in the previous formula. Recall that R ≥ −1 at the initial time t = 0 for the normalized initial manifold. Recall that the surgeries occur at the parts where the scalar curvatures are very large. Thus we can apply the maximum principle to conclude that the solution with surgery still satisfies R ≥ −1 everywhere in space-time. This implies (7.4.4)
3
L+ (γ) ≤ L(γ) + (2ε2 ) 2 .
By applying the assertion (7.4.2), we now choose δ˜ > 0 (depending on r, ε and rem ) ˜ such that as each δ-cutoff surgery at the time interval t ∈ [(m − 1)ε2 , T ] has δ ≤ δ, every barely admissible curve γ from (x0 , t0 ) to a point (x, t) (with t ∈ [(m − 1)ε2 , t0 )) has √ L+ (γ) ≥ 22 2. Thus if the Li-Yau-Perelman distance from (x0 , t0 ) to a point (x, t) (with t ∈ [(m − 1)ε2 , t0 )) is achieved by a space-time curve which is not admissible, then its Li-YauPerelman distance has 3
(7.4.5)
l≥
L+ − (2ε2 ) 2 √ > 10ε−1 . 2 2ε
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
437
We also observe that the absolute value of l(x0 , τ ) is very small as τ closes to zere. Thus the maximum principle argument in Corollary 3.2.6 still works for our solutions with surgery because barely admissible curves do not attain the minimum. So we conclude that 3 lmin (¯ τ ) = min{l(x, τ¯) | x lies in the solution manifold at t0 − τ¯} ≤ 2 for τ¯ ∈ (0, t0 − (m − 1)ε2 ]. In particular, there exists a minimizing curve γ of lmin(t0 − (m − 1)ε2 ), defined on τ ∈ [0, t0 − (m − 1)ε2 ] with γ(0) = x0 , such that √ 3 √ · 2 2ε + 2 2ε3 2 ≤ 5ε,
L+ (γ) ≤
(7.4.6)
since 0 < ε ≤ ε¯0 with ε¯0 sufficiently small (to be further determined). Consequently, there exists a point (¯ x, t¯) on the minimizing curve γ with t¯ ∈ [(m − 1)ε2 + 41 ε2 , (m − 3 2 2 1)ε + 4 ε ] (i.e., τ ∈ [t0 − (m − 1)ε2 − 43 ε2 , t0 − (m − 1)ε2 − 14 ε2 ]) such that −2 R(¯ x, t¯) ≤ 25e rm .
(7.4.7) Otherwise, we have L+ (γ) ≥
Z
t0 −(m−1)ε2 − 41 ε2
t0 −(m−1)ε2 − 34 ε2
−2 > 25e rm
r
1 2 ε 4
> 5ε,
√ τ R(γ(τ ), t0 − τ )dτ
1 2 ε 2
since 0 < rem < ε. This contradicts (7.4.6). Next we want to get a lower bound for Perelman’s reduced volume of a ball around x ¯ of radius about rem at some time slightly before t¯. 1 −1 1 −1 η and θ2 = 64 η , where η is the universal positive constant Denote by θ1 = 16 in the gradient estimates (7.3.4). Since the solution satisfies the canonical neighborhood assumption on the time interval [(m − 1)ε2 , mε2 ), it follows from the gradient estimates (7.3.4) that (7.4.8)
−2 R(x, t) ≤ 400e rm
2 2 ¯ for those (x, t) ∈ P (¯ x, t¯, θ1 rem , −θ2 rem ) , {(x′ , t′ ) | x′ ∈ Bt′ (¯ x, θ1 rem ), t′ ∈ [t¯− θ2 rem , t]}, for which the solution is defined. And since the scalar curvature at the points where e −2 re−2 , the δ-cutoff surgeries occur in the time interval [(m − 1)ε2 , mε2 ) is at least (δ) m 2 the solution is well-defined on the whole parabolic region P (¯ x, t¯, θ1 rem , −θ2 rem ) (i.e., this parabolic region is unaffected by surgery). Thus by combining (7.4.6) and (7.4.8), we know that the Li-Yau-Perelman distance from (x0 , t0 ) to each point of the ball 2 (¯ Bt¯−θ2 rem x, θ1 rem ) is uniformly bounded by some universal constant. Let us define 2 (¯ Perelman’s reduced volume of the ball Bt¯−θ2 rem x, θ1 rem ), by 2 (Bt 2 (¯ x, θ1 rem )) Vet0 −t¯+θ2 rem ¯−θ2 r em Z 3 2 (4π(t0 − t¯ + θ2 rem ))− 2 =
em ) x,θ1 r Bt¯−θ re2 (¯ 2 m
2 2 (q), · exp(−l(q, t0 − t¯ + θ2 rem ))dVt¯−θ2 rem
438
H.-D. CAO AND X.-P. ZHU
where l(q, τ ) is the Li-Yau-Perelman distance from (x0 , t0 ). Hence by the κm noncollapsing assumption on the time interval [(m − 1)ε2 , mε2 ), we conclude that 2 (¯ Perelman’s reduced volume of the ball Bt¯−θ2 rem x, θ1 rem ) is bounded from below by a positive constant depending only on κm and rem . Finally we want to apply a local version of the monotonicity of Perelman’s reduced volume to get a lower bound estimate for the volume of the ball Bt0 (x0 , r0 ).
We have seen that the Li-Yau-Perelman distance from (x0 , t0 ) to each point of the 2 (¯ ball Bt¯−θ2 rem x, θ1 rem ) is uniformly bounded by some universal constant. Now we can choose a sufficiently small (universal) positive constant ε¯0 such that when 0 < ε ≤ ε¯0 , 2 (¯ x, θ1 rem ) can be connected to (x0 , t0 ) by (7.4.5), all the points in the ball Bt¯−θ2 rem by shortest L-geodesics, and all of these L-geodesics are admissible (i.e., they stay in the region unaffected by surgery). The union of all shortest L-geodesics from (x0 , t0 ) 2 (¯ 2 (¯ to the ball Bt¯−θ2 rem x, θ1 rem ) defined by CBt¯−θ2 rem x, θ1 rem ) = {(x, t) | (x, t) lies in 2 (¯ a shortest L-geodesic from (x0 , t0 ) to a point in Bt¯−θ2 rem x, θ1 rem )}, forms a conelike subset in space-time with the vertex (x0 , t0 ). Denote B(t) by the intersection of 2 (¯ the cone-like subset CBt¯−θ2 rem x, θ1 rem ) with the time-slice at t. Perelman’s reduced volume of the subset B(t) is given by Vet0 −t (B(t)) =
Z
3
B(t)
(4π(t0 − t))− 2 exp(−l(q, t0 − t))dVt (q).
2 (¯ Since the cone-like subset CBt¯−θ2 rem x, θ1 rem ) lies entirely in the region unaffected by surgery, we can apply Perelman’s Jacobian comparison theorem (Theorem 3.2.7) to conclude that
(7.4.9)
2 (Bt 2 (¯ x, θ1 rem )) Vet0 −t (B(t)) ≥ Vet0 −t¯+θ2 rem ¯−θ2 r em
≥ c(κm , rem ),
2 for all t ∈ [t¯− θ2 rem , t0 ], where c(κm , rem ) is some positive constant depending only on κm and rem . 1
Set ξ = r0−1 Vol t0 (Bt0 (x0 , r0 )) 3 . Our purpose is to give a positive lower bound 2 . for ξ. Without loss of generality, we may assume ξ < 14 , thus 0 < ξr02 < t0 − ¯t + θ2 rem 2 2 e 0 − ξr ) the subset of the time-slice {t = t0 − ξr } of which every Denote by B(t 0 0 point can be connected to (x0 , t0 ) by an admissible shortest L-geodesic. Clearly, e 0 − ξr2 ). We now argue as in the proof of Theorem 3.3.2 to bound B(t0 − ξr02 ) ⊂ B(t 0 e 0 − ξr2 ) from above. Perelman’s reduced volume of B(t 0 1 ˜ ε, rem ) sufficiently small, the whole region P (x0 , t0 , r0 , Since r0 ≥ 2η r and δ˜ = δ(r, 2 −r0 ) is unaffected by surgery. Then by exactly the same argument as in deriving (3.3.5), we see that there exists a universal positive constant ξ0 such that when 0 < ξ ≤ ξ0 , there holds
(7.4.10)
L exp
{|υ|≤ 41 ξ
−1 2
}
(ξr02 ) ⊂ Bt0 (x0 , r0 ).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
439
e 0 − ξr2 ) is given by Perelman’s reduced volume of B(t 0 (7.4.11)
e 0 − ξr02 )) Veξr02 (B(t Z 3 (4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) = e 0 −ξr02 ) B(t
=
Z
3
e 0 −ξr02 )∩L exp B(t
+
Z
{|υ|≤ 1 ξ 4
−1 2}
(ξr02 )
(4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) 3
e
B(t0 −ξr02 )\L exp (ξr02 ) −1 {|υ|≤ 1 ξ 2 } 4
(4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q).
The first term on the RHS of (7.4.11) can be estimated by Z 3 (7.4.12) (4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) e 0 −ξr02 )∩L exp B(t
{|υ|≤ 1 ξ 4 3 2
3
−1 2}
(ξr02 )
≤ eCξ (4π)− 2 · ξ
for some universal constant C, as in deriving (3.3.7). While as in deriving (3.3.8), the second term on the RHS of (7.4.11) can be estimated by Z 3 (7.4.13) (4πξr02 )− 2 exp(−l(q, ξr02 ))dVt0 −ξr02 (q) e 0 −ξr02 )\L exp B(t
≤
{|υ|≤ 1 ξ 4
Z
−1 2}
(ξr02 )
3
1 {|υ|> 41 ξ − 2 3
= (4π)− 2
Z
(4πτ )− 2 exp(−l(τ ))J (τ )|τ =0 dυ }
1 {|υ|> 14 ξ − 2
exp(−|υ|2 )dυ, }
where we have used Perelman’s Jacobian comparison theorem (Theorem 3.2.7) in the first inequality. Hence the combination of (7.4.9), (7.4.11), (7.4.12) and (7.4.13) bounds ξ from below by a positive constant depending only on κm and rem . Therefore we have completed the proof of the lemma. Now we can prove the proposition.
Proof of Proposition 7.4.1. The proof of the proposition is by induction: having constructed our sequences for 1 ≤ j ≤ m, we make one more step, defining rem+1 , κm+1 , δem+1 , and redefining δem = δem+1 . In view of the previous lemma, we only need to define rem+1 and δem+1 . In Theorem 7.1.1 we have obtained the canonical neighborhood structure for smooth solutions. When adapting the arguments in the proof of Theorem 7.1.1 to the present surgical solutions, we will encounter the new difficulty of how to take a limit for the surgically modified solutions. The idea to overcome the difficulty consists of two parts. The first part, due to Perelman [104], is to choose δem and δem+1 small enough to push the surgical regions to infinity in space. (This is the reason why we need to redefine δem = δem+1 .) The second part is to show that solutions are smooth on some small, but uniform, time intervals (on compact subsets) so that we can apply Hamilton’s compactness theorem, since we only have curvature bounds;
440
H.-D. CAO AND X.-P. ZHU
otherwise Shi’s interior derivative estimate may not be applicable. In fact, the second part is more crucial. That is just concerned with the question of whether the surgery times accumulate or not. Our argument will use the canonical neighborhood characterization of the standard solution in Lemma 7.3.5. We now start to prove the proposition by contradiction. Suppose for sequence of 1 positive numbers rα and δeαβ , satisfying rα → 0 as α → ∞ and δeαβ ≤ α·β (→ 0), there
αβ exist sequences of solutions gij to the Ricci flow with surgery, where each of them has only a finite number of cutoff surgeries and has a compact orientable normalized three-manifold as initial data, so that the following two assertions hold: (i) each δ-cutoff at a time t ∈ [(m − 1)ε2 , (m + 1)ε2 ] satisfies δ ≤ δeαβ ; and (ii) the solutions satisfy the statement of the proposition on [0, mε2 ], but violate the canonical neighborhood assumption (with accuracy ε) with r = rα on [mε2 , (m + 1)ε2 ]. αβ For each solution gij , we choose t¯ (depending on α and β) to be the nearly first time for which the canonical neighborhood assumption (with accuracy ε) is violated. More precisely, we choose t¯ ∈ [mε2 , (m + 1)ε2 ] so that the canonical neighborhood assumption with r = rα and with accuracy parameter ε is violated at some (¯ x, t¯), however the canonical neighborhood assumption with accuracy parameter 2ε holds on t ∈ [mε2 , t¯]. After passing to subsequences, we may assume each δeαβ is less than the δe in Lemma 7.4.2 with r = rα when α is fixed. Then by Lemma 7.4.2 we have uniform κ-noncollapsing on all scales less than ε on [0, ¯t] with some κ > 0 independent of α, β. Slightly abusing notation, we will often drop the indices α and β. αβ Let geij be the rescaled solutions around (¯ x, t¯) with factors R(¯ x, t¯)(≥ r−2 → +∞) and shift the times t¯ to zero. We hope to take a limit of the rescaled solutions for subsequences of α, β → ∞ and show the limit is an orientable ancient κ-solution, which will give the desired contradiction. We divide our arguments into the following six steps. αβ e tˆ) ≤ A (for Step 1. Let (y, tˆ) be a point on the rescaled solution e gij with R(y, 2 some A ≥ 1) and tˆ ∈ [−(t¯ − (m − 1)ε )R(¯ x, t¯), 0]. Then we have estimate
(7.4.14)
e t) ≤ 10A R(x,
1 for those (x, t) in the parabolic neighborhood P (y, tˆ, 12 η −1 A− 2 , − 18 η −1 A−1 ) , et′ (y, 1 η −1 A− 21 ), t′ ∈ [tˆ − 1 η −1 A−1 , tˆ]}, for which the rescaled so{(x′ , t′ ) | x′ ∈ B 2 8 lution is defined. Indeed, as in the first step of the proof of Theorem 7.1.1, this follows directly from the gradient estimates (7.3.4) in the canonical neighborhood assumption with parameter 2ε.
Step 2. In this step, we will prove three time extension results. Assertion 1. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ and 0 ≤ B < 12 ε2 (rα )−2 − 81 η −1 C −1 , there is a β0 = β0 (ε, A, B, C) (independent of α) αβ e0 (¯ such that if β ≥ β0 and the rescaled solution e gij on the ball B x, A) is defined on a time interval [−b, 0] with 0 ≤ b ≤ B and the scalar curvature satisfies e t) ≤ C, R(x,
e0 (¯ on B x, A) × [−b, 0],
αβ e0 (¯ then the rescaled solution e gij on the ball B x, A) is also defined on the extended 1 −1 −1 time interval [−b − 8 η C , 0].
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
441
Before giving the proof, we make a simple observation: once a space point in the Ricci flow with surgery is removed by surgery at some time, then it never appears for later time; if a space point at some time t cannot be defined before the time t , then either the point lies in a gluing cap of the surgery at time t or the time t is the initial time of the Ricci flow. Proof of Assertion 1. Firstly we claim that there exists β0 = β0 (ε, A, B, C) such αβ e0 (¯ that when β ≥ β0 , the rescaled solution geij on the ball B x, A) can be defined before e0 (¯ the time −b (i.e., there are no surgeries interfering in B x, A) × [−b − ǫ′ , −b] for some ′ ǫ > 0). e0 (¯ We argue by contradiction. Suppose not, then there is some point x˜ ∈ B x, A) αβ such that the rescaled solution e gij at x ˜ cannot be defined before the time −b. By the above observation, there is a surgery at the time −b such that the point x ˜ lies in the instant gluing cap. 1 ˜ (= R(¯ Let h x, t¯) 2 h) be the cut-off radius at the time −b for the rescaled ˜ ≤ R(˜ ˜ e x, −b)− 21 ≤ Dh. solution. Clearly, there is a universal constant D such that D−1 h By Lemma 7.3.4 and looking at the rescaled solution at the time −b, the gluing 1 ˜ constitute a (δeαβ ) 2 -cap K. For any fixed cap and the adjacent δ-neck, of radius h, small positive constant δ ′ (much smaller than ε), we see that 1 −1 e e(−b) (˜ B x, (δ ′ ) R(˜ x, −b)− 2 ) ⊂ K
when β large enough. We first verify the following
˜ >0 Claim 1. For any small constants 0 < θ˜ < 1, δ ′ > 0, there exists a β(δ ′ , ε, θ) ˜ we have such that when β ≥ β(δ ′ , ε, θ), −1 ˜ αβ e(−b) (˜ (i) the rescaled solution geij over B x, (δ ′ ) h) is defined on the time interval 2 ˜ ˜ [−b, 0] ∩ [−b, −b + (1 − θ)h ]; 1 −1 ˜ e(−b) (˜ (ii) the ball B x, (δ ′ ) h) in the (δeαβ ) 2 -cap K evolved by the Ricci flow on ˜h ˜ 2 ] is, after scaling with factor h ˜ −2 , the time interval [−b, 0] ∩ [−b, −b + (1 − θ) δ ′ -close (in the C [δ solution.
′−1
]
topology) to the corresponding subset of the standard
This claim essentially follows from Lemma 7.3.6. Indeed, suppose there is a ˜h ˜ 2 ] which removes some point surgery at some time t˜˜ ∈ [−b, 0] ∩ (−b, −b + (1 − θ) −1 ˜ ˜ We assume t˜ ∈ (−b, 0] is the first time with that property. e(−b) (˜ ˜˜ ∈ B x x, (δ ′ ) h). ¯ ′ , ε, θ) ˜ such that if δeαβ < δ, ¯ then the ball Then by Lemma 7.3.6, there is a δ¯ = δ(δ 1
−1 ˜ e(−b) (˜ B x, (δ ′ ) h) in the (δeαβ ) 2 -cap K evolved by the Ricci flow on the time interval ˜ −2 , δ ′ -close to the corresponding subset of the [−b, ˜t˜) is, after scaling with factor h −1 ˜ e(−b) (˜ standard solution. Note that the metrics for times in [−b, t˜˜) on B x, (δ ′ ) h) are −1 ˜ keeps looking like a cap e(−b) (˜ equivalent. By Lemma 7.3.6, the solution on B x, (δ ′ ) h) for t ∈ [−b, ˜˜t). On the other hand, by the definition, the surgery is always done along the middle two-sphere of a δ-neck with δ < δeαβ . Then for β large, all the points in −1 ˜ e(−b) (˜ B x, (δ ′ ) h) are removed (as a part of a capped horn) at the time t˜˜. But x˜ (near the tip of the cap) exists past the time t˜˜. This is a contradiction. Hence we have proved ˜ is defined on the time interval [−b, 0] ∩ [−b, −b + (1 − θ) ˜h ˜ 2 ]. e(−b) (˜ that B x, (δ ′ )−1 h)
442
H.-D. CAO AND X.-P. ZHU
˜h ˜ 2 ]) e(−b) (˜ The δ ′ -closeness of the solution on B x, (δ ′ ) h)×([−b, 0]∩[−b, −b+(1− θ) with the corresponding subset of the standard solution follows from Lemma 7.3.6. Then we have proved Claim 1. We next verify the following −1
˜ ˜h ˜ 2 when β large. Claim 2. There is θ˜ = θ(CB), 0 < θ˜ < 1, such that b ≤ (1 − θ) Note from Proposition 7.3.3, there is a universal constant D′ > 0 such that the standard solution satisfies the following curvature estimate R(y, s) ≥
2D′ . 1−s
We choose θ˜ = D′ /2(D′ + CB). Then for β large enough, the rescaled solution satisfies (7.4.15)
e t) ≥ R(x,
D′ ˜ −2 1 − (t + b)h
˜ −2 h
−1 ˜ ˜h ˜ 2 ]). e(−b) (˜ on B x, (δ ′ ) h) × ([−b, 0] ∩ [−b, −b + (1 − θ) ˜h ˜ 2 . Then by combining with the assumption R(˜ e x, t) ≤ C for Suppose b ≥ (1 − θ) 2 ˜ ˜ t = (1 − θ)h − b, we have
C≥
D′ ˜ −2 , h ˜ −2 1 − (t + b)h
and then ′ ˜ 1+ D . 1 ≥ (1 − θ) CB This is a contradiction. Hence we have proved Claim 2. The combination of the above two claims shows that there is a positive constant ˜ ˜ such 0 < θ˜ = θ(CB) < 1 such that for any small δ ′ > 0, there is a positive β(δ ′ , ε, θ) ′ 2 ˜ ˜ ˜ that when β ≥ β(δ , ε, θ), we have b ≤ (1 − θ)h and the rescaled solution in the ball ˜ on the time interval [−b, 0] is, after scaling with factor h ˜ −2 , δ ′ -close e(−b) (˜ B x, (δ ′ )−1 h) ′ −1 ( in the C [(δ ) ] topology) to the corresponding subset of the standard solution. e ≤ C on B e0 (¯ By (7.4.15) and the assumption R x, A) × [−b, 0], we know that the ˜ cut-off radius h at the time −b for the rescaled solution satisfies r D′ ˜ h≥ . C Let δ ′ > 0 be much smaller than ε and min{A−1 , A}. Since d˜0 (˜ x, x ¯) ≤ A, it follows ˜ depending only on θ˜ such that d˜(−b) (˜ ˜ ≪ that there is constant C(θ) x, x ¯) ≤ C(θ)A ′ −1 ˜ (δ ) h. We now apply Lemma 7.3.5 with the accuracy parameter ε/2. Let C(ε/2) be the positive constant in Lemma 7.3.5. Without loss of generality, we may assume the positive constant C1 (ε) in the canonical neighborhood assumption is larger than 4C(ε/2). When δ ′ (> 0) is much smaller than ε and min{A−1 , A}, the point x¯ at the time t¯ has a neighborhood which is either a 43 ε-cap or a 34 ε-neck. Since the canonical neighborhood assumption with accuracy parameter ε is violated at (¯ x, t¯), the neighborhood of the point x¯ at the new time zero for the rescaled
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
443
solution must be a 34 ε-neck. By Lemma 7.3.5 (b), we know the neighborhood is the slice at the time zero of the parabolic neighborhood 4 e x, 0)− 21 , − min{R(¯ e x, 0)−1 , b}) P (¯ x, 0, ε−1 R(¯ 3
e x, 0) = 1) which is 3 ε-close (in the C [ 3 ε ] topology) to the corresponding (with R(¯ 4 subset of the evolving standard cylinder S2 × R over the time interval [− min{b, 1}, 0] with scalar curvature 1 at the time zero. If b ≥ 1, the 34 ε-neck is strong, which is a contradiction. While if b < 1, the 34 ε-neck at time −b is contained in the union of the gluing cap and the adjacent δ-neck where the δ-cutoff surgery took place. Since ε is small (say ε < 1/100), it is clear that the point x¯ at time −b is the center of an ε-neck which is entirely contained in the adjacent δ-neck. By the proof of Lemma 7.3.2, the adjacent δ-neck approximates an ancient κ-solution. This implies the point x ¯ at the time t¯ has a strong ε-neck, which is also a contradiction. Hence we have proved that there exists β0 = β0 (ε, A, B, C) such that when β ≥ β0 , e0 (¯ the rescaled solution on the ball B x, A) can be defined before the time −b. αβ αβ Let [tA , 0] ⊃ [−b, 0] be the largest time interval so that the rescaled solution geij αβ 1 −1 −1 e0 (¯ can be defined on B x, A) × [tαβ C for A , 0]. We finally claim that tA ≤ −b − 8 η β large enough. Indeed, suppose not, by the gradient estimates as in Step 1, we have the curvature estimate 4 −1
e t) ≤ 10C R(x,
e0 (¯ on B x, A) × [tαβ A , −b]. Hence we have the curvature estimate e t) ≤ 10C R(x,
e0 (¯ on B x, A) × [tαβ By the above argument there is a β0 = β0 (ε, A, B + A , 0]. 1 −1 −1 e0 (¯ C , 10C) such that for β ≥ β0 , the solution in the ball B x, A) can be de8η αβ fined before the time tA . This is a contradiction. Therefore we have proved Assertion 1. Assertion 2. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞ and 1 −1 0 < B < 21 ε2 (rα )−2 − 50 η , there is a β0 = β0 (ε, A, B, C) (independent of α) such αβ e0 (¯ that if β ≥ β0 and the rescaled solution e gij on the ball B x, A) is defined on a time 1 −1 ′ ′ and the scalar curvature interval [−b + ǫ , 0] with 0 < b ≤ B and 0 < ǫ < 50 η satisfies e t) ≤ C R(x,
e0 (¯ on B x, A) × [−b + ǫ′ , 0],
e0 (¯ e −b + ǫ′ ) ≤ 3 , then the rescaled and there is a point y ∈ B x, A) such that R(y, 2 αβ 1 −1 η , 0] and solution geij at y is also defined on the extended time interval [−b − 50 satisfies the estimate for t ∈ [−b −
1 −1 , −b + ǫ′ ]. 50 η
e t) ≤ 15 R(y,
Proof of Assertion 2. We imitate the proof of Assertion 1. If the rescaled solution αβ 1 −1 η , −b + ǫ′ ), then there is a geij at y cannot be defined for some time in [−b − 50
444
H.-D. CAO AND X.-P. ZHU
1 −1 surgery at some time t˜˜ ∈ [−b − 50 η , −b + ǫ′ ] such that y lies in the instant gluing 1 ˜ (= R(¯ cap. Let h x, t¯) 2 h) be the cutoff radius at the time t˜˜ for the rescaled solution. ˜ By ˜ ≤ R(y, e t˜˜)− 21 ≤ Dh. Clearly, there is a universal constant D > 1 such that D−1 h the gradient estimates as in Step 1, the cutoff radius satisfies
˜ ≥ D−1 15− 21 . h
As in Claim 1 (i) in the proof of Assertion 1, for any small constants 0 < θ˜ < 12 , ˜ > 0 such that for β ≥ β(δ ′ , ε, θ), ˜ there is no surgery δ ′ > 0, there exists a β(δ ′ , ε, θ) ˜ ˜ ˜ ′ −1 2 ˜ × ([t˜, (1 − θ) ˜h ˜ + t˜] ∩ (t˜, 0]). Without loss of generality, e˜(y, (δ ) h) interfering in B t˜ we may assume that the universal constant η is much larger than D. Then we have ˜h ˜ 2 + t˜ ˜ > −b + 1 η −1 . As in Claim 2 in the proof of Assertion 1, we can use (1 − θ) 50 ˜ ˜h ˜ 2 + t˜˜ ≥ 0; the curvature bound assumption to choose θ˜ = θ(B, C) such that (1 − θ) otherwise C≥
D′ θ˜˜h2
for some universal constant D′ > 1, and |t˜˜ + b| ≤
1 −1 η , 50
which implies D′ ˜ 1+ 1 ≥ (1 − θ) 1 −1 C B + 50 η
!
.
1 −1 This is a contradiction if we choose θ˜ = D′ /2(D′ + C(B + 50 η )). ˜ ˜ So there is a positive constant 0 < θ = θ(B, C) < 1 such that for any δ ′ > 0, ˜ such that when β ≥ β(δ ′ , ε, θ), ˜ we have −t˜˜ ≤ (1 − θ) ˜h ˜2 there is a positive β(δ ′ , ε, θ) ˜ ′ −1 ˜ e and the solution in the ball Bt˜˜(˜ x, (δ ) h) on the time interval [t˜, 0] is, after scaling ′−1 −2 ′ ˜ with factor h , δ -close (in the C [δ ] topology) to the corresponding subset of the standard solution. Then exactly as in the proof of Assertion 1, by using the canonical neighborhood structure of the standard solution in Lemma 7.3.5, this gives the desired contradiction with the hypothesis that the canonical neighborhood assumption with accuracy parameter ε is violated at (¯ x, t¯), for β sufficiently large. The curvature estimate at the point y follows from Step 1. Therefore the proof of Assertion 2 is complete. Note that the standard solution satisfies R(x1 , t) ≤ D′′ R(x2 , t) for any t ∈ [0, 12 ] and any two points x1 , x2 , where D′′ ≥ 1 is a universal constant.
Assertion 3. For arbitrarily fixed α, 0 < A < +∞, 1 ≤ C < +∞, there is a 1 β0 = β0 (ε, AC 2 ) such that if any point (y0 , t0 ) with 0 ≤ −t0 < 21 ε2 (rα )−2 − 18 η −1 C −1 αβ e 0 , t0 ) ≤ C , then either the rescaled of the rescaled solution e gij for β ≥ β0 satisfies R(y 1 −1 −1 η C , t0 ] and the rescaled scalar solution at y0 can be defined at least on [t0 − 16 curvature satisfies i h e 0 , t) ≤ 10C for t ∈ t0 − 1 η −1 C −1 , t0 , R(y 16
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
445
or we have e 1 , t0 ) ≤ 2D′′ R(x e 2 , t0 ) R(x
et0 (y0 , A), where D′′ is the above universal constant. for any two points x1 , x2 ∈ B
αβ Proof of Assertion 3. Suppose the rescaled solution geij at y0 cannot be defined 1 −1 −1 for some t ∈ [t0 − 16 η C , t0 ); then there is a surgery at some time t˜ ∈ [t0 − 1 1 −1 −1 ˜ (= R(¯ C , t0 ] such that y0 lies in the instant gluing cap. Let h x, t¯) 2 h) be the 16 η αβ cutoff radius at the time t˜ for the rescaled solution geij . By the gradient estimates as in Step 1, the cutoff radius satisfies
˜ ≥ D−1 10− 21 C − 12 , h
where D is the universal constant in the proof of the Assertion 1. Since we assume η ˜ 2 + t˜ > t0 . As in Claim 1 (ii) in the is suitably larger than D as before, we have 12 h ′ proof of Assertion 1, for arbitrarily small δ > 0, we know that for β large enough the ˜ −2 , δ ′ -close e˜(y0 , (δ ′ )−1 ˜h) × [t˜, t0 ] is, after scaling with factor h rescaled solution on B t ′ −1 [(δ ) ] (in the C topology) to the corresponding subset of the standard solution. Since ˜ ≫ A for β large enough, Assertion 3 follows from the curvature estimate of (δ ′ )−1 h standard solution in the time interval [0, 21 ]. Step 3. For any subsequence (αk , βk ) of (α, β) with rαk → 0 and δ αk βk → 0 as k → ∞, we next argue as in the second step of the proof of Theorem 7.1.1 to αk βk show that the curvatures of the rescaled solutions g˜ij at the new times zero (after shifting) stay uniformly bounded at bounded distances from x ¯ for all sufficiently large k. More precisely, we will prove the following assertion: αk βk Assertion 4. Given any subsequence of the rescaled solutions g˜ij with rαk → αk βk 0 and δ → 0 as k → ∞, then for any L > 0, there are constants C(L) > 0 and αk βk k(L) such that the rescaled solutions g˜ij satisfy ˜ 0) ≤ C(L) for all points x with d˜0 (x, x¯) ≤ L and all k ≥ 1; (i) R(x, ˜0 (¯ (ii) the rescaled solutions over the ball B x, L) are defined at least on the time 1 −1 η C(L)−1 , 0] for all k ≥ k(L). interval [− 16
Proof of Assertion 4. For each ρ > 0, set n o ˜ 0) | k ≥ 1 and d˜0 (x, x¯) ≤ ρ in the rescaled solutions g˜αk βk M (ρ) = sup R(x, ij
and
ρ0 = sup{ρ > 0 | M (ρ) < +∞}. Note that the estimate (7.4.14) implies that ρ0 > 0. For (i), it suffices to prove ρ0 = +∞. We argue by contradiction. Suppose ρ0 < +∞. Then there is a sequence of αk βk ˜ 0) → +∞. points y in the rescaled solutions g˜ij with d˜0 (¯ x, y) → ρ0 < +∞ and R(y, ˜0 (¯ Denote by γ a minimizing geodesic segment from x ¯ to y and denote by B x, ρ0 ) the αk βk open geodesic ball centered at x ¯ of radius ρ0 on the rescaled solution g˜ij . First, we claim that for any 0 < ρ < ρ0 with ρ near ρ0 , the rescaled solutions 1 −1 ˜0 (¯ on the balls B x, ρ) are defined on the time interval [− 16 η M (ρ)−1 , 0] for all large
446
H.-D. CAO AND X.-P. ZHU
k. Indeed, this follows from Assertion 3 or Assertion 1. For the later purpose in Step 6, we now present an argument by using Assertion 3. If the claim is not true, 1 −1 η M (ρ)−1 , 0] such that some point then there is a surgery at some time t˜ ∈ [− 16 ˜0 (¯ y˜ ∈ B x, ρ) lies in the instant gluing cap. We can choose sufficiently small δ ′ > 0 1 ˜ where h ˜ ≥ D−1 20− 12 M (ρ)− 21 is the cutoff radius of the such that 2ρ0 < (δ ′ )− 2 h, rescaled solutions at t˜. By applying Assertion 3 with (˜ y , 0) = (y0 , t0 ), we see that there is a k(ρ0 , M (ρ)) > 0 such that when k ≥ k(ρ0 , M (ρ)), e 0) ≤ 2D′′ R(x,
e0 (¯ for all x ∈ B x, ρ). This is a contradiction as ρ → ρ0 . Since for each fixed 0 < ρ < ρ0 with ρ near ρ0 , the rescaled solutions are defined on 1 −1 ˜0 (¯ η M (ρ)−1 , 0] for all large k, by Step 1 and Shi’s derivative estimate, B x, ρ) × [− 16 we know that the covariant derivatives and higher order derivatives of the curvatures 1 −1 ˜0 (¯ on B x, ρ − (ρ02−ρ) ) × [− 32 η M (ρ)−1 , 0] are also uniformly bounded. By the uniform κ-noncollapsing property and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked seαk βk ∞ ˜0 (¯ quence (B x, ρ0 ), geij ,x ¯) converges in the Cloc topology to a marked (noncomplete) ∞ manifold (B∞ , e gij , x ¯) and the geodesic segments γ converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from x ¯. Clearly, the limit has nonnegative sectional curvature by the pinching assumption. Consider a tubular neighborhood along γ∞ defined by [ e∞ (q0 ))− 21 ), V = B∞ (q0 , 4π(R q0 ∈γ∞
e∞ denotes the scalar curvature of the limit and where R e∞ (q0 ))− 12 ) B∞ (q0 , 4π(R
e∞ (q0 ))− 21 . Let B ¯∞ denote the is the ball centered at q0 ∈ B∞ with the radius 4π(R ∞ ¯ completion of (B∞ , e gij ), and y∞ ∈ B∞ the limit point of γ∞ . Exactly as in the second step of the proof of Theorem 7.1.1, it follows from the canonical neighborhood ∞ assumption with accuracy parameter 2ε that the limiting metric geij is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ and then the metric space V¯ = V ∪ {y∞ } by adding the point y∞ has nonnegative curvature in the Alexandrov sense. Consequently we have a three-dimensional non-flat tangent cone Cy∞ V¯ at y∞ which is a metric cone with aperture ≤ 20ε. On the other hand, note that by the canonical neighborhood assumption, the canonical 2ε-neck neighborhoods are strong. Thus at each point q ∈ V near y∞ , the ∞ limiting metric e gij actually exists on the whole parabolic neighborhood V
\
P
1 e∞ (q))− 12 , − 1 η −1 (R e∞ (q))−1 , q, 0, η −1 (R 3 10
and is a smooth solution of the Ricci flow there. Pick z ∈ Cy∞ V¯ with distance one from the vertex y∞ and it is nonflat around z. By definition the ball B(z, 12 ) ⊂ Cy∞ V¯ is the Gromov-Hausdorff convergent limit of the scalings of a sequence of balls ∞ ∞ B∞ (zℓ , σℓ )(⊂ (V, e gij )) where σℓ → 0. Since the estimate (7.4.14) survives on (V, e gij ) for all A < +∞, and the tangent cone is three-dimensional and nonflat around z,
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
447
∞ we see that this convergence is actually in the Cloc topology and over some ancient 1 time interval. Since the limiting B∞ (z, 2 )(⊂ Cy∞ V¯ ) is a piece of nonnegatively curved nonflat metric cone, we get a contradiction with Hamilton’s strong maximum principle (Theorem 2.2.1) as before. So we have proved ρ0 = ∞. This proves (i). By the same proof of Assertion 1 in Step 2, we can further show that for any L, ˜0 (¯ the rescaled solutions on the balls B x, L) are defined at least on the time interval 1 −1 [− 16 η C(L)−1 , 0] for all sufficiently large k. This proves (ii).
Step 4. For any subsequence (αk , βk ) of (α, β) with rαk → 0 and δeαk βk → 0 as k → ∞, by Step 3, the κ-noncollapsing property and Hamilton’s compactness αk βk ∞ theorem, we can extract a Cloc convergent subsequence of g˜ij over some space-time open subsets containing the slice {t = 0}. We now want to show any such limit has bounded curvature at t = 0. We prove by contradiction. Suppose not, then there is a sequence of points zℓ divergent to infinity in the limiting metric at time zero with curvature divergent to infinity. Since the curvature at zℓ is large (comparable to one), zℓ has a canonical neighborhood which is a 2ε-cap or strong 2ε-neck. Note that the boundary of 2ε-cap lies in some 2ε-neck. So we get a sequence of 2ε-necks with radius going to zero. Note also that the limit has nonnegative sectional curvature. Without loss of generality, we may assume 2ε < ε0 , where ε0 is the positive constant in Proposition 6.1.1. Thus this arrives at a contradiction with Proposition 6.1.1. Step 5. In this step, we will choose some subsequence (αk , βk ) of (α, β) so that αk βk we can extract a complete smooth limit of the rescaled solutions geij to the Ricci flow with surgery on a time interval [−a, 0] for some a > 0. Choose αk , βk → ∞ so that rαk → 0, δeαk βk → 0, and Assertion 1, 2, 3 hold with α = αk , β = βk for all A ∈ {p/q | p, q = 1, 2, . . . , k}, and B, C ∈ {1, 2, . . . , k}. By Step αk βk ∞ 3, we may assume the rescaled solutions geij converge in the Cloc topology at the time t = 0. Since the curvature of the limit at t = 0 is bounded by Step 4, it follows from Assertion 1 in Step 2 and the choice of the sequence (αk , βk ) that the limiting ∞ (M∞ , geij (·, t)) is defined at least on a backward time interval [−a, 0] for some positive constant a and is a smooth solution to the Ricci flow there.
Step 6. We further want to extend the limit in Step 5 backwards in time to αk βk infinity to get an ancient κ-solution. Let e gij be the convergent sequence obtained in the above Step 5. Denote by n tmax = sup t′ |we can take a smooth limit on (−t′ , 0] (with bounded curvature at each time slice) from a subsequence of o αk βk the rescaled solutions geij .
αk βk We first claim that there is a subsequence of the rescaled solutions geij which con∞ ∞ verges in the Cloc topology to a smooth limit (M∞ , e gij (·, t)) on the maximal time interval (−tmax , 0]. Indeed, let tℓ be a sequence of positive numbers such that tℓ → tmax and there exist smooth limits (M∞ , geℓ∞ (·, t)) defined on (−tℓ , 0]. For each ℓ, the limit has nonnegative sectional curvature and has bounded curvature at each time slice. Moreover by the gradient estimate in canonical neighborhood assumption with accuracy parameter 2ε, e the limit has bounded curvature on each subinterval [−b, 0] ⊂ (−tℓ , 0]. Denote by Q
448
H.-D. CAO AND X.-P. ZHU
e is independent of ℓ). the scalar curvature upper bound of the limit at time zero (Q Then we can apply Li-Yau-Hamilton inequality (Corollary 2.5.5) to get eℓ∞ (x, t) ≤ R
tℓ e Q, t + tℓ
e∞ (x, t) are the scalar curvatures of the limits (M∞ , ge∞ (·, t)). Hence by the where R ℓ ℓ definition of convergence and the above curvature estimates, we can find a subsequence αk βk ∞ of the rescaled solutions e gij which converges in the Cloc topology to a smooth limit ∞ (M∞ , geij (·, t)) on the maximal time interval (−tmax , 0]. We need to show −tmax = −∞. Suppose −tmax > −∞, there are only the following two possibilities: either ∞ (1) The curvature of the limiting solution (M∞ , geij (·, t)) becomes unbounded as t ց −tmax ; or (2) For each small constant θ > 0 and each large integer k0 > 0, there is some k ≥ αk βk k0 such that the rescaled solution geij has a surgery time Tk ∈ [−tmax − θ, 0] and a surgery point xk lying in a gluing cap at the times Tk so that d2Tk (xk , x¯) is uniformly bounded from above by a constant independent of θ and k0 . We next claim that the possibility (1) always occurs. Suppose not; then the ∞ curvature of the limiting solution (M∞ , e gij (·, t)) is bounded on M∞ × (−tmax , 0] by ˆ some positive constant C. In particular, for any A > 0, there is a sufficiently large αk βk integer k1 > 0 such that any rescaled solution geij with k ≥ k1 on the geodesic 1 −1 ˆ −1 e η C , 0] and its scalar ball B0 (¯ x, A) is defined on the time interval [−tmax + 50 curvature is bounded by 2Cˆ there. (Here, without loss of generality, we may assume 1 −1 ˆ −1 that the upper bound Cˆ is so large that −tmax + 50 η C < 0.) By Assertion 1 in αk βk e0 (¯ Step 2, for k large enough, the rescaled solution e gij over B x, A) can be defined 1 −1 ˆ −1 on the extended time interval [−tmax − 50 η C , 0] and has the scalar curvature 1 −1 ˆ −1 e ≤ 10C ˆ on B e0 (¯ R x, A) × [−tmax − 50 η C , 0]. So we can extract a smooth limit from the sequence to get the limiting solution which is defined on a larger time interval 1 −1 ˆ −1 [−tmax − 50 η C , 0]. This contradicts the definition of the maximal time −tmax . It remains to exclude the possibility (1). By using Li-Yau-Hamilton inequality (Corollary 2.5.5) again, we have e∞ (x, t) ≤ R
tmax e Q. t + tmax
So we only need to control the curvature near −tmax . Exactly as in Step 4 in the proof of Theorem 7.1.1, it follows from Li-Yau-Hamilton inequality that q ˜ ˜ ˜ e (7.4.16) d0 (x, y) ≤ dt (x, y) ≤ d0 (x, y) + 30tmax Q
for any x, y ∈ M∞ and t ∈ (−tmax , 0]. Since the infimum of the scalar curvature is nondecreasing in time, we have some 1 −1 e∞ (y∞ , t∞ ) < η such that R point y∞ ∈ M∞ and some time −tmax < t∞ < −tmax + 50 e0 > 0 such that d˜t (¯ e0 /2 for all 5/4. By (7.4.16), there is a constant A x , y∞ ) ≤ A t ∈ (−tmax , 0]. αk βk Now we come back to the rescaled solution e gij . Clearly, for arbitrarily given ′ small ǫ > 0, when k large enough, there is a point yk in the underlying manifold of αk βk geij at time 0 satisfying the following properties (7.4.17)
e k , t∞ ) < 3 , R(y 2
e0 det (¯ x , yk ) ≤ A
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
449
for t ∈ [−tmax + ǫ′ , 0]. By the definition of convergence, we know that for any fixed e0 , for k large enough, the rescaled solution over B e0 (¯ A0 ≥ 2A x, A0 ) is defined on the time interval [t∞ , 0] and satisfies e e t) ≤ 2tmax Q R(x, t + tmax
e0 (¯ on B x, A0 ) × [t∞ , 0]. Then by Assertion 2 of Step 2, we have proved that there is a αk βk sufficiently large integer k¯0 such that when k ≥ k¯0 , the rescaled solutions geij at yk 1 −1 can be defined on [−tmax − 50 η , 0], and satisfy e k , t) ≤ 15 R(y
1 −1 η , t∞ ]. for t ∈ [−tmax − 50 We now prove a statement analogous to Assertion 4 (i) of Step 3. αk βk Assertion 5. For the above rescaled solutions geij and k¯0 , we have that for αk βk any L > 0, there is a positive constant ω(L) such that the rescaled solutions e gij satisfy
e t) ≤ ω(L) R(x,
for all (x, t) with d˜t (x, yk ) ≤ L and t ∈ [−tmax −
1 −1 , t∞ ], 50 η
and for all k ≥ k¯0 .
Proof of Assertion 5. We slightly modify the argument in the proof of Assertion 4 (i). Let n e t)|d˜t (x, yk ) ≤ ρ and t ∈ [−tmax − 1 η −1 , t∞ ] M (ρ) = sup R(x, 50 o αk βk in the rescaled solutions e g , k ≥ k¯0 ij
and
ρ0 = sup{ρ > 0 | M (ρ) < +∞}. Note that the estimate (7.4.14) implies that ρ0 > 0. We only need to show ρ0 = +∞. We argue by contradiction. Suppose ρ0 < +∞. Then, after passing to αk βk a subsequence, there is a sequence (˜ yk , tk ) in the rescaled solutions e gij with 1 −1 ˜ e yk , tk ) → +∞. Detk ∈ [−tmax − 50 η , t∞ ] and dtk (yk , y˜k ) → ρ0 < +∞ such that R(˜ note by γk a minimizing geodesic segment from yk to y˜k at the time tk and denote by et (yk , ρ0 ) the open geodesic ball centered at yk of radius ρ0 on the rescaled solution B k αk βk geij (·, tk ). For any 0 < ρ < ρ0 with ρ near ρ0 , by applying Assertion 3 as before, we get et (yk , ρ) are defined on the time interval that the rescaled solutions on the balls B k 1 −1 −1 [tk − 16 η M (ρ) , tk ] for all large k. By Step 1 and Shi’s derivative estimate, we further know that the covariant derivatives and higher order derivatives of the curvaet (yk , ρ− (ρ0 −ρ) )×[tk − 1 η −1 M (ρ)−1 , tk ] are also uniformly bounded. Then tures on B k 2 32 by the uniform κ-noncollapsing property and Hamilton’s compactness theorem (Theorem 4.1.5), after passing to a subsequence, we can assume that the marked sequence αk βk ∞ ˜t (yk , ρ0 ), e (B gij (·, tk ), yk ) converges in the Cloc topology to a marked (noncomplete) k
450
H.-D. CAO AND X.-P. ZHU
∞ manifold (B∞ , e gij , y∞ ) and the geodesic segments γk converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from y∞ . Clearly, the limit also has nonnegative sectional curvature by the pinching assumption. Then by repeating the same argument as in the proof of Assertion 4 (i) in the rest, we derive a contradiction with Hamilton’s strong maximum principle. This proves Assertion 5. We then apply the second estimate of (7.4.17) and Assertion 5 to conclude that for any large constant 0 < A < +∞, there is a positive constant C(A) such that for αk βk any small ǫ′ > 0, the rescaled solutions geij satisfy
(7.4.18)
e t) ≤ C(A), R(x,
e0 (¯ for all x ∈ B x, A) and t ∈ [−tmax + ǫ′ , 0], and for all sufficiently large k. Then by αk βk applying Assertion 1 in Step 2, we conclude that the rescaled solutions e gij on the e geodesic balls B0 (¯ x, A) are also defined on the extended time interval [−tmax + ǫ′ − 1 −1 −1 C(A) , 0] for all sufficiently large k. Furthermore, by the gradient estimates as 8η in Step 1, we have e t) ≤ 10C(A), R(x,
e0 (¯ for x ∈ B x, A) and t ∈ [−tmax + ǫ′ − 18 η −1 C(A)−1 , 0]. Since ǫ′ > 0 is arbitrarily small and the positive constant C(A) is independent of ǫ′ , we conclude that the αk βk e0 (¯ rescaled solutions e gij on B x, A) are defined on the extended time interval [−tmax − 1 −1 −1 C(A) , 0] and satisfy 16 η (7.4.19)
e t) ≤ 10C(A), R(x,
1 −1 e0 (¯ for x ∈ B x, A) and t ∈ [−tmax − 16 η C(A)−1 , 0], and for all sufficiently large k. αk βk Now, by taking convergent subsequences from the rescaled solutions geij , we see that the limit solution is defined smoothly on a space-time open subset of M∞ × (−∞, 0] containing M∞ × [−tmax , 0]. By Step 4, we see that the limiting metric ∞ geij (·, −tmax ) at time −tmax has bounded curvature. Then by combining with the canonical neighborhood assumption of accuracy 2ε, we conclude that the curvature of the limit is uniformly bounded on the time interval [−tmax , 0]. So we have excluded the possibility (1). Hence we have proved a subsequence of the rescaled solutions converges to an orientable ancient κ-solution. Finally by combining with the canonical neighborhood theorem (Theorem 6.4.6), we see that (¯ x, t¯) has a canonical neighborhood with parameter ε, which is a contradiction. Therefore we have completed the proof of the proposition. Summing up, we have proved that for any ε > 0, (without loss of generality, we e may assume ε ≤ ε¯0 ), there exist nonincreasing (continuous) positive functions δ(t) and re(t), defined on [0, +∞) with 1 e ¯ δ(t) ≤ δ(t) = min , δ0 , 2e2 log(1 + t)
e such that for arbitrarily given (continuous) positive function δ(t) with δ(t) < δ(t) on [0, +∞), and arbitrarily given a compact orientable normalized three-manifold as
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
451
initial data, the Ricci flow with surgery has a solution on [0, T ) obtained by evolving the Ricci flow and by performing δ-cutoff surgeries at a sequence of times 0 < t1 < e i ) at each time ti , so that the pinching t2 < · · · < ti < · · · < T , with δ(ti ) ≤ δ ≤ δ(t assumption and the canonical neighborhood assumption (with accuracy ε) with r = re(t) are satisfied. (At this moment we still do not know whether the surgery times ti are discrete.) Since the δ-cutoff surgeries occur at the points lying deeply in the ε-horns, the minimum of the scalar curvature Rmin (t) of the solution to the Ricci flow with surgery at each time-slice is achieved in the region unaffected by the surgeries. Thus we know from the evolution equation of the scalar curvature that d 2 2 Rmin (t) ≥ Rmin (t). dt 3 In particular, the minimum of the scalar curvature Rmin (t) is nondecreasing in time. Also note that each δ-cutoff surgery decreases volume. Then the upper derivative of the volume in time satisfies ¯ d V (t + △t) − V (t) V (t) , lim sup dt △t △t→0
(7.4.20)
≤ −Rmin (0)V (t)
which implies that V (t) ≤ V (0)e−Rmin (0)t . On the other hand, by Lemma 7.3.2 and the δ-cutoff procedure given in the previous section, we know that at each time ti , each δ-cutoff surgery cuts down the volume at least at an amount of h3 (ti ) with h(ti ) depending only on δ(ti ) and re(ti ). Thus the surgery times ti cannot accumulate in any finite interval. When the solution becomes extinct at some finite time T , the solution at time near T is entirely covered by canonical neighborhoods and then the initial manifold is diffeomorphic to a connected sum of a finite copies of S2 ×S1 and S3 /Γ (the metric quotients of round three-sphere). So we have proved the following long-time existence result which was proposed by Perelman in [104]. Theorem 7.4.3 (Long-time existence theorem). For any fixed constant ε > 0, e there exist nonincreasing (continuous) positive functions δ(t) and re(t), defined on [0, +∞), such that for an arbitrarily given (continuous) positive function δ(t) with e on [0, +∞), and arbitrarily given a compact orientable normalized threeδ(t) ≤ δ(t) manifold as initial data, the Ricci flow with surgery has a solution with the following properties: either (i) it is defined on a finite interval [0, T ) and obtained by evolving the Ricci flow and by performing a finite number of cutoff surgeries, with each δ-cutoff at a time t ∈ (0, T ) having δ = δ(t), so that the solution becomes extinct at the finite time T , and the initial manifold is diffeomorphic to a connected sum of a finite copies of S2 ×S1 and S3 /Γ (the metric quotients of round three-sphere) ; or (ii) it is defined on [0, +∞) and obtained by evolving the Ricci flow and by performing at most countably many cutoff surgeries, with each δ-cutoff at a time t ∈ [0, +∞) having δ = δ(t), so that the pinching assumption and the canonical neighborhood assumption (with accuracy ε) with r = re(t) are satisfied, and there exist at most a finite number of surgeries on every finite time interval.
452
H.-D. CAO AND X.-P. ZHU
In particular, if the initial manifold has positive scalar curvature, say R ≥ a > 0, then by (7.4.20), the solution becomes extinct at T ≤ 32 a. Hence we have the following topological description of compact three-manifolds with nonnegative scalar curvature which improves the well-known work of Schoen-Yau [109], [110]. Corollary 7.4.4 (Perelman [104]). Let M be a compact orientable threemanifold with nonnegative scalar curvature. Then either M is flat or it is diffeomorphic to a connected sum of a finite copies of S2 × S1 and S3 /Γ (the metric quotients of the round three-sphere). The famous Poincar´ e conjecture states that every compact three-manifold with trivial fundamental group is diffeomorphic to S3 . Developing tools to attack the conjecture formed the basis for much of the works in three-dimensional topology over the last one hundred years. Now we use the Ricci flow to discuss the Poincar´ e conjecture. Let M be a compact three-manifold with trivial fundamental group. In particular, the three-manifold M is orientable. Arbitrarily given a Riemannian metric on M , by scaling we may assume the metric is normalized. With this normalized metric as initial data, we consider the solution to the Ricci flow with surgery. If one can show the solution becomes extinct in finite time, it will follow from Theorem 7.4.3 (i) that the three-manifold M is diffeomorphic to the three-sphere S3 . Such finite extinction time result was first proposed by Perelman in [105], and recently, Colding-Minicozzi has published a proof to it in [42]. So the combination of Theorem 7.4.3 (i) and Colding-Minicozzi’s finite extinction result gives a complete proof of the Poincar´ e conjecture. We also remark that the above long-time existence result has been extended to compact four-manifolds with positive isotropic curvature by Chen and the second author in [34]. As a consequence it gave a complete proof of the following classification theorem of compact four-manifolds, with no essential incompressible space-form and with a metric of positive isotropic curvature. The theorem was first proved by Hamilton in ([64]), though it was later found that the proof contains some gaps (see for example the comment of Perelman in Page 1, the second paragraph, of [104]). Theorem 7.4.5. A compact four-manifold with no essential incompressible spaceform and with a metric of positive isotropic curvature is diffeomorphic to S4 , or RP4 , e 1 (the Z2 quotient of S3 × S1 where Z2 flips S3 antipodally and or S3 × S1 , or S3 ×S 1 0 rotates S by 180 ), or a connected sum of them. 7.5. Curvature Estimates for Surgically Modified Solutions. In this section we will generalize the curvature estimates for smooth solutions in Section 7.2 to that of solutions with cutoff surgeries. We first state and prove a version of Theorem 7.2.1. Theorem 7.5.1 (Perelman [104]). For any ε > 0 and 1 ≤ A < +∞, one can find κ = κ(A, ε) > 0, K1 = K1 (A, ε) < +∞, K2 = K2 (A, ε) < +∞ and r¯ = r¯(A, ε) > 0 such that for any t0 < +∞ there exists δ¯A = δ¯A (t0 ) > 0 (depending also on ε), nonincreasing in t0 , with the following property. Suppose we have a solution, constructed by Theorem 7.4.3 with the nonincreasing (continuous) positive functions e and re(t), to the Ricci flow with δ-cutoff surgeries on time interval [0, T ] and with δ(t) a compact orientable normalized three-manifold as initial data, where each δ-cutoff e on [0, T ] and δ = δ(t) ≤ δ¯A on [ t0 , t0 ]; assume at a time t satisfies δ = δ(t) ≤ δ(t) 2
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
453
that the solution is defined on the whole parabolic neighborhood P (x0 , t0 , r0 , −r02 ) , {(x, t) | x ∈ Bt (x0 , r0 ), t ∈ [t0 − r02 , t0 ]}, 2r02 < t0 , and satisfies |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ), and
Vol t0 (Bt0 (x0 , r0 )) ≥ A−1 r03 .
Then (i) the solution is κ-noncollapsed on all scales less than r0 in the ball Bt0 (x0 , Ar0 ); (ii) every point x ∈ Bt0 (x0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighbor1 hood B, with Bt0 (x, σ) ⊂ B ⊂ Bt0 (x, 2σ) for some 0 < σ < C1 (ε)R− 2 (x, t0 ), which is √ either a strong ε-neck or an ε-cap; (iii) if r0 ≤ r¯ t0 then R ≤ K2 r0−2 in Bt0 (x0 , Ar0 ). Here C1 (ε) is the positive constant in the canonical neighborhood assumption. Proof. Without loss of generality, we may assume 0 < ε ≤ ε¯0 , where ε¯0 is the sufficiently small (universal) positive constant in Lemma 7.4.2. (i) This is analog of no local collapsing theorem II (Theorem 3.4.2). In comparison with the no local collapsing theorem II, this statement gives κ-noncollapsing property no matter how big the time is and it also allows the solution to be modified by surgery. Let η(≥ 10) be the universal constant in the definition of the canonical neighborhood assumption. Recall that we had removed every component which has positive sectional curvature in our surgery procedure. By the same argument as in the first part of the proof of Lemma 7.4.2, the canonical neighborhood assumption of the so1 lution implies the κ-noncollapsing on the scales less than 2η re(t0 ) for some positive constant κ depending only on C1 (ε) and C2 (ε) (in the definition q of the canonical neighborhood assumption). So we may assume
scales ρ,
1 e(t0 ) 2η r
1 e(t0 ) 2η r
≤ r0 ≤
t0 2,
and study the
≤ ρ ≤ r0 . Let x ∈ Bt0 (x0 , Ar0 ) and assume that the solution satisfies |Rm| ≤ ρ−2
for those points in P (x, t0 , ρ, −ρ2 ) , {(y, t) | y ∈ Bt (x, ρ), t ∈ [t0 − ρ2 , t0 ]} for which the solution is defined. We want to bound the ratio Vol t0 (Bt0 (x, ρ))/ρ3 from below. Recall that a space-time curve is called admissible if it stays in the region unaffected by surgery, and a space-time curve on the boundary of the set of admissible curves is called a barely admissible curve. Consider any barely admissible curve γ, parametrized by t ∈ [tγ , t0 ], t0 − r02 ≤ tγ ≤ t0 , with γ(t0 ) = x. The same proof for the assertion (7.4.2) (in the proof of Lemma 7.4.2) shows that for arbitrarily large L > 0 ¯ t0 , re(t0 ), re( t0 ), ε) > 0 (to be determined later), one can find a sufficiently small δ(L, 2 t0 ¯ such that when each δ-cutoff in [ 2 , t0 ] satisfies δ ≤ δ(L, t0 , re(t0 ), re( t20 ), ε), there holds (7.5.1)
Z
t0
tγ
√
2 t0 − t(R+ (γ(t), t) + |γ(t)| ˙ )dt ≥ Lr0 .
From now on, we assume that each δ-cutoff of the solution in the time interval ¯ t0 , re(t0 ), re( t0 ), ε). [ t20 , t0 ] satisfies δ ≤ δ(L, 2
454
H.-D. CAO AND X.-P. ZHU
Let us scale the solution, still denoted by gij (·, t), to make r0 = 1 and the time as t0 = 1. By the maximum principle, it is easy to see that the (rescaled) scalar curvature satisfies R≥−
3 2t
on (0, 1]. Let us consider the time interval [ 12 , 1] and define a function of the form √ ¯ τ) + 2 τ) h(y, t) = φ(dt (x0 , y) − A(2t − 1))(L(y, where τ = 1 − t, φ is the function of one variable chosen in the proof of Theorem 1 1 1 ), rapidly increasing to infinity on ( 20 , 10 ), and 3.4.2 which is equal to one on (−∞, 20 (φ′ )2 ′′ ′ ¯ satisfies 2 φ − φ ≥ (2A + 300)φ − C(A)φ for some constant C(A) < +∞, and L is the function defined by n √ Z τ√ ¯ τ ) = inf 2 τ L(q, s(R + |γ| ˙ 2 )ds | (γ(s), s), s ∈ [0, τ ] 0 o is a space-time curve with γ(0) = x and γ(τ ) = q . Note that
(7.5.2)
√ ¯ τ) ≥ 2 τ L(y,
Z
τ
√
sRds
0
≥ −4τ 2 √ > −2 τ
since R ≥ −3 and 0 < τ ≤ 21 . This says h is positive for t ∈ [ 12 , 1]. Also note that (7.5.3)
∂ ¯ ¯≤6 L + △L ∂τ
¯ is achieved by admissible curves. Then as long as the shortest L-geodesics as long as L from (x0 , 0) to (y, τ ) are admissible, there holds at y and t = 1 − τ , √ ∂ ∂ ¯ + 2 τ) − △ h ≥ φ′ − △ dt − 2A − φ′′ · (L ∂t ∂t 1 ¯ − 6+ √ φ − 2h∇φ, ∇Li. τ Firstly, we may assume the constant L in (7.5.1) is not less than 2 exp(C(A) + 100). We claim that Lemma 3.4.1(i) is applicable for d = dt (·, x0 ) ¯ τ ) is achieved by admissible curves at y and t = 1 − τ (with τ ∈ [0, 12 ]) whenever L(y, and satisfies the estimate √ ¯ τ ) ≤ 3 τ exp(C(A) + 100). L(y, Indeed, since the solution is defined on the whole neighborhood P (x0 , t0 , r0 , −r02 ) with r0 = 1 and t0 = 1, the point x0 at the time t = 1 − τ lies on the region unaffected ¯ τ ) is achieved by admissible by surgery. Note that R ≥ −3 √ for t ∈ [ 21 , 1]. When L(y, ¯ τ ) ≤ 3 τ exp(C(A) + 100), the estimate (7.5.1) implies that curves and satisfies L(y,
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
455
the point y at the time t = 1 − τ does not lie in the collars of the gluing caps. Thus any minimal geodesic (with respect to the metric gij (·, t) with t = 1 − τ ) connecting x0 and y also lies in the region unaffected by surgery; otherwise the geodesic is not minimal. Then from the proof of Lemma 3.4.1(i), we see that it is applicable. Assuming the minimum of h at a time, say t = 1 − τ , is achieved at a point, ¯ τ ) is achieved by admissible curves and satisfies L(y, ¯ τ) ≤ say y, and assuming L(y, √ 3 τ exp(C(A) + 100), we have √ ¯ + 2 τ )∇φ = −φ∇L, ¯ (L and then by the computations and estimates in the proof of Theorem 3.4.2, ∂ −△ h ∂t √ (φ′ )2 ∂ ¯ + 2 τ ) − 6 + √1 φ − △ dt − 2A − φ′′ + 2 · (L ≥ φ′ ∂t φ τ h 1 √ , ≥ −C(A)h − 6 + √ τ (2 τ − 4τ 2 ) at y and t = 1 − τ . Here we used (7.5.2) and Lemma 3.4.1(i). As before, denoting by hmin (τ ) = minz h(z, 1 − τ ), we obtain √ d hmin (τ ) 6 τ +1 1 √ √ − (7.5.4) log ≤ C(A) + 2 dτ τ 2τ − 4τ τ 2τ 50 ≤ C(A) + √ , τ ¯ ≤ 3√τ exp(C(A)+ as long as the associated shortest L-geodesics are admissible with L 100). On the other hand, by definition, we have (7.5.5)
lim+
τ →0
hmin (τ ) √ ≤ φ(d1 (x0 , x) − A) · 2 = 2. τ
The combination of (7.5.4) and (7.5.5) gives the following assertion: ¯ s) | dt (x0 , y) ≤ A(2t − 1) + Let τ ∈ [0, 21 ]. If for each s ∈ [0, τ ], inf{L(y, with s = 1 − t} is achieved by admissible curves, then we have ¯ τ ) | dt (x0 , y) ≤ A(2t − 1) + 1 with τ = 1 − t inf L(y, (7.5.6) 10 √ ≤ 2 τ exp(C(A) + 100). 1 10
Note again that R ≥ −3 for t ∈ [ 21 , 1]. By combining with (7.5.1), we know that any barely admissible curve γ, parametrized by s ∈ [0, τ ], 0 ≤ τ ≤ 12 , with γ(0) = x, satisfies Z τ √ 7 s(R + |γ| ˙ 2 )ds ≥ exp(C(A) + 100), 4 0 by assuming L ≥ 2 exp(C(A) + 100).
456
H.-D. CAO AND X.-P. ZHU
1 re(t0 ) (and t0 = 1) Since |Rm| ≤ ρ−2 on P (x, t0 , ρ, −ρ2 ) with ρ ≥ 2η t0 ¯ and δ(L, t0 , re(t0 ), re( 2 ), ε) > 0 is sufficiently small, the parabolic neighborhood P (x, 1, ρ, −ρ2 ) around the point (x, 1) is contained in the region unaffected by the 1 ¯ can be bounded from inf L surgery. Thus as τ = 1 − t is sufficiently close to zero, 2√ τ ¯ τ ) | dt (x0 , y) ≤ above by a small positive constant and then the infimum inf{L(y, 1 A(2t − 1) + 10 with τ = 1 − t} is achieved by admissible curves. Hence we conclude that for each τ ∈ [0, 21 ], any minimizing curve γτ of ¯ τ )| dt (x0 , y) ≤ A(2t − 1) + 1 with τ = 1 − t} is admissible and satisfies inf{L(y, 10 Z τ √ s(R + |γ˙ τ |2 )ds ≤ exp(C(A) + 100). 0
Now we come back to the unrescaled solution. It then follows that the Li-YauPerelman distance l from (x, t0 ) satisfies the following estimate 1 1 ≤ exp(C(A) + 100), (7.5.7) min l y, t0 − r02 y ∈ Bt0 − 12 r02 x0 , r0 2 10
by noting the (parabolic) scaling invariance of the Li-Yau-Perelman distance. By the assumption that |Rm| ≤ r0−2 on P (x0 , t0 , r0 , −r02 ), exactly as before, for any q ∈ Bt0 −r02 (x0 , r0 ), we can choose a path γ parametrized by τ ∈ [0, r02 ] with 1 γ(0) = x, γ(r02 ) = q, and γ( 21 r02 ) = y ∈ Bt0 − 21 r02 (x0 , 10 r0 ), where γ|[0, 21 r02 ] achieves the 1 1 2 minimum min{l(y, t0 − 2 r0 ) | y ∈ Bt0 − 12 r02 (x0 , 10 r0 )} and γ|[ 12 r02 ,r02 ] is a suitable curve satisfying γ|[ 21 r02 ,r02 ] (τ ) ∈ Bt0 −τ (x0 , r0 ), for each τ ∈ [ 12 r02 , r02 ], so that the L-length of γ is uniformly bounded from above by a positive constant (depending only on A) multiplying r0 . This implies that the Li-Yau-Perelman distance from (x, t0 ) to the ball Bt0 −r02 (x0 , r0 ) is uniformly bounded by a positive constant L(A) (depending only on A). Now we can choose the constant L in (7.5.1) by L = max{2L(A), 2 exp(C(A) + 100)}. Thus every shortest L-geodesic from (x, t0 ) to the ball Bt0 −r02 (x0 , r0 ) is necessarily admissible. By combining with the assumption that Vol t0 (Bt0 (x0 , r0 )) ≥ A−1 r03 , we conclude that Perelman’s reduced volume of the ball Bt0 −r02 (x0 , r0 ) satisfies the estimate Z 3 e (7.5.8) Vr02 (Bt0 −r02 (x0 , r0 )) = (4πr02 )− 2 exp(−l(q, r02 ))dVt0 −r02 (q) 2 (x0 ,r0 ) 0 −r0
Bt
≥ c(A)
for some positive constant c(A) depending only on A. We can now argue as in the last part of the proof of Lemma 7.4.2 to get a lower bound estimate for the volume of the ball Bt0 (x, ρ). The union of all shortest Lgeodesics from (x, t0 ) to the ball Bt0 −r02 (x0 , r0 ), defined by CBt0 −r02 (x0 , r0 ) = {(y, t) | (y, t) lies in a shortest L-geodesic from
(x, t0 ) to a point in Bt0 −r02 (x0 , r0 )},
forms a cone-like subset in space-time with vertex (x, t0 ). Denote by B(t) the intersection of the cone-like subset CBt0 −r02 (x0 , r0 ) with the time-slice at t. Perelman’s
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
457
reduced volume of the subset B(t) is given by Z 3 e (4π(t0 − t))− 2 exp(−l(q, t0 − t))dVt (q). Vt0 −t (B(t)) = B(t)
Since the cone-like subset CBt0 −r02 (x0 , r0 ) lies entirely in the region unaffected by surgery, we can apply Perelman’s Jacobian comparison Theorem 3.2.7 and the estimate (7.5.8) to conclude that Vet0 −t (B(t)) ≥ Ver02 (Bt0 −r02 (x0 , r0 ))
(7.5.9)
≥ c(A)
for all t ∈ [t0 − r02 , t0 ]. 1 As before, denoting by ξ = ρ−1 Vol t0 (Bt0 (x, ρ)) 3 , we only need to get a positive e 0 − ξρ2 ), the subset lower bound for ξ. Of course we may assume ξ < 1. Consider B(t 2 at the time-slice {t = t0 − ξρ } where every point can be connected to (x, t0 ) by an e 0 − ξρ2 ) is given by admissible shortest L-geodesic. Perelman’s reduced volume of B(t (7.5.10)
e 0 − ξρ2 )) Veξρ2 (B(t Z 3 (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) = e −ξρ2 ) B(t Z 0 3 (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) = e 0 −ξρ2 )∩L exp B(t
+
{|υ|≤ 1 ξ 4
Z
−1 2}
(ξρ2 )
3
e 0 −ξρ2 )\L exp B(t
{|υ|≤ 1 ξ 4
−1 2}
(ξρ2 )
(4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q).
Note that the whole region P (x, t0 , ρ, −ρ2 ) is unaffected by surgery because ρ ≥ 1 ¯ t0 , re(t0 ), re( t0 ), ε) > 0 is sufficiently small. Then exactly as before, e(t0 ) and δ(L, 2η r 2 there is a universal positive constant ξ0 such that when 0 < ξ ≤ ξ0 , there holds L exp
1
{|υ|≤ 14 ξ − 2 }
(ξρ2 ) ⊂ Bt0 (x, ρ)
and the first term on RHS of (7.5.10) can be estimated by Z 3 (7.5.11) (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) e 0 −ξρ2 )∩L exp B(t 3
≤ eCξ (4π)− 2 ξ
{|υ|≤ 1 ξ 4 3 2
−1 2}
(ξρ2 )
for some universal constant C; while the second term on RHS of (7.5.10) can be estimated by Z 3 (7.5.12) (4πξρ2 )− 2 exp(−l(q, ξρ2 ))dVt0 −ξρ2 (q) e 0 −ξρ2 )\L exp B(t 3
≤ (4π)− 2
Z
{|υ|≤ 1 ξ 4
1
−1 2}
(ξρ2 )
exp(−|υ|2 )dυ.
{|υ|> 14 ξ − 2 }
e 0 − ξρ2 ), the combination of (7.5.9)-(7.5.12) bounds ξ from Since B(t0 − ξρ2 ) ⊂ B(t below by a positive constant depending only on A. This proves the statement (i).
458
H.-D. CAO AND X.-P. ZHU
(ii) This is analogous to the claim in the proof of Theorem 7.2.1. We argue by contradiction. Suppose that for some A < +∞ and a sequence K1α → ∞, there exists ¯αβ > 0 with δ¯αβ → 0 for fixed α, we a sequence tα 0 such that for any sequences δ αβ have sequences of solutions gij to the Ricci flow with surgery and sequences of points αβ xαβ 0 , of radii r0 , which satisfy the assumptions but violate the statement (ii) at some αβ α αβ −2 xαβ ∈ Btα0 (x0 , Ar0αβ ) with R(xαβ , tα . Slightly abusing notation, we 0 ) ≥ K1 (r0 ) will often drop the indices α, β in the following argument. Exactly as in the proof of Theorem 7.2.1, we need to adjust the point (x, t0 ). More r2 ¯, precisely, we claim that there exists a point (¯ x, t¯) ∈ Bt¯(x0 , 2Ar0 ) ×[t0 − 20 , t0 ] with Q R(¯ x, t¯) ≥ K1 r0−2 such that the point (¯ x, t¯) does not satisfy the canonical neighborhood ¯ does, where P¯ is the set of all statement, but each point (y, t) ∈ P¯ with R(y, t) ≥ 4Q 1 1 −1 ′ ′ ′ ′ ¯ ≤ t ≤ t¯, dt′ (x0 , x ) ≤ dt¯(x0 , x ¯ − 12 . Indeed as ¯) + K12 Q (x , t ) satisfying t¯ − 4 K1 Q before, the point (¯ x, t¯) is chosen by an induction argument. We first choose (x1 , t1 ) = (x, t0 ) which satisfies dt1 (x0 , x1 ) ≤ Ar0 and R(x1 , t1 ) ≥ K1 r0−2 , but does not satisfy the canonical neighborhood statement. Now if (xk , tk ) is already chosen and is not the desired (¯ x, t¯), then some point (xk+1 , tk+1 ) satisfies tk − 41 K1 R(xk , tk )−1 ≤ tk+1 ≤ tk , 1
1
dtk+1 (x0 , xk+1 ) ≤ dtk (x0 , xk ) + K12 R(xk , tk )− 2 , and R(xk+1 , tk+1 ) ≥ 4R(xk , tk ), but (xk+1 , tk+1 ) does not satisfy the canonical neighborhood statement. Then we have R(xk+1 , tk+1 ) ≥ 4k R(x1 , t1 ) ≥ 4k K1 r0−2 , 1
dtk+1 (x0 , xk+1 ) ≤ dt1 (x0 , x1 ) + K12
k X i=1
1
R(xi , ti )− 2 ≤ Ar0 + 2r0 ,
and k X 1 1 t0 ≥ tk+1 ≥ t0 − K1 R(xi , ti )−1 ≥ t0 − r02 . 4 2 i=1
So the sequence must be finite and its last element is the desired (¯ x, t¯). −2 ¯ ¯ Rescale the solutions along (¯ x, t) with factor R(¯ x, t)(≥ K1 r0 ) and shift the times t¯ to zero. We will adapt both the proof of Proposition 7.4.1 and that of Theorem αβ 7.2.1 to show that a sequence of the rescaled solutions geij converges to an ancient κ-solution, which will give the desired contradiction. Since we only need to consider the scale of the curvature less than re(t¯)−2 , the present situation is much easier than that of Proposition 7.4.1. Firstly as before, we need to get a local curvature estimate. 1 −1 ¯ −1 ¯ η Q , t] For each adjusted (¯ x, t¯), let [t′ , t¯] be the maximal subinterval of [t¯ − 20 so that for each sufficiently large α and then sufficiently large β, the canonical 1 1 ¯ − 12 , t′ − t¯) = {(x, t) | x ∈ neighborhood statement holds for any (y, t) in P (¯ x, t¯, 10 K12 Q 1 1 ¯ − 12 ), t ∈ [t′ , t¯]} with R(y, t) ≥ 4Q, ¯ where η is the universal positive Bt (¯ x, 10 K12 Q constant in the definition of canonical neighborhood assumption. We want to show (7.5.13)
t′ = t¯ −
1 −1 ¯ −1 η Q . 20
Consider the scalar curvature R at the point x ¯ over the time interval [t′ , t¯]. If ′ ¯ we let t˜ be the first of such time from there is a time t˜ ∈ [t , t¯] satisfying R(¯ x, t˜) ≥ 4Q, t¯. Since the chosen point (¯ x, t¯) does not satisfy the canonical neighborhood statement,
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
459
we know R(¯ x, t¯) ≤ re(t¯)−2 . Recall from our designed surgery procedure that if there is a cutoff surgery at a point x at a time t, the scalar curvature at (x, t) is at least αβ (δ¯αβ )−2 re(t)−2 . Then for each fixed α, for β large enough, the solution gij (·, t) around 1 −1 ¯ −1 ˜ the point x ¯ over the time interval [t˜ − 20 η Q , t] is well defined and satisfies the following curvature estimate ¯ R(¯ x, t) ≤ 8Q, 1 −1 ¯ −1 ¯ for t ∈ [t˜ − 20 η Q , t] (or t ∈ [t′ , t¯] if there is no such time t˜). By the assumption 2 that t0 > 2r0 , we have
t0 x, t¯) t¯R(¯ x, t¯) ≥ R(¯ 2 ≥ r02 (K1 r0−2 )
= K1 → +∞.
Thus by using the pinching assumption and the gradient estimates in the canonical neighborhood assumption, we further have ¯ |Rm(x, t)| ≤ 30Q, 1 −1 ¯ − 21 1 −1 ¯ −1 ¯ for all x ∈ Bt (¯ x, 10 η Q ) and t ∈ [t˜− 20 η Q , t] (or t ∈ [t′ , t¯]) and all sufficiently large α and β. Observe that Lemma 3.4.1 (ii) is applicable for dt (x0 , x ¯) with t ∈ 1 −1 ¯ −1 ¯ [t˜− 20 η Q , t] (or t ∈ [t′ , t¯]) since any minimal geodesic, with respect to the metric gij (·, t), connecting x0 and x¯ lies in the region unaffected by surgery; otherwise the geodesic is not minimal. After having obtained the above curvature estimate, we can argue as deriving (7.2.2) and (7.2.3) in the proof of Theorem 7.2.1 to conclude that 1 1 1 −1 ¯ −1 ¯ − 12 , η Q ≤ t ≤ t¯ (or t ∈ [t′ , t¯]) and dt (x, x¯) ≤ 10 K12 Q any point (x, t), with t˜ − 20 satisfies
1 1 ¯− 1 2, dt (x, x0 ) ≤ dt¯(¯ x, x0 ) + K12 Q 2 for all sufficiently large α and β. Then by combining with the choice of the points 1 −1 ¯ −1 (¯ x, t¯), we prove t′ = t¯ − 20 η Q (i.e., the canonical neighborhood statement holds 1 1 ¯ − 12 ,− 1 η −1 Q ¯ −1 ) with for any point (y, t) in the parabolic neighborhood P (¯ x, t¯, 10 K12 Q 20 ¯ R(y, t) ≥ 4Q) for all sufficiently large α and then sufficiently large β. Now it follows from the gradient estimates in the canonical neighborhood assumpαβ tion that the scalar curvatures of the rescaled solutions e gij satisfy e t) ≤ 40 R(x,
1 −1 1 −1 1 −1 et′ (¯ η , − 20 η ) , {(x′ , t′ ) | x′ ∈ B x, 10 η ), t′ ∈ for those (x, t) ∈ P (¯ x, 0, 10 1 −1 e [− 20 η , 0]}, for which the rescaled solution is defined. (Here Bt′ denotes the geodesic ball in the rescaled solution at time t′ ). Note again that R(¯ x, t¯) ≤ re(t¯)−2 and recall from our designed surgery procedure that if there is a cutoff surgery at a point x at a time t, the scalar curvature at (x, t) is at least (δ¯αβ )−2 re(t)−2 . Then for each fixed αβ sufficiently large α, for β large enough, the rescaled solution geij is defined on the 1 −1 1 −1 whole parabolic neighborhood P (¯ x, 0, 10 η , − 20 η ). More generally, for arbitrarily e < +∞, there is a positive integer α0 so that for each α ≥ α0 we can fixed 0 < K
460
H.-D. CAO AND X.-P. ZHU
find β0 > 0 (depending on α) such that if β ≥ β0 and (y, 0) is a point on the rescaled αβ e 0) ≤ K e and de0 (y, x¯) ≤ K, e we have estimate solution geij with R(y,
(7.5.14)
e t) ≤ 40K e R(x,
1 −1 e − 12 1 −1 e −1 et′ (y, for (x, t) ∈ P (y, 0, 10 η K , − 20 η K ) , {(x′ , t′ ) | x′ ∈ B 1 −1 e −1 1 −1 e − 12 ′ K ), t ∈ [− 20 η K , 0]}. In particular, the rescaled solution is defined 10 η 1 −1 e − 21 1 −1 e −1 on the whole parabolic neighborhood P (y, 0, 10 η K , − 20 η K ). Next, we want to show the curvature of the rescaled solutions at the new times zero (after shifting) stay uniformly bounded at bounded distances from x¯ for some subsequences of α and β. Let αm , βm → +∞ be chosen so that the estimate (7.5.14) e = m. For all ρ ≥ 0, set holds with K
and
e 0) | m ≥ 1, d0 (x, x¯) ≤ ρ in the rescaled solutions geαm βm } M (ρ) = sup{R(x, ij ρ0 = sup{ρ ≥ 0 | M (ρ) < +∞}.
Clearly the estimate (7.5.14) yields ρ0 > 0. As we consider the unshifted time t¯, by combining with the assumption that t0 > 2r02 , we have (7.5.15)
t0 t¯R(¯ x, t¯) ≥ R(¯ x, t¯) 2 ≥ r02 (K1 r0−2 )
= K1 → +∞.
It then follows from the pinching assumption that we only need to show ρ0 = +∞. As before, we argue by contradiction. Suppose we have a sequence of points ym in αm βm e m , 0) → +∞. Dethe rescaled solutions e gij with de0 (¯ x, ym ) → ρ0 < +∞ and R(y e0 (¯ note by γm a minimizing geodesic segment from x ¯ to ym and denote by B x, ρ0 ) the open geodesic balls centered at x ¯ of radius ρ0 of the rescaled solutions. By applying the assertion in statement (i), we have uniform κ-noncollapsing at the points (¯ x, t¯). By combining with the local curvature estimate (7.5.14) and Hamilton’s compactness theorem, we can assume that, after passing to a subsequence, the marked sequence αm βm ∞ e0 (¯ (B x, ρ0 ), geij ,x ¯) converges in the Cloc topology to a marked (noncomplete) man∞ ifold (B∞ , e gij , x∞ ) and the geodesic segments γm converge to a geodesic segment (missing an endpoint) γ∞ ⊂ B∞ emanating from x∞ . Moreover, by the pinching assumption and the estimate (7.5.15), the limit has nonnegative sectional curvature. Then exactly as before, we consider the tubular neighborhood along γ∞ [ e∞ (q0 ))− 21 ) V = B∞ (q0 , 4π(R q0 ∈γ∞
¯∞ of (B∞ , ge∞ ) with y∞ ∈ B ¯∞ the limit point of γ∞ . As and the completion B ij ∞ before, by the choice of the points (¯ x, t¯), we know that the limiting metric e gij is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ . Then by the same reason as before the metric space V¯ = V ∪ {y∞ } has nonnegative curvature in Alexandrov sense, and we have a three-dimensional nonflat tangent cone Cy∞ V¯ at y∞ . Pick z ∈ Cy∞ V¯ with distance one from the vertex and it is nonflat around z. By
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
461
definition B(z, 12 )(⊂ Cy∞ V¯ ) is the Gromov-Hausdoff convergent limit of the scalings ∞ of a sequence of balls B∞ (zk , σk )(⊂ (V, e gij )) with σk → 0. Since the estimate (7.5.14) ∞ e survives on (V, geij ) for all K < +∞, we know that this convergence is actually in ∞ the Cloc topology and over some time interval. Since the limit B(z, 21 )(⊂ Cy∞ V¯ ) is a piece of a nonnegatively curved nonflat metric cone, we get a contradiction with Hamilton’s strong maximum Principle (Theorem 2.2.1) as before. Hence we have αm βm proved that a subsequence of the rescaled solution e gij has uniformly bounded curvatures at bounded distance from x ¯ at the new times zero. Further, by the uniform κ-noncollapsing at the points (¯ x, t¯) and the estimate ∞ ∞ (7.5.14) again, we can take a Cloc limit (M∞ , e gij , x∞ ), defined on a space-time subset which contains the time slice {t = 0} and is relatively open in M∞ × (−∞, 0], for the subsequence of the rescaled solutions. The limit is a smooth solution to the Ricci flow, and is complete at t = 0, as well as has nonnegative sectional curvature by the pinching assumption and the estimate (7.5.15). Thus by repeating the same argument as in the Step 4 of the proof Proposition 7.4.1, we conclude that the curvature of the limit at t = 0 is bounded. Finally we try to extend the limit backwards in time to get an ancient κ-solution. Since the curvature of the limit is bounded at t = 0, it follows from the estimate (7.5.14) that the limit is a smooth solution to the Ricci flow defined at least on a backward time interval [−a, 0] for some positive constant a. Let (t∞ , 0] be the maximal time interval over which we can extract a smooth limiting solution. It suffices to show t∞ = −∞. If t∞ > −∞, there are only two possibilities: either there exist surgeries in finite distance around the time t∞ or the curvature of the limiting solution becomes unbounded as t ց t∞ . 1 −1 η . Note again that the Let c > 0 be a positive constant much smaller than 100 infimum of the scalar curvature is nondecreasing in time. Then we can find some point e∞ (y∞ , t∞ + θ) ≤ 2. y∞ ∈ M∞ and some time t = t∞ + θ with 0 < θ < 3c such that R αm βm Consider the (unrescaled) scalar curvature R of gij (·, t) at the point x¯ over ¯ −1 , t¯]. Since the scalar curvature R∞ of the limit on the time interval [t¯ + (t∞ + 2θ )Q M∞ × [t∞ + 3θ , 0] is uniformly bounded by some positive constant C, we have the curvature estimate ¯ R(¯ x, t) ≤ 2C Q ¯ −1 , t¯] and all sufficiently large m. For each fixed m and αm , for all t ∈ [t¯ + (t∞ + 2θ )Q we may require the chosen βm to satisfy αm −2 t αm βm −2 m −2 ¯ ¯ ≥ me r (tα ≥ mQ. (δ ) re 0 0 ) 2
When m is large enough, we observe again that Lemma 3.4.1 (ii) is applicable for ¯ −1 , t¯]. Then by repeating the argument as in the dt (x0 , x¯) with t ∈ [t¯ + (t∞ + θ2 )Q derivation of (7.2.1), (7.2.2) and (7.2.3), we deduce that for all sufficiently large m, the canonical neighborhood statement holds for any (y, t) in the parabolic neighborhood 1 1 ¯ − 12 , (t∞ + θ )Q ¯ −1 ). K12 Q P (¯ x, t¯, 10 2 ¯ ¯ Let (ym , t + (t∞ + θm )Q−1 ) be a sequence of associated points and times in the αm βm (unrescaled) solutions gij (·, t) so that after rescaling, the sequence converges to (y∞ , t∞ + θ) in the limit. Clearly θ2 ≤ θm ≤ 2θ for all sufficiently large m. Then by the argument as in the derivation of (7.5.13), we know that for all sufficiently large
462
H.-D. CAO AND X.-P. ZHU
αm βm m, the solution gij (·, t) at ym is defined on the whole time interval [t¯+ (t∞ + θm − 1 −1 ¯ −1 ¯ ¯ −1 ] and satisfies the curvature estimate )Q , t + (t∞ + θm )Q 20 η
¯ R(ym , t) ≤ 8Q there; moreover the canonical neighborhood statement holds for any (y, t) with 1 ¯ −1 ). ¯ in the parabolic neighborhood P (ym , t¯, 1 K 2 Q ¯ − 12 , (t∞ − c )Q R(y, t) ≥ 4Q 10
1
3
αm βm We now consider the rescaled sequence geij (·, t) with the marked points re¯ −1 , t¯ + (t∞ + c )Q ¯ −1 ]. placed by ym and the times replaced by sm ∈ [t¯ + (t∞ − 4c )Q 4 As before the Li-Yau-Hamilton inequality implies the rescaling limit around (ym , sm ) agrees with the original one. Then the arguments in previous paragraphs imply the limit is well-defined and smooth on a space-time open neighborhood of the maximal time slice {t = t∞ }. Particularly this excludes the possibility of existing surgeries in finite distance around the time t∞ . Moreover, the limit at t = t∞ also has bounded curvature. By using the gradient estimates in the canonical neighborhood assump1 1 ¯ − 12 , (t∞ − c )Q ¯ −1 ), we see that tion on the parabolic neighborhood P (ym , t¯, 10 K12 Q 3 the second possibility is also impossible. Hence we have proved a subsequence of the rescaled solutions converges to an ancient κ-solution. Therefore we have proved the canonical neighborhood statement (ii).
(iii) This is analogous to Theorem 7.2.1. We also argue by contradiction. Suppose for some A < +∞ and sequences of positive numbers K2α → +∞, r¯α → 0 there exists ¯αβ > 0 with δ¯αβ → 0 for fixed α, we a sequence of times tα 0 such that for any sequences δ αβ have sequences of solutions gij to the Ricci flow with surgery and sequences of points p αβ αβ xαβ ¯α tα 0 which satisfy the assumptions, but 0 , of positive constants r0 with r0 ≤ r for all α, β there hold (7.5.16)
αβ α αβ −2 R(xαβ , tα , for some xαβ ∈ Btα0 (xαβ 0 ) > K2 (r0 ) 0 , Ar0 ).
α ¯ We may assume that δ¯αβ ≤ δ¯4A (tα 0 ) for all α, β, where δ4A (t0 ) is chosen so that αβ αβ αβ the statements (i) and (ii) hold on Btα0 (x0 , 4Ar0 ). Let gˆij be the rescaled solutions αβ αβ −2 of gij around the origins xαβ and shift the times tα 0 to zero. Then 0 with factor (r0 ) by applying the statement (ii), we know that the regions, where the scalar curvature αβ of the rescaled solutions gˆij is at least K1 (= K1 (4A)), are canonical neighborhood regions. Note that canonical ε-neck neighborhoods are strong. Also note that the pinching assumption and the assertion αβ −2 ≥ (¯ rα )−2 → +∞, as α → +∞, tα 0 (r0 ) αβ imply that any subsequent limit of the rescaled solutions gˆij must have nonnegative sectional curvature. Thus by the above argument in the proof of the statement (ii) (or the argument in Step 2 of the proof of Theorem 7.1.1), we conclude that there exist subsequences α = αm , β = βm such that the curvatures of the rescaled solutions αm βm m βm gˆij stay uniformly bounded at distances from the origins xα not exceeding 2A. 0 This contradicts (7.5.16) for m sufficiently large. This proves the statements (iii). Clearly for fixed A, after defining the δ¯A (t0 ) for each t0 , one can adjust the δ¯A (t0 ) so that it is nonincreasing in t0 . Therefore we have completed the proof of the theorem. e so that it is also less than δ¯2(t+1) (2t) From now on we redefine the function δ(t) and then the above theorem always holds for A ∈ [1, 2(t0 + 1)]. Particularly, we still
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
463
have e ≤ δ(t) ¯ = min{ δ(t)
1 2e2 log(1
+ t)
, δ0 },
which tends to zero as t → +∞. We may also require that re(t) tends to zero as t → +∞. The next result is a version of Theorem 7.2.2 for solutions with surgery. Theorem 7.5.2 (Perelman [104]). For any ε > 0 and w > 0, there exist τ = τ (w, ε) > 0, K = K(w, ε) < +∞, r¯ = r¯(w, ε) > 0, θ = θ(w, ε) > 0 and T = T (w) < +∞ with the following property. Suppose we have a solution, constructed by e and re(t), Theorem 7.4.3 with the nonincreasing (continuous) positive functions δ(t) to the Ricci flow with surgery on the time interval [0, t0 ] with a compact orientable normalized three-manifold as initial data, where each δ-cutoff at a time t ∈ [0, t0 ] has e re(2t)}. Let r0 , t0 satisfy θ−1 h ≤ r0 ≤ r¯√t0 and t0 ≥ T , where h δ = δ(t) ≤ min{δ(t), is the maximal cutoff radius for surgeries in [ t20 , t0 ] (if there is no surgery in the time interval [ t20 , t0 ], we take h = 0), and assume that the solution on the ball Bt0 (x0 , r0 ) satisfies Rm(x, t0 ) ≥ −r0−2 , on Bt0 (x0 , r0 ), and
Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 .
Then the solution is well defined and satisfies R(x, t) < Kr0−2 in the whole parabolic neighborhood n o r0 r0 P x0 , t0 , , −τ r02 = (x, t) | x ∈ Bt x0 , , t ∈ [t0 − τ r02 , t0 ] . 4 4 Proof. We are given that Rm(x, t0 ) ≥ −r0−2 for x ∈ Bt0 (x0 , r0 ), and Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 . The same argument in the derivation of (7.2.7) and (7.2.8) (by using the Alexandrov space theory) implies that there exists a ball Bt0 (x′ , r′ ) ⊂ Bt0 (x0 , r0 ) with (7.5.17)
Vol t0 (Bt0 (x′ , r′ )) ≥
1 α3 (r′ )3 2
and with (7.5.18)
r′ ≥ c(w)r0
for some small positive constant c(w) depending only on w, where α3 is the volume of the unit ball in R3 . As in (7.1.2), we can rewrite the pinching assumption (7.3.3) as Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R, where y = f (x) = x(log x − 3),
for e2 ≤ x < +∞,
464
H.-D. CAO AND X.-P. ZHU
is increasing and convex with range −e2 ≤ y < +∞, and its inverse function is also increasing and satisfies lim f −1 (y)/y = 0.
y→+∞
Note that t0 r0−2 ≥ r¯−2 by the hypotheses. We may require T (w) ≥ 8c(w)−1 . Then by applying Theorem 7.5.1 (iii) with A = 8c(w)−1 and combining with the pinching assumption, we can reduce the proof of the theorem to the special case w = 12 α3 . In the following we simply assume w = 21 α3 . Let us first consider the case r0 < re(t0 ). We claim that R(x, t0 ) ≤ C02 r0−2 on Bt0 (x0 , r30 ), for some sufficiently large positive constant C0 depending only on ε. If not, then there is a canonical neighborhood around (x, t0 ). Note that the type (c) canonical neighborhood has already been ruled out by our design of cutoff surgeries. Thus (x, t0 ) belongs to an ε-neck or an ε-cap. This tells us that there is a nearby point y, 1 with R(y, t0 ) ≥ C2−1 R(x, t0 ) > C2−1 C02 r0−2 and dt0 (y, x) ≤ C1 R(x, t0 )− 2 ≤ C1 C0−1 r0 , 1 which is the center of the ε-neck Bt0 (y, ε−1 R(y, t0 )− 2 ). (Here C1 , C2 are the positive constants in the definition of canonical neighborhood assumption). Clearly, when we choose C0 to be much larger than C1 , C2 and ε−1 , the whole ε-neck 1 Bt0 (y, ε−1 R(y, t0 )− 2 ) is contained in Bt0 (x0 , r20 ) and we have 1
(7.5.19)
Vol t0 (Bt0 (y, ε−1 R(y, t0 )− 2 )) 1
(ε−1 R(y, t0 )− 2 )3
≤ 8πε2 .
Without loss of generality, we may assume ε > 0 is very small. Since we have assumed that Rm ≥ −r0−2 on Bt0 (x0 , r0 ) and Vol t0 (Bt0 (x0 , r0 )) ≥ 12 α3 r03 , we then get a contradiction by applying the standard Bishop-Gromov volume comparison. Thus we have the desired curvature estimate R(x, t0 ) ≤ C02 r0−2 on Bt0 (x0 , r30 ). Furthermore, by using the gradient estimates in the definition of canonical neigh1 −1 −2 borhood assumption, we can take K = 10C02 , τ = 100 η C0 and θ = 15 C0−1 in this −2 case. And since r0 ≥ θ−1 h, we have R < 10C02 r0 ≤ 21 h−2 and the surgeries do not interfere in P (x0 , t0 , r40 , −τ r02 ). √ We now consider the remaining case re(t0 ) ≤ r0 ≤ r¯ t0 . Let us redefine τ¯0 1 −1 −2 τ = min , η C0 , 2 100 ¯ 2B 2 ¯ , 25C0 , K = max 2 C + τ¯0 and θ=
1 − 12 K 2
¯ = B(w) and C¯ = C(w) are the positive constants in Theorem where τ¯0 = τ0 (w), B 1 6.3.3(ii) with w = 2 α3 , and C0 is the positive constant chosen above. We will show there is a sufficiently small r¯ > 0 such that the conclusion of the theorem for w = 21 α3 holds for the chosen τ, K and θ. Argue by contradiction. Suppose not, then there exist a sequence of r¯α → 0, α α and a sequence of solutions gij with points (xα radii r0α such that the as0 , t0 ) and pα α α α sumptions of the theorem do hold with re(t0 ) ≤ r0 ≤ r¯ t0 whereas the conclusion
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
465
does not. Similarly as in the proof of Theorem 7.2.2, we claim that we may assume that for all sufficiently large α, any other point (xα , tα ) and radius rα > 0 α α with that property has either tα > tα = tα ≥ r0α ; moreover tα 0 or t 0 with r α tends to +∞ as α → +∞. Indeed, for fixed α and the solution gij , let tα min be α α α the infimum of all possible times t with some point x and some radius r hav√ ing that property. Since each such tα satisfies r¯α tα ≥ rα ≥ re(tα ), it follows that when α is large, tα min must be positive and very large. Clearly for each fixed sufficiently large α, by passing to a limit, there exist some point xα min and some radius α α rmin (≥ re(tα ) > 0) so that all assumptions of the theorem still hold for (xα min min , tmin ) α α and rmin , whereas the conclusion of the theorem does not hold with R ≥ K(rmin )−2 1 α α α 2 somewhere in P (xα min , tmin , 4 rmin , −τ (rmin ) ) for all sufficiently large α. Here we 1 α α −2 α α 2 used the fact that if R < K(rmin ) on P (xα min , tmin , 4 rmin , −τ (rmin ) ), then there is no δ-cutoff surgery there; otherwise there must be a point there with scalar curtα tα tα 1 −2 α −2 −2 α min min )(e r ( )) ≥ (e r (t )) (e r ( )) ≫ K(rmin )−2 since vature at least 21 δ −2 ( min min 2 2 2 2 p α r¯α tα e(tα ¯α → 0, which is a contradiction. min ) and r min ≥ rmin ≥ r After choosing the first time tα min , by passing to a limit again, we can then choose α α α rmin to be the smallest radius for all possible (xα min , tmin )’s and rmin ’s with that property. Thus we have verified the claim. For simplicity, we will drop the index α in the following arguments. By the assumption and the standard volume comparison, we have 1 Vol t0 Bt0 x0 , r0 ≥ ξ0 r03 2 for some universal positive ξ0 . As in deriving (7.2.7) and (7.2.8), we can find a ball Bt0 (x′0 , r0′ ) ⊂ Bt0 (x0 , r20 ) with Vol t0 (Bt0 (x′0 , r0′ )) ≥
1 1 α3 (r0′ )3 and r0 ≥ r0′ ≥ ξ0′ r0 2 2
for some universal positive constant ξ0′ . Then by what we had proved in the previous case and by the choice of the first time t0 and the smallest radius r0 , we know that r′ the solution is defined in P (x′0 , t0 , 40 , −τ (r0′ )2 ) with the curvature bound R < K(r0′ )−2 ≤ K(ξ0′ )−2 r0−2 .
√ Since r¯ t0 ≥ r0 ≥ re(t0 ) and r¯ → 0 as α → ∞, we see that t0 → +∞ and t0 r0−2 → +∞ ¯ for some suitable large universal positive as α → +∞. Define T (w) = 8c(w)−1 + ξ, ¯ Then for α sufficiently large, we can apply Theorem 7.5.1(iii) and the constant ξ. pinching assumption to conclude that τ (7.5.20) R ≤ K ′ r0−2 , on P x0 , t0 , 4r0 , − (ξ0′ )2 r02 , 2 for some positive constant K ′ depending only on K and ξ0′ . Furthermore, by combining with the pinching assumption, we deduce that when α sufficiently large, (7.5.21)
Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R ≥ −r0−2
466
H.-D. CAO AND X.-P. ZHU
on P (x0 , t0 , r0 , − τ2 (ξ0′ )2 r02 ). So by applying Theorem 6.3.3(ii) with w = 12 α3 , we have that when α sufficiently large, (7.5.22)
Vol t (Bt (x0 , r0 )) ≥ ξ1 r03 ,
for all t ∈ [t0 − τ2 (ξ0′ )2 r02 , t0 ], and (7.5.23)
R≤
¯ 2B 1 C¯ + r0−2 ≤ Kr0−2 τ¯0 2
on P (x0 , t0 , r40 , − τ2 (ξ0′ )2 r02 ), where ξ1 is some universal positive constant. Next we want to extend the estimate (7.5.23) backwards in time. Denote by t1 = t0 − τ2 (ξ0′ )2 r02 . The estimate (7.5.22) gives Vol t1 (Bt1 (x0 , r0 )) ≥ ξ1 r03 . By the same argument in the derivation of (7.2.7) and (7.2.8) again, we can find a ball Bt1 (x1 , r1 ) ⊂ Bt1 (x0 , r0 ) with Vol t1 (Bt1 (x1 , r1 )) ≥
1 α3 r13 2
and with r1 ≥ ξ1′ r0 for some universal positive constant ξ1′ . Then by what we had proved in the previous case and by the lower bound (7.5.21) at t1 and the choice of the first time t0 , we know that the solution is defined on P (x1 , t1 , r41 , −τ r12 ) with the curvature bound R < Kr1−2 . By applying Theorem 7.5.1(iii) and the pinching assumption again we get that for α sufficiently large, R ≤ K ′′ r0−2
(7.5.20)′
on P (x0 , t1 , 4r0 , − τ2 (ξ1′ )2 r02 ), for some positive constant K ′′ depending only on K and ξ1′ . Moreover, by combining with the pinching assumption, we have (7.5.21)′
Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R ≥ −r0−2
on P (x0 , t1 , 4r0 , − τ2 (ξ1′ )2 r02 ), for α sufficiently large. So by applying Theorem 6.3.3 (ii) with w = 12 α3 again, we have that for α sufficiently large, (7.5.22)′
Vol t (Bt (x0 , r0 )) ≥ ξ1 r03 ,
for all t ∈ [t0 − τ2 (ξ0′ )2 r02 − τ2 (ξ1′ )2 r02 , t0 ], and (7.5.23)
′
¯ 2B 1 ¯ R≤ C+ r0−2 ≤ Kr0−2 τ¯0 2
on P (x0 , t0 , r40 , − τ2 (ξ0′ )2 r02 − τ2 (ξ1′ )2 r02 ).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
467
Note that the constants ξ0 , ξ0′ , ξ1 and ξ1′ are universal, independent of the time t1 and the choice of the ball Bt1 (x1 , r1 ). Then we can repeat the above procedure as many times as we like, until we reach the time t0 − τ r02 . Hence we obtain the estimate (7.5.23)′′
R ≤ (C¯ +
¯ −2 1 2B )r0 ≤ Kr0−2 τ¯0 2
on P (x0 , t0 , r40 , −τ r02 ), for sufficiently large α. This contradicts the choice of the point (x0 , t0 ) and the radius r0 which make R ≥ Kr0−2 somewhere in P (x0 , t0 , r40 , −τ r02 ). Therefore we have completed the proof of the theorem. Consequently, we have the following result which is analog of Corollary 7.2.4. Corollary 7.5.3. For any ε > 0 and w > 0, there exist r¯ = r¯(w, ε) > 0, θ = θ(w, ε) > 0 and T = T (w) with the following property. Suppose we have a e and re(t), to solution, constructed by Theorem 7.4.3 with the positive functions δ(t) the Ricci flow with surgery with a compact orientable normalized three-manifold as e re(2t)}. If initial data, where each δ-cutoff at a time t has δ = δ(t)√ ≤ min{δ(t), Bt0 (x0 , r0 ) is a geodesic ball at time t0 , with θ−1 h ≤ r0 ≤ r¯ t0 and t0 ≥ T , where h is the maximal cutoff radii for surgeries in [ t20 , t0 ] (if there is no surgery in the time interval [ t20 , t0 ], we take h = 0), and satisfies min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , then Vol t0 (Bt0 (x0 , r0 )) < wr03 .
Proof. We argue by contradiction. Let θ = θ(w, ε) and T = 2T (w), where θ(w, ε) and T (w) are the positive constant in Theorem 7.5.2. Suppose for any r¯ > 0 there is a solution and a geodesic ball Bt0 (x0 , r0 ) satisfying the assumptions of the corollary √ with θ−1 h ≤ r0 ≤ r¯ t0 and t0 ≥ T , and with min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} = −r0−2 , but Vol t0 (Bt0 (x0 , r0 )) ≥ wr03 . Without loss of generality, we may assume that r¯ is less than the corresponding constant in Theorem 7.5.2. We can then apply Theorem 7.5.2 to get R(x, t) ≤ Kr0−2 whenever t ∈ [t0 −τ r02 , t0 ] and dt (x, x0 ) ≤ r40 , where τ and K are the positive constants in Theorem 7.5.2. Note that t0 r0−2 ≥ r¯−2 → +∞ as r¯ → 0. By combining with the pinching assumption we have Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R 1 ≥ − r0−2 2
468
H.-D. CAO AND X.-P. ZHU
in the region P (x0 , t0 , r40 , −τ r02 ) = {(x, t) | x ∈ Bt (x0 , r40 ), t ∈ [t0 − τ r02 , t0 ]}, provided r¯ > 0 is sufficiently small. Thus we get the estimate |Rm| ≤ K ′ r0−2 in P (x0 , t0 , r40 , −τ r02 ), where K ′ is a positive constant depending only on w and ε. We can now apply Theorem 7.5.1 (iii) to conclude that e −2 R(x, t) ≤ Kr 0
e is a positive constant depending whenever t ∈ [t0 − τ2 r02 , t0 ] and dt (x, x0 ) ≤ r0 , where K only on w and ε. By using the pinching assumption again we further have 1 Rm(x, t) ≥ − r0−2 2
in the region P (x0 , t0 , r0 , − τ2 r02 ) = {(x, t) | x ∈ Bt (x0 , r0 ), t ∈ [t0 − τ2 r02 , t0 ]}, as long as r¯ is sufficiently small. In particular, this would imply min{Rm(x, t0 ) | x ∈ Bt0 (x0 , r0 )} > −r0−2 , which is a contradiction. Remark 7.5.4. In section 7.3 of [104], Perelman claimed a stronger statement than the above Corollary 7.5.3 that allows r0 < θ−1 h in the assumptions. Nevertheless, the above weaker statement is sufficient to deduce the geometrization result. 7.6. Long Time Behavior. In Section 5.3, we obtained the long time behavior for smooth (compact) solutions to the three-dimensional Ricci flow with bounded normalized curvature. The purpose of this section is to adapt the arguments there to solutions of the Ricci flow with surgery and to drop the bounded normalized curvature assumption. Recall from Corollary 7.4.4 that we have completely understood the topological structure of a compact, orientable three-manifold with nonnegative scalar curvature. From now on we assume that our initial manifold does not admit any metric with nonnegative scalar curvature, and that once we get a compact component with nonnegative scalar curvature, it is immediately removed. Furthermore, if a solution to the Ricci flow with surgery becomes extinct in a finite time, we have also obtained the topological structure of the initial manifold. So in the following we only consider those solutions to the Ricci flow with surgery which exist for all times t ≥ 0. Let gij (t), 0 ≤ t < +∞, be a solution to the Ricci flow with δ-cutoff surgeries, constructed by Theorem 7.4.3 with normalized initial data. Let 0 < t1 < t2 < · · · < tk < · · · be the surgery times, where each δ-cutoff at a time tk has δ = δ(tk ) ≤ e k ), re(2tk )}. On each time interval (tk−1 , tk ) (denote by t0 = 0), the scalar min{δ(t curvature satisfies the evolution equation ◦ ∂ 2 R = ∆R + 2| Ric |2 + R2 ∂t 3
(7.6.1)
◦
where Ric is the trace-free part of Ric . Then Rmin (t), the minimum of the scalar curvature at the time t, satisfies 2 2 d Rmin (t) ≥ Rmin (t) dt 3
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
469
for t ∈ (tk−1 , tk ), for each k = 1, 2, . . .. Since our surgery procedure had removed all components with nonnegative scalar curvature, the minimum Rmin (t) is negative for all t ∈ [0, +∞). Also recall that the cutoff surgeries were performed only on δ-necks. Thus the surgeries do not occur at the parts where Rmin (t) are achieved. So the differential inequality 2 2 d Rmin (t) ≥ Rmin (t) dt 3 holds for all t ≥ 0, and then by normalization, Rmin (0) ≥ −1, we have (7.6.2)
3 1 Rmin (t) ≥ − · , for all t ≥ 0. 2 t + 23
Meanwhile, on each time interval (tk−1 , tk ), the volume satisfies the evolution equation Z d V = − RdV dt and then by (7.6.2), d 3 1 V ≤ · V. dt 2 (t + 32 ) Since the cutoff surgeries do not increase volume, we thus have − 32 ! 3 d ≤0 (7.6.3) log V (t) t + dt 2 3
for all t ≥ 0. Equivalently, the function V (t)(t + 32 )− 2 is nonincreasing on [0, +∞). 3 We can now use the monotonicity of the function V (t)(t + 32 )− 2 to extract the information of the solution at large times. On each time interval (tk−1 , tk ), we have − 23 ! d 3 log V (t) t + dt 2 ! Z 1 3 + = − Rmin (t) + (Rmin (t) − R)dV. V M 2 t + 32
Then by noting that the cutoff surgeries do not increase volume, we get ( Z t V (0) 3 V (t) (7.6.4) − Rmin (t) + dt 3 ≤ 3 exp 2(t + 23 ) (t + 32 ) 2 ( 23 ) 2 0 ) Z t Z 1 − (R − Rmin (t))dV dt 0 V M
for all t > 0. Now by this inequality and the equation (7.6.1), we obtain the following consequence. Lemma 7.6.1. Let gij (t) be a solution to the Ricci flow with surgery, constructed by Theorem 7.4.3 with normalized initial data. If for a fixed 0 < r < 1 and a sequence of times tα → ∞, the rescalings of the solution on the parabolic neighborhoods
470
H.-D. CAO AND X.-P. ZHU
√ √ P (xα , tα , r tα , −r2 tα ) = {(x, t) | x ∈ Bt (xα , r tα ), t ∈ [tα − r2 tα , tα ]}, with factor (tα )−1 and shifting the times tα to 1, converge in the C ∞ topology to some smooth limiting solution, defined in an abstract parabolic neighborhood P (¯ x, 1, r, −r2 ), then this limiting solution has constant sectional curvature −1/4t at any time t ∈ [1 − r2 , 1]. In the previous section we obtained several curvature estimates for the solutions to the Ricci flow with surgery. Now we combine the curvature estimates with the above lemma to derive the following asymptotic result for the curvature. Lemma 7.6.2 (Perelman [104]). For any ε > 0, let gij (t), 0 ≤ t < +∞, be a solution to the Ricci flow with surgery, constructed by Theorem 7.4.3 with normalized initial data. (i) Given w > 0, r > 0, ξ > 0, √ one can find T = T (w, r, ξ, ε) < +∞ such that if the geodesic ball Bt0 (x0 , r t0 ) at some time t0 ≥ T has volume at least 3
wr3 t02 and the sectional curvature at least −r−2 t−1 0 , then the curvature at x0 at time t = t0 satisfies (7.6.5)
|2tRij + gij | < ξ.
(ii) Given in addition 1 ≤ A < ∞ and √ allowing T to depend on A, we can ensure (7.6.5) for all points in Bt0 (x0 , Ar t0 ). (iii) The same √ is true for all points in the forward parabolic neighborhood √ P (x0 , t0 , Ar t0 , Ar2 t0 ) , {(x, t) | x ∈ Bt (x0 , Ar t0 ), t ∈ [t0 , t0 + Ar2 t0 ]}. Proof. (i) By the assumptions and the standard volume comparison, we have Vol t0 (Bt0 (x0 , ρ)) ≥ cwρ3
√ for all 0 < ρ ≤ r t0 , where c is a universal positive constant. Let r¯ = r¯(cw, √ ε) be the positive constant in Theorem 7.5.2 and set r = min{r, r ¯ }. On B (x , r t0 )(⊂ 0 t 0 0 0 √ Bt0 (x0 , r t0 )), we have √ Rm ≥ −(r0 t0 )−2 (7.6.6) √ √ and Vol t0 (Bt0 (x0 , r0 t0 )) ≥ cw(r0 t0 )3 . √ √ Obviously, there holds θ−1 h ≤ r0 t0 ≤ r¯ t0 when t0 is large enough, where θ = θ(cw, ε) is the positive constant in Theorem 7.5.2 and h is the maximal cutoff radius for surgeries in [ t20 , t0 ] (if there is no surgery in the time interval [ t20 , t0 ], we take h = 0). Then it follows from Theorem 7.5.2 that the solution is defined and satisfies √ R < K(r0 t0 )−2 √ √ on whole parabolic neighborhood P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ). Here τ = τ (cw, ε) and K = K(cw, ε) are the positive constants in Theorem 7.5.2. By combining with the pinching assumption we have
Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R √ ≥ −const. K(r0 t0 )−2
√ √ in the region P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ). Thus we get the estimate √ (7.6.7) |Rm| ≤ K ′ (r0 t0 )−2
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
471
√ √ on P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ), for some positive constant K ′ = K ′ (w, ε) depending only on w and ε. The curvature estimate (7.6.7) and the volume estimate (7.6.6) ensure that as t0 → +∞ we can take smooth (subsequent) limits for the rescalings of the solution √ √ r0 t0 −1 with factor (t0 ) on parabolic neighborhoods P (x0 , t0 , 4 , −τ (r0 t0 )2 ). Then by applying Lemma 7.6.1, we can find T = T (w, r, ξ, ε) < +∞ such that when t0 ≥ T , there holds
|2tRij + gij | < ξ,
(7.6.8)
√ on P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ), in particular, √
|2tRij + gij |(x0 , t0 ) < ξ. This proves the assertion (i). (ii) In view √ of the above argument, to get the estimate (7.6.5) for all points bound for the scalar curvature on in Bt0 (x0 , Ar t0 ), the key point is to get √ a upper √ the parabolic neighborhood P (x0 , t0 , Ar t0 , −τ (r0 t0 )2 ). After having the estimates (7.6.6) and (7.6.7), one would like to use Theorem 7.5.1(iii) to obtain the desired scalar curvature estimate. Unfortunately it does not work since our r0 may be much larger than the constant r¯(A, ε) there when A is very large. In the following we will use Theorem 7.5.1(ii) to overcome the difficulty. Given 1 ≤ A < +∞, based on (7.6.6) and (7.6.7), we can use Theorem 7.5.1(ii) √ to find a positive constant K1 = K1 (w, r, A, ε) such that each point in B (x , 2Ar t0 ) t0 0 √ with its scalar curvature at least K1 (r t0 )−2 has a canonical neighborhood. We claim that there exists T = T (w, r, A, ε) < +∞ so that when t0 ≥ T , we have √ √ (7.6.9) R < K1 (r t0 )−2 , on Bt0 (x0 , 2Ar t0 ). Argue by contradiction. Suppose not; then there exist a sequence of times tα 0 → pα α α α α α +∞ and sequences of points x0 , x with x ∈ Bt0 (x0 , 2Ar t0 ) and R(xα , tα ) = 0 p −2 ) . Since there exist canonical neighborhoods (ε-necks or ε-caps) around K1 (r tα 0 the points (xα , tα 0 ), there exist positive constants c1 , C2 depending only on ε such that − 12 p α 3 −1 p Vol tα0 (Btα0 (xα , K1 2 (r tα 0 ))) ≥ c1 (K1 (r t0 ))
and
C2−1 K1 (r −1
on Btα0 (xα , K1 2 (r have
−1
p
on Btα0 (xα , K1 2 (r proved that
p
−2 tα ≤ R(x, tα 0 ) ≤ C2 K1 (r 0)
p
−2 tα , 0)
tα 0 )), for all α. By combining with the pinching assumption we
p
Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R p −2 , ≥ −const. C2 K1 (r tα 0)
tα 0 )), for all α. It then follows from the assertion (i) we just lim |2tRij + gij |(xα , tα 0 ) = 0.
α→+∞
472
H.-D. CAO AND X.-P. ZHU
In particular, we have α α tα 0 R(x , t0 ) < −1
for αp sufficiently large. This contradicts our assumption that R(xα , tα 0) = −2 α K1 (r t0 ) . So we have proved assertion (7.6.9). Now by combining (7.6.9) with the pinching assumption as before, we have (7.6.10)
√ Rm ≥ −K2 (r t0 )−2
√ on Bt0 (x0 , 2Ar t0 ), where K2 = K2 (w, r, A, ε) is some positive constant depending only on w, r, A and ε. Thus by (7.6.9) and (7.6.10) we have (7.6.11)
√ √ |Rm| ≤ K1′ (r t0 )−2 , on Bt0 (x0 , 2Ar t0 ),
for some positive constant K1′ = K1′ (w, r, A, ε) depending only on w, r, A and ε. This √ gives us the curvature estimate on Bt0 (x0 , 2Ar t0 ) for all t0 ≥ T (w, r, A, ε). From the arguments in proving the above assertion (i), we have the estimates (7.6.6) and (7.6.7) and the solution is well-defined on the whole parabolic neighbor√ √ hood P (x0 , t0 , r0 4 t0 , −τ (r0 t0 )2 ) for all t0 ≥ T (w, r, A, ε). Clearly we may assume √ 1 that (K1′ )− 2 r < min{ r40 , τ r0 }. Thus by combining with the curvature estimate (7.6.11), we can apply Theorem 7.5.1(i) to get the following volume control (7.6.12)
1 √ 1 √ Vol t0 (Bt0 (x, (K1′ )− 2 r t0 )) ≥ κ((K1′ )− 2 r t0 )3
√ for any x ∈ Bt0 (x0 , Ar t0 ), where κ = κ(w, r, A, ε) is some positive constant depending only on w, r, A and ε. So by using the assertion (i), we see that for t0 ≥ T with T = T (w, r,√ξ, A, ε) large enough, the curvature estimate (7.6.5) holds for all points in Bt0 (x0 , Ar t0 ). (iii) We next want to extend the curvature √ estimate (7.6.5) to all points in the forward parabolic neighborhood P (x0 , t0 , Ar t0 , Ar2 t0 ). Consider the time interval [t0 , t0 + Ar2 t0 ] in the parabolic neighborhood. In assertion (ii), we have obtained the desired estimate (7.6.5) at t = t0 . Suppose estimate (7.6.5) holds on a maximal time interval [t0 , t′ ) with t′ ≤ t0 + Ar2 t0 . This says that we have (7.6.13)
|2tRij + gij | < ξ
√ √ on P (x0 , t0 , Ar t0 , t′ − t0 ) , {(x, t) | x ∈√Bt (x0 , Ar t0 ), t ∈ [t0 , t′ )} so that either there exists a surgery in the√ball Bt′ (x0 , Ar t0 ) at t = t′ , or there holds |2tRij +gij | = ξ somewhere in Bt′ (x0 , Ar t0 ) at t = t′ . Since the Ricci curvature is near − 2t1′ in the geodesic ball, the surgeries cannot occur there. Thus we only need to consider the latter possibility. Recall that the evolution of the length of a curve γ and the volume of a domain Ω are given by Z d Lt (γ) = − Ric (γ, ˙ γ)ds ˙ t dt γ Z d and V olt (Ω) = − RdVt . dt Ω
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
473
By substituting the curvature estimate (7.6.13) into the above two evolution equations and using the volume lower bound (7.6.6), it is not hard to see √ 3 (7.6.14) Vol t′ (Bt′ (x0 , t′ )) ≥ κ′ (t′ ) 2 for some positive constant κ′ = κ′ (w, r, ξ, A, ε) depending only on w, r, ξ, A and ε. Then by the above assertion (ii), the combination of the curvature estimate (7.6.13) and the volume lower bound (7.6.14) √ implies that the curvature estimate (7.6.5) still holds for all points in Bt′ (x0 , Ar t0 ) provided T = T (w, r, ξ, A, ε) is chosen large enough. This is a contradiction. Therefore we have proved assertion (iii). We now state and prove the important Thick-thin decomposition theorem. A more general version (without the restriction on ε) was implicitly claimed by Perelman in [103] and [104]. Theorem 7.6.3 (The Thick-thin decomposition theorem). For any w > 0 and 0 < ε ≤ 21 w, there exists a positive constant ρ = ρ(w, ε) ≤ 1 with the following property. Suppose gij (t) (t ∈ [0, +∞)) is a solution, constructed by Theorem 7.4.3 with e and re(t), to the Ricci flow with the nonincreasing (continuous) positive functions δ(t) surgery and with a compact orientable normalized three-manifold as initial data, where e re(2t)}. Then for any arbitrarily each δ-cutoff at a time t has δ = δ(t) ≤ min{δ(t), fixed ξ > 0, for t large enough, the manifold Mt at time t admits a decomposition Mt = Mthin (w, t) ∪ Mthick(w, t) with the following properties: √ (a) For √ every x ∈ Mthin(w, t), there exists some r = r(x, t) > 0, with 0 < r t < ρ t, such that √ √ Rm ≥ −(r t)−2 on Bt (x, r t), and √ √ Vol t (Bt (x, r t)) < w(r t)3 . (b) For every x ∈ Mthick (w, t), we have
√ |2tRij + gij | < ξ on Bt (x, ρ t), and √ √ 1 w(ρ t)3 . Vol t (Bt (x, ρ t)) ≥ 10
Moreover, if we take any sequence of points xα ∈ Mthick (w, tα ), tα → +∞, then the scalings of gij (tα ) around xα with factor (tα )−1 converge smoothly, along a subsequence of α → +∞, to a complete hyperbolic manifold of finite volume with constant sectional curvature − 41 . Proof. Let r¯ = r¯(w, ε), θ = θ(w, ε) and h be the positive constants obtained in Corollary 7.5.3. We may assume ρ ≤ r¯ ≤ e−3 . For any point x ∈ Mt , there are two cases: either √ √ (i) min{Rm | Bt (x, ρ t)} ≥ −(ρ t)−2 , or (ii)
√ √ min{Rm | Bt (x, ρ t)} < −(ρ t)−2 .
474
H.-D. CAO AND X.-P. ZHU
√ Let us first consider Case (i). If Vol t (Bt (x, ρ t)) < choose r slightly less than ρ so that √ √ Rm ≥ −(ρ t)−2 ≥ −(r t)−2 √ √ on Bt (x, r t)(⊂ Bt (x, ρ t)), and
1 10 w(ρ
√
t)3 , then we can
√ √ √ 1 w(ρ t)3 < w(r t)3 ; Vol t (Bt (x, r t)) < 10 √ √ 1 w(ρ t)3 , we can apply Lemma 7.6.2(ii) thus x ∈ Mthin (w, t). If Vol t (Bt (x, ρ t)) ≥ 10 to conclude that for t large enough, √ |2tRij + gij | < ξ on Bt (x, ρ t); thus x ∈ Mthick (w, t). Next we consider Case (ii). By continuity, there exists 0 < r = r(x, t) < ρ such that √ √ (7.6.15) min{Rm | Bt (x, r t)} = −(r t)−2 . √ √ If θ−1 h ≤ r t (≤ r¯ t), we can apply Corollary 7.5.3 to conclude √ √ Vol t (Bt (x, r t)) < w(r t)3 ; thus x ∈ Mthin (w, t). √ We now consider the difficult subcase r t < θ−1 h. By the pinching assumption, we have √ √ R ≥ (r t)−2 (log[(r t)−2 (1 + t)] − 3) √ ≥ (log r−2 − 3)(r t)−2 √ ≥ 2(r t)−2 ≥ 2θ2 h−2
√ somewhere in Bt (x, r t). Since h is the maximal cutoff radius for surgeries in [ 2t , t], by the design of the δ-cutoff surgery, we have t ,t h ≤ sup δ 2 (s)e r (s) | s ∈ 2 t t re(t)e r . ≤ δe 2 2
e t ) → 0 as t → +∞. Thus from the canonical neighborhood assumption, Note also δ( 2 √ we see that for t large enough, there exists a point in the ball Bt (x, r t) which has a canonical neighborhood. We claim that for t sufficiently large, the point x satisfies R(x, t) ≥
1 √ −2 (r t) , 2
and then the above argument shows that the point x also has a canonical ε-neck√or ε-cap neighborhood. Otherwise, by continuity, we can choose a point x∗ ∈ Bt (x, r t)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
475
√ with R(x∗ , t) = 12 (r t)−2 . Clearly the new point x∗ has a canonical neighborhood B ∗ by the above argument. In particular, there holds C2−1 (ε)R ≤
1 √ −2 (r t) ≤ C2 (ε)R 2
on the canonical neighborhood B ∗ . By the definition of canonical neighborhood assumption, we have Bt (x∗ , σ ∗ ) ⊂ B ∗ ⊂ Bt (x∗ , 2σ ∗ ) 1
for some σ ∗ ∈ (0, C1 (ε)R− 2 (x∗ , t)). Clearly, without loss of generality, we may assume 1 (in the definition of canonical neighborhood assumption) that σ ∗ > 2R− 2 (x∗ , t). Then 1 −1 C (ε)r−2 2 2 1 ≥ C2−1 (ε)ρ−2 2
R(1 + t) ≥
√ on Bt (x∗ , 2r t). Thus when we choose ρ = ρ(w, ε) small enough, it follows from the pinching assumption that Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R 1 √ ≥ − (r t)−2 , 2 √ on Bt (x∗ , 2r t). This is a contradiction with (7.6.15). We have seen that tR(x, t) ≥ 21 r−2 (≥ 21 ρ−2 ). Since r−2 > θ2 h−2 t in this subcase, we conclude that for arbitrarily given A < +∞, (7.6.16)
tR(x, t) > A2 ρ−2
as long as t is large enough. Let B, with Bt (x, σ) ⊂ B ⊂ Bt (x, 2σ), be the canonical ε-neck or ε-cap neighborhood of (x, t). By the definition of the canonical neighborhood assumption, we have 1
0 < σ < C1 (ε)R− 2 (x, t), C2−1 (ε)R ≤ R(x, t) ≤ C2 (ε)R, on B, and (7.6.17)
Vol t (B) ≤ εσ 3 ≤
1 3 wσ . 2
1
Choose 0 < A < C1 (ε) so that σ = AR− 2 (x, t). For sufficiently large t, since R(1 + t) ≥ C2−1 (ε)(tR(x, t)) 1 ≥ C2−1 (ε)ρ−2 , 2
476
H.-D. CAO AND X.-P. ZHU
on B, we can require ρ = ρ(w, ε) to be smaller still, and use the pinching assumption to conclude (7.6.18)
Rm ≥ −[f −1 (R(1 + t))/(R(1 + t))]R 1
≥ −(AR− 2 (x, t))−2 = −σ −2 ,
on B. For sufficiently large t, we adjust √ (7.6.19) r = σ( t)−1
√ 1 = (AR− 2 (x, t))( t)−1 < ρ,
by (7.6.16). Then the combination of (7.6.17), (7.6.18) and (7.6.19) implies that x ∈ Mthin (w, t). The last statement in (b) follows directly from Lemma 7.6.2. (Here we also used Bishop-Gromov volume comparison, Theorem 7.5.2 and Hamilton’s compactness theorem to take a subsequent limit.) Therefore we have completed the proof of the theorem. To state the long-time behavior of a solution to the Ricci flow with surgery, we first recall some basic terminology in three-dimensional topology. A three-manifold M is called irreducible if every smooth two-sphere embedded in M bounds a threeball in M . If we have a solution (Mt , gij (t)) obtained by Theorem 7.4.3 with a compact, orientable and irreducible three-manifold (M, gij ) as initial data, then at each time t > 0, by the cutoff surgery procedure, the solution manifold Mt consists of a finite number of components where one of the components, called the essential (1) component and denoted by Mt , is diffeomorphic to the initial manifold M while the rest are diffeomorphic to the three-sphere S3 . The main result of this section is the following generalization of Theorem 5.3.4. A more general version of the result (without the restriction on ε) was implicitly claimed by Perelman in [104]. Theorem 7.6.4 (Long-time behavior of the Ricci flow with surgery). Let w > 0 and 0 < ε ≤ 21 w be any small positive constants and let (Mt , gij (t)), 0 < t < +∞, be a solution to the Ricci flow with surgery, constructed by Theorem 7.4.3 with e and re(t) and with a compact, the nonincreasing (continuous) positive functions δ(t) orientable, irreducible and normalized three-manifold M as initial data, where each e re(2t)}. Then one of the following holds: δ-cutoff at a time t has δ = δ(t) ≤ min{δ(t), either (i) for all sufficiently large t, we have Mt = Mthin(w, t); or (ii) there exists a sequence of times tα → +∞ such that the scalings of gij (tα ) on (1) the essential component Mtα , with factor (tα )−1 , converge in the C ∞ topology to a hyperbolic metric on the initial compact manifold M with constant sectional curvature − 41 ; or (iii) we can find a finite collection of complete noncompact hyperbolic threemanifolds H1 , . . . , Hm , with finite volume, and compact subsets K1 , . . . , Km of H1 , . . . , Hm respectively obtained by truncating each cusp of the hyperbolic manifolds along constant mean curvature torus of small area, and for all t beyond some time T < +∞ we can find diffeomorphisms ϕl , 1 ≤ l ≤ m, of Kl
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
477
into Mt so that as long as t is sufficiently large, the metric t−1 ϕ∗l (t)gij (t) is as close to the hyperbolic metric as we like on the compact sets K1 , . . . , Km ; moreover, the complement Mt \(ϕ1 (K1 ) ∪ · · · ∪ ϕm (Km )) is contained in the thin part Mthin (w, t), and the boundary tori of each Kl are incompressible in the sense that each ϕl injects π1 (∂Kl ) into π1 (Mt ). Proof. The proof of the theorem follows, with some modifications, essentially from the same argument of Hamilton as in the proof of Theorem 5.3.4. Clearly we may assume that the thick part Mthick (w, t) is not empty for a sequence tα → +∞, since otherwise we have case (i). If we take a sequence of points xα ∈ Mthick (w, tα ), then by Theorem 7.6.3(b) the scalings of gij (tα ) around xα with factor (tα )−1 converge smoothly, along a subsequence of α → +∞, to a complete hyperbolic manifold of finite volume with constant sectional curvature − 14 . The limits may be different for different choices of (xα , tα ). If a limit is compact, we have case (ii). Thus we assume that all limits are noncompact. Consider all the possible hyperbolic limits of the solution, and among them choose one such complete noncompact hyperbolic three-manifold H with the least possible number of cusps. Denote by hij the hyperbolic metric of H. For all small a > 0 we can truncate each cusp of H along a constant mean curvature torus of area a which is uniquely determined; we denote the remainder by Ha . Fix a > 0 so small that Lemma 5.3.7 is applicable for the compact set K = Ha . Pick an integer l0 sufficiently large and an ǫ0 sufficiently small to guarantee from Lemma 5.3.8 that the identity map Id is the only harmonic map F from Ha to itself with taking ∂Ha to itself, with the normal derivative of F at the boundary of the domain normal to the boundary of the target, and with dC l0 (Ha ) (F, Id) < ǫ0 . Then choose a positive integer q0 and a small number δ0 > 0 from Lemma 5.3.7 such that if Fe is a diffeomorphism of Ha e e into another complete noncompact hyperbolic three-manifold (H, hij ) with no fewer cusps (than H), of finite volume and satisfying kFe ∗e hij − hij kC q0 (Ha ) ≤ δ0 ,
e such that then there exists an isometry I of H to H
dC l0 (Ha ) (Fe , I) < ǫ0 .
By Lemma 5.3.8 we further require q0 and δ0 to guarantee the existence of a harmonic diffeomorphism from (Ha , geij ) to (Ha , hij ) for any metric e gij on Ha with ke gij − hij kC q0 (Ha ) ≤ δ0 . Let xα ∈ Mthick (w, tα ), tα → +∞, be a sequence of points such that the scalings of gij (tα ) around xα with factor (tα )−1 converge to hij . Then there exist a marked point x∞ ∈ Ha and a sequence of diffeomorphisms Fα from Ha into Mtα such that Fα (x∞ ) = xα and k(tα )−1 Fα∗ gij (tα ) − hij kC m (Ha ) → 0 as α → ∞ for all positive integers m. By applying Lemma 5.3.8 and the implicit function theorem, we can change Fα by an amount which goes to zero as α → ∞ so as to make Fα a harmonic diffeomorphism taking ∂Ha to a constant mean curvature hypersurface Fα (∂Ha ) of (Mtα , (tα )−1 gij (tα )) with the area a and satisfying the free boundary condition that the normal derivative of Fα at the boundary of the domain is normal to the boundary of the target; and by combining with Lemma 7.6.2 (iii),
478
H.-D. CAO AND X.-P. ZHU
we can smoothly continue each harmonic diffeomorphism Fα forward in time a little to a family of harmonic diffeomorphisms Fα (t) from Ha into Mt with the metric t−1 gij (t), with Fα (tα ) = Fα and with the time t slightly larger than tα , where Fα (t) takes ∂Ha into a constant mean curvature hypersurface of (Mt , t−1 gij (t)) with the area a and also satisfies the free boundary condition. Moreover, since the surgeries do not take place at the points where the scalar curvature is negative, by the same argument as in Theorem 5.3.4 for an arbitrarily given positive integer q ≥ q0 , positive number δ < δ0 , and sufficiently large α, we can ensure the extension Fα (t) satisfies kt−1 Fα∗ (t)gij (t) − hij kC q (Ha ) ≤ δ on a maximal time interval tα ≤ t ≤ ω α (or tα ≤ t < ω α when ω α = +∞), and with k(ω α )−1 Fα∗ (ω α )gij (ω α ) − hij kC q (Ha ) = δ, when ω α < +∞. Here we have implicitly used the fact that Fα (ω α )(∂Ha ) is still strictly concave to ensure the map Fα (ω α ) is diffeomorphic. We further claim that there must be some α such that ω α = +∞; in other words, at least one hyperbolic piece persists. Indeed, suppose that for each large enough α we can only continue the family Fα (t) on a finite interval tα ≤ t ≤ ω α < +∞ with k(ω α )−1 Fα∗ (ω α )gij (ω α ) − hij kC q (Ha ) = δ. Consider the new sequence of manifolds (Mωα , gij (ω α )). Clearly by Lemma 7.6.1, the scalings of gij (ω α ) around the new origins Fα (ω α )(x∞ ) with factor (ω α )−1 converge smoothly (by passing to a subsequence) to a complete noncompact hyperbolic threee with the metric e manifold H hij and the origin x e∞ and with finite volume. By the e choice of the old limit H, the new limit H has at least as many cusps as H. By the eα exhausting definition of convergence, we can find a sequence of compact subsets U ∞ e e eα H and containing x e , and a sequence of diffeomorphisms Fα of neighborhood of U ∞ α ∞ e e e into Mωα with Fα (e x ) = Fα (ω )(x ) such that for each compact subset U of H and each integer m, k(ω α )−1 Feα∗ (gij (ω α )) − e hij kC m (Ue ) → 0
as α → +∞. Thus for sufficiently large α, we have the map such that
e Gα = Feα−1 ◦ Fα (ω α ) : Ha → H kG∗α e hij − hij kC q (Ha ) < δe
for any fixed δe > δ. Then a subsequence of Gα converges at least in the C q−1 (Ha ) e which is a harmonic map from Ha into H e and topology to a map G∞ of Ha into H e e takes ∂Ha to a constant mean curvature hypersurface G∞ (∂Ha ) of (H, hij ) with the area a, as well as satisfies the free boundary condition. Clearly, G∞ is at least a local diffeomorphism. Since G∞ is the limit of diffeomorphisms, the only possibility of overlap is at the boundary. Note that G∞ (∂Ha ) is still strictly concave. So G∞ is still a diffeomorphism. Moreover by using the standard regularity result of elliptic partial differential equations (see for example [48]), we also have (7.6.20)
kG∗∞e hij − hij kC q (Ha ) = δ.
e with Now by Lemma 5.3.7 we deduce that there exists an isometry I of H to H dC l0 (Ha ) (G∞ , I) < ǫ0 .
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
479
Thus I −1 ◦ G∞ is a harmonic diffeomorphism of Ha to itself which satisfies the free boundary condition and dC l0 (Ha ) (I −1 ◦ G∞ , Id) < ǫ0 . However the uniqueness in Lemma 5.3.8 concludes that I −1 ◦ G∞ = Id which contradicts (7.6.20). So we have shown that at least one hyperbolic piece persists and the metric t−1 Fα∗ (t)gij (t), for ω α ≤ t < ∞, is as close to the hyperbolic metric hij as we like. We can continue to form other persistent hyperbolic pieces in the same way as long as there is a sequence of points y β ∈ Mthick(w, tβ ), tβ → +∞, lying outside the chosen 3 pieces. Note that V (t)(t + 23 )− 2 is nonincreasing on [0, +∞). Therefore by combining with Margulis lemma (see for example [55] or [76]), we have proved that there exists a finite collection of complete noncompact hyperbolic three-manifolds H1 , . . . , Hm with finite volume, a small number a > 0 and a time T < +∞ such that for all t beyond T we can find diffeomorphisms ϕl (t) of (Hl )a into Mt , 1 ≤ l ≤ m, so that as long as t is sufficiently large, the metric t−1 ϕ∗l (t)gij (t) is as close to the hyperbolic metrics as we like and the complement Mt \(ϕ1 (t)((H1 )a ) ∪ · · · ∪ ϕm (t)((Hm )a )) is contained in the thin part Mthin (w, t). It remains to show the boundary tori of any persistent hyperbolic piece are incompressible. Let B be a small positive number and assume the above positive number a is much smaller than B. Let Ma (t) = ϕl (t)((Hl )a ) (1 ≤ l ≤ m) be such a persistent hyperbolic piece of the manifold Mt truncated by boundary tori of area at ◦
with constant mean curvature, and denote by Mac (t) = Mt \ M a (t) the part of Mt exterior to Ma (t). Thus there exists a family of subsets MB (t) ⊂ Ma (t) which is a persistent hyperbolic piece of the manifold Mt truncated by boundary tori of area Bt ◦
with constant mean curvature. We also denote by MBc (t) = Mt \ M B (t). By Van Kampen’s Theorem, if π1 (∂MB (t)) injects into π1 (MBc (t)) then it injects into π1 (Mt ) also. Thus we only need to show π1 (∂MB (t)) injects into π1 (MBc (t)). As before we will use a contradiction argument to show π1 (∂MB (t)) injects into π1 (MBc (t)). Let T be a torus in ∂MB (t) and suppose π1 (T ) does not inject into π1 (MBc (t)). By Dehn’s Lemma we know that the kernel is a cyclic subgroup of π1 (T ) generated by a primitive element. Consider the normalized metric geij (t) = t−1 gij (t) on Mt . Then by the work of Meeks-Yau [86] or Meeks-Simon-Yau [87], we know that among all disks in MBc (t) whose boundary curve lies in T and generates the kernel of π1 (T ), there is a smooth embedded disk normal to the boundary which has the least possible area (with respect to the normalized metric e gij (t)). Denote by D the e = A(t) e its area. We will show that A(t) e decreases at a certain minimal disk and A rate which will arrive at a contradiction. We first consider the case that there exist no surgeries at the time t. Exactly as e comes from the in Part III of the proof of Theorem 5.3.4, the change of the area A(t) change in the metric and the change in the boundary. For the change in the metric, we choose an orthonormal frame X, Y, Z at a point x in the disk D so that X and Y are tangent to the disk D while Z is normal. Since the normalized metric e gij evolves by ∂ eij ), geij = −t−1 (e gij + 2R ∂t
480
H.-D. CAO AND X.-P. ZHU
the (normalized) area element de σ of the disk D around x satisfies ∂ g (X, X) + Ric g (Y, Y ))de de σ = −t−1 (1 + Ric σ. ∂t
eij is very small for For the change in the boundary, we notice that the tensor geij + 2R the persistent hyperbolic piece. Then by using the Gauss-Bonnet theorem as before, we obtain the rate of change of the area ! Z Z e e 1 R 1 2π 1 e dA e ≤− + de σ+ kde s− +o L, (7.6.21) dt t 2t t ∂D t t D
e is the length of the boundary where e k is the geodesic curvature of the boundary and L e ≥ −3t/2(t + 3 ) for curve ∂D (with respect to the normalized metric geij (t)). Since R 2 all t ≥ 0 by (7.6.2), the first term on the RHS of (7.6.21) is bounded above by ! Z e R 1 1 1 e + de σ≤− − o(1) A; − t 2t t 4 D while the second term on the RHS of (7.6.21) can be estimated exactly as before by Z 1 1 1 e e kde s≤ + o(1) L. t ∂D t 4
Thus we obtain (7.6.22)
e dA 1 ≤ dt t
1 1 e e + o(1) L − − o(1) A − 2π . 4 4
Next we show that these arguments also work for the case that there exist surgeries at the time t. To this end, we only need to check that the embedded minimal disk D lies in the region which is unaffected by surgery. Our surgeries for the irreducible three-manifold took place on δ-necks in ε-horns, where the scalar curvatures are at least δ −2 (e r (t))−1 , and the components with nonnegative scalar curvature have been removed. So the hyperbolic piece is not affected by the surgeries. In particular, the boundary ∂D is unaffected by the surgeries. Thus if surgeries occur on the minimal disk, the minimal disk has to pass through a long thin neck before it reaches the surgery regions. Look at the intersections of the embedding minimal disk with a generic center two-sphere S2 of the long thin neck; these are circles. Since the twosphere S2 is simply connected, we can replace the components of the minimal disk D outside the center two-sphere S2 by some corresponding components on the center two-sphere S2 to form a new disk which also has ∂D as its boundary. Since the metric on the long thin neck is nearly a product metric, we could choose the generic center two-sphere S2 properly so that the area of the new disk is strictly less than the area of the original disk D. This contradiction proves the minimal disk lies entirely in the region unaffected by surgery. Since a is much smaller than B, the region within a long distance from ∂MB (t) into MBc (t) will look nearly like a hyperbolic cusplike collar and is unaffected by the surgeries. So we can repeat the arguments in the last part of the proof of Theorem e by the area A e and to conclude 5.3.4 to bound the length L e π dA ≤− dt t
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
481
for all sufficiently large times t, which is impossible because the RHS is not integrable. This proves that the boundary tori of any persistent hyperbolic piece are incompressible. Therefore we have proved the theorem. 7.7. Geometrization of Three-manifolds. In the late 70’s and early 80’s, Thurston [122], [123] [124] proved a number of remarkable results on the existence of geometric structures on a class of three-manifolds: Haken manifolds (i.e. each of them contains an incompressible surface of genus ≥ 1). These results motivated him to formulate a profound conjecture which roughly says every compact three-manifold admits a canonical decomposition into domains, each of which has a canonical geometric structure. To give a detailed description of the conjecture, we recall some terminology as follows. An n-dimensional complete Riemannian manifold (M, g) is called a homogeneous manifold if its group of isometries acts transitively on the manifold. This means that the homogeneous manifold looks the same metrically at everypoint. For example, the round n-sphere Sn , the Euclidean space Rn and the standard hyperbolic space Hn are homogeneous manifolds. A Riemannian manifold is said to be modeled on a given homogeneous manifold (M, g) if every point of the manifold has a neighborhood isometric to an open set of (M, g). And an n-dimensional Riemannian manifold is called a locally homogeneous manifold if it is complete and is modeled on a homogeneous manifold. By a theorem of Singer [119], the universal cover of a locally homogeneous manifold (with the pull-back metric) is a homogeneous manifold. In dimension three, every locally homogeneous manifold with finite volume is modeled on one of the following eight homogeneous manifolds (see for example Theorem 3.8.4 of [125]): (1) S3 , the round three-sphere; (2) R3 , the Euclidean space ; (3) H3 , the standard hyperbolic space; (4) S2 × R; (5) H2 × R; (6) N il, the three-dimensional nilpotent Heisenberg group (consisting of upper triangular 3 × 3 matrices with diagonal entries 1); ] (7) P SL(2, R), the universal cover of the unit sphere bundle of H2 ; (8) Sol, the three-dimensional solvable Lie group. A three-manifold M is called prime if it is not diffeomorphic to S3 and if every (topological) S2 ⊂ M , which separates M into two pieces, has the property that one of the two pieces is diffeomorphic to a three-ball. Recall that a three-manifold is irreducible if every embedded two-sphere bounds a three-ball in the manifold. Clearly an irreducible three-manifold is either prime or is diffeomorphic to S3 . Conversely, an orientable prime three-manifold is either irreducible or is diffeomorphic to S2 × S1 (see for example [69]). One of the first results in three-manifold topology is the following prime decomposition theorem obtained by Kneser [79] in 1929 (see also Theorem 3.15 of [69]). Prime Decomposition Theorem. Every compact orientable three-manifold admits a decomposition as a finite connected sum of orientable prime three-manifolds. In [90], Milnor showed that the factors involved in the above Prime Decomposition are unique. Based on the prime decomposition, the question about topology
482
H.-D. CAO AND X.-P. ZHU
of compact orientable three-manifolds is reduced to the question about prime threemanifolds. Thurston’s Geometrization Conjecture is about prime three-manifolds. Thurston’s Geometrization Conjecture. Let M be a compact, orientable and prime three-manifold. Then there is an embedding ` of a finite number of disjoint unions, possibly empty, of incompressible two-tori i Ti2 ⊂ M such that every component of the complement admits a locally homogeneous Riemannian metric of finite volume. We remark that the existence of a torus decomposition, also called JSJdecomposition, was already obtained by Jaco-Shalen [74] and Johannsen [75]. The JSJ-decomposition states that any compact, orientable, and prime three-manifold has a finite collection, possibly empty, of disjoint incompressible embedding two-tori {Ti2} which separate the manifold into a finite collection of compact three-manifolds (with toral boundary), each of which is either a graph manifold or is atoroidal in the sense that any subgroup of its fundamental group isomorphic to Z × Z is conjugate into the fundamental group of some component of its boundary. A compact three-manifold X, possibly with boundary, is called a graph manifold if there is S a finite collection of disjoint embedded tori Ti ⊂ X such that each component of X \ Ti is an S1 bundle over a surface. Thus the point of the conjecture is that the components should all be geometric. The geometrization conjecture for a general compact orientable 3-manifold is the statement that each of its prime factors satisfies the above conjecture. We say a compact orientable three-manifold is geometrizable if it satisfies the geometric conjecture. We also remark that the Poincar´e conjecture can be deduced from Thurston’s geometrization conjecture. Indeed, suppose that we have a compact simply connected three-manifold that satisfies the conclusion of the geometrization conjecture. If it were not diffeomorphic to the three-sphere S3 , there would be a prime factor in the prime decomposition of the manifold. Since the prime factor still has vanishing fundamental group, the (torus) decomposition of the prime factor in the geometrization conjecture must be trivial. Thus the prime factor is a compact homogeneous manifold model. From the list of above eight models, we see that the only compact three-dimensional model is S3 . This is a contradiction. Consequently, the compact simply connected three-manifold is diffeomorphic to S3 . Now we apply the Ricci flow to discuss Thurston’s geometrization conjecture. Let M be a compact, orientable and prime three-manifold. Since a prime orientable three-manifold is either irreducible or is diffeomorphic to S2 × S1 , we may thus assume the manifold M is irreducible also. Arbitrarily given a (normalized) Riemannian metric for the manifold M , we use it as initial data to evolve the metric by the Ricci flow with surgery. From Theorem 7.4.3, we know that the Ricci flow with surgery has a long-time solution on a maximal time interval [0, T ) which satisfies the a priori assumptions and has a finite number of surgeries on each finite time interval. Furthermore, from the long-time behavior theorem (Theorem 7.6.4), we have wellunderstood geometric structures on the thick part. Whereas, to understand the thin part, Perelman announced the following assertion in [104]. α Perelman’s Claim ([104]). Suppose (M α , gij ) is a sequence of compact orientable three-manifolds, closed or with convex boundary, and wα → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα (x), 0 < ρ < 1, not exceeding the diameter of the manifold, such that the ball B(x, ρ) in the
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
483
α metric gij has volume at most wα ρ3 and sectional curvatures at least −ρ−2 ; (2) each component of the boundary of M α has diameter at most wα , and has a (topologically trivial) collar of length one, where the sectional curvatures are between −1/4 − ǫ and −1/4 + ǫ. Then M α for sufficiently large α are diffeomorphic to graph manifolds.
The topology of graph manifolds is well understood; in particular, every graph manifold is geometrizable (see [126]). The proof of Perelman’s Claim promised in [104] is still not available in literature. Nevertheless, recently in [118], Shioya and Yamaguchi provided a proof of Perelman’s α Claim for the special case that all the manifolds (M α , gij ) are closed. That is, they proposed a proof for the following weaker assertion. Weaker Assertion (Theorem 8.1 of Shioya-Yamaguchi [118]). Suppose α (M α , gij ) is a sequence of compact orientable three-manifolds without boundary, and wα → 0. Assume that for each point x ∈ M α there exists a radius ρ = ρα (x), not exceeding the diameter of the manifold, such that the ball B(x, ρ) in the metric α gij has volume at most wα ρ3 and sectional curvatures at least −ρ−2 . Then M α for sufficiently large α are diffeomorphic to graph manifolds. Based on the the long-time behavior theorem (Theorem 7.6.4) and assuming the above Weaker Assertion, we can now give a proof for Thurston’s geometrization conjecture . We remark that if we assume the above Perelman’s Claim, then we does not need to use Thurston’s theorem for Haken manifolds in the proof of Theorem 7.7.1. Theorem 7.7.1. Thurston’s geometrization conjecture is true. Proof. Let M be a compact, orientable, and prime three-manifold (without boundary). Without loss of generality, we may assume that the manifold M is irreducible also. Recall that the theorem of Thurston (see for example Theorem A and Theorem B in Section 3 of [94], see also [85] and [102]) says that any compact, orientable, and irreducible Haken three-manifold (with or without boundary) is geometrizable. Thus in the following, we may assume that the compact three-manifold M (without boundary) is atoroidal, and then the fundamental group π1 (M ) contains no noncyclic, abelian subgroup. Arbitrarily given a (normalized) Riemannian metric on the manifold M , we use it as initial data for the Ricci flow. Arbitrarily take a sequence of small positive constants wα → 0 as α → +∞. For each fixed α, we set ε = wα /2 > 0. Then by α Theorem 7.4.3, the Ricci flow with surgery has a long-time solution (Mtα , gij (t)) on α a maximal time interval [0, T ), which satisfies the a priori assumptions (with the accuracy parameter ε = wα /2) and has a finite number of surgeries on each finite time interval. Since the initial manifold is irreducible, by the surgery procedure, we know that for each α and each t > 0 the solution manifold Mtα consists of a finite number of components where the essential component (Mtα )(1) is diffeomorphic to the initial manifold M and the others are diffeomorphic to the three-sphere S3 . α0 If for some α = α0 the maximal time T α0 is finite, then the solution (Mtα0 , gij (t)) α0 becomes extinct at T and the (irreducible) initial manifold M is diffeomorphic to S3 /Γ (the metric quotients of round three-sphere); in particular, the manifold M is geometrizable. Thus we may assume that the maximal time T α = +∞ for all α. We now apply the long-time behavior theorem (Theorem 7.6.4). If there is some α such that case (ii) of Theorem 7.6.4 occurs, then for some sufficiently large time
484
H.-D. CAO AND X.-P. ZHU
t, the essential component (Mtα )(1) of the solution manifold Mtα is diffeomorphic to a compact hyperbolic space, so the initial manifold M is geometrizable. Whereas if there is some sufficiently large α such that case (iii) of Theorem 7.6.4 occurs, then it follows that for all sufficiently large t, there is an embedding of a (nonempty) finite ` number of disjoint unions of incompressible two-tori i Ti2 in the essential component (Mtα )(1) of Mtα . This is a contradiction since we have assumed the initial manifold M is atoroidal. It remains to deal with the situation that there is a sequence of positive αk → αk +∞ such that the solutions (Mtαk , gij (t)) always satisfy case (i) of Theorem 7.6.4. αk That is, for each αk , Mt = Mthin (wαk , t) when the time t is sufficiently large. By the Thick-thin decomposition theorem (Theorem 7.6.3), there is a positive constant, αk 0 < ρ(wαk ) ≤ 1, such that as long as t is sufficiently √ large,α for √ every x ∈ Mt = αk k Mthin (w , t), we have some r = r(x, t), with 0 < r t < ρ(w ) t, such that √ √ (7.7.1) Rm ≥ −(r t)−2 on Bt (x, r t), and (7.7.2)
√ √ Vol t (Bt (x, r t)) < wαk (r t)3 .
Clearly we only need to consider the essential component (Mtαk )(1) . We divide the discussion into the following two cases: (1) there is a positive constant 1 < C < +∞ such that for each αk there is a sufficiently large time tk > 0 such that √ (7.7.3) r(x, tk ) tk < C · diam (Mtαkk )(1)
for all x ∈ (Mtαkk )(1) ⊂ Mthin (wαk , tk ); (2) there are a subsequence αk (still denoted by αk ), and sequences of positive constants Ck → +∞ and times Tk < +∞ such that for each t ≥ Tk , we have √ (7.7.4) r(x(t), t) t ≥ Ck · diam (Mtαk )(1)
for some x(t) ∈ (Mtαk )(1) , k = 1, 2, . . . . Here we denote by diam ((Mtα )(1) ) the diamα eter of the essential component (Mtα )(1) with the metric gij (t) at the time t. αk (1) Let us first consider case (1). For each point x ∈ (M ) ⊂ Mthin (wαk , tk ), we tk √ −1 denote by ρk (x) = C r(x, tk ) tk . Then by (7.7.1), (7.7.2) and (7.7.3), we have ρk (x) < diam (Mtαkk )(1) , √ Vol tk (Btk (x, ρk (x))) ≤ Vol tk (Btk (x, r(x, tk ) tk )) < C 3 wαk (ρk (x))3 , and √ Rm ≥ −(r(x, tk ) tk )−2 ≥ −(ρk (x))−2
on Btk (x, ρk (x)). Then it follows from the above Weaker Assertion that (Mtαkk )(1) , for sufficiently large k, are diffeomorphic to graph manifolds. This implies that the (irreducible) initial manifold M is diffeomorphic to a graph manifold. So the manifold M is geometrizable in case (1).
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
485
We next consider case (2). Clearly, for each αk and the chosen Tk , we may assume that the estimates (7.7.1) and (7.7.2) hold for all t ≥ Tk and x ∈ (Mtαk )(1) . The combination of (7.7.1) and (7.7.4) gives (7.7.5)
Rm ≥ −Ck−2 (diam ((Mtαk )(1) ))−2
on (Mtαk )(1) ,
for all t ≥ Tk . If there are a subsequence αk (still denoted by αk ) and a sequence of times tk ∈ (Tk , +∞) such that (7.7.6)
Vol tk ((Mtαkk )(1) ) < wk′ (diam ((Mtαkk )(1) ))3
for some sequence wk′ → 0, then it follows from the Weaker Assertion that (Mtαkk )(1) , for sufficiently large k, are diffeomorphic to graph manifolds which implies the initial manifold M is geometrizable. Thus we may assume that there is a positive constant w′ such that (7.7.7)
Vol t ((Mtαk )(1) ) ≥ w′ (diam ((Mtαk )(1) ))3
for each k and all t ≥ Tk . In view of the estimates (7.7.5) and (7.7.7), we now want to use Theorem 7.5.2 to get a uniform upper bound for the curvatures of the essential components αk ((Mtαk )(1) , gij (t)) with sufficiently large time t. Note that the estimate in Theorem 7.5.2 depends on the parameter ε and our ε’s depend on wαk with 0 < ε = wαk /2; so it does not work in the present situation. Fortunately we notice that the curvature estimate for smooth solutions in Corollary 7.2.3 is independent of ε. In the following we try to use Corollary 7.2.3 to obtain the desired curvature estimate. We first claim that for each k, there is a sufficiently large Tk′ ∈ (Tk , +∞) such αk that the solution, when restricted to the essential component ((Mtαk )(1) , gij (t)), has ′ no surgery for all t ≥ Tk . Indeed, for each fixed k, if there is a δ(t)-cutoff surgery αk at a sufficiently large time t, then the manifold ((Mtαk )(1) , gij (t)) would contain a −1 αk (1) − 21 δ(t)-neck Bt (y, δ(t) R(y, t) ) for some y ∈ (Mt ) with the volume ratio (7.7.8)
Vol t (Bt (y, δ(t)
−1
1
R(y, t)− 2 )) − 21
(δ(t)−1 R(y, t)
)3
2
≤ 8πδ(t) .
On the other hand, by (7.7.5) and (7.7.7), the standard Bishop-Gromov volume comparison implies that Vol t (Bt (y, δ(t) (δ(t)
−1
−1
1
R(y, t)− 2 )) 1
R(y, t)− 2 )3
≥ c(w′ )
for some positive constant c(w′ ) depending only on w′ . Since δ(t) is very small when t is large, this arrives at a contradiction with (7.7.8). So for each k, the essential αk component ((Mtαk )(1) , gij (t)) has no surgery for all sufficiently large t. αk For each k, we consider any fixed time t˜k > 3Tk′ . Let us scale the solution gij (t) αk (1) on the essential component (Mt ) by αk αk g˜ij (·, s) = (t˜k )−1 gij (·, t˜k s).
Note that (Mtαk )(1) is diffeomorphic to M for all t. By the above claim, we see that αk the rescaled solution (M, g˜ij (·, s)) is a smooth solution to the Ricci flow on the time 1 interval s ∈ [ 2 , 1]. Set p −1 t˜k diam (Mt˜αk )(1) . r˜k = k
486
H.-D. CAO AND X.-P. ZHU
Then by (7.7.4), (7.7.5) and (7.7.7), we have r˜k ≤ Ck−1 → 0,
and
g ≥ −C −2 (˜ Rm rk )−2 , k
as k → +∞, on B1 (x(t˜k ), r˜k ),
Vol 1 (B1 (x(t˜k ), r˜k )) ≥ w′ (˜ rk )3 , g is the rescaled curvature, x(t˜k ) is the point given by (7.7.4) and where Rm B1 (x(t˜k ), r˜k ) is the geodesic ball of rescaled solution at the time s = 1. Moreover, the αk closure of B1 (x(t˜k ), r˜k ) is the whole manifold (M, g˜ij (·, 1)). Note that in Theorem 7.2.1, Theorem 7.2.2 and Corollary 7.2.3, the condition about normalized initial metrics is just to ensure that the solutions satisfy the αk Hamilton-Ivey pinching estimate. Since our solutions (Mtαk , gij (t)) have already satisfied the pinching assumption, we can then apply Corollary 7.2.3 to conclude g |Rm(x, s)| ≤ K(w′ )(˜ rk )−2 ,
αk whenever s ∈ [1 − τ (w′ )(˜ rk )2 , 1], x ∈ (M, g˜ij (·, s)) and k is sufficiently large. Here ′ ′ K(w ) and τ (w ) are positive constants depending only on w′ . Equivalently, we have the curvature estimates
(7.7.9)
|Rm(·, t)| ≤ K(w′ )(diam ((Mt˜αk )(1) ))−2 ,
on M,
k
whenever t ∈ [t˜k − τ (w′ )(diam ((Mt˜αk )(1) ))2 , t˜k ] and k is sufficiently large. k
αk For each k, let us scale ((Mtαk )(1) , gij (t)) with the factor (diam((Mt˜αk )(1) ))−2 and shift k the time t˜k to the new time zero. By the curvature estimate (7.7.9) and Hamilton’s compactness theorem (Theorem 4.1.5), we can take a subsequential limit (in the C ∞ topology) and get a smooth solution to the Ricci flow on M × (−τ (w′ ), 0]. Moreover, by (7.7.5), the limit has nonnegative sectional curvature on M × (−τ (w′ ), 0]. Recall that we have removed all compact components with nonnegative scalar curvature. By combining this with the strong maximum principle, we conclude that the limit is a flat metric. Hence in case (2), M is diffeomorphic to a flat manifold and then it is also geometrizable. Therefore we have completed the proof of the theorem.
REFERENCES [1] U. Abresch and W.T. Meyer, Injectivity radius estimates and sphere theorems, Comparison Geometry, MSRI Publications, 30 (1997), pp. 1–48. [2] S. Aloff and N. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structure, Bull. Amer. Math. Soc., 81 (1975), pp. 93–97. [3] W. Ambrose and I.M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc., 75 (1953), pp. 428–443. [4] M. Anderson, Geometrization of 3-manifolds via the Ricci flow, Notices Amer. Math. Soc., 51:2 (2004), pp. 184–193. [5] S. Bando, On the classification of three-dimensional compact K¨ ahler manifolds of nonnegative bisectional curvature, J. Differential Geom., 19:2 (1984), pp. 283–297. [6] J. Bartz, M. Struwe, and R. Ye, A new approach to the Ricci flow on S 2 , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21:3 (1994), pp. 475–482.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
487
[7] M. Berger, Sur les groupes d’holonomie des vari´ et´ es ` a connexion affine et les vari´ et´ es riemanniennes, Bull. Soc. Math. France, 83 (1955), pp. 279–330. [8] Y. Burago, M. Gromov, and G. Perelman, A.D. Alexandrov spaces with curvatures bounded below, Russian Math. Surveys, 47 (1992), pp. 1–58. [9] D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, Amer. Math. Soc. Providence, Rhode Island. [10] R. Bryant, Local existence of gradient Ricci solitons, preprint. [11] H.-D. Cao, Deformation of K¨ ahler metrics to K¨ ahler-Einstein metrics on compact K¨ ahler manifolds, Invent. Math., 81:2 (1985), pp. 359–372. [12] , On Harnack’s inequalities for the K¨ ahler-Ricci flow, Invent. Math., 109:2 (1992), pp. 247–263. , Existence of gradient K¨ ahler-Ricci solitons, Elliptic and Parabolic Methods in Geom[13] etry (Minneapolis, MN, 1994), pp. 1–16, A K Peters, Wellesley, MA, 1996. [14] , Limits of Solutions to the K¨ ahler-Ricci flow, J. Differential Geom., 45 (1997), pp. 257– 272. [15] H.-D. Cao, B.L. Chen, and X.P. Zhu, Ricci flow on compact K¨ ahler manifolds of positive bisectional curvature, C.R. Math. Acad. Sci. Paris, 337:12 (2003), pp. 781–784. [16] H.-D. Cao and B. Chow, Recent developments on the Ricci flow, Bull. Amer. Math. Soc (N.S.), 36:1 (1999), pp. 59–74. [17] H.-D. Cao, B. Chow, S.C. Chu, and S.-T. Yau, Collected Papers on Ricci Flow, Series in Geometry and Topology, 37, International Press, 2003. [18] H.-D. Cao, R.S. Hamilton, and T. Ilmanen, Gaussian densities and stability for some Ricci solitons, arXiv:DG.math.DG/0404165. [19] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math., 92 (1970), pp. 61–74. [20] J. Cheeger and T.H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), 144:1 (1996), pp. 189–237. [21] , On the structure of the spaces with Ricci curvature bounded below, I, J. Differential Geom., 46 (1997), pp. 406–480. [22] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, 1975. [23] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math., 46 (1972), pp. 413–433. [24] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, J. Differential Geom., 23:3 (1986), pp. 309–346. , Collapsing Riemannian manifolds while keeping their curvature bounded, II, J. Dif[25] ferential Geom., 32:1 (1990), pp. 269–298. [26] J. Cheeger, M. Gromov, and K. Fukaya, Mikhael Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc., 5:2 (1992), pp. 327–372. [27] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom., 17 (1982), pp. 15–53. [28] J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Comm. on Pure and Appl. Math., 34 (1981), pp. 465–480. [29] B.L. Chen, S.H. Tang, and X.P. Zhu, A uniformization theorem of complete noncompact K¨ ahler surfaces with positive bisectional curvature, J. Differential Geometry, 67 (2004), pp. 519–570. [30] B.L. Chen and X.P. Zhu, Complete Riemannian manifolds with pointwise pinched curvature, Invent. Math., 140:2 (2000), pp. 423–452. [31] , On complete noncompact K¨ ahler manifolds with positive bisectional curvature, Math. Ann., 327 (2003), pp. 1–23. , Volume growth and curvature decay of positively curved K¨ ahler manifolds, Quarterly [32] Journal of Pure and Applied Mathematics, 1:1 (2005), pp. 68–108. [33] , Uniqueness of the Ricci Flow on Complete Noncompact Manifolds, arXiv:DG (AP) math. DG/0505447, May 2005. , Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature, [34] arXiv:math.DG/0504478, April 2005. [35] S.Y. Cheng, P. Li, and S.-T. Yau, On the upper estimate of the heat kernel of complete Riemannian manifold, Amer. J. Math., 103:5 (1981), pp. 1021–1063. [36] B. Chow, The Ricci flow on 2-sphere, J. Differential Geom., 33 (1991), pp. 325–334. [37] , On the entropic estimate for the Ricci flow on compact 2-orbifolds, J. Differential Geom., 33 (1991), pp. 597–600.
488
H.-D. CAO AND X.-P. ZHU
[38] B. Chow and S.C. Chu, A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow, Math. Res. Lett., 2 (1995), pp. 701–718. [39] B. Chow and D. Knopf, The Ricci flow: An introduction, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 2004. [40] B. Chow and P. Lu, The maximum principle for systems of parabolic equations subject to an avoidance set, Pacific J. Math., 214:2 (2004), pp. 201–222. [41] B. Chow, P. Lu, and L. Ni, Hamilton’s Ricci flow, American Mathematical Society and Science Press, to appear. [42] T.H. Colding and W.P. Minicozzi, II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, J. Amer. Math. Soc., 18:3 (2005), pp. 561–569. [43] D. De Turck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom., 18 (1983), pp. 157–162. [44] G. Drees, Asymptotically flat manifold of nonnegative curvature, Diff. Geom. Appl., 4 (1994), pp. 77–90. [45] J. Eschenburg, V. Schroeder, and M. Strake, Curvature at infinity of open nonnegatively curved manifold, J. Differential Geom., 30 (1989), pp. 155–166. [46] M. Feldman, T. Ilmanen, and D. Knopf, Rotationally symmetric shrinking and expanding K¨ ahler-Ricci solitons, J. Differential Geom., 65 (2003), pp. 169–209. [47] T. Frankel, Manifolds with positive curvature, Pacific J. Math., 11 (1961), pp. 165–174. [48] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [49] R.E. Greene and H. Wu, On a new gap phenomonon in Riemannian geometry, Proc. Nat. Acad. Sci. U.S.A., 79:2 (1982), pp. 714–715. , Gap theorems for noncompact Riemannian manifolds, Duke Math. J., 49 (1982), [50] pp. 731–756. [51] , Lipschitz convergence of Riemannian manifolds, Pacific J. Math., 131 (1988), pp. 119–141. [52] D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math., 90 (1969), pp. 75–90. [53] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Edited by J. LaFontaine and P. Pansu, English translation by Sean Michael Bates, Birkh¨ auser, 1998. , Almost flat manifolds, J. Differential Geom., 13 (1978), pp. 231–241. [54] [55] , Hyperbolic manifolds (according to Thurston and Jφrgensen), Bourbaki Seminar, Vol. 1979/80, pp. 40–53, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981. [56] M. Gromov and H.B. Lawson, Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2), 111:3 (1980), pp. 423–434. [57] , Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, ´ Inst. Hautes Etudes Sci. Publ. Math., 58 (1983), pp. 83–196. [58] R.S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), pp. 255–306. [59] , Four–manifolds with positive curvature operator, J. Differential Geom., 24 (1986), pp. 153–179. , The Ricci flow on surfaces, Contemporary Mathematics, 71 (1988), pp. 237–261. [60] [61] , The Harnack estimate for the Ricci flow, J. Differential Geom., 37 (1993), pp. 225243. [62] , A compactness property for solution of the Ricci flow, Amer. J. Math., 117 (1995), pp. 545–572. , The formation of singularities in the Ricci flow, Surveys in Differential Geometry, 2, [63] pp. 7–136, International Press, 1995. [64] , Four manifolds with positive isotropic curvature, Comm. in Analysis and Geometry, 5 (1997), pp. 1–92. , Non-singular solutions to the Ricci flow on three manifolds, Comm. Anal. Geom., 7 [65] (1999), pp. 695–729. [66] , Eternal solutions to the Ricci flow, J. Differential Geom., 38:1 (1993), pp. 1–11. [67] , Lectures on the Ricci flow at Columbia University, Fall, 2003. [68] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962 (Second edition, 1978). [69] J. Hempel, 3-manifolds, Reprint of the 1976 original, AMS Chelsea Publishing, Providence, RI, 2004. [70] M.W. Hirsch, Differential Topology, Springer-Verlag, 1976.
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
489
[71] G. Huisken, Ricci deformation of the metric on a Riemanian manifold, J. Differential Geometry, 21 (1985), pp. 47–62. , Asymptotic behavior for singularities of the mean curvature flow. J. Differential [72] Geometry, 31 (1990), pp. 285–299. [73] T. Ivey, Ricci solitons on compact three-manifolds, Diff. Geom. Appl., 3 (1993), pp. 301–307. [74] W. Jaco and P. Shalen, Seifert fibered spaces in 3-manifolds, Men. Amer. Math. Soc., 220 (1979). [75] K. Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Math., 761 (1979). [76] D.A. Kazdan and G. Margulis, A simple proof of Selberg’s hypothesis, Math. Sb. (117), 75 (1968), pp. 163–168. [77] J.L. Kazdan and F.W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2), 99 (1974), pp. 14–47. [78] B. Kleiner and J. Lott, Note on Perelman’s paper, http://www.math.lsa.umich.edu /research /ricciflow /perelman.html. [79] H. Kneser, Geschlossene Fl¨ achen in dreidmensionalen mannigfaltigkeiten, Jahresbericht der Deut. Math. Verein., 38 (1929), pp. 248–260. [80] N. Koiso, On rotationally symmmetric Hamilton’s equation for K¨ ahler-Einstein metrics, Recent Topics in Diff. Anal. Geom. Adv. Studies Pure Math., 18-I (1990), pp. 327–337. [81] O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Uralceva, Linear and quasilinear equations of parabolic type, Transl. Amer. Math. Soc., 23 (1968). [82] P. Li and S.-T. Yau, On the parabolic kernel of the Schr¨ odinger operator, Acta Math., 156:3-4 (1986), pp. 153–201. [83] C. Margerin, Une caract´ erisation optimale de la structure diff´ erentielle standard de la sph` ere en terme de courbure pour (presque) toutes les dimensions. I. Les ´ enonc´ es, (French) [A sharp curvature-type characterization of the smooth sphere in all but a few dimensions. I. Statement of the theorems] C.R. Acad. Sci. Paris S´ er. I Math., 319:6 (1994), pp. 605–607. [84] , A sharp characterization of the smooth 4-sphere in curvature terms, Comm. Anal. Geom., 6:1 (1998), pp. 21–65. [85] C. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies 12, Princeton U. Press, 1996. [86] W.W. Meeks, III and S.-T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z., 179:2 (1982), pp. 151–168. [87] W.W. Meeks, III, L. Simon, and S.-T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math., 116 (1982), pp. 621–659. [88] M. Micallef and J.D. Moore, J Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2), 127 (1988), pp. 199– 227. [89] A.D. Milka, Metric structure of some class of spaces containing straight lines, Ukrain. Geometrical. Sbornik, 4 (1967), pp. 43–48. [90] J. Milnor, A unique factorization theorem for 3-manifolds, Amer. J. Math., 84 (1962), pp. 1– 7. , Towards the Poincar´ e conjecture and the classification of 3-manifolds, Notices Amer. [91] Math. Soc., 50:10 (2003), pp. 1226–1233. [92] N. Mok, The uniformization theorem for compact K¨ ahler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom., 27 (1988), pp. 179–214. [93] N. Mok, Y.T. Siu, and S.-T. Yau, The Poincar´ e–Lelong equation on complete K¨ ahler manifolds, Compositio Math., 44 (1981), pp. 183–218. [94] J.W. Morgan, On Thurston’s uniformization theorem for three-dimensional manifolds, The Smith Conjecture (Bass and Morgan, eds.), Pure Appl. Math., 112, Academic Press, (1984), pp. 37–125. [95] , Recent progress on the Poincar´ e conjecture and the classification of 3-manifolds, Bull.of the A.M.S., 42:1 (2005), pp. 57–78. [96] S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math., 100 (1979), pp. 593– 606. [97] G.D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. [98] L. Ni, Ancient solution to Kahler-Ricci flow, arXiv:math.DG.0502494, February 2005, preprint. , A note on Perelman’s LYH inequality, arXiv:math.DG.0602337, February 2006, [99] preprint. [100] L. Ni and L.F. Tam, Plurisubharmonic functions and the structure of complete K¨ ahler man-
490
H.-D. CAO AND X.-P. ZHU
ifolds with nonnegative curvature, J. Differential Geom., 64:3 (2003), pp. 457–524. [101] S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), pp. 343–352, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986. [102] J.-P. Otal, Le th´ eor´ eme d’hyperbolisation pir les vari´ et´ es fibr´ ees de dimension 3, At´ erisque 235, 1996. [103] G Perelmann, The entropy formula for the Ricci flow and its geometric applications, arXiv:math. DG/0211159 v1 November 11, 2002, preprint. [104] , Ricci flow with surgery on three manifolds, arXiv:math.DG/0303109 v1 March 10, 2003, preprint. , Finite extinction time to the solutions to the Ricci flow on certain three manifolds, [105] arXiv:math.DG/0307245 July 17, 2003, preprint. [106] S. Peters, Convergence of Riemannian mainfolds, Comp. Math., 62 (1987), pp. 3–16. [107] G. Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math., 21 (1973), pp. 255–286. [108] O.S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schr¨ odinger opreators, J. Funct. Anal., 42:1 (1981), pp. 110–120. [109] R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta math., 28 (1979), pp. 159–183. [110] , Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2), 110:1 (1979), pp. 127– 142. , Complete three dimensional manifolds with positive Ricci curvature and scalar cur[111] vature, Seminor on Differential Geometry, Princeton University Press, 1982, pp. 209–228. , Lectures on differential geometry, in ‘Conference Proceedings and Lecture Notes in [112] Geometry and Topology’, 1, International Press Publications, 1994. [113] N. Sesum, Limiting behaviour of the Ricci flow, arXiv:math.DG/0402194, February 2004. [114] W.X. Shi, Deforming the metric on complete Riemannian manifold, J. Differential Geometry, 30 (1989), pp. 223–301. [115] , Ricci deformation of the metric on complete noncompact K¨ ahler manifolds, Ph.D. Thesis, Harvard University, 1990. [116] , Complete noncompact K¨ ahler manifolds with positive holomorphic bisectional curvature, Bull. Amer Math. Soc., 23 (1990), pp. 437–440. , Ricci flow and the uniformization on complete noncompact K¨ ahler manifolds, J. [117] Differential Geom., 45:1 (1997), pp. 94–220. [118] T. Shioya and T. Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound, Math. Ann., 333 (2005), pp. 131–155. [119] I.M. Singer, Infinitesimallly homogeneous spaces, Comm. Pure Appl. Math., 13 (1960), pp. 685–697. [120] Y.T. Siu and S.-T. Yau, Complete K¨ ahler manifolds with non–positive curvature of faster than quadratic decay, Ann. of Math., 105 (1977), pp. 225–264. [121] M. Struwe, Curvature flows on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1:2 (2002), pp. 247–274. [122] W.P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), 6:3 (1982), pp. 357–381. [123] , Hyperbolic structures on 3-manifolds. I: Deformation of acylindrical manifolds, Ann. of Math. (2), 124:2 (1986), pp. 203–246. , Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber [124] over the circle, preprint, 1988. [125] , Three-dimensional geometry and topology, Vol. 1, Edited by Silvio Levy, Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, 1997. [126] F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, II, Invent. Math., 3 (1967), pp. 308–333; 4 (1967), pp. 87–117. [127] X.J. Wang, and X.H. Zhu, K¨ ahler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., 188:1 (2004), pp. 87–103. [128] S.-T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J., 25 (1976), pp. 659–670. , The role of partial differential equations in differential geometry, in ‘Proceedings of [129] the International Congress of Mathematicians’, Helsinki, 1978, pp. 237–250. [130] , Perspectives on geometric analysis, to appear in Survey in Differential Geometry, Vol. X. (arXiv: math.DG/0602363, February, 2006)
THE HAMILTON-PERELMAN THEORY OF RICCI FLOW
491
INDEX Ti converges to a tensor T , 267 δ-cutoff surgery, 424 κ-noncollapsed, 255, 433 λ-remote, 360 L-Jacobian, 252 L-Jacobian field, 248 L-distance function, 245 L-exponential map with parameter τ¯, 252 L-geodesic, 244 curve, 244 equation, 244 L-length, 244 ε-cap, 414 ε-horn, 414 ε-neck, 414 strong, 416 ε-neck of radius r, 357 ε-tube, 414 k-jet distance, 343 space, 343 (almost) maximum points, 291 a priori assumptions (with accuracyε), 416 admissible curve, 435 ancient κ-solution, 357 solution, 233 asymptotic scalar curvature ratio, 303, 362 asymptotic volume ratio, 373 atoroidal, 482 barely admissible curve, 435 be modeled, 481 Berger’s rigidity theorem, 321 breather, 199 expanding, 199 shrinking, 199 steady, 199 canonical neighborhood assumption (with accuracy ε), 416 canonical neighborhood theorem, 396 capped ε-horn, 414 center of an evolving ε-neck, 394
Cheeger’s lemma, 288 classical sphere theorems, 321 classification of three-dimensional shrinking solitons, 384 collapsed, 338 compactness of ancient κ0 -solutions, 393 conjugate heat equation, 235 converges to a marked manifold, 267 converges to an evolving marked manifold, 282 curvature β-bump, 360 degree, 308 double ε-horn, 414 Einstein manifold, 174 metric, 174 elliptic type estimate, 391 essential component, 476 evolving ε-cap, 396 evolving ε-neck, 396 exceptional part, 338 finite bump theorem, 360 free boundary condition, 345 geometrizable, 482 gradient shrinking Ricci soliton, 384 graph manifold, 482 Gromoll-Meyer injectivity radius estimate, 286 Haken, 481 Hamilton’s advanced maximum principle, 218 Hamilton’s compactness theorem, 285 Hamilton’s strong maximum principle, 213 Hamilton-Ivey pinching estimate, 224, 336 homogeneous manifold, 481 locally, 481 incompressible, 338 injectivity radius, 286 condition, 291
492
H.-D. CAO AND X.-P. ZHU
irreducible, 476 Jacobian comparison theorem, 253 justification of the canonical neighborhood assumption, 433 justification of the pinching assumption, 424 K¨ahler-Ricci flow, 176 K¨ahler-Ricci soliton expanding, 176 shrinking, 176 steady, 176 Klingenberg’s lemma, 286 Li-Yau-Hamilton estimate, 226, 230 Li-Yau-Hamilton quadratic, 230 Li-Yau-Perelman distance, 238, 250 Little Loop Lemma, 290 marked Riemannian manifold, 267 marking, 267 maximal solution, 291 Mostow type rigidity, 343 no local collapsing theorem I, 255, 256 no local collapsing theorem I′ , 259 no local collapsing theorem II, 263 noncollapsing limit, 338 normalized, 398 normalized Ricci flow, 307 origin, 267 Perelman’s claim, 482 Perelman’s reduced volume, 243, 252 element, 254 pinching assumption, 416 Poincar´e conjecture, 452 prime, 481 decomposition theorem, 481 regular, 411 Ricci flow, 173 Ricci flow with surgery, 416 Ricci soliton expanding, 175 shrinking, 175 gradient , 175 steady , 175 Shi’s derivative estimate, 192
singularity model, 292 singularity structure theorem, 399 soliton cigar, 177 steady, 175 solution ancient, 233 nonsingular, 336 standard, 420 solution becomes extinct, 417 solution develops a singularity, 267 standard capped infinite cylinder, 420 support function, 220 surgery times, 416 surgically modified solution, 416 tangent cone, 219 thick-thin decomposition theorem, 473 Thurston’s geometrization conjecture, 482, 483 type I, 291–293 type II, 293 (a), 291, 294 (b), 292, 295 type III, 293 (a), 292, 296 (b), 292 universal noncollapsing, 388 Weaker Assertion, 483
arXiv:math/0605667v3 [math.DG] 10 Oct 2008
NOTES ON PERELMAN’S PAPERS BRUCE KLEINER AND JOHN LOTT
1. Introduction These are notes on Perelman’s papers “The Entropy Formula for the Ricci Flow and its Geometric Applications” [51] and “Ricci Flow with Surgery on Three-Manifolds’ [52]. In these two remarkable preprints, which were posted on the ArXiv in 2002 and 2003, Grisha Perelman announced a proof of the Poincar´e Conjecture, and more generally Thurston’s Geometrization Conjecture, using the Ricci flow approach of Hamilton. Perelman’s proofs are concise and, at times, sketchy. The purpose of these notes is to provide the details that are missing in [51] and [52], which contain Perelman’s arguments for the Geometrization Conjecture. Among other things, we cover the construction of the Ricci flow with surgery of [52]. We also discuss the long-time behavior of the Ricci flow with surgery, which is needed for the full Geometrization Conjecture. The papers [24, 53], which are not covered in these notes, each provide a shortcut in the case of the Poincar´e Conjecture. Namely, these papers show that if the initial manifold is simply-connected then the Ricci flow with surgery becomes extinct in a finite time, thereby removing the issue of the long-time behavior. Combining this claim with the proof of existence of Ricci flow with surgery gives the shortened proof in the simply-connected case. These notes are intended for readers with a solid background in geometric analysis. Good sources for background material on Ricci flow are [22, 23, 33, 66]. The notes are selfcontained but are designed to be read along with [51, 52]. For the most part we follow the format of [51, 52] and use the section numbers of [51, 52] to label our sections. We have done this in order to respect the structure of [51, 52] and to facilitate the use of the present notes as a companion to [51, 52]. In some places we have rearranged Perelman’s arguments or provided alternative arguments, but we have refrained from an overall reorganization. Besides providing details for Perelman’s proofs, we have included some expository material in the form of overviews and appendices. Section 3 contains an overview of the Ricci flow approach to geometrization of 3-manifolds. Sections 4 and 57 contain overviews of [51] and [52], respectively. The appendices discuss some background material and techniques that are used throughout the notes. Regarding the proofs, the papers [51, 52] contain some incorrect statements and incomplete arguments, which we have attempted to point out to the reader. (Some of the mistakes in [51] were corrected in [52].) We did not find any serious problems, meaning problems that cannot be corrected using the methods introduced by Perelman. Date: September 19, 2008. Research supported by NSF grants DMS-0306242 and DMS-0204506, and the Clay Mathematics Institute. 1
2
BRUCE KLEINER AND JOHN LOTT
We will refer to Section X.Y of [51] as I.X.Y, and Section X.Y of [52] as II.X.Y. A reader may wish to start with the overviews, which explain the logical structure of the arguments and the interrelations between the sections. It may also be helpful to browse through the appendices before delving into the main body of the material. These notes have gone through various versions, which were posted at [39]. An initial version with notes on [51] was posted in June 2003. A version covering [51, 52] was posted in September 2004. After the May 2006 version of these notes was posted on the ArXiv, expositions of Perelman’s work appeared in [15] and [45]. Acknowledgements. In the preparation of the September 2004 version of our notes, we benefited from a workshop on Perelman’s surgery procedure that was held in August 2004, at Princeton. We thank the participants of that meeting, as well as the Clay Mathematics Institute for supporting it. We are grateful for comments and corrections that we have received regarding earlier versions of these notes. We especially thank G´erard Besson for comments on Theorem 13.3 and Lemma 80.4, Peng Lu for comments on Lemma 74.1 and Bernhard Leeb for discussions on the issue of uniqueness for the standard solution. Along with these people we thank Mike Anderson, Albert Chau, Ben Chow, Xianzhe Dai, Jonathan Dinkelbach, Sylvain Maillot, John Morgan, Joan Porti, Gang Tian, Peter Topping, Bing Wang, Guofang Wei, Hartmut Weiss, Jon Wolfson, Rugang Ye and Maciej Zworski. We thank the referees for their detailed and helpful comments on this paper. We thank the Clay Mathematics Institute for supporting the writing of the notes on [52]. Contents 1. Introduction
1
2. A reading guide
6
3. An overview of the Ricci flow approach to 3-manifold geometrization
7
3.1. The definition of Ricci flow, and some basic properties
7
3.2. A rough outline of the Ricci flow proof of the Poincar´e Conjecture
8
3.3. Outline of the proof of the Geometrization Conjecture
9
3.4. Claim 3.2 and the structure of singularities
11
3.5. The proof of Claims 3.3 and 3.4
13
4. Overview of The Entropy Formula for the Ricci Flow and its Geometric Applications [51]
13
4.1. I.1-I.6
14
4.2. I.7-I.10
15
4.3. I.11
16
4.4. I.12-I.13
17
NOTES ON PERELMAN’S PAPERS
5. I.1.1. The F -functional and its monotonicity
3
18
6. Basic example for I.1
21
7. I.2.2. The λ-invariant and its applications
22
8. I.2.3. The rescaled λ-invariant
24
9. I.2.4. Gradient steady solitons on compact manifolds
25
10. Ricci flow as a gradient flow
26
11. The W-functional
27
12. I.3.1. Monotonicity of the W-functional
28
13. I.4. The no local collapsing theorem I
30
14. I.5. The W-functional as a time derivative
32
15. I.7. Overview of reduced length and reduced volume
33
16. Basic example for I.7
34
17. Remarks about L-Geodesics and L exp
34
18. I.(7.3)-(7.6). First derivatives of L
37
19. I.(7.7). Second variation of L
39
20. I.(7.8)-(7.9). Hessian bound for L
41
21. I.(7.10). The Laplacian of L
42
22. I.(7.11). Estimates on L-Jacobi fields 23. Monotonicity of the reduced volume Ve
42 43
24. I.(7.15). A differential inequality for L
44
25. I.7.2. Estimates on the reduced length
46
26. I.7.3. The no local collapsing theorem II
47
27. I.8.3. Length distortion estimates
50
28. I.8.2. No local collapsing propagates forward in time and to larger scales
52
29. I.9. Perelman’s differential Harnack inequality
54
30. The statement of the pseudolocality theorem
57
31. Claim 1 of I.10.1. A point selection argument
58
32. Claim 2 of I.10.1. Getting parabolic regions
59
33. Claim 3 of I.10.1. An upper bound on the integral of v
60
34. Theorem I.10.1. Proof of the pseudolocality theorem
62
35. I.10.2. The volumes of future balls
65
36. I.10.4. κ-noncollapsing at future times
66
37. I.10.5. Diffeomorphism finiteness
66
4
BRUCE KLEINER AND JOHN LOTT
38. I.11.1. κ-solutions
67
39. I.11.2. Asymptotic solitons
68
40. I.11.3. Two dimensional κ-solutions
70
41. I.11.4. Asymptotic scalar curvature and asymptotic volume ratio
70
42. In a κ-solution, the curvature and the normalized volume control each other
74
43. An alternate proof of Corollary 40.1 using Proposition 41.13 and Corollary 42.1 76 44. I.11.5. A volume bound
77
45. I.11.6. Curvature bounds for Ricci flow solutions with nonnegative curvature operator, assuming a lower volume bound
78
46. I.11.7. Compactness of the space of three-dimensional κ-solutions
80
47. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - weak version
82
48. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - strong version
83
49. More properties of κ-solutions
85
50. I.11.9. Getting a uniform value of κ
85
51. II.1.2. Three-dimensional noncompact κ-noncollapsed gradient shrinkers are standard
86
52. I.12.1. Canonical neighborhood theorem
89
53. I.12.2. Later scalar curvature bounds on bigger balls from curvature and volume bounds 96 54. I.12.3. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds
99
55. I.12.4. Small balls with strongly negative curvature are volume-collapsed
101
56. I.13.1. Thick-thin decomposition for nonsingular flows
102
57. Overview of Ricci Flow with Surgery on Three-Manifolds [52]
103
57.1. II.1-II.3
104
57.2. II.4-II.5
104
57.3. II.6-II.8
107
58. II. Notation and terminology
108
59. II.1. Three-dimensional κ-solutions
110
60. II.2. Standard solutions
113
61. Claim 2 of II.2. The blow-up time for a standard solution is ≤ 1
115
62. Claim 4 of II.2. The blow-up time of a standard solution is 1
115
63. Claim 5 of II.2. Canonical neighborhood property for standard solutions
116
NOTES ON PERELMAN’S PAPERS
5
64. Compactness of the space of standard solutions
118
65. Claim 1 of II.2. Rotational symmetry of standard solutions
118
66. Claim 3 of II.2. Uniqueness of the standard solution
120
67. II.3. Structure at the first singularity time
122
68. Ricci flow with surgery: the general setting
127
69. II.4.1. A priori assumptions
130
70. II.4.2. Curvature bounds from the a priori assumptions
131
71. II.4.3. δ-necks in ǫ-horns
132
72. Surgery and the pinching condition
134
73. II.4.4. Performing surgery and continuing flows
138
74. II.4.5. Evolution of a surgery cap
142
75. II.4.6. Curves that penetrate the surgery region
144
76. II.4.7. A technical estimate
145
77. II.5. Statement of the the existence theorem for Ricci flow with surgery
146
78. The L-function of I.7 and Ricci flows with surgery
148
79. Establishing noncollapsing in the presence of surgery
152
80. Construction of the Ricci flow with surgery
157
81. II.6. Double sided curvature bound in the thick part
160
82. II.6.5. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and a later volume bound
161
83. II.6.6. Locating small balls whose subballs have almost Euclidean volume
163
84. II.6.8. Proof of the double sided curvature bound in the thick part, modulo two propositions 164 85. II.6.3. Canonical neighborhoods and later curvature bounds on bigger balls from curvature and volume bounds
165
86. II.6.4. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds, in the presence of possible surgeries
169
87. II.7.1. Noncollapsed pointed limits are hyperbolic
171
88. II.7.2. Noncollapsed regions with a lower curvature bound are almost hyperbolic on a large scale 173 89. II.7.3. Thick-thin decomposition
175
90. Hyperbolic rigidity and stabilization of the thick part
177
91. Incompressibility of cuspidal tori
182
92. II.7.4. The thin part is a graph manifold
186
93. II.8. Alternative proof of cusp incompressibility
191
6
BRUCE KLEINER AND JOHN LOTT
93.1. The approach using the σ-invariant
191
93.2. The approach using the λ-invariant
193
Appendix A. Maximum principles
199
Appendix B. φ-almost nonnegative curvature
201
Appendix C. Ricci solitons
202
Appendix D. Local derivative estimates
205
Appendix E. Convergent subsequences of Ricci flow solutions
205
Appendix F. Harnack inequalities for Ricci flow
206
Appendix G. Alexandrov spaces
208
Appendix H. Finding balls with controlled curvature
209
Appendix I. Statement of the geometrization conjecture
209
References
210
2. A reading guide Perelman’s papers contain a wealth of results about Ricci flow. We cover all of these results, whether or not they are directly relevant to the Poincar´e and Geometrization Conjectures. Some readers may wish to take an abbreviated route that focuses on the proof of the Poincar´e Conjecture or the Geometrization Conjecture. Such readers can try the following itinerary. Begin with the overviews in Sections 3 and 4. Then review Hamilton’s compactness theorem and its variants, as described in Appendix E; an exposition is in [66, Chapter 7]. Next, read I.7 (Sections 15-26), followed by I.8.3(b) (Section 27). After reviewing the theory of Riemannian manifolds and Alexandrov spaces of nonnegative sectional curvature (Appendix G and references therein), proceed to I.11 (Sections 38-50), followed by II.1.2 and I.12.1 (Sections 51-52). At this point, the reader should be ready for the overview of Perelman’s second paper in Section 57, and can proceed with II.1-II.5 (Sections 58-80). In conjunction with one of the finite extinction time results [24, 25, 53], this completes the proof of the Poincar´e Conjecture. To proceed with the rest of the proof of the Geometrization Conjecture, the reader can begin with the large-time estimates for nonsingular Ricci flows, which appear in I.12.2-I.12.4 (Sections 53-55). The reader can then go to II.6 and II.7 (Sections 81-92). The main topics that are missed by such an abbreviated route are the F and W functionals (Sections 5-14), Perelman’s differential Harnack inequality (Section 29), pseudolocality (Sections 30-37) and Perelman’s alternative proof of cusp incompressibility (Section 93).
NOTES ON PERELMAN’S PAPERS
7
3. An overview of the Ricci flow approach to 3-manifold geometrization This section is an overview of the Ricci flow approach to 3-manifold geometrization. We make no attempt to present the history of the ideas that go into the argument. We caution the reader that for the sake of readability, in many places we have suppressed technical points and deliberately oversimplified the story. The overview will introduce the argument in three passes, with successively greater precision and detail : we start with a very crude sketch, then expand this to a step-by-step outline of the strategy, and then move on to more detailed commentary on specific points. Other overviews may be found in [14, 44]. The primary objective of our exposition is to prepare the reader for a more detailed study of Perelman’s work. We refer the reader to Appendix I for the statement of the geometrization conjecture. By convention, all manifolds and Riemannian metrics in this section will be smooth. We follow the notation of [51, 52] for pointwise quantities: R denotes the scalar curvature, Ric the Ricci curvature, and | Rm | the largest absolute value of the sectional curvatures. An inequality such as Rm ≥ C means that all of the sectional curvatures at a point or in a region, depending on the context, are bounded below by C. In this section, we will specialize to three dimensions. 3.1. The definition of Ricci flow, and some basic properties. Let M be a compact 3-manifold and let {g(t)}t∈[a,b] be a smoothly varying family of Riemannian metrics on M. Then g(·) satisfies the Ricci flow equation if ∂g (t) = − 2 Ric(g(t)) ∂t holds for every t ∈ [a, b]. Hamilton showed in [35] that for any Riemannian metric g0 on M, there is a T ∈ (0, ∞] with the property that there is a (unique) solution g(·) to the Ricci flow equation defined on the time interval [0, T ) with g(0) = g0 , so that if T < ∞ then the curvature of g(t) becomes unbounded as t → T . We refer to this maximal solution as the Ricci flow with initial condition g0 . If T < ∞ then we call T the blow-up time. A basic example is the shrinking round 3-sphere, with g0 = r02 gS 3 and g(t) = (r02 − 4t) gS 3 , in which r2 case T = 40 . (3.1)
Suppose that M is simply-connected. Based on the round 3-sphere example, one could hope that every Ricci flow on M blows up in finite time and becomes round while shrinking to a point, as t approaches the blow-up time T . If so, then by rescaling and taking a limit as t → T , one would show that M admits a metric of constant positive sectional curvature and therefore, by a classical theorem, is diffeomorphic to S 3 . The analogous argument does work in two dimensions [22, Chapter 5]. Furthermore, if the initial metric g0 has positive Ricci curvature then Hamilton showed in [33] that this is the correct picture: the manifold shrinks to a point in finite time and becomes round as it shrinks. One is then led to ask what can happen if M is simply-connected but g0 does not have positive Ricci curvature. Here a new phenomenon can occur — the Ricci flow solution may become singular before it has time to shrink to a point, due to a possible neckpinch. A neckpinch is modeled by a product region (−c, c) × S 2 in which one or many S 2 -fibers
8
BRUCE KLEINER AND JOHN LOTT
separately shrink to a point at time T , due to the positive curvature of S 2 . The formation of neckpinch (and other) singularities prevents one from continuing the Ricci flow. In order to continue the evolution some intervention is required, and this is the role of surgery. Roughly speaking, the idea of surgery is to remove a neighborhood diffeomorphic to (−c′ , c′ ) × S 2 containing the shrinking 2-spheres, and cap off the resulting boundary components by gluing in 3-balls. Of course the topology of the manifold changes during surgery – for instance it may become disconnected – but it changes in a controlled way. The postsurgery Riemannian manifold is smooth, so one may restart the Ricci flow using it as an initial condition. When continuing the flow one may encounter further neckpinches, which give rise to further surgeries, etc. One hopes that eventually all of the connected components shrink to points while becoming round, i.e. that the Ricci flow solution has a finite extinction time. 3.2. A rough outline of the Ricci flow proof of the Poincar´ e Conjecture. We now give a step-by-step glimpse of the proof, stating the needed steps as claims. One starts with a compact orientable 3-manifold M with an arbitrary metric g0 . For the moment we do not assume that M is simply-connected. Let g(·) be the Ricci flow with initial condition g0 , defined on [0, T ). Suppose that T < ∞. Let Ω ⊂ M be the set of points x ∈ M for which limt→T − R(x, t) exists and is finite. Then M − Ω is the part of M that is going singular. (For example, in the case of a single standard neckpinch, M − Ω is a 2-sphere.) The first claim says what M looks like near this singularity set. Claim 3.2. [52] The set Ω is open and as t → T , the evolving metric g(·) converges smoothly on compact subsets of Ω to a Riemannian metric g. There is a geometrically defined neighborhood U of M − Ω such that each connected component of U is either A. Compact and diffeomorphic to S 1 × S 2 , S 1 ×Z2 S 2 or S 3 /Γ, where Γ is a finite subgroup of SO(4) that acts freely and isometrically on the round S 3 . (In writing S 1 ×Z2 S 2 , the generator of Z2 acts on S 1 by complex conjugation and on S 2 by the antipodal map. Then S 1 ×Z2 S 2 is diffeomorphic to RP 3 #RP 3 .) or
B. Noncompact and diffeomorphic to R×S 2 , R3 or the twisted line bundle R×Z2 S 2 over RP 2 . In Case B, the connected component meets Ω in geometrically controlled collar regions diffeomorphic to R × S 2 . Thus Claim 3.2 provides a topological description of a neighborhood U of the region M − Ω where the Ricci flow is going singular, along with some geometric control on U. Claim 3.3. [52] There is a well-defined way to perform surgery on M, which yields a smooth post-surgery manifold M ′ with a Riemannian metric g ′ . Claim 3.3 means that there is a well-defined procedure for specifying the part of M that will be removed, and for gluing caps on the resulting manifold with boundary. The discarded part corresponds to the neighborhood U in Claim 3.2. The procedure is required to satisfy a number of additional conditions which we do not mention here.
NOTES ON PERELMAN’S PAPERS
9
Undoing the surgery, i.e. going from the postsurgery manifold to the presurgery manifold, amounts to restoring some discarded components (as in Case A of Claim 3.2) and performing connected sums of some of the components of the postsurgery manifold, along with some possible connected sums with a finite number of new S 1 × S 2 and RP 3 factors. The S 1 × S 2 comes from the case when a surgery does not disconnect the connected component where it is performed. The RP 3 factors arise from the twisted line bundle components in Case B of Claim 3.2. After performing a surgery one lets the new manifold evolve under the Ricci flow until one encounters the next blowup time (if there is one). One then performs further surgery, lets the new manifold evolve, and repeats the process. Claim 3.4. [52] One can arrange the surgery procedure so that the surgery times do not accumulate. If the surgery times were to accumulate, then one would have trouble continuing the flow further, effectively killing the whole program. Claim 3.4 implies that by alternating Ricci flow and surgery, one obtains an evolutionary process that is defined for all time (though the manifold may become the empty set from some time onward). We call this Ricci flow with surgery. Claim 3.5. [24, 25, 53] If the original manifold M is simply-connected then any Ricci flow with surgery on M becomes extinct in finite time. Having a finite extinction time means that from some time onwards, the manifold is the empty set. More generally, the same proof shows that if the prime decomposition of the original manifold M has no aspherical factors, then every Ricci flow with surgery on M becomes extinct in finite time. (Recall that a connected manifold X is aspherical if πk (X) = 0 for all k > 1 or, equivalently, if its universal cover is contractible.) The Poincar´e Conjecture follows immediately from the above claims. From Claim 3.5, after some finite time the manifold is the empty set. From Claims 3.2, 3.3 and 3.4, the original manifold M is diffeomorphic to a connected sum of factors that are each S 1 × S 2 or a standard quotient S 3 /Γ of S 3 . As we are assuming that M is simply-connected, van Kampen’s theorem implies that M is diffeomorphic to a connected sum of S 3 ’s, and hence is diffeomorphic to S 3 . 3.3. Outline of the proof of the Geometrization Conjecture. We now drop the assumption that M is simply-connected. The main difference is that Claim 3.5 no longer applies, so the Ricci flow with surgery may go on forever in a nontrivial way. (We remark that Claim 3.5 is needed only for a shortened proof of the Poincar´e Conjecture; the proof in the general case is logically independent of Claim 3.5 and also implies the Poincar´e Conjecture.) The possibility that there are infinitely many surgery times is not excluded, although it is not known whether this can actually happen. A simple example of a Ricci flow that does not become extinct is when M = H 3 /Γ, where Γ is a freely-acting cocompact discrete subgroup of the orientation-preserving isometries of hyperbolic space H 3 . If ghyp denotes the metric on M of constant sectional curvature −1
10
BRUCE KLEINER AND JOHN LOTT
and g0 = r02 ghyp then g(t) = (r02 + 4t) ghyp . Putting b g (t) = 1t g(t), one finds that limt→∞ b g (t) 1 is the metric on M of constant sectional curvature − 4 , independent of r0 .
Returning to the general case, let Mt denote the time-t manifold in a Ricci flow with surgery. (If t is a surgery time then we consider the postsurgery manifold.) If for some t a component of Mt admits a metric with nonnegative scalar curvature then one can show that the component becomes extinct or admits a flat metric; either possiblity is good enough when we are trying to prove the Geometrization Conjecture for the initial manifold M. So we will assume that for every t, each component of Mt has a point with strictly negative scalar curvature.
Motivated by the hyperbolic example, we consider the metric gb(t) = 1t g(t) on Mt . Given x ∈ Mt , define the intrinsic scale ρ(x, t) to be the radius ρ such that inf B(x,ρ) Rm = − ρ−2 , where Rm denotes the sectional curvature of b g (t); this is well-defined because the scalar curvature is negative somewhere in the connected component of Mt containing x. Given w > 0, define the w-thick part of Mt by (3.6)
M + (w, t) = {x ∈ Mt : vol(B(x, ρ(x, t))) > w ρ(x, t)3 }.
It is not excluded that M + (w, t) = Mt or M + (w, t) = ∅. The next claim says that for any w > 0, as time goes on, M + (w, t) approaches the w-thick part of a manifold of constant sectional curvature − 14 . Claim 3.7. [52] There is a finite collection {(Hi, xi )}ki=1 of complete pointed finite-volume 3-manifolds with constant sectional curvature − 41 and, for large t, a decreasing function α(t) tending to zero and a family of maps (3.8)
ft :
k G
i=1
Hi ⊃
k G
i=1
1 → Mt , B xi , α(t)
such that 1. ft is α(t)-close to being an isometry. 2. The image of ft contains M + (α(t), t). 3. The image under ft of a cuspidal torus of {Hi }ki=1 is incompressible in Mt . The proof of Claim 3.7 uses earlier work by Hamilton [34]. S Claim 3.9. [52, 64] Let Yt be the truncation of ki=1 Hi obtained by removing horoballs at 1 distance approximately 2α(t) from the basepoints xi . Then for large t, Mt − ft (Yt ) is a graph manifold. Claim 3.9 reduces to a statement in Riemannian geometry about 3-manifolds that are locally volume-collapsed with a lower bound on sectional curvature. Claims 3.7 and 3.9, along with Claims 3.2-3.4, imply the geometrization conjecture, cf. Appendix I. In the remainder of this section, we will discuss some of the claims in more detail.
NOTES ON PERELMAN’S PAPERS
11
3.4. Claim 3.2 and the structure of singularities. Claim 3.2 is derived from a more localized statement, which says that near points of large scalar curvature, the Ricci flow looks very special : it is well-approximated by a special kind of model Ricci flow, called a κ-solution. Claim 3.10. Suppose that we have a given Ricci flow solution on a finite time interval. If x ∈ M and the scalar curvature R(x, t) is large then in the time-t slice, there is a ball 1 centered at x of radius comparable to R(x, t)− 2 in which the geometry of the Ricci flow is close to that of a ball in a κ-solution. 1
The quantity R(x, t)− 2 is sometimes called the curvature scale at (x, t), because it scales like a distance. We will define κ-solutions below, but mention here that they are Ricci flows with nonnegative sectional curvature, and they are ancient, i.e. defined on a time interval of the form (−∞, a). The strength of Claim 3.10 comes from the fact that there is a good description of κsolutions. Claim 3.11. [52] Any three-dimensional oriented κ-solution (M∞ , g∞ (·)) falls into one of the following types : (a) A finite isometric quotient of the round shrinking 3-sphere. (b) A Ricci flow on a manifold diffeomorphic to S 3 or RP 3 . (c) A standard shrinking round neck on R × S 2 (d) A Ricci flow on a manifold diffeomorphic to R3 , each time slice of which is asymptotically necklike at infinity. (e) The Z2 -quotient R ×Z2 S 2 of a shrinking round neck. Together, Claims 3.10 and 3.11 say that where the scalar curvature is large, there is a region of diameter comparable to the curvature scale where one sees either a closed manifold of known topology (cases (a) and (b)), a neck region (case (c)), a neck region capped off by a 3-ball (case (d)), or a neck region capped off by a twisted line bundle over RP 2 (case (e)). Applying this statement to every point of large scalar curvature at a time t just prior to the blow-up time T , one obtains a cover of M by regions with special geometry and topology. Any overlaps occur in neck-like regions, permitting one to splice them together to form the connected components with known topology whose existence is asserted in Claim 3.2. Claim 3.10 is proved using a rescaling (or blow-up) argument. This is a standard technique in geometric analysis and PDE’s for treating scale-invariant equations, such as the Ricci flow equation. The claim is equivalent to the statement that if {(xi , ti )}∞ i=1 is a sequence of spacetime points for which limi→∞ R(xi , ti ) = ∞, then by rescaling the Ricci flow and passing to a subsequence, one obtains a new sequence of Ricci flows which converges to a κ-solution. More precisely, view (xi , ti ) as a new spacetime basepoint and spatially expand 1 the solution around (xi , ti ) by R(xi , ti ) 2 . For dimensional reasons, in order for rescaling to produce a new Ricci flow solution one must also expand the time factor by R(xi , ti ). The new Ricci flow solution, with time parameter s, is given by (3.12) g i (s) = R(xi , ti ) g R(xi , ti )−1 s + ti . The new time interval for s is [− R(xi , ti ) ti , 0]. One would then hope to take an appropriate limit (M∞ , g ∞ ) of a subsequence of these rescaled solutions {(M, g i (·))}∞ i=1 . (Technically
12
BRUCE KLEINER AND JOHN LOTT
speaking, one uses smooth convergence of sequences of Ricci flows with basepoints; this notion of convergence allows us to focus on what happens near the spacetime points (xi , ti ).) Any such limit solution (M∞ , g ∞ ) will be an ancient solution, since limi→∞ −R(xi , ti ) ti = −∞. Furthermore, from a 3-dimensional result of Hamilton and Ivey (see Appendix B), any limit solution will have nonnegative sectional curvature. Although this sounds promising, a major problem was to show that a limit solution actually exists. To prove this, one would like to invoke Hamilton’s compactness theorem [32]. In the present situation, the compactness theorem says that the sequence of rescaled Ricci flows {(M, g i (·))}∞ i=1 has a smoothly convergent subsequence provided two conditions are met: A. For every r > 0 and sufficiently large i, the sectional curvature of gi is bounded uniformly independent of i at each point (x, s) in spacetime such that x lies in the gi -ball B0 (xi , r) and s ∈ [−r 2 , 0] and B. The injectivity radii inj(xi , 0) in the time-0 slices of the gi ’s have a uniform positive lower bound. For the moment, we ignore the issue of verifying condition A, and simply assume that it holds for the sequence {(M, g i (·))}∞ i=1 . In the presence of the sectional curvature bounds in condition A, a lower bound on the injectivity radius is known to be equivalent to a lower bound on the volume of metric balls. In terms of the original Ricci flow solution, this becomes the condition that (3.13)
r −3 vol(Bt (x, r)) ≥ κ > 0,
where Bt (x, r) is an arbitrary metric r-ball in a time-t slice, and the curvature bound | Rm | ≤ r12 holds in Bt (x, r). The number κ could depend on the given Ricci flow solution, but the bound (3.13) should hold for all t ∈ [0, T ) and all r < ρ, where ρ is a relevant scale.
One of the outstanding achievements of [51] is to prove that for an arbitrary Ricci flow defined on a finite time interval, equation (3.13) does hold with appropriate values of κ and ρ. In fact, the proof works in arbitrary dimension. This result is called a “no local collapsing theorem” because it excludes the phenomenon of Cheeger-Gromov collapse, in which a sequence of Riemannian manifolds has uniformly bounded curvature, but fails to converge because the injectivity radii tend to zero.
One can then apply the no local collapsing theorem to the preceding rescaling argument, provided that one has the needed sectional curvature bounds, in order to construct the ancient solution (M∞ , g ∞ ). In the blowup limit the condition that r < ρ goes away, and so we can say that (M∞ , g ∞ ) is κ-noncollapsed (i.e. satisfies (3.13)) at all scales. In addition, in the three-dimensional case one can show that (M∞ , g ∞ ) has bounded sectional curvature. To summarize, (M∞ , g ∞ ) is a κ-solution, meaning that it is an ancient Ricci flow solution with nonnegative curvature operator on each time slice and bounded sectional curvature on compact time intervals, which is κ-noncollapsed at all scales. With the no local collapsing theorem in place, most of the proof of Claim 3.10 is concerned with showing that in the rescaling argument, we effectively have the needed curvature bounds
NOTES ON PERELMAN’S PAPERS
13
of condition A. The argument is a tour-de-force with many ingredients, including earlier work of Hamilton and the theory of Riemannian manifolds with nonnegative sectional curvature. 3.5. The proof of Claims 3.3 and 3.4. Claim 3.2 allows one to take the limit of the evolving metric g(·) as t → T , on the open set Ω where the metric is not becoming singular. It also provides geometrically defined regions – the connected components of the open set U – which one removes during surgery. Each boundary component of the resulting manifold is a nearly round 2-sphere with a nearly cylindrical collar, because the collar regions in Case B of Claim 3.2 have a neck-like geometry. This enables one to glue in 3-balls with a standard metric, using a partition of unity construction. The Ricci flow starting with the postsurgery metric may also go singular after a finite time. If so, one can appeal to Claim 3.2 again to perform surgery. However, the elapsed time between successive surgeries will depend on the scales at which surgeries are performed. Unless one performs the surgeries very carefully, the surgery times may accumulate. The way to rule out an accumulation of surgery times is to arrange the surgery procedure so that a surgery at time t removes a definite amount of volume v(t). That is, a surgery at time t should be performed at a definite scale h(t). In order to guarantee that this is possible, one needs to establish a quantitative version of Claim 3.2 for a Ricci flow with surgery, which applies not just at the first surgery time T but also at a later surgery time T ′ . The output of this quantitative version can depend on the surgery time T ′ and the time-zero metric, but it should be independent of whether or when surgeries occur before time T ′ . The general idea of the proof is similar to that of Claim 3.2, except that one has to carefully prescribe the surgery procedure in order to control the effect of the earlier surgeries. In particular, one of Perelman’s remarkable achievements is a version of the no local collapsing theorem for Ricci flows with surgery. We refer the reader to Section 57 for a more detailed overview of the proof of Claim 3.4, and for further discussion of Claims 3.7 and 3.9. 4. Overview of The Entropy Formula for the Ricci Flow and its Geometric Applications [51] The paper [51] deals with nonsingular Ricci flows; the surgery process is considered in [52]. In particular, the final conclusion of [51] concerns Ricci flows that are singularity-free and exist for all positive time. It does not apply to compact 3-manifolds with finite fundamental group or positive scalar curvature. The purpose of the present overview is not to give a comprehensive summary of the results of [51]. Rather we indicate its organization and the interdependence of its sections, for the convenience of the reader. Some of the remarks in the overview may only become clear once the reader has absorbed a portion of the detailed notes. Sections I.1-I.10, along with the first part of I.11, deal with Ricci flow on n-dimensional manifolds. The second part of I.11, and Sections I.12-I.13, deal more specifically with Ricci flow on 3-dimensional manifolds. The main result is that geometrization holds if a
14
BRUCE KLEINER AND JOHN LOTT
compact 3-manifold admits a Riemannian metric which is the initial metric of a smooth Ricci flow. This was previously shown in [34] under the additional assumption that the sectional curvatures in the Ricci flow are O(t−1 ) as t → ∞. The paper [51] can be divided into four main parts.
Sections I.1-I.6 construct certain entropy-type functionals F and W that are monotonic under Ricci flow. The functional W is used to prove a no local collapsing theorem. Sections I.7-I.10 introduce and apply another monotonic quantity, the reduced volume V˜ . It is also used to prove a no local collapsing theorem. The construction of V˜ uses a modified notion of a geodesic, called an L-geodesic. For technical reasons the reduced volume V˜ seems to be easier to work with than the W-functional, and is used in most of the sequel. A reader who wants to focus on the Poincar´e Conjecture or the Geometrization Conjecture could in principle start with I.7. Section I.11 is concerned with κ-solutions, meaning nonflat ancient solutions that are κ-noncollapsed at all scales (for some κ > 0) and have bounded nonnegative curvature operator on each time slice. In three dimensions, a blowup limit of a finite-time singularity will be a κ-solution. Sections I.12-I.13 are about three-dimensional Ricci flow solutions. It is shown that highscalar-curvature regions are modeled by rescalings of κ-solutions. A decomposition of the time-t manifold into “thick” and “thin” pieces is described. It is stated that as t → ∞, the thick piece becomes more and more hyperbolic, with incompressible cuspidal tori, and the thin piece is a graph manifold. More details of these assertions appear in [52], which also deals with the necessary modifications if the solution has singularities. We now describe each of these four parts in a bit more detail. 4.1. I.1-I.6. In these sections M is assumed to be a closed n-dimensional manifold. A functional F (g) of Riemannian metrics g is said to be monotonic under Ricci flow if F (g(t)) is nondecreasing in t whenever g(·) is a Ricci flow solution. Monotonic quantities are an important tool for understanding Ricci flow. One wants to have useful monotonic quantities, in particular with a characterization of the Ricci flows for which F (g(t)) is constant in t. Formally thinking of Ricci flow as a flow on the space of metrics, one way to get a monotonic quantity would be if the Ricci flow were the gradient flow of a functional F . In Sections I.1-I.2, a functional F is introduced whose gradient flow is not quite Ricci flow, but only differs from the Ricci flow by the action of diffeomorphisms. (If one formally considers the Ricci flow as a flow on the space of metrics modulo diffeomorphisms then it turns out to be the gradient flow of a functional λ1 .) The functional F actually depends on a Riemannian metric g and a function f . If g(·) satisfies the Ricci flow equation and e−f (·) satisfies a conjugate or “backward” heat equation, in terms of g(·), then F (g(t), f (t)) is nondecreasing in t. Furthermore, it is constant in t if and only if g(·) is a gradient steady R −f soliton with associated function f (·). Minimizing F (g, f ) over all functions f with e dV = 1 gives the monotonic quantity λ1 (g), which turns out to be the lowest M eigenvalue of − 4 △ + R.
NOTES ON PERELMAN’S PAPERS
15
In Section I.3 a modified “entropy” functional W(g, f, τ ) is introduced. It is nondecreasing n in t provided that g(·) is a Ricci flow, τ = t0 − t and (4πτ )− 2 e−f satisfies the conjugate heat equation. The functional W is constant on a Ricci flow if and only if the flow is a gradient shrinking soliton that terminates at time t0 . Because of this, W is more suitable than F when one wants information that is localized in spacetime. In Section I.4 the entropy functional W is used to prove a no local collapsing theorem. The statement is that if g(·) is a given Ricci flow on a finite time interval [0, T ) then for any (scale) ρ > 0, there is a number κ > 0 so that if Bt (x, r) is a time-t ball with radius r less than ρ, on which | Rm | ≤ r12 , then vol(Bt (x, r)) ≥ κ rn . The method of proof is to show that if r −n vol(Bt (x, r)) is very small then the evaluation of W at time t is very negative, which contradicts the monotonicity of W. The significance of a no local collapsing theorem is that it allows one to use Hamilton’s compactness theorem to construct blowup limits of finite time singularities, and more generally to understand high curvature regions.
Section I.5 and I.6 are not needed in the sequel. Section I.5 gives some thermodynamiclike equations in which W appears as an entropy. Section I.6 motivates the construction of the reduced volume of Section I.7. 4.2. I.7-I.10. A new monotonic quantity, the reduced volume V˜ , is introduced in I.7. It is defined in terms of so-called L-geodesics. Let (p, t0 ) be a fixed spacetime point. Define backward time by τ = t0 − t. Given a curve γ(τ ) in M defined for 0 ≤ τ ≤ τ (i.e. going backward in real time) with γ(0) = p, its L-length is Z τ √ τ |γ(τ ˙ )|2 + R(γ(τ ), t0 − τ ) dτ. (4.1) L(γ) = 0
One can compute the first and second variations of L, in analogy to what is done in Riemannian geometry.
= p and γ(τ ) = q. Put l(q, τ ) = Let L(q, τ ) be the infimum of L(γ) over curves γ with R γ(0) −n −l(q,τ ) ˜ 2 e dvol(q). The remarkable The reduced volume is defined by V (τ ) = M τ ˜ fact is that if g(·) is a Ricci flow solution then V is nonincreasing in τ , i.e. nondecreasing in real time t. Furthermore, it is constant if and only if g(·) is a gradient shrinking soliton that terminates at time t0 . The proof of monotonicity uses a subtle cancellation between the τ derivative of l(γ(τ ), τ ) along an L-geodesic and the Jacobian of the so-called L-exponential map. L(q,τ ) √ . 2 τ
Using a differential inequality, it is shown that for each τ there is some point q(τ ) ∈ M so that l(q(τ ), τ ) ≤ n2 . This is then used to prove a no local collapsing theorem : Given a Ricci flow solution g(·) defined on a finite time interval [0, t0 ] and a scale ρ > 0 there is a number κ > 0 with the following property. For r < ρ, suppose that | Rm | ≤ r12 on the “parabolic” ball {(x, t) : distt0 (x, p) ≤ r, t0 − r 2 ≤ t ≤ t0 }. Then vol(Bt0 (p, r)) ≥ κ rn . The number κ can be chosen to depend on ρ, n, t0 and bounds on the geometry of the initial metric g(0). The hypotheses of the no local collapsing theorem proved using V˜ are more stringent than those of the no local collapsing theorem proved using W, but the consequences turn out to
16
BRUCE KLEINER AND JOHN LOTT
be the same. The no local collapsing theorem is used extensively in Sections I.11 and I.12 when extracting a convergent subsequence from a sequence of Ricci flow solutions. Theorem I.8.2 of Section I.8 says that under appropriate assumptions, local κ-noncollapsing extends forwards in time to larger distances. This will be used in I.12 to analyze long-time behavior. The statement is that for any A < ∞ there is a κ = κ(A) > 0 with the following property. Given r0 > 0 and x0 ∈ M, suppose that g(·) is defined for t ∈ [0, r02], with | Rm(x, t)| ≤ r0−2 for all (x, t) satisfying dist0 (x, x0 ) < r0 , and the volume of the time-zero ball B0 (x0 , r0 ) is at least A−1 r0n . Then the metric g(t) cannot be κ-collapsed on scales less than r0 at a point (x, r02 ) with distr02 (x, x0 ) ≤ Ar0 . A localized version of the W-functional appears in I.9. Section I.10, which is not needed for the sequel but is of independent interest, shows if a point in a time slice lies in a ball with quantitatively bounded geometry then at nearby later times, the curvature at the point is quantitatively bounded. That is, there is a damping effect with regard to exterior high-curvature regions. The “bounded geometry” assumptions on the initial ball are a lower bound on its scalar curvature and an assumption that the isoperimetric constants of subballs are close to the Euclidean value.
4.3. I.11. Section I.11 contains an analysis of κ-solutions. As mentioned before, in three dimensions κ-solutions arise as blowup limits of finite-time singularities and, more generally, as limits of rescalings of high-scalar-curvature regions. In addition to the no local collapsing theorem, some of the tools used to analyze κ-solutions are Hamilton’s Harnack inequality for Ricci flows with nonnegative curvature operator, and the comparison geometry of nonnegatively curved manifolds. The first result is that any time slice of a κ-solution has vanishing asymptotic volume ratio limr→∞ r −n vol(Bt (p, r)). This apparently technical result is used to show that if a κ-solution (M, g(·)) has scalar curvature normalized to equal one at some spacetime point (p, t) then there is an a priori upper bound on the scalar curvature R(q, t) at other points q in terms of distt (p, q). Using the curvature bound, it is shown that a sequence {(Mi , pi , gi(·))}∞ i=1 of pointed n-dimensional κ-solutions, normalized so that R(pi , 0) = 1 for each i, has a convergent subsequence whose limit satisfies all of the requirements to be a κ-solution, except possibly the condition of having bounded sectional curvature on each time slice. In three dimensions this statement is improved by showing that the sectional curvature will be bounded on each compact time interval, so the space of pointed 3-dimensional κsolutions (M, p, g(·)) with R(p, 0) = 1 is in fact compact. This is used to draw conclusions about the global geometry of 3-dimensional κ-solutions. If M is a compact 3-dimensional κ-solution then Hamilton’s theorem about compact 3manifolds with nonnegative Ricci curvature implies that M is diffeomorphic to a spherical space form. If M is noncompact then assuming that M is oriented, it follows easily that M is diffeomorphic to R3 , or isometric to the round shrinking cylinder R ×S 2 or its Z2 -quotient R ×Z2 S 2 . In the noncompact case it is shown that after rescaling, each time slice is neck-like at infinity. More precisely, considering a given time-t slice, for each ǫ > 0 there is a compact
NOTES ON PERELMAN’S PAPERS
17
subset Mǫ ⊂ M so that / Mǫ then the pointed manifold (M, y, R(y, t)g(t)) is ǫ-close to if 1y ∈ 1 the standard cylinder − ǫ , ǫ × S 2 of scalar curvature one. 4.4. I.12-I.13. Sections I.12 and I.13 deal with three-dimensional Ricci flows. Theorem I.12.1 uses the results of Section I.11 to model the high-scalar-curvature regions of a Ricci flow. Let us assume a pinching condition of the form Rm ≥ − Φ(R) for an appropriate function Φ with lims→∞ Φ(s) = 0. (This will eventually follow from Hamiltons Ivey pinching, cf. Appendix B.) Theorem I.12.1 says that given numbers ǫ, κ > 0, one can find r0 > 0 with the following property. Suppose that g(·) is a Ricci flow solution defined on some time interval [0, T ] that satisfies the pinching condition and is κ-noncollapsed at scales less than one. Then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , after scaling by the factor Q, the solution in the region {(x, t) : dist2t0 (x, x0 ) ≤ (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is ǫ-close to the corresponding subset of a κ-solution. Theorem I.12.1 says in particular that near a first singularity, the geometry is modeled by a κ-solution, for some κ. This fact is used in [52]. Although Theorem I.12.1 is not used directly in [51], its method of proof is used in Theorem I.12.2.
The method of proof of Theorem I.12.1 is by contradiction. If it were not true then there (i) would be a sequence r0 → 0 and a sequence(Mi , gi (·)) of Ricci flow solutions that satisfy (i) (i) the assumptions, each with a spacetime point x0 , t0 that does not satisfy the conclusion. (i) (i) To consider first a special case, suppose that each point x0 , t0 is the first point at which (i) (i) a certain curvature threshold Ri is achieved, i.e. R(y, t) ≤ R x0 , t0 for each y ∈ Mi h i (i) (i) (i) and t ∈ 0, t0 . Then after rescaling the Ricci flow gi (·) by Qi = R x0 , t0 and shifting (i)
the time parameter, one has the curvature bounds on the time interval [−Qi t0 , 0] that form part of the hypotheses of Hamilton’s compactness theorem. Furthermore, the no local collapsing theorem gives the lower injectivity radius bound needed to apply Hamilton’s theorem and take a convergent subsequence of the pointed rescaled solutions. The limit will be a κ-solution, giving the contradiction. In the general case, one effectively proceedsby induction on the size of the scalar curvature. (i) (i) By modifying the choice of points x0 , t0 , one can assume that the conclusion of the (i) (i) theorem holds for all of the points (y, t) in a large spacetime neighborhood of x0 , t0 that have R(y, t) > 2Qi . One then shows that one has the curvature bounds needed to form the time-zero slice of the putative κ-solution. One shows that this “time-zero” metric can be extended backward in time to form a κ-solution, thereby giving the contradiction. The rest of Section I.12 begins the analysis of the long-time behaviour of a nonsingular 3-dimensional Ricci flow. There are two main results, Theorems I.12.2 and I.12.3. They extend curvature bounds forward and backward in time, respectively. Theorem I.12.2 roughly says that if one has | Rm | ≤ r0−2 on a spacetime region of spatial size r0 and temporal size r02 , and if one has a lower bound on the volume of the initial time face of the region, then one gets scalar curvature bounds on much larger spatial balls at
18
BRUCE KLEINER AND JOHN LOTT
the final time. More precisely, for any A < ∞ there are numbers K = K(A) < ∞ and ρ = ρ(A) < ∞ so that with the hypotheses and notation of Theorem I.8.2, if in addition r02 Φ(r0−2 ) < ρ then R(x, r02 ) ≤ Kr0−2 for points x lying in the ball of radius Ar0 around x0 at time r02 . Theorem I.12.3 says that if one has a lower bound on volume and sectional curvature on a ball at a certain time then one obtains an upper scalar curvature bound on a smaller ball at an earlier time. More precisely, given w > 0 there exist τ = τ (w) > 0, ρ = ρ(w) > 0 and K = K(w) < ∞ with the following property. Suppose that a ball B(x0 , r0 ) at time t0 has volume bounded below by wr03 and sectional curvature bounded below by − r0−2. Then R(x, t) < Kr0−2 for t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) < 41 r0 , provided that φ(r0−2 ) < ρ.
Applying a back-and-forth argument using Theorems I.12.2 and I.12.3, along with the pinching condition, one concludes, roughly speaking, that if a metric ball of small radius r has infimal sectional curvature exactly equal to − r −2 then the ball has a small volume compared to r 3 . Such a ball can be said to be locally volume-collapsed with respect to a lower sectional curvature bound. Section I.13 defines the thick-thin decomposition of a large-time slice of a nonsingular Ricci flow and shows the geometrization. Rescaling the metric to b g (t) = t−1 g(t), there is a universal function Φ so that for large t, the metric b g (t) satisfies the Φ-pinching condition. In terms of the original unscaled metric, given x ∈ M let rb(x, t) > 0 be the unique number such that inf Rm Bt (x,br) = − rb−2 .
Given w > 0, define the w-thin part Mthin (w, t) of the time-t slice to be the points x ∈ M so that vol(Bt (x, rb(x, t))) < w rb(x, t)3 . That is, a point in Mthin (w, t) lies in a ball that is locally volume-collapsed with respect to a lower sectional curvature bound. Put Mthick (w, t) = M − Mthin (w, t). One shows that for large t, the subset Mthick (w, t) has bounded geometry in the sense that >0√ and K = K(w) < ∞ √ there are numbers √ ρ = ρ(w) 1 w (ρ t))3 , whenever x ∈ so that | Rm | ≤ Kt−1 on B(x, ρ t) and vol(B(x, ρ t)) ≥ 10 Mthick (w, t).
Invoking arguments of Hamilton (that are written out in more detail in [52]) one can take a sequence t → ∞ and w → 0 so that Mthick (w, t) converges to a complete finite-volume manifold with constant sectional curvature − 14 , whose cuspidal tori are incompressible in M. On the other hand, a result from Riemannian geometry implies that for large t and small w, Mthin (w, t) is homeomorphic to a graph manifold; again a more precise statement appears in [52]. The conclusion is that M satisfies the geometrization conjecture. Again, one is assuming in I.13 that the Ricci flow is nonsingular for all times. 5. I.1.1. The F -functional and its monotonicity The goal of this section is to show that in an appropriate sense, Ricci flow is a gradient flow on the space of metrics. We introduce the entropy functional F . We compute its formal variation and show that the corresponding gradient flow is a modified Ricci flow. In Sections 5 through 14 of these notes, M is a closed manifold. We will use the Einstein summation convention freely. We also follow Perelman’s convention that a condition like a > 0 means that a should be considered to be a small parameter, while a condition like
NOTES ON PERELMAN’S PAPERS
19
A < ∞ means that A should be considered to be a large parameter. This convention is only for pedagogical purposes and may be ignored by a logically minded reader. Let M denote the space of smooth Riemannian metrics g on M. We think of M formally as an infinite-dimensional manifold. The tangent space Tg M consists of the symmetric covariant 2-tensors vij on M. Similarly, C ∞ (M) is an infinite-dimensional manifold with Tf C ∞ (M) = C ∞ (M). The diffeomorphism group Diff(M) acts on M and C ∞ (M) by pullback. Let dV denote the Riemannian volume density associated to a metric g. We use the convention that △ = div grad. Definition 5.1. The F -functional F : M × C ∞ (M) → R is given by Z (5.2) F (g, f ) = R + |∇f |2 e−f dV. M
Given vij ∈ Tg M and h ∈ Tf C ∞ (M), the evaluation of the differential dF on (vij , h) is written as δF (vij , h). Put v = g ij vij . Proposition 5.3. (cf. I.1.1) We have Z i h v 2 −f − h (2△f − |∇f | + R) dV. (5.4) δF (vij , h) = e − vij (Rij + ∇i ∇j f ) + 2 M Proof. From a standard formula, δR = − △v + ∇i ∇j vij − Rij vij .
(5.5) As
|∇f |2 = g ij ∇i f ∇j f,
(5.6) we have (5.7)
δ|∇f |2 = − v ij ∇i f ∇j f + 2 h∇f, ∇hi.
p
det(g) dx1 . . . dxn , we have δ(dV ) = v2 dV , so v (5.8) δ e−f dV = − h e−f dV. 2 Putting this together gives Z (5.9) δF = e−f [− △v + ∇i ∇j vij − Rij vij − vij ∇i f ∇j f + M v i 2 h∇f, ∇hi + (R + |∇f |2 ) − h dV. 2
As dV =
The goal now is to rewrite the right-hand side of (5.9) so that vij and h appear algebraically, i.e. without derivatives. As △e−f = (|∇f |2 − △f ) e−f ,
(5.10) we have (5.11)
Z
M
−f
e
[− △v] dV = −
Z
−f
(△e M
) v dV =
Z
M
e−f (△f − |∇f |2 ) v dV.
20
Next, (5.12)
BRUCE KLEINER AND JOHN LOTT
Z
−f
e
M
∇i ∇j vij dV = =
Z
−f
ZM M
Finally, (5.13)
2
Z
−f
e
M
(∇i ∇j e
) vij dV = −
Z
M
∇i (e−f ∇j f ) vij dV
e−f (∇i f ∇j f − ∇i ∇j f ) vij dV. Z
−f
Z
h∇f, ∇hi dV = − 2 h∇e , ∇hi dV = 2 (△e−f ) h dV M Z M = 2 e−f (|∇f |2 − △f ) h dV. M
Then
Z
− h (2△f − 2|∇f |2) − vij (Rij + ∇i ∇j f ) 2 M v i 2 + − h (R + |∇f | ) dV 2 Z i h v 2 −f − h (2△f − |∇f | + R) dV. = e − vij (Rij + ∇i ∇j f ) + 2 M This proves the proposition. We would like to get rid of the v2 − h (2△f − |∇f |2 + R) term in (5.14). We can do this by restricting our variations so that v2 − h = 0. From (5.8), this amounts to assuming that assuming e−f dV is fixed. We now fix a smooth measure dm on M and relate f to g by requiring that e−f dV = dm. Equivalently, we define a section s : M → M × C ∞ (M) dV by s(g) = g, ln dm . Then the composition F m = F ◦ s is a function on M and its differential is given by Z m (5.15) dF (vij ) = e−f [− vij (Rij + ∇i ∇j f )] dV. (5.14)
δF =
e−f
h v
M
Defining a formal Riemannian metric on M by Z 1 (5.16) hvij , vij ig = v ij vij dm, 2 M the gradient flow of F m on M is given by
(5.17)
(gij )t = −2 (Rij + ∇i ∇j f ).
The induced flow equation for f is dV 1 ∂ ln = g ij (gij )t = − △f − R. (5.18) ft = ∂t dm 2 As with any gradient flow, the function F m is nondecreasing along the flow line with its derivative being given by the length squared of the gradient, i.e. Z m (5.19) Ft = 2 |Rij + ∇i ∇j f |2 dm, M
as follows from (5.14) and (5.17)
NOTES ON PERELMAN’S PAPERS
21
We now perform time-dependent diffeomorphisms to transform (5.17) into the Ricci flow equation. If V (t) is the time-dependent generating vector field of the diffeomorphisms then the new equations for g and f become (5.20)
(gij )t = −2 (Rij + ∇i ∇j f ) + LV g, ft = − △f − R + LV f.
Taking V = ∇f gives (5.21)
(gij )t = −2 Rij ,
ft = − △f − R + |∇f |2.
As F (g, f ) is unchanged by a simultaneous pullback of g and f , and the right-hand side of (5.19) is also unchanged under a simultaneous pullback, it follows that under the new evolution equations (5.21) we still have Z d (5.22) F (g(t), f (t)) = 2 |Rij + ∇i ∇j f |2 e−f dV dt M (This can also be checked directly). Because of the diffeomorphisms that we applied, g and f are no longer related by e−f dV = R −f dm. We do have that M e dV is constant in t, as e−f dV is related to dm by a diffeomorphism.
The relation between g and f is as follows: we solve (or assume that we have a solution for) the first equation in (5.21), with some initial metric. Then given the solution g(t), we require that f satisfy the second equation in (5.21) (which is in terms of g(t)). The second equation in (5.21) can be written as
∂ −f e = − △e−f + R e−f . ∂t As this is a backward heat equation, we cannot solve for f forward in time starting with an arbitrary smooth function. Instead, (5.21) will be applied by starting with a solution for (gij )t = −2 Rij on some time interval [t1 , t2 ] and then solving (5.23) backwards in time on [t1 , t2 ] (which can always be done) starting with some initial f (t2 ). Having done this, the solution (g(t), f (t)) on [t1 , t2 ] will satisfy (5.22). (5.23)
6. Basic example for I.1 In this section we compute F in a Euclidean example.
Consider Rn with the standard metric, constant in time. Fix t0 > 0. Put τ = t0 − t and (6.1)
f (t, x) =
|x|2 n + ln(4πτ ), 4τ 2
so (6.2)
e−f = (4πτ )−n/2 e−
|x|2 4τ
.
22
BRUCE KLEINER AND JOHN LOTT
This is the standard heat kernel when considered for τ going from 0 to t0 , i.e. for t going from t0 to 0. One can check that (g, f ) solves (5.21). As Z |x|2 (6.3) e− 4τ dV = (4πτ )n/2 , Rn
2
x and |∇f |2 = |x| . Differentiating (6.3) with f is properly normalized. Then ∇f = 2τ 4τ 2 respect to τ gives Z |x|2 − |x|2 n (6.4) e 4τ dV = (4πτ )n/2 , 2 2τ Rn 4τ so Z n . (6.5) |∇f |2 e−f dV = 2τ Rn
Then F (t) =
n 2τ
=
n . 2(t0 −t)
In particular, this is nondecreasing as a function of t ∈ [0, t0 ).
7. I.2.2. The λ-invariant and its applications In this section we define λ(g) and show that it is nondecreasing under Ricci flow. We use this to show that a steady breather on a compact manifold is a gradient steady soliton. Proposition Given a metric g, there is a unique minimizer f of F (g, f ) under the R 7.1. −f constraint M e dV = 1.
Proof. Write
F =
(7.2) −f /2
Putting Φ = e (7.3)
,
F =
Z
2
Z
M
Re−f + 4|∇e−f /2 |2 dV. 2
Z
4|∇Φ| + R Φ dV = Φ (− 4△Φ + RΦ) dV. M R The constraint equation becomes M Φ2 dV = 1. Then λ is the smallest eigenvalue of −4△ + R and e−f /2 is a corresponding normalized eigenvector. As the operator is a Schr¨odinger operator, there is a unique normalized positive eigenvector [55, Chapter XIII.12]. M
Definition 7.4. The λ-functional is given by λ(g) = F (g, f). If g(t) is a smooth family of metrics then it follows from eigenvalue perturbation theory that λ(g(t)) and f(t) are smooth in t [55, Chapter XII]. Proposition 7.5. (cf. I.2.2) If g(·) is a Ricci flow solution then λ(g(t)) is nondecreasing in t. Proof. Consider a time interval [t1 , t2 ], and the minimizer f (t2 ). In particular, λ(t2 ) = F (g(t2), f (t2 )). Put u(t2 ) = e−f (t2 ) . Solve the backward heat equation (7.6)
∂u = − △u + Ru ∂t
NOTES ON PERELMAN’S PAPERS
23
backward on [t1 , t2 ]. We claim that u(x′ , t′ ) > 0 for all x′ ∈ M and t′ ∈ [t1 , t2 ]. To see this, we take t′ ∈ = △h on (t′ , t2 ] with [t1 , t2 ), and let h be the solution to the forward heat equation ∂h ∂t limt→t′ h(t) = δx′ . We have Z Z d (7.7) u(t) h(t) dV = [(∂t u + △u − Ru) v + u (∂t h − △h)] dV = 0. dt M M One knows that h(t) > 0 for all t ∈ (t′ , t2 ]. Then Z Z Z ′ ′ ′ (7.8) u(x , t ) = u(x, t ) δx′ (x) dV (x) = lim′ u(t) h(t) dV = u(t2 ) h(t2 ) dV > 0. t→t
M
M
M
For t ∈ [t1 , t2 ], define f (t) by u(t) = e−f (t) . By (5.22), F (g(t1), f (t1 )) ≤ F (g(t2), f (t2 )). By the of λ, λ(tR1 ) = F (g(t1 ), f(t1 )) ≤ F (g(t1), f (t1 )). (We are using the fact R definition −f (t1 ) that M e dV (t1 ) = M e−f (t2 ) dV (t2 ) = 1.) Thus λ(t1 ) ≤ λ(t2 ).
Definition 7.9. A steady breather is a Ricci flow solution on an interval [t1 , t2 ] that satisfies the equation g(t2 ) = φ∗ g(t1 ) for some φ ∈ Diff(M). Steady soliton solutions are steady breathers.
Again, we are assuming that M is compact. The next result is not essential for the sequel, but gives a good illustration of how a monotonicity formula is used. Proposition 7.10. (cf. I.2.2) A steady breather is a gradient steady soliton. Proof. We have λ(g(t2 )) = λ(φ∗ g(t1 )) = λ(g(t1)). Thus we have equality in Proposition 7.5. Tracing through the proof, F (g(t), f (t)) must be constant in t. From (5.22), Rij + ∇i ∇j f = 0. Then R + △f = 0 and so (5.21) becomes (C.5). One can sharpen Proposition 7.5. Lemma 7.11. (cf. Proposition I.1.2) 2 2 dλ ≥ λ (t). dt n
(7.12)
Proof. Given a time interval [t1 , t2 ], with the notation of the proof of Proposition 7.5 we have Z t2 Z (7.13) λ(t1 ) ≤ F (g(t1 ), f (t1 )) = F (g(t2), f (t2 )) − 2 |Rij + ∇i ∇j f |2 e−f dV dt = λ(t2 ) − 2 Then (7.14)
Z
t2
t1
Z
t1
M
M
|Rij + ∇i ∇j f |2 e−f dV dt.
λ(t2 ) − λ(t1 ) dλ = lim− ≥ 2 dt t=t2 t2 − t1 t1 →t2
where the right-hand side is evaluated at time t2 .
Z
M
|Rij + ∇i ∇j f |2 e−f dV,
24
BRUCE KLEINER AND JOHN LOTT
Hence for all t, dλ ≥ 2 dt
(7.15) and so
Z
M
dλ 2 ≥ dt n
(7.16)
|Rij + ∇i ∇j f |2 e−f dV Z
M
(R + △f)2 e−f dV.
R From the Cauchy-Schwarz inequality and the fact that M e−f dV = 1, Z 2 Z −f ≤ (R + △f)2 e−f dV. (7.17) (R + △f ) e dV M
M
Finally, (5.10) gives Z Z −f (7.18) (R + △f) e dV = (R + |∇f |2 ) e−f dV = F (g(t), f(t)) = λ(t). M
M
This proves the lemma.
8. I.2.3. The rescaled λ-invariant In this section we show the monotonicity of a scale-invariant version of λ. This will be used in Section 93. We then show that an expanding breather on a compact manifold is a gradient expanding soliton. 2
Put λ(g) = λ(g)V (g) n . As λ is scale-invariant, it is constant in t along a steady, shrinking or expanding soliton solution. Proposition 8.1. (cf. Claim of I.2.3) If g(·) is a Ricci flow solution and λ(g(t)) ≤ 0 for some t then dtd λ(g(t)) ≥ 0. Proof. We have 2−n 2 dλ 2 dλ = V (t) n − V (t) n λ(t) dt dt n
(8.2) From (7.15), (8.3)
2 dλ ≥ V (t) n dt
Z
R dV.
M
Z Z 2 −1 2 −f R dV . 2 |Rij + ∇i ∇j f | e dV − V (t) λ(t) n M M
Using the spatially-constant function ln V (t) as a test function for F gives Z −1 (8.4) λ(t) ≤ V (t) R dV. M
The assumption that λ(t) ≤ 0 gives (8.5)
2
− λ(t) ≤ − V (t)
and so (8.6)
2 dλ ≥ V (t) n dt
−1
λ(t)
Z
R dV
M
Z 2 2 −f 2 2 |Rij + ∇i ∇j f| e dV − λ(t) . n M
NOTES ON PERELMAN’S PAPERS
Next, (8.7)
|Rij
25
2 1 1 + ∇i ∇j f | ≥ Rij + ∇i ∇j f − (R + △f) gij + (R + △f)2 . n n 2
Using (7.18), one obtains (8.8) "Z 2 Z −f 2 1 dλ 1 Rij + ∇i ∇j f − (R + △f) gij e dV + ≥ 2 V (t) n (R + △f )2 e−f dV dt n n M M Z 2 # 1 − (R + △f) e−f dV . n M R As M e−f dV = 1, the Cauchy-Schwarz inequality implies that the right-hand side of (8.8) is nonnegative. Corollary 8.9. If λ is a constant nonpositive number on an interval [t1 , t2 ] then the Ricci flow solution is a gradient soliton. Proof. From equation (8.8), we obtain that R + △f = α(t) for some function α that is spatially constant, and Rij + ∇i ∇j f = α(t) gij . Thus g evolves by diffeomorphisms and n dilations. After a shift of the time parameter, α(t) is proportionate to t cf. [22, Lemma 2.4]. This is the gradient soliton equation of Appendix C. Definition 8.10. An expanding breather is a Ricci flow solution on [t1 , t2 ] that satisfies g(t2 ) = c φ∗ g(t1) for some c > 1 and φ ∈ Diff(M). Expanding soliton solutions are expanding breathers. Again, we are assuming that M is compact. Proposition 8.11. An expanding breather is a gradient expanding soliton. Proof. First, λ(t2 ) = λ(t1 ). As V (t2 ) > V (t1 ), we must have dV > 0 for some t ∈ [t1 , t2 ]. dt R dV From (8.4), dt = − M R dV ≤ − λ(t) V (t), so λ(t) must be negative for some t ∈ [t1 , t2 ]. Proposition 8.1 implies that λ(t1 ) < 0. Then as λ(t2 ) = λ(t1 ), it follows that λ is a negative constant on [t1 , t2 ]. From Corollary 8.9, the solution is a gradient expanding soliton. 9. I.2.4. Gradient steady solitons on compact manifolds It was shown in Section 7 that a steady breather on a compact manifold is a gradient steady soliton. We now show that it is in fact Ricci flat. This was previously shown in [33, Theorem 20.1]. Proposition 9.1. A gradient steady solution on a compact manifold is Ricci flat. Proof. As we are in the equality case of Proposition 7.5, the function f (t) must be the minimizer of F (g(t), ·) for all t. That is, (9.2)
f
f
(− 4△ + R) e− 2 = λ e− 2
26
BRUCE KLEINER AND JOHN LOTT
for all t, where λ is constant in t. Equivalently, 2△f −|∇f |2 +RR= λ. As R+△f = R0, we have △f −|∇f |2 = λ. Then △e−f = −λe−f . Integrating gives 0 = M △e−f dV = −λ M e−f dV , 2 R R so λ = 0. Then 0 = − M e−f △e−f dV = M ∇e−f dV , so f is constant and g is Ricci flat. A similar argument shows that a gradient expanding soliton on a compact manifold comes from an Einstein metric with negative Ricci curvature. 10. Ricci flow as a gradient flow We have shown in Section 5 that the modified Ricci flow is the gradient flow for the functional F m on the space of metrics M. One can ask if the unmodified Ricci flow is a gradient flow. This turns out to be true provided that one considers it as a flow on the space M/ Diff(M).
As mentioned in II.8, Ricci flow is the gradient flow for the function λ. More precisely, this statement is valid on M/ Diff(M), with the latter being equipped with an appropriate metric. To see this, we first consider λ as a function on the space of metrics M. Here the formal Riemannian metric on M comes from saying that for vij ∈ Tg M, Z 1 (10.1) hvij , vij i = v ij vij Φ2 dV (g), 2 M where Φ = Φ(g) is the unique normalized positive eigenvector corresponding to λ(g). Lemma 10.2. The formal gradient flow of λ is ∂gij = − 2 (Rij − 2 ∇i ∇j ln Φ) . (10.3) ∂t Proof. We set (10.4)
λ(g) =
inf R
f ∈C ∞ (M ) :
M
e−f dV = 1
F (g, f ).
To calculate the variation in λ due to a variation δgij = vij , we let h = δf be the variation induced by letting f be the minimizer in (10.4). Then Z Z v −f (10.5) 0=δ e dV = − h e−f dV. M M 2 Now equation (5.14) gives
(10.6)
δλ(vij ) =
f
As Φ = e− 2 satisfies (10.7) it follows that (10.8)
Z
e−f [− vij (Rij + ∇i ∇j f ) + M i v 2 − h (2△f − |∇f | + R) dV. 2 − 4 △Φ + RΦ = λΦ, 2 △f − |∇f |2 + R = λ.
NOTES ON PERELMAN’S PAPERS
27
Hence the last term in (10.6) vanishes, and (10.6) becomes Z (10.9) δλ(vij ) = − e−f vij (Rij + ∇i ∇j f ) dV. M
The corresponding gradient flow is ∂gij (10.10) = − 2 (Rij + ∇i ∇j f ) = − 2 (Rij − 2 ∇i ∇j ln Φ) . ∂t We note that it follows from (10.8) that (10.11)
∇j (Rij + ∇i ∇j f ) e−f
= 0.
This implies that the gradient vector field of λ is perpendicular to the infinitesimal diffeomorphisms at g, as one would expect. In the sense of [10], the quotient space M/ Diff(M) is a stratified infinite-dimensional Riemannian manifold, with the strata corresponding to the possible isometry groups Isom(M, g). We give it the quotient Riemannian metric coming from (10.1). The modified Ricci flow (10.10) on M projects to a flow on M/ Diff(M) that coincides with the projection of the unmodified Ricci flow dg = − 2 Ric. The upshot is that the Ricci flow, as a flow on dt M/ Diff(M), is the gradient flow of λ, the latter now being considered as a function on M/ Diff(M). One sees an intuitive explanation for Proposition 7.10. If a gradient flow on a finitedimensional manifold has a periodic orbit then it must be a fixed-point. Applying this principle formally to the Ricci flow on M/ Diff(M), one infers that a steady breather only evolves by diffeomorphisms. 11. The W-functional Definition 11.1. The W-functional W : M × C ∞ (M) × R+ → R is given by Z n (11.2) W(g, f, τ ) = τ |∇f |2 + R + f − n (4πτ )− 2 e−f dV. M
The W-functional is a scale-invariant variant of F . It has the symmetries W(φ∗ g, φ∗f, τ ) = W(g, f, τ ) for φ ∈ Diff(M), and W(cg, f, cτ ) = W(g, f, τ ) for c > 0. Hence it is constant in t = − τ along a gradient shrinking soliton defined for t ∈ (−∞, 0), as in Appendix C. In this sense, W is constant on gradient shrinking solitons just as F is constant on gradient steady solitons. As an example of a gradient shrinking soliton, consider Rn with the flat metric, constant in time t ∈ (−∞, 0). Put τ = − t and (11.3)
f (t, x) =
|x|2 , 4τ
so (11.4)
e−f = e−
|x|2 4τ
.
28
BRUCE KLEINER AND JOHN LOTT
One can check that (g(t), f (t), τ (t)) satisfies (12.12) and (12.13). Now |x|2 |x|2 |x|2 + − n = − n. 4τ 2 4τ 2τ It follows from (6.3) and (6.4) that W(t) = 0 for all t. τ (|∇f |2 + R) + f − n = τ ·
(11.5)
12. I.3.1. Monotonicity of the W-functional In this section we compute the variation of W, in analogy with the computation in Section 5 of the variation of F . We then show that a shrinking breather on a compact manifold is a gradient shrinking soliton. As in Section 5, we write δgij = vij and δf = h. Put σ = δτ . Proposition 12.1. We have (12.2) Z δW(vij , h, σ) = σ(R + |∇f |2 ) − τ vij (Rij + ∇i ∇j f ) + h + M v nσ i 2 τ (2△f − |∇f | + R) + f − n − h − (4πτ )−n/2 e−f dV. 2 2τ Proof. One finds
δ (4πτ )−n/2 e−f dV
(12.3) Writing
W =
(12.4)
Z
M
we can use (5.14) to obtain
=
v
2
− h −
nσ (4πτ )−n/2 e−f dV. 2τ
τ (R + |∇f |2 ) + f − n (4πτ )−n/2 e−f dV
(12.5)
Z h v 2 δW = σ(R + |∇f | ) + τ − h (2△f − 2|∇f |2) − τ vij (Rij + ∇i ∇j f ) + 2 M v nσ i (12.6) − h − (4πτ )−n/2 e−f dV. h + τ (R + |∇f |2) + f − n 2 2τ Then (5.10) gives (12.7)
Z
σ(R + |∇f |2) − τ vij (Rij + ∇i ∇j f ) + h + M v nσ i τ (2△f − |∇f |2 + R) + f − n − h − (4πτ )−n/2 e−f dV. 2 2τ This proves the proposition. δW =
We now fix a smooth measure dm on M with mass 1 and relate f to g and τ by requiring that (4πτ )−n/2 e−f dV = dm. Then v2 − h − nσ = 0 and 2τ Z (12.8) δW = σ(R + |∇f |2 ) − τ vij (Rij + ∇i ∇j f ) + h (4πτ )−n/2 e−f dV. M
NOTES ON PERELMAN’S PAPERS
We now consider
dW dt
when (gij )t = − 2 (Rij + ∇i ∇j f ), n ft = − △f − R + , 2τ τt = −1.
(12.9)
To apply (12.8), we put
vij = − 2 (Rij + ∇i ∇j f ), n h = − △f − R + , 2τ σ = −1.
(12.10)
v 2
− h − nσ = 0. Then from (12.8), 2τ Z dW = − (R + |∇f |2) + 2 τ |Rij + ∇i ∇j f |2 dt M ni − △f − R + (4πτ )−n/2 e−f dV 2τ Z h ni (4πτ )−n/2 e−f dV = − 2(R + △f ) + 2 τ |Rij + ∇i ∇j f |2 + 2τ ZM 1 gij |2 (4πτ )−n/2 e−f dV. 2 τ |Rij + ∇i ∇j f − = 2τ M
We do have (12.11)
29
Adding a Lie derivative to the right-hand side of (12.9) gives the new flow equations (12.12)
(gij )t = − 2 Rij ,
ft = − △f + |∇f |2 − R +
n , 2τ
τt = −1,
with (12.11) still holding. We no longer have (4πτ )−n/2 e−f dV = dm, but we do have Z (12.13) (4πτ )−n/2 e−f dV = 1. M
We now wantR to look at the variational problem of minimizing W(g, f, τ ) under the constraint that M (4πτ )−n/2 e−f dV = 1. We write Z (12.14) µ(g, τ ) = inf {W(g, f, τ ) : (4πτ )−n/2 e−f dV = 1}. f
M
− f2
Making the change of variable Φ = e , we are minimizing Z −n/2 (12.15) (4πτ ) τ (4|∇Φ|2 + RΦ2 ) − 2Φ2 log Φ − n Φ2 dV M
R under the constraint (4πτ ) Φ2 dV = 1. From [56, Section 1] the infimum is finite M and there is a positive continuous minimizer Φ. It will be a weak solution of the variational equation −n/2
(12.16)
τ (−4△ + R)Φ = 2Φ log Φ + (µ(g, τ ) + n)Φ.
30
BRUCE KLEINER AND JOHN LOTT
From elliptic theory, Φ is smooth. Then f = − 2 log Φ is also smooth.
As in Section 7, it follows that µ(g(t), t0 −t) is nondecreasing in t for a Ricci flow solution, where t0 is any fixed number and t < t0 . If it is constant in t then the solution must be a gradient shrinking soliton that goes singular at time t0 . Definition 12.17. A shrinking breather is a Ricci flow solution on [t1 , t2 ] that satisfies g(t2 ) = c φ∗ g(t1) for some c < 1 and φ ∈ Diff(M). Gradient shrinking soliton solutions are shrinking breathers. Again, we are assuming that M is compact. Proposition 12.18. A shrinking breather is a gradient shrinking soliton. t2 −ct1 . 1−c
Then if τ1 = t0 − t1 and τ2 = t0 − t2 , we have τ2 = cτ1 . Hence τ2 ∗ φ g(t1 ), τ2 = µ (φ∗ g(t1 ), τ1 ) = µ (g(t1 ), τ1 ) . (12.19) µ(g(t2), τ2 ) = µ τ1 It follows that the solution is a gradient shrinking soliton. Proof. Put t0 =
13. I.4. The no local collapsing theorem I In this section we prove the no local collapsing theorem. Definition 13.1. A smooth Ricci flow solution g(·) on a time interval [0, T ) is said to be locally collapsing at T if there is a sequence of times tk → T and a sequence of metric balls Bk = B(pk , rk ) at times tk such that rk2 /tk is bounded, | Rm |(g(tk )) ≤ rk−2 in Bk and limk→∞ rk−n vol(Bk ) = 0. Remark 13.2. In the definition of noncollapsing, T could be infinite. This is why it is written that rk2 /tk stays bounded, while if T < ∞ then this is obviously the same as saying that rk stays bounded. Theorem 13.3. (cf. Theorem I.4.1) If M is closed and T < ∞ then g(·) is not locally collapsing at T . Proof. We first sketch the idea of the proof. In Section 11 we showed that in the case of flat n
−f
−
|x|2 4τ
rk2
−fk
−
|x|2 4r 2 k
R , taking e (x) = e , , we get W(g, f, τ ) = 0. So putting τ = and e (x) = e we have W(g, fk , rk2 ) = 0. In the collapsing case, the idea is to use a test function fk so that (13.4)
−
e−fk (x) ∼ e−ck e
distt (x,pk )2 k 4r 2 k
,
where ck is determined by the normalization condition Z (13.5) (4πrk2 )−n/2 e−fk dV = 1. M
The main difference between computing (13.5) in M and in Rn comes from the difference in 1 volumes, which means that e−ck ∼ r−n vol(B . In particular, as k → ∞, we have ck → −∞. ) k
k
NOTES ON PERELMAN’S PAPERS
31
Now that fk is normalized correctly, the main difference between computing W(g(tk ), fk , rk2 ) in M, and the analogous computation for the Gaussian in Rn , comes from the f term in the integrand of W. Since fk ∼ ck , this will drive W(g(tk ), fk , rk2 ) to −∞ as k → ∞, so µ(g(tk ), rk2 ) → −∞; by the monotonicity of µ(g(t), t0 − t) it follows that µ(g(0), tk + rk2 ) → −∞ as k → ∞. This contradicts the fact that µ(g(0), τ ) is a continuous function of τ . To write this out precisely, let us put Φ = e−f /2 , so that Z −n/2 (13.6) W(g, f, τ ) = (4πτ ) 4τ |∇Φ|2 + (τ R − 2 ln Φ − n) Φ2 dV. M
For the argument, it is enough to obtain small values of W for positive Φ. Since lims→0 (−2 ln s)s2 = 0, by an approximation it is enough to obtain small values of W for nonnegative Φ, where the integrand is declared to be 4τ |∇Φ|2 at points where Φ vanishes. Take Φk (x) = e−ck /2 φ(disttk (x, pk )/rk ),
(13.7)
where φ : [0, ∞) → [0, 1] is a monotonically nonincreasing function such that φ(s) = 1 if s ∈ [0, 1/2], φ(s) = 0 if s ≥ 1 and |φ′ (s)| ≤ 10 for s ∈ [1/2, 1]. The function Φk is a priori only Lipschitz, but by smoothing it slightly we can use Φk in the variational formula to bound W from above. The constant ck is determined by Z ck (13.8) e = (4πrk2 )−n/2 φ2 (disttk (x, pk )/rk ) dV ≤ (4πrk2 )−n/2 vol(Bk ). M
Thus ck → −∞. Next, (13.9)
W(g(tk ), fk , rk2 )
=
(4πrk2 )−n/2
Z
M
4rk2 |∇Φk |2 + (rk2 R − 2 ln Φk − n) Φ2k dV.
Let Ak (s) be the mass of the distance sphere S(pk , rk s) around pk . Put Z 2 −1 (13.10) Rk (s) = rk Ak (s) R d area . S(pk ,rk s)
We can compute the integral in (13.9) radially to get (13.11) R1 ′ 4(φ (s))2 + (Rk (s) + ck − 2 ln φ(s) − n) φ2 (s) Ak (s) ds 2 0 . W(g(tk ), fk , rk ) = R1 2 (s) A (s) ds φ k 0
The expression 4(φ′ (s))2 − 2 ln φ(s) φ2 (s) vanishes if s ∈ / [1/2, 1], and is bounded above by −1 400 + e if s ∈ [1/2, 1]. Then the lower bound on the Ricci curvature and the BishopGromov inequality give R1 [4(φ′ (s))2 − 2 ln φ(s) φ2 (s)] Ak (s) ds vol(B(pk , rk )) − vol(B(pk , rk /2)) ≤ 401 (13.12) 0 R1 vol(B(pk , rk /2)) φ2 (s) Ak (s) ds 0 ! R1 n−1 sinh (s) ds 0 − 1 . ≤ 401 R 1/2 n−1 sinh (s) ds 0 Next, from the upper bound on scalar curvature, Rk (s) ≤ n(n − 1) for s ∈ [0, 1]. Putting this together gives W(g(tk ), fk , rk2 ) ≤ const. + ck and so W(g(tk ), fk , rk2 ) → −∞ as k → ∞.
32
BRUCE KLEINER AND JOHN LOTT
Thus µ(g(tk ), rk2 ) → −∞. For any t0 > t, µ(g(t), t0 − t) is nondecreasing in t. Hence µ(g(0), tk + rk2 ) ≤ µ(g(tk ), rk2 ), so µ(g(0), tk + rk2 ) → −∞. Since T is finite, tk and rk2 are uniformly bounded, and tk uniformly positive, which contradicts the fact that µ(g(0), τ ) is a continuous function of τ . Remark 13.13. In the preceding argument we only used the upper bound on scalar curvature and the lower bound on Ricci curvature, i.e. in the definition of local collapsing one could have assumed that R(gij (tk )) ≤ n(n − 1) rk−2 in Bk and Ric(gij (tk )) ≥ − (n − 1) rk−2 in Bk . In fact, one can also remove the lower bound on Ricci curvature (observation of Perelman, communicated by Gang Tian). The necessary ingredients of the preceding argument were that 1. rk−n vol(B(pk , rk )) → 0, 2. rk2 R is uniformly bounded above on B(pk , rk ) and vol(B(pk ,rk )) is uniformly bounded above. 3. vol(B(p k ,rk /2)) Suppose only that rk−n vol(B(pk , rk )) → 0 and for all k, rk2 R ≤ n(n − 1) on B(pk , rk ). If vol(B(pk ,rk )) < 3n for all k then we are done. If not, suppose that for a given k, vol(B(p ≥ k ,rk /2)) −n ′ ′ −n ′ 3 . Putting rk = rk /2, we have that (rk ) vol(B(pk , rk )) ≤ rk vol(B(pk , rk )) and vol(B(pk ,rk )) (rk′ )2 R ≤ n(n − 1) on B(pk , rk′ ). We replace rk by rk′ . If now vol(B(p < 3n then we k ,rk /2)) stop. If not then we repeat the process and replace rk by rk /2. Eventually we will achieve vol(B(pk ,rk )) < 3n . Then we can apply the preceding argument to this new sequence that vol(B(p k ,rk /2)) of pairs {(pk , rk )}∞ k=1 . vol(B(pk ,rk )) vol(B(pk ,rk /2)) n
Definition 13.14. (cf. Definition I.4.2) We say that a metric g is κ-noncollapsed on the scale ρ if every metric ball B of radius r < ρ, which satisfies | Rm(x)| ≤ r −2 for every x ∈ B, has volume at least κrn . Remark 13.15. We caution the reader that this definition differs slightly from the definition of noncollapsing that is used from section I.7 onwards. Note that except for the overall scale ρ, the κ-noncollapsed condition is scale-invariant. From the proof of Theorem 13.3 we extract the following statement. Given a Ricci flow defined on an interval [0, T ), with T < ∞, and a scale ρ, there is some number κ = κ(g(0), T, ρ) so that the solution is κ-noncollapsed on the scale ρ for all t ∈ [0, T ). We note that the estimate on κ deteriorates as T → ∞, as there are Ricci flow solutions that collapse at long time.
14. I.5. The W-functional as a time derivative We will only discuss one formula from I.5, showing that along a Ricci flow, W is itself the time-derivative of an integral expression.
NOTES ON PERELMAN’S PAPERS
33
Again, we put τ = − t. Consider the evolution equations (12.9), with (4πτ )−n/2 e−f dV = dm. Then Z Z Z d n n n (14.1) τ f − dm = f − dm + τ △f + R − dm dτ 2 2 2τ M M M Z Z n n dm + τ |∇f |2 + R − dm = f − 2 2τ M M = W(g(t), f (t), τ ). With respect to the evolution equations (12.12) obtained by performing diffeomorphisms, we get Z d n −n/2 −f (14.2) τ f − (4πτ ) e dV = W(g(t), f (t), τ ). dτ 2 M Similarly, with respect to (5.21), Z d −f − f e dV = F (g(t), f (t)). (14.3) dt M 15. I.7. Overview of reduced length and reduced volume We first give a brief summary of I.7. In I.7, the variable τ = t0 − t is used and so the corresponding Ricci flow equation is (gij )τ = 2Rij . The goal is to prove a no local collapsing theorem by means of the L-lengths of curves γ : [τ1 , τ2 ] → M, defined by Z τ2 2 √ (15.1) L(γ) = τ R(γ(τ )) + γ(τ ˙ ) dτ, τ1
where the scalar curvature R(γ(τ )) and the norm |γ(τ ˙ )| are evaluated using the metric at dγ time t0 − τ . Here τ1 ≥ 0. With X = dτ , the corresponding L-geodesic equation is
1 1 ∇R + X + 2 Ric(X, ·) = 0, 2 2τ where again the connection and curvature are taken at the corresponding time, and the 1-form Ric(X, ·) has been identified with the corresponding dual vector field. √ Fix p ∈ M. Taking τ1 = 0 and γ(0) = p, the vector v = limτ →0 τ X(τ ) is welldefined in Tp M and is called the initial vector of the geodesic. The L-exponential map Lexpτ : Tp M → M sends v to γ(τ ). (15.2)
∇X X −
The function L(q, τ ) is the infimal L-length of curves γ with γ(0) = p and γ(τ ) = q. Defining the reduced length by (15.3)
l(q, τ ) =
and the reduced volume by (15.4)
V˜ (τ ) =
Z
M
L(q, τ ) √ 2 τ
n
τ − 2 e−l(q,τ ) dq,
34
BRUCE KLEINER AND JOHN LOTT
the goal is to show that V˜ (τ ) is nonincreasing in τ , i.e. nondecreasing in t. To do this, one uses the L-exponential map to write V˜ (τ ) as an integral over Tp M: Z n ˜ (15.5) V (τ ) = τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) χτ (v) dv, Tp M
where J (v, τ ) = det d (Lexpτ )v is the Jacobian factor in the change of variable and χτ is a cutoff function related to the L-cut locus of p. To show that V˜ (τ ) is nonincreasing in τ n it suffices to show that τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) is nonincreasing in τ or, equivalently, that − n2 ln(τ ) − l(Lexpτ (v), τ ) + ln J (v, τ ) is nonincreasing in τ . Hence it is necessary to (v,τ ) . The computation of the latter will involve the L-Jacobi compute dl(Lexpdττ (v),τ ) and dJdτ fields. The fact that V˜ (τ ) is nonincreasing in τ is then used to show that the Ricci flow solution cannot be collapsed near p. 16. Basic example for I.7 In this section we say what the various expressions of I.7 become in the model case of a flat Euclidean Ricci solution. If M is flat Rn and p = ~0 then the unique L-geodesic γ with γ(0) = ~0 and γ(τ ) = ~q is τ 21 1 (16.1) γ(τ ) = ~q = 2 τ 2 ~v . τ The function L is given by 1 1 (16.2) L(q, τ ) = τ − 2 |q|2 2 and the reduced length (15.3) is given by (16.3)
l(q, τ ) =
|q|2 . 4τ
1
The function L(q, τ ) = 2 τ 2 L(q, τ ) is L(q, τ ) = |q|2.
(16.4) Then (16.5)
V˜ (τ ) =
Z
n
|q|2
n
τ − 2 e− 4τ dn q = (4π) 2
Rn
is constant in τ .
17. Remarks about L-Geodesics and L exp In this section we discuss the variational equation corresponding to (15.1). To derive the L-geodesic equation, as in Riemannian geometry we consider a 1-parameter family of curves γs : [τ1 , τ2 ] → M, parametrized by s ∈ (−ǫ, ǫ). Equivalently, we have a map γ γ and Y = ∂˜ , we have [X, Y ] = 0. γ˜ (s, τ ) with s ∈ (−ǫ, ǫ) and t ∈ [τ1 , τ2 ]. Putting X = ∂˜ ∂τ ∂s
NOTES ON PERELMAN’S PAPERS
35
Then ∇X Y = ∇Y X. Restricting to the curve γ(τ ) = γ˜ (0, τ ) and writing δY as shorthand d for ds , we have (δY γ)(τ ) = Y (τ ) and (δY X)(τ ) = (∇X Y )(τ ). Then s=0 Z τ2 √ τ (hY, ∇Ri + 2 h∇X Y, Xi) dτ. (17.1) δY L = τ1
Using the fact that
dgij dτ
(17.2)
dhY, Xi = h∇X Y, Xi + hY, ∇X Xi + 2 Ric(Y, X). dτ
Then
Z
= 2Rij , we have
τ2
√ τ (hY, ∇Ri + 2 h∇X Y, Xi) dτ = τ1 Z τ2 √ d τ hY, ∇Ri + 2 hY, Xi − 2hY, ∇X Xi − 4 Ric(Y, X) dτ = dτ τ1 Z τ2 τ2 √ √ 1 τ Y, ∇R − 2∇X X − 4 Ric(X, ·) − X dτ. 2 τ hX, Y i τ1 + τ τ1 Hence the L-geodesic equation is 1 1 (17.4) ∇X X − ∇R + X + 2 Ric(X, ·) = 0. 2 2τ
(17.3)
We now discuss some technical issues about L-geodesics and the L-exponential map. We are assuming that (M, g(·)) is a Ricci flow, where the curvature operator of M is uniformly bounded on a τ -interval [τ1 , τ2 ], and each τ -slice (M, g(τ )) is complete for τ ∈ [τ1 , τ2 ]. By Appendix D, for every τ ′ < τ2 there is a constant D < ∞ such that D (17.5) |∇R(x, τ )| < √ τ2 − τ for all x ∈ M, τ ∈ [τ1 , τ2 ). √ Making the change of variable s = τ in the formula for L-length, we get Z s2 2 1 dγ 2 (17.6) L(γ) = 2 + s R(γ(s)) ds. 4 ds s1 The Euler-Lagrange equation becomes ˆ − 2s2 ∇R + 4s Ric(X, ˆ ·) = 0, (17.7) ∇Xˆ X ˆ = dγ = 2sX. Putting s1 = √τ1 , it follows from standard existence theory for where X ds ODE’s that for each p ∈ M and v ∈ Tp M, there is a unique solution γ(s) to (17.7), defined on an interval [s1 , s1 + ǫ), with γ(s1 ) = p and √ dγ 1 ′ γ (s1 ) = lim τ = v. (17.8) τ →τ1 2 dτ If γ(s) is defined for s ∈ [s1 , s′ ] then d ˆ 2 d ˆ ˆ ˆ X) ˆ + 2h∇ ˆ X, ˆ Xi ˆ (17.9) |X| = hX, Xi = 4s Ric(X, X ds ds ˆ X) ˆ + 4s2 h∇R, Xi ˆ = −4s Ric(X,
36
BRUCE KLEINER AND JOHN LOTT
ˆ and so if X(s) 6= 0 then (17.10)
1 d ˆ 2 d ˆ ˆ Ric |X| = |X| = −2s|X| ˆ ds ds 2|X|
ˆ X ˆ X , ˆ |X| ˆ |X|
!
+ 2s2
*
ˆ X ∇R, ˆ |X|
+
.
By (17.5), d ˆ ˆ + √ C2 |X| ≤ C1 |X| ds s2 − s √ for appropriate constants C1 and C2 , where s2 = τ2 . Since the metrics g(τ ) are uniformly comparable for τ ∈ [τ1 , τ2 ], we conclude (by a continuity argument in s ) that the L-geodesic γv with 12 γv′ (s1 ) = v is defined on the whole interval [s1 , s2 ]. In particular, in terms of the original variable τ , for each τ ∈ [τ1 , τ2 ] and each p ∈ M, we get a globally defined and smooth L-exponential √ dγ map L expτ : Tp M → M which takes each v ∈ Tp M to γ(τ ), where v = limτ ′ →τ1 τ ′ dτ ′ . Note that unlike in the case of Riemannian geometry, L expτ (0) may not be p, because of the ∇R term in (17.4). (17.11)
We now fix p ∈ M, take τ1 = 0, and let L(q, τ¯) be the minimizer function as in Section 15. We can imitate the traditional Riemannian geometry proof that geodesics minimize √ for a short time. Using the change of variable s = τ and the implicit function theorem, there is an r = r(p) > 0 (which varies continuously with p) such that for every q ∈ M with d(q, p) ≤ 10r at τ = 0, and every 0 < τ¯ ≤ r 2 , there is a unique L-geodesic γ(q,¯τ ) : [0, τ¯] → M, starting at p and ending at q, which remains within the ball B(p, 100r) (in the τ = 0 slice (M, g(0))), and γ(q,¯τ ) varies smoothly with (q, τ¯). Thus, the L-length of γ(q,¯τ ) varies smoothly ˆ τ¯) near (p, 0). We claim that L ˆ = L near (p, 0). with (q, τ¯), and defines a function L(q, Suppose that q ∈ B(p, r) and let α : [0, τ¯] → M be a smooth curve whose L-length is close to L(q, τ¯). If r is small, relative to the assumed curvature bound, then α must stay within B(p, 10r). Equations (18.2) and (18.6) below imply that (17.12) dα 2 √ √ √ d ˆ d dα 2 L(α(τ ), τ ) = h2 τ X, i + τ (R−|X| ) ≤ τ R + = Llength(α [0,τ ] ) . dτ dτ dτ dτ Thus γ(q,¯τ ) minimizes when (q, τ¯) is close to (p, 0).
We can now deduce that for all (q, τ¯), there is an L-geodesic γ : [0, τ¯] → M which has infimal L-length among all piecewise smooth curves starting at p and ending at q (with domain [0, τ¯]). This can be done by imitating the usual broken geodesic argument, using the fact that for x, y in a given small ball of M and for sufficiently small time intervals R τ ′ +ǫ √ [τ ′ , τ ′ + ǫ] ⊂ [0, τ ], there is a unique minimizer γ for τ ′ τ R(γ(τ )) + |γ(τ ˙ )|2 dτ with √ γ(τ ′ ) = x and γ(τ ′ + ǫ) = y. Alternatively, using the change of variable s = τ , one can take a minimizer of L among H 1,2-regular curves.
Another technical issue is the justification of the change of variables from M to Tp M in the proof of monotonicity of reduced volume. Fix p ∈ M and τ > 0, and let L expτ : Tp M → M be the√map which takes v ∈ Tp M to γv (τ ), where γv : [0, τ ] → M is the unique L-geodesic v with τ ′ dγ → v as τ ′ → 0. Let B ⊂ M be the set of points which are either endpoints of dτ ′ more than one minimizing L-geodesic, or which are the endpoint of a minimizing geodesic γv : [0, τ ] → M where v ∈ Tp M is a critical point of L expτ . We will call B the time-τ L-cut
NOTES ON PERELMAN’S PAPERS
37
locus of p. It is a closed subset of M. Let G ⊂ M be the complement of B and let Ωτ ⊂ Tp M be the corresponding set of initial conditions for minimizing L-geodesics. Then Ωτ is an open set, and L exp maps it diffeomorphically onto G. We claim that B has measure zero. By Sard’s theorem, to prove this it suffices to prove that the set B′ of points q ∈ B which are regular values of L expτ , has measure zero. Pick q ∈ B′ , and distinct points v1 , v2 ∈ Tp M such that γvi : [0, τ ] → M are both minimizing geodesics ending at q. Then L expτ is a local diffeomorphism near each vi . The first variation formula and the implicit function theorem then show that there are neighborhoods Ui of vi , and a smooth hypersurface H passing through q, such that if we have points wi ∈ Ui with (17.13)
q ′ = L expτ (w1 ) = L expτ (w2 ) and L length(γw1 ) = L length(γw2 ),
then q ′ lies on H. Thus B′ is contained in a countable union of hypersurfaces, and hence has measure zero. Therefore one may compute the integral of any integrable function on M by pulling it back to Ωτ ⊂ Tp M and using the change of variables formula. Note that if τ ≤ τ ′ then Ωτ ′ ⊂ Ωτ . 18. I.(7.3)-(7.6). First derivatives of L In this section we do some preliminary calculations leading up to the computation of the second variation of L. A remark about the notation : L is a function of a point q and a time τ . The notation Lτ refers to the partial derivative with respect to τ , i.e. differentiation while keeping q fixed. The notation dτd refers to differentiation along an L-geodesic, i.e. simultaneously varying both the point and the time. If q is not in the time-τ L-cut locus of p, let γ : [0, τ ] → M be the unique minimizing Lgeodesic from p to q, with length L(q, τ ). If c : (−ǫ, ǫ) → M is a short curve with c(0) = q, consider the 1-parameter family of minimizing L-geodesics γ˜ (s, τ ) with γ˜ (s, 0) = p and ∂˜ γ (s,τ ) γ˜ (s, τ ) = c(s). Putting Y (τ ) = ∂s , equation (17.3) gives s=0
(18.1)
Hence (18.2) and (18.3)
h∇L, c′ (0)i =
√ dL(c(s), τ ) = 2 τ hX(τ), Y (τ )i. ds s=0
(∇L)(q, τ ) = 2
√ τ X(τ )
|∇L|2 (q, τ ) = 4 τ |X(τ )|2 = − 4 τ R(q) + 4 τ R(q) + |X(τ )|2 .
If we simply extend the L-geodesic γ in τ , we obtain √ dL(γ(τ ), τ ) (18.4) = τ R(γ(τ )) + |X(τ )|2 . dτ As dL(γ(τ ), τ ) (18.5) = Lτ (q, τ ) + h(∇L)(q, τ ), X(τ)i, dτ
38
BRUCE KLEINER AND JOHN LOTT
equations (18.2) and (18.4) give √ Lτ (q, τ ) = τ R(q) + |X(τ )|2 − h(∇L)(q, τ ), X(τ )i (18.6) √ √ = 2 τ R(q) − τ R(q) + |X(τ)|2 .
When computing in I.(7.3), (18.7)
dl(γ(τ ),τ ) , dτ
2 it will be useful to have a formula for R(γ(τ )) + X(τ ) . As
2 d R(γ(τ )) + X(τ ) = Rτ + h∇R, Xi + 2h∇X X, Xi + 2 Ric(X, X). dτ
Using the L-geodesic equation (17.4) gives
(18.8) 2 1 1 d R(γ(τ )) + X(τ ) = Rτ + R + 2h∇R, Xi − 2 Ric(X, X) − (R + |X|2 ) dτ τ τ 1 = − H(X) − (R + |X|2), τ where (18.9)
H(X) = − Rτ −
1 R − 2h∇R, Xi + 2 Ric(X, X) τ 3
is the expression of (F.9) after the change τ = −t and X → −X. Multiplying (18.8) by τ 2 and integrating gives Z τ 2 3 d R(γ(τ )) + X(τ ) dτ = − K − L(q, τ ), (18.10) τ2 dτ 0 where
(18.11)
K =
Z
τ
3
τ 2 H(X(τ )) dτ.
0
Then integrating the left-hand side of (18.10) by parts gives (18.12)
τ
3 2
2 1 R(γ(τ )) + X(τ ) = − K + L(q, τ ). 2
Plugging this back into (18.6) and (18.3) gives (18.13)
√ 1 1 Lτ (q, τ ) = 2 τ R(q) − L(q, τ ) + K 2τ τ
and (18.14)
2 4 |∇L|2 (q, τ ) = − 4 τ R(q) + √ L(q, τ ) − √ K. τ τ
NOTES ON PERELMAN’S PAPERS
39
19. I.(7.7). Second variation of L In this section we compute the second variation of L. We use it to compute the Hessian of L on M. To compute the second variation δY2 L, we start with the first variation equation Z τ √ τ (hY, ∇Ri + 2 h∇Y X, Xi) dτ. (19.1) δY L = 0
Recalling that δY γ(τ ) = Y (τ ) and δY X(τ ) = (∇Y X)(τ ), the second variation is (19.2) δY2
L = = =
Z
Z
Z
τ
0 τ
0 τ 0
√
τ (Y · Y · R + 2 h∇Y ∇Y X, Xi + 2 h∇Y X, ∇Y Xi) dτ
2 √ τ Y · Y · R + 2 h∇Y ∇X Y, Xi + 2 ∇X Y dτ
2 √ τ Y · Y · R + 2 h∇X ∇Y Y, Xi + 2 hR(Y, X)Y, Xi + 2 ∇X Y dτ,
where the notation Z · u refers to the directional derivative, i.e. Z · u = iZ du. In order to deal with the h∇X ∇Y Y, Xi term, we have to compute dτd h∇Y Y, Xi.
From the general equation for the Levi-Civita connection in terms of the metric [17, dg ˙ = d∇ , then (1.29)], if g(τ ) is a 1-parameter family of metrics, with g˙ = dτ and ∇ dτ
(19.3)
˙ X Y, Zi = (∇X g)(Y, 2h∇ ˙ Z) + (∇Y g)(Z, ˙ X) − (∇Z g)(X, ˙ Y ).
In our case g˙ = 2 Ric and so (19.4)
d ˙ Y Y, Xi h∇Y Y, Xi = h∇X ∇Y Y, Xi + h∇Y Y, ∇X Xi + 2 Ric(∇Y Y, X) + h∇ dτ = h∇X ∇Y Y, Xi + h∇Y Y, ∇X Xi + 2 Ric(∇Y Y, X) + 2(∇Y Ric)(Y, X) − (∇X Ric)(Y, Y ).
(Although we will not need it, we can write (19.5) 2Y · Ric(Y, X) − X · Ric(Y, Y ) = 2(∇Y Ric)(Y, X) + 2 Ric(∇Y Y, X) + 2 Ric(Y, ∇Y X) − (∇X Ric)(Y, Y ) − 2 Ric(∇X Y, Y )
= 2 Ric(∇Y Y, X) + 2(∇Y Ric)(Y, X)
− (∇X Ric)(Y, Y ) − 2 Ric([X, Y ], Y ). We are assuming that the variation field Y satisfies [X, Y ] = 0 (this was used in deriving the L-geodesic equation). Hence one obtains the formula (19.6) of I.7.)
d h∇Y Y, Xi = h∇X ∇Y Y, Xi + h∇Y Y, ∇X Xi + 2Y · Ric(Y, X) − X · Ric(Y, Y ) dτ
40
BRUCE KLEINER AND JOHN LOTT
Next, using (19.4), (19.7) Z τ √ d √ 2 τ h∇Y Y, Xi = 2 τ h∇Y Y, Xi dτ 0 dτ Z τ √ 1 d = τ h∇Y Y, Xi + 2 h∇Y Y, Xi dτ τ dτ 0 Z τ √ 1 τ = h∇Y Y, Xi + 2h∇X ∇Y Y, Xi + 2h∇Y Y, ∇X Xi + τ 0 4 Ric(∇Y Y, X) + 4(∇Y Ric)(Y, X) − 2(∇X Ric)(Y, Y )] dτ Z τ √ τ [2h∇X ∇Y Y, Xi + (∇Y Y )R + = 0
−
Z
4(∇Y Ric)(Y, X) − 2(∇X Ric)(Y, Y )] dτ τ √ 1 τ ∇Y Y, ∇R − 2∇X X − 4 Ric(X, ·) − X dτ. τ 0
(Of course the last term vanishes if γ is an L-geodesic, but we do not need to assume this here.) The quadratic form Q representing the Hessian of L on the path space is given by (19.8)
Q(Y, Y ) = δY2 L − δ∇Y Y L √ = δY2 L − 2 τ h∇Y Y, Xi − Z τ √ 1 τ ∇Y Y, ∇R − 2∇X X − 4 Ric(X, ·) − X . τ 0
It follows that (19.9)
Z
τ
√ τ [Y · Y · R − (∇Y Y )R + 2hR(Y, X)Y, Xi + 0 2 |∇X Y |2 − 4(∇Y Ric)(Y, X) + 2(∇X Ric)(Y, Y ) dτ Z τ √ τ HessR (Y, Y ) + 2hR(Y, X)Y, Xi + 2 |∇X Y |2 =
Q(Y, Y ) =
0
− 4(∇Y Ric)(Y, X) + 2(∇X Ric)(Y, Y )] dτ.
There √ is an associated second-order differential operator T on√vector fields Y given by saying that 2 τ hY, T Y i equals the integrand of (19.9) minus 2 dτd ( τ h∇X Y, Y i). Explicitly, (19.10) T Y = − ∇X ∇X Y −
1 1 ∇X Y + HessR (Y, ·) − 2 (∇Y Ric)(X, ·) − 2 Ric(∇Y X, ·). 2τ 2
Then (19.11)
Q(Y, Y ) = 2
Z
0
τ
√
τ hY, T Y i dτ + 2
√ τ h∇X Y (τ ), Y (τ )i.
An L-Jacobi field along an L-geodesic is a field Y (τ ) that is annihilated by T .
NOTES ON PERELMAN’S PAPERS
41
The Hessian of the function L(·, τ ) can be computed as follows. Assume that q ∈ M is outside of the time-τ L-cut locus. Let γ : [0, τ ] → M be the minimizing L-geodesic with γ(0) = p and γ(τ ) = q. Given w ∈ Tq M, take a short geodesic c : (−ǫ, ǫ) → M with c(0) = q and c′ (0) = w. Form the 1-parameter family of L-geodesics γ˜ (s, τ ) with (s,τ ) is an L-Jacobi field Y along γ γ˜ (s, 0) = p and γ˜ (s, τ ) = c(s). Then Y (τ ) = ∂˜γ∂s s=0
with Y (0) = 0 and Y (τ ) = w. We have √ d2 L(c(s)) τ h∇X Y (τ ), Y (τ )i. (19.12) HessL (w, w) = = Q(Y, Y ) = 2 ds2 s=0
From (19.11), a minimizer of Q(Y, Y ), among fields Y with given values at the endpoints, is an L-Jacobi field. It follows that HessL (w, w) ≤ Q(Y, Y ) for any field Y along γ satisfying Y (0) = 0 and Y (τ ) = w. 20. I.(7.8)-(7.9). Hessian bound for L In this section we use a test variation field in order to estimate the Hessian of L. If Y (τ ) is a unit vector at γ(τ ), solve for Ye (τ ) in the equation
1 e ∇X Ye = − Ric(Ye , ·) + Y, 2τ on the interval 0 < τ ≤ τ with the endpoint condition Ye (τ ) = Y (τ ). (For this section, we change notation from the Y in I.7 to Ye .) Then (20.1)
d e e 1 hY , Y i = 2 Ric(Ye , Ye ) + 2h∇X Ye , Ye i = hYe , Ye i, dτ τ so hYe (τ ), Ye (τ )i = ττ . Thus we can extend Ye continuously to the interval [0, τ ] by putting Ye (0) = 0. Substituting into (19.9) gives (20.2)
(20.3)
"
2 1 τ HessR (Ye , Ye ) + 2hR(Ye , X)Ye , Xi + 2 − Ric(Ye , ·) + Q(Ye , Ye ) = Ye 2τ 0 i − 4(∇Ye Ric)(Ye , X) + 2(∇X Ric)(Ye , Ye ) dτ Z τ √ h = τ HessR (Ye , Ye ) + 2hR(Ye , X)Ye , Xi + 2(∇X Ric)(Ye , Ye ) 0 1 2 2 dτ. − 4(∇Ye Ric)(Ye , X) + 2 | Ric(Ye , ·)| − Ric(Ye , Ye ) + τ 2τ τ Z
τ
√
From (20.4) d Ric(Ye (τ ), Ye (τ )) = Ricτ (Ye , Ye ) + (∇X Ric)(Ye , Ye ) + 2 Ric(∇X Ye , Ye ) dτ 1 = Ricτ (Ye , Ye ) + (∇X Ric)(Ye , Ye ) + Ric(Ye , Ye ) − 2| Ric(Ye , ·)|2, τ
42
BRUCE KLEINER AND JOHN LOTT
one obtains (20.5) Z τ √ d √ τ Ric(Ye , Ye ) dτ = − 2 τ Ric(Y (τ ), Y (τ )) = − 2 0 dτ Z τ √ 1 2 2 − τ Ric(Ye , Ye ) + 2 Ricτ (Ye , Ye ) + 2(∇X Ric)(Ye , Ye ) + Ric(Ye , Ye ) − 4| Ric(Ye , ·)| . τ τ 0
Combining (20.3) and (20.5) gives
Z τ √ √ 1 e e √ (20.6) HessL (Y (τ ), Y (τ )) ≤ Q(Y , Y ) = τ H(X, Ye ) dτ, − 2 τ Ric(Y (τ ), Y (τ )) − τ 0 where (20.7) e e e e e e e e H(X, Y ) = − HessR (Y , Y ) − 2hR(Y , X)Y , Xi − 4 ∇X Ric(Y , Y ) − ∇Ye Ric(Y , X) 2 1 Ric(Ye , Ye ). − 2 Ricτ (Ye , Ye ) + 2 Ric(Ye , ·) − τ is the expression appearing in (F.4), after the change τ = −t and X → −X. Note that H(X, Ye ) is a quadratic form in Ye . For its relation to the expression H(X) from (18.9), see Appendix F. 21. I.(7.10). The Laplacian of L In this section we estimate △L.
Let {Yi (τ )}ni=1 be an orthonormal basis of Tγ(τ ) M. Solve for Yei (τ ) from (20.1). Putting 1/2 ei (τ ), the vectors {ei (τ )}ni=1 form an orthonormal basis of Tγ(τ ) M. SubstiYei (τ ) = ττ tuting into (20.6) and summing over i gives Z τ X √ n 1 (21.1) △L ≤ √ − 2 τ R − τ 3/2 H(X, ei ) dτ. τ 0 τ i Then from (F.8), (21.2)
Z τ √ n 1 △L ≤ √ − 2 τ R − τ 3/2 H(X) dτ τ τ 0 √ 1 n = √ − 2 τ R − K. τ τ 22. I.(7.11). Estimates on L-Jacobi fields
In this section we estimate the growth rate of an L-Jacobi field.
Given an L-Jacobi field Y , we have (22.1)
d |Y |2 = 2 Ric(Y, Y ) + 2h∇X Y, Y i = 2 Ric(Y, Y ) + 2h∇Y X, Y i. dτ
NOTES ON PERELMAN’S PAPERS
43
Thus (22.2)
d|Y |2 1 = 2 Ric(Y (τ ), Y (τ )) + √ HessL (Y (τ ), Y (τ )), dτ τ =τ τ
where we have used (19.12). Let Y˜ be a field along γ as in Section 20, satisfying (20.1) with Y˜ (τ ) = Y (τ ) and |Y (τ )| = 1. Then from (20.6), Z τ √ 1 1 1 √ HessL (Y (τ ), Y (τ )) ≤ (22.3) τ H(X, Y˜ ) dτ. − 2 Ric(Y (τ ), Y (τ )) − √ τ τ τ 0 Thus
(22.4)
Z τ √ d|Y |2 1 1 τ H(X, Y˜ ) dτ. ≤ − √ dτ τ =τ τ τ 0
The inequality is sharp if and only if the first inequality in (20.6) is an equality. This is the case if and only if Y˜ is actually the L-Jacobi field Y , in which case 1 1 d|Y |2 d|Y˜ |2 (22.5) = = 2 Ric(Y (τ ), Y (τ )) + √ HessL (Y (τ ), Y (τ )). = τ dτ τ =τ dτ τ =τ τ
τ.
23. Monotonicity of the reduced volume Ve
In this section we show that the reduced volume Ve (τ ) is monotonically nonincreasing in
Fix p ∈ M. Define l(q, τ ) as in (15.3). In order to show that V˜ (τ ) is well-defined in the noncompact case, we will need a lower bound on l(q, τ ). For later use, we prove something slightly more general. Recall that we are assuming that we have bounded curvature on compact time intervals, and that time slices are complete.
Lemma 23.1. Given 0 < τ 1 ≤ τ 2 , there constants C1 , C2 > 0 so that for all τ ∈ [τ 1 , τ 2 ] and all q ∈ M, we have (23.2)
l(q, τ ) ≥ C1 d(p, q)2 − C2 .
Proof. We write L in the form (17.6). Given an L-geodesic γ with γ(0) = p and γ(τ ) = q, we obtain Z √τ 2 dγ 1 ds − const. (23.3) L(γ) ≥ 2 0 ds
As the multiplicative change in the metric between times τ 1 and τ 2 is bounded by a factor econst. (τ 2 −τ 1 ) , it follows that L(q, τ ) ≥ const. d(p, q)2 − const., where the distance is measured at time τ . The lemma follows. Define V˜ (τ ) as in (15.4). As the volume of time-τ balls in M increases at most exponentially fast in the radius, it follows that V˜ (τ ) is well-defined. From the discussion in Section
44
BRUCE KLEINER AND JOHN LOTT
17, we can write (23.4)
V˜ (τ ) =
Z
n
Tp M
τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) χτ (v) dv,
where J (v, τ ) = det d (Lexpτ )v is the Jacobian factor in the change of variable and χτ is the characteristic function of the time-τ domain Ωτ of Section 17. We first show that for each v, the expression − n2 ln(τ ) − l(Lexpτ (v), τ ) + ln J (v, τ ) is nonincreasing in τ . Let γ be the L-geodesic with initial vector v ∈ Tp M. From (18.4) and (18.12), 3 dl(γ(τ ), τ ) 1 1 1 (23.5) = − l(γ(τ )) + R(γ(τ )) + |X(τ )|2 = − τ − 2 K. dτ τ =τ 2τ 2 2
Next, let {Yi }ni=1 be a basis for the Jacobi fields along γ that vanish at τ = 0. We can write (23.6)
ln J (v, τ )2 = ln det ((d (Lexpτ )v )∗ d (Lexpτ )v ) = ln det(S(τ )) + const.,
where S is the matrix (23.7) Then (23.8)
Sij (τ ) = hYi (τ ), Yj (τ )i. d ln J (v, τ ) 1 −1 dS = Tr S . dτ 2 dτ
To compute the derivative at τ = τ , we can choose a basis so that S(τ ) = In , i.e. hYi (τ ), Yj (τ )i = δij . Then using (22.4) and computing as in Section 21, n 3 1 d ln J (v, τ ) 1 X d|Yi|2 n − τ − 2 K. (23.9) = ≤ dτ 2 i=1 dτ τ =τ 2τ 2 τ =τ If we have equality then (22.5) holds for each Yi , i.e. 2 Ric +
√1 τ
HessL =
g τ
at γ(τ ).
n
From (23.5) and (23.9), we deduce that τ − 2 e−l(Lexpτ (v),τ ) J (v, τ ) is nonincreasing in τ . Finally, recall that if τ ≤ τ ′ then Ωτ ′ ⊂ Ωτ , so χτ (v) is nonincreasing in τ . Hence V˜ (τ ) is nonincreasing in τ . If it is not strictly decreasing then we must have (23.10)
g(τ ) 1 . 2 Ric(τ ) + √ HessL(τ ) = τ τ
on all of M. Hence we have a gradient shrinking soliton solution. 24. I.(7.15). A differential inequality for L In this section we discuss an important differential inequality concerning the reduced length l. We use the differential inequality to estimate min l(·, τ ) from above. We then give a lower bound on l. √ With L(q, τ ) = 2 τ L(q, τ ), equations (18.13) and (21.2) imply that (24.1)
Lτ + △L ≤ 2n
NOTES ON PERELMAN’S PAPERS
45
away from the time-τ L-cut locus of p. We will eventually apply the maximum principle to this differential inequality. However to do so, we must discuss several senses in which the inequality can be made global on M, i.e. how it can be interpreted on the cut locus. The first sense is that of a barrier differential inequality. Given f ∈ C(M) and a function g on M, one says that △f ≤ g in the sense of barriers if for all q ∈ M and ǫ > 0, there is a neighborhood V of q and some uǫ ∈ C 2 (V ) so that uǫ (q) = f (q), uǫ ≥ f on V and ≤ g [26]. △uǫ ≤ g + ǫ on V [13]. There is a similar spacetime definition for △f − ∂f ∂t The point of barrier differential inequalities is that they allow one to apply the maximum principle just as with smooth solutions. We illustrate this by constructing a barrier function for L in (24.1). Given the spacetime point (q, τ ), let γ : [0, τ ] → M be a minimizing L-geodesic with γ(0) = p and γ(τ ) = q. Given a small ǫ > 0, let uǫ (q ′ , τ ′ ) be the minimum of Z τ′ Z ǫ 2 2 √ √ (24.2) τ R(γ2 (τ )) + γ˙2 (τ ) dτ + τ R(γ(τ )) + γ(τ ˙ ) dτ ǫ
0
′
among curves γ2 : [ǫ, τ ] → M with γ2 (ǫ) = γ(ǫ) and γ2 (τ ′ ) = q ′ . Because the new basepoint (γ(ǫ), ǫ) is moved in along γ from p, the minimizer γ2 will be unique and will vary smoothly with q ′ , when q ′ is close to q; otherwise a second minimizer or a “conjugate point” would imply that γ was not minimizing. Thus √ the function uǫ is smooth in a spacetime ′ ′ neighborhood V of (q, τ ). Put Uǫ (q , τ ) = 2 τ ′ uǫ (q ′ , τ ′ ). By construction, Uǫ ≥ L in V and Uǫ (q, τ ) = L(q, τ ). For small ǫ, (Uǫ )τ ′ + △Uǫ will be bounded above on V by something close to 2n. Hence L satisfies (24.1) globally on M in the barrier sense. As we are assuming bounded curvature on compact time intervals, we can now apply the maximum principle of Appendix A to conclude that the minimum of L(·, τ ) − 2nτ is nonincreasing in τ . (Note that from Lemma 23.1, the minimum of L(·, τ ) − 2nτ exists.) Lemma 24.3. For small positive τ , we have min L(·, τ) − 2nτ < 0. Proof. Consider the static curve at the point p. Then for small τ , we have L(·, τ ) ≤ const. τ 2 , from which the claim follows. (Being a bit more careful with the estimates in the proof of Lemma 23.1, one sees that limτ →0 min L(·, τ) = 0.) Then for τ > 0, we must have min L(·, τ ) ≤ 2nτ , so min l(·, τ ) ≤ n2 . The other sense of a differential inequality is the distributional sense, i.e. △f ≤ g if for every nonnegative compactly-supported smooth function φ on M, Z Z (24.4) (△φ) f dV ≤ φ g dV. M
M
A general fact is that a barrier differential inequality implies a distributional differential inequality [37, 67]. We illustrate this by giving an alternative proof that V˜ (τ ) is nonincreasing in τ . From (18.13), (18.14) and (21.2), one finds that in the barrier sense (and hence in the distributional sense as well) n (24.5) lτ − △l + |∇l|2 − R + ≥ 0 2τ
46
BRUCE KLEINER AND JOHN LOTT
or, equivalently, that n
(∂τ − △) τ − 2 e−l dV
(24.6)
≤ 0.
Then for all nonnegative φ ∈ Cc∞ (M) and 0 < τ 1 ≤ τ 2 , one obtains (24.7) Z
−n τ2 2
φ M Z τ2 Z
Z
τ1 τ2 τ1
Z
−l(·,τ 2 )
e
n
M
n
−n 2
e−l(·,τ 1 ) dV (τ 1 ) = M Z τ2 Z n −l(·,τ ) e dV (τ ) dτ + (△φ) τ − 2 e−l(·,τ ) dV (τ ) dτ ≤
dV (τ 2 ) −
φ (∂τ − △) τ − 2
Z
φ τ1
τ1
M
(△φ) τ − 2 e−l(·,τ ) dV (τ ) dτ. M
We can find a sequence {φi }∞ i=1 of such functions φ with range [0, 1] so that φi is one on B(p, i), vanishes outside of B(p, i2 ), and supM |△φi | ≤ i−1 , uniformly in τ ∈ [τ 1 , τ 2 ]. Then to finish the argument it suffices to have a good upper bound on e−l(·,τ ) in terms of d(p, ·), uniformly in τ ∈ [τ 1 , τ 2 ]. This is given by Lemma 23.1. The monotonicity of V˜ follows. We also note the equation
(24.8)
2△l − |∇l|2 + R +
l−n ≤ 0, τ
which follows from (18.14) and (21.2). Finally, suppose that the Ricci flow exists on a time interval τ ∈ [0, τ0 ]. From the maximum principle of Appendix A, R(·, τ ) ≥ − 2(τ0n−τ ) . Then one obtains a lower bound on l as before, using the better lower bound for R. 25. I.7.2. Estimates on the reduced length In this section we suppose that our solution has nonnegative curvature operator. We use this to derive estimates on the reduced length l. We refer to Appendix F for Hamilton’s differential Harnack inequality. We consider a Ricci flow defined on a time interval t ∈ [0, τ0 ], with bounded nonnegative curvature operator, and put τ = τ0 − t. The differential Harnack inequality gives the nonnegativity of the expression in (F.4). Comparing this with the formula for H(X, Y ) in (20.7), we can write the nonnegativity as Ric(Y, Y ) Ric(Y, Y ) + ≥ 0, (25.1) H(X, Y ) + τ τ0 − τ or 1 1 (25.2) H(X, Y ) ≥ − Ric(Y, Y ) + . τ τ0 − τ
Then
(25.3)
H(X) ≥ − R
1 1 + τ τ0 − τ
.
NOTES ON PERELMAN’S PAPERS
47
As long as τ ≤ (1 − c) τ0 , equations (18.3) and (18.12) give Z τ 4 2 4τ |∇l| = − 4τ R + 4l − √ (25.4) τ˜3/2 H(X) d˜ τ τ 0 Z τ 4 1 1 3/2 ≤ − 4τ R + 4l + √ τ˜ R + d˜ τ τ 0 τ˜ τ0 − τ˜ Z τ√ τ0 4 τ˜R = − 4τ R + 4l + √ d˜ τ τ 0 τ0 − τ˜ Z τ√ 4 ≤ − 4τ R + 4l + √ τ˜R d˜ τ c τ 0 8l ≤ − 4τ R + 4l + , c where the last line uses (15.1). Thus Cl (25.5) |∇l|2 + R ≤ τ for a constant C = C(c). One shows similarly that 1 d ln |Y |2 ≤ (Cl + 1) (25.6) dτ τ for an L-Jacobi field Y , using (22.4). 26. I.7.3. The no local collapsing theorem II In this section we use the reduced volume to prove a no-local-collapsing theorem for a Ricci flow on a finite time interval. Definition 26.1. We now say that a Ricci flow solution g(·) defined on a time interval [0, T ) is κ-noncollapsed on the scale ρ if for each r < ρ and all (x0 , t0 ) ∈ M × [0, T ) with t0 ≥ r 2 , whenever it is true that | Rm(x, t)| ≤ r −2 for every x ∈ Bt0 (x0 , r) and t ∈ [t0 − r 2 , t0 ], then we also have vol(Bt0 (x0 , r)) ≥ κrn . Definition 26.1 differs from Definition 13.14 by the requirement that the curvature bound holds in the entire parabolic region Bt0 (x0 , r) × [t0 − r 2 , t0 ] instead of just on the ball Bt0 (x0 , r) in the final time slice. Therefore a Ricci flow which is κ-noncollapsed in the sense of Definition 13.14 is also κ-noncollapsed in the sense of Definition 26.1. Theorem 26.2. Given numbers n ∈ Z+ , T < ∞ and ρ, K, c > 0, there is a number κ = κ(n, K, c, ρ, T ) > 0 with the following property. Let (M n , g(·)) be a Ricci flow solution defined on a time interval [0, T ) with T < ∞, such that the curvature | Rm | is bounded on every compact subinterval [0, T ′ ] ⊂ [0, T ). Suppose that (M, g(0)) is a complete Riemannian manifold with | Rm | ≤ K and inj(M, g(0)) ≥ c > 0. Then the Ricci flow solution is κ-noncollapsed on the scale ρ, in the sense of Definition 26.1. Furthermore, with the other constants fixed, we can take κ to be nonincreasing in T . Proof. We first observe that the existence of L-geodesics and the monotonicity of the reduced volume are valid in this setting; see Section 17.
48
BRUCE KLEINER AND JOHN LOTT
Suppose that the theorem were false. Then for given T < ∞ and ρ, K, c > 0, there are : 1. A sequence {(Mk , gk (·))}∞ k=1 of Ricci flow solutions, each defined on the time interval [0, T ), with | Rm | ≤ K on (Mk , gk (0)) and inj(Mk , gk (0)) ≥ c, 2. Spacetime points (pk , tk ) ∈ Mk × [0, T ) and 3. Numbers rk ∈ (0, ρ) having the following property : tk ≥ rk2 and if we put Bk = Btk (pk , rk ) ⊂ Mk then 1 | Rm |(x, t) ≤ rk−2 whenever x ∈ Bk and t ∈ [tk − rk2 , tk ], but ǫk = rk−1 vol(Bk ) n → 0 as k → ∞. From short-time curvature estimates along with the assumed bounded geometry at time zero, there is some t > 0 so that we have uniformly bounded geometry on the time interval [0, t]. In particular, we may assume that each tk is greater than t. We define V˜k using curves going backward in real time from the basepoint (pk , tk ), i.e. forward in τ -time from τ = 0. The first step is to show that V˜k (ǫk rk2 ) is small. Note that τ = ǫk rk2 corresponds to a real time of tk − ǫk rk2 , which is very close to tk . Given an L-geodesic γ(τ ) with γ(0) = pk and velocity vector X(τ ) = dγ , its initial dτ √ −1/2 vector is v = limτ →0 τ X(τ ) ∈ Tpk Mk . We first want to show that if |v| ≤ .1 ǫk then 2 γ does not escape from Bk in time ǫk rk . We have (26.3)
d hX(τ ), X(τ )i = 2 Ric(X, X) + 2hX, ∇X Xi dτ = 2 Ric(X, X) + hX, ∇R − = −
1 X − 4 Ric(X, ·)i τ
|X|2 − 2 Ric(X, X) + hX, ∇Ri, τ
so d τ |X|2 = − 2 τ Ric(X, X) + τ hX, ∇Ri. dτ Letting C denote a generic n-dependent constant, for x ∈ B(pk , rk /2) and t ∈ [tk − rk2 /2, tk ], the fact that gk satisfies the Ricci flow gives an estimate |∇R|(x, t) ≤ Crk−3 , as follows from the case l = 0, m = 1 of Appendix D. Then in terms of dimensionless variables, d 2 τ |X| ≤ C τ |X|2 + C (τ /rk2 )1/2 (τ |X|2 )1/2 . (26.5) 2 d(τ /rk )
(26.4)
Equivalently,
(26.6)
1/2 √ √ d τ |X| . ≤ C τ |X| + C τ /rk2 2 d(τ /rk )
Let us rewrite this as 1/2 d 1√ 1√ τ 2 2 2 ǫ τ |X| ≤ C ǫk ǫk τ |X| + C ǫk . (26.7) d( ǫkτr2 ) k ǫk rk2 k
We are interested in the time range when ǫkτr2 ∈ [0, 1] and the initial condition satisfies k 1 √ 2 limτ →0 ǫk τ |X|(τ ) ≤ .1. Then because of the ǫk -factors on the right-hand side, it follows
NOTES ON PERELMAN’S PAPERS
49
1 √ from (26.7) that for large k, we will have ǫk2 τ |X|(τ ) ≤ .11 for all τ ∈ [0, ǫk rk2 ]. Next, Z ǫk r 2 Z ǫk r 2 1 Z ǫk r 2 k k k dτ √ dτ − 12 − 12 2 √ = .22 rk . (26.8) |X(τ )| dτ = ǫk ǫk τ |X|(τ ) √ ≤ .11 ǫk τ τ 0 0 0
From the Ricci flow equation gτ = 2 Ric, it follows that the metrics g(τ ) between τ = 0 and τ = ǫk rk2 are eCǫk -biLipschitz close to each other. Then for ǫk small, the length of γ, as measured with the metric at time tk , will be at most .3 rk . This shows that γ does not leave Bk within time ǫk rk2 . 1
˜k (ǫk r 2 ) coming from vectors v ∈ Tp Mk with |v| ≤ .1 ǫ− 2 is at Hence the contribution to V k k k R n 2 most Bk (ǫk rk2 )− 2 e− l(q,ǫk rk ) dq. We now want to give a lower bound on l(q, ǫk rk2 ) for q ∈ Bk . Given the L-geodesic γ : [0, ǫk rk2 ] → Mk with γ(0) = pk and γ(ǫk rk2 ) = q, we have Z ǫk rk2 Z ǫk rk2 3 √ √ 2 (26.9) L(γ) ≥ τ R(γ(τ )) dτ ≥ − τ n(n − 1) rk−2 dτ = − n(n − 1) ǫk2 rk . 3 0 0
Then
l(q, ǫk rk2 ) ≥ −
(26.10)
1 n(n − 1) ǫk . 3 1
− Thus the contribution to V˜k (ǫk rk2 ) coming from vectors v ∈ Tpk Mk with |v| ≤ .1 ǫk 2 is at most
(26.11)
e
1 3
n(n−1) ǫk
n (ǫk rk2 )− 2
vol
tk −ǫk rk2
n
(Bk ) ≤ e
1 3
n(n−1) ǫk
const.
e
1 r2 k
ǫk rk2
n
ǫk2 ,
which is less than 2ǫk2 for large k. 1
− To estimate the contribution to V˜k (ǫk rk2 ) coming from vectors v ∈ Tpk Mk with |v| > .1ǫk 2 , we can use the previously-shown monotonicity of the integrand in τ . As τ → 0, the Euclidean 2 calculation of Section 16 shows that τ −n/2 e− l(Lexpτ (v),τ ) J (v, τ ) → 2n e− |v| . Then for all τ > 0 and all v ∈ Ωτ , 2
τ −n/2 e− l(Lτ (v),τ ) J (v, τ ) ≤ 2n e− |v| ,
(26.12) giving (26.13) Z
τ −1/2
Tpk Mk −B(0,.1 ǫk
)
−n/2 − l(Lτ (v),τ )
e
n
J(v, τ ) χτ d v ≤ 2
n
Z
2
−1/2
Tpk Mk −B(0,.1 ǫk
)
−
e− |v| dn v ≤ e
1 10ǫk
for k large. The conclusion is that limk→∞ V˜k (ǫk rk2 ) = 0. We now claim that there is a uniform positive lower bound on V˜k (tk ). To estimate V˜k (tk ) (where τ = tk corresponds to t = 0), we choose a point qk at time t = 2t , i.e. at τ = tk − 2t , for which l(qk , tk − t/2) ≤ n2 ; see Section 24. Then we consider (k) (k) the concatenation of a fixed curve γ1 : [0, tk − t/2] → Mk , having γ1 (0) = pk and (k) (k) (k) γ1 (tk − t/2) = qk , with a fan of curves γ2 : [tk − t/2, tk ] → Mk having γ2 (tk − t/2) = qk . Because of the uniformly bounded geometry in the spacetime region with t ∈ [0, t/2], we
50
BRUCE KLEINER AND JOHN LOTT
can get an upper bound in this way for l(·, tk ) in a region around qk . Integrating e−l(·,tk ) , we get a positive lower bound on V˜k (tk ) that is uniform in k. As ǫk rk2 → 0, the monotonicity of V˜ implies that V˜k (tk ) ≤ V˜k (ǫk rk2 ) for large k, which is a contradiction. 27. I.8.3. Length distortion estimates The distortion of distances under Ricci flow can be estimated in terms of the Ricci tensor. We first mention a crude estimate. Lemma 27.1. If Ric ≤ (n − 1) K then for t1 > t0 , (27.2)
distt1 (x0 , x1 ) ≥ e−(n−1)K(t1 −t0 ) . distt0 (x0 , x1 )
Proof. For any curve γ : [0, a] → M, we have Z a s Z a d dγ dγ d dγ dγ ds dγ ≥ − (n − 1) K L(γ). (27.3) L(γ) = , ds = − Ric , dt dt 0 ds ds ds ds 0 ds
Integrating gives (27.4)
L(γ) t1 ≥ e−(n−1)K(t1 −t0 ) . L(γ) t0
The lemma follows by taking γ to be a minimal geodesic at time t1 between x0 and x1 . Remark 27.5. By a similar argument, if Ric ≥ − (n − 1) K then for t1 > t0 , (27.6)
distt1 (x0 , x1 ) ≤ e(n−1)K(t1 −t0 ) . distt0 (x0 , x1 )
We can write the conclusion of Lemma 27.1 as (27.7)
d distt (x0 , x1 ) ≥ − (n − 1)K distt (x0 , x1 ), dt
where the derivative is interpreted in the sense of forward difference quotients. The estimate in Lemma 27.1 is multiplicative. We now give an estimate that is additive in the distance. Lemma 27.8. (cf. Lemma I.8.3(b)) Suppose distt0 (x0 , x1 ) ≥ 2r0 , and Ric(x, t0 ) ≤ (n−1)K for all x ∈ Bt0 (x0 , r0 ) ∪ Bt0 (x1 , r0 ). Then 2 d −1 distt (x0 , x1 ) ≥ − 2 (n − 1) K r0 + r 0 (27.9) dt 3 at time t = t0 .
NOTES ON PERELMAN’S PAPERS
51
Proof. If γ is a normalized minimal geodesic from x0 to x1 with velocity field X(s) = dγ ds then for any piecewise-smooth normal vector field V along γ that vanishes at the endpoints, the second variation formula gives Z
(27.10)
d(x0 ,x1 )
0
2 ∇X V + hR(V, X)V, Xi ds ≥ 0.
Let {ei (s)}n−1 i=1 be a parallel orthonormal frame along γ that is perpendicular to X. Put Vi (s) = f (s) ei (s), where s r0 f (s) = 1 d(x0 ,x1 )−s
(27.11)
r0
Then ∇X Vi = |f ′ (s)| and Z
(27.12)
d(x0 ,x1 )
0
Next,
(27.13)
Z
if 0 ≤ s ≤ r0 , if r0 ≤ s ≤ d(x0 , x1 ) − r0 , if d(x0 , x1 ) − r0 ≤ s ≤ d(x0 , x1 ).
Z 2 ∇X Vi ds = 2
0
d(x0 ,x1 )
hR(Vi , X)Vi , Xi ds =
0
r0
Z
r0
0
Z
2 1 ds = . 2 r0 r0
s2 hR(ei , X)ei , Xi ds + r02
d(x0 ,x1 )−r0
r0 Z d(x0 ,x1 )
d(x0 ,x1 )−r0
hR(ei , X)ei, Xi ds + (d(x0 , x1 ) − s)2 hR(ei , X)ei , Xi ds. r02
Then
(27.14)
0 ≤
n−1 Z X i=1
d(x0 ,x1 ) 0
2 ∇X Vi + hR(Vi , X)Vi, Xi ds
Z d(x0 ,x1 ) Z r0 2(n − 1) s2 − Ric(X, X) ds + 1 − 2 Ric(X, X) ds + = r0 r0 0 0 Z d(x0 ,x1 ) (d(x0 , x1 ) − s)2 1− Ric(X, X) ds. r02 d(x0 ,x1 )−r0
52
BRUCE KLEINER AND JOHN LOTT
This gives (27.15)
d distt (x0 , x1 ) = − dt
Z
d(x0 ,x1 )
Ric(X, X) ds Z r0 2(n − 1) s2 ≥ − − 1 − 2 Ric(X, X) ds r0 r0 0 Z d(x0 ,x1) (d(x0 , x1 ) − s)2 Ric(X, X) ds − 1− r02 d(x0 ,x1 )−r0
≥ −
0
2 2(n − 1) − 2(n − 1)K · r0 , r0 3
which proves the lemma.
We now give an additive version of Lemma 27.1. Corollary 27.16. [33, Theorem 17.2] If Ric ≤ K with K > 0 then for all x0 , x1 ∈ M, (27.17)
d distt (x0 , x1 ) ≥ − const.(n) K 1/2 . dt
Proof. Put r0 = K −1/2 . If distt (x0 , x1 ) ≤ 2r0 then the corollary follows from (27.7). If distt (x0 , x1 ) > 2r0 then it follows from Lemma 27.8. The proof of the next lemma is similar to that of Lemma 27.8 and is given in I.8. Lemma 27.18. (cf. Lemma I.8.3(a)) Suppose that Ric(x, t0 ) ≤ (n − 1) K on Bt0 (x0 , r0 ). Then the distance function d(x, t) = distt (x, x0 ) satisfies 2 −1 (27.19) dt − △d ≥ − (n − 1) K r0 + r 0 3
at time t = t0 , outside of Bt0 (x0 , r0 ). The inequality must be understood in the barrier sense (see Section 24) if necessary. 28. I.8.2. No local collapsing propagates forward in time and to larger scales This section is concerned with a localized version of the no-local-collapsing theorem. The main result, Theorem 28.2, says that noncollapsing propagates forward in time and to a larger distance scale. We first give a local version of Definition 26.1. Definition 28.1. (cf. Definition of I.8.1) A Ricci flow solution is said to be κ-collapsed at (x0 , t0 ), on the scale r > 0, if | Rm |(x, t) ≤ r −2 for all (x, t) ∈ Bt0 (x0 , r) × [t0 − r 2 , t0 ], but vol(Bt0 (x0 , r 2 )) ≤ κrn . Theorem 28.2. (cf. Theorem I.8.2) For any 0 < A < ∞, there is some κ = κ(A) > 0 with the following property. Let g(·) be a Ricci flow solution defined for t ∈ [0, r02], having complete time slices and uniformly bounded sectional curvature.. Suppose that vol(B0 (x0 , r0 )) ≥
NOTES ON PERELMAN’S PAPERS
53
A−1 r0n and that | Rm |(x, t) ≤ nr1 2 for all (x, t) ∈ B0 (x0 , r0 )×[0, r02 ]. Then the solution cannot 0 be κ-collapsed on a scale less than r0 at any point (x, r02 ) with x ∈ Br02 (x0 , Ar0 ). Remark 28.3. In [51, Theorem I.8.2] the assumption is that | Rm |(x, t) ≤ r0−2 . We make the slightly stronger assumption that | Rm |(x, t) ≤ nr1 2 . The extra factor of n is needed in order 0 1 to assert in the proof that the region {(y, t) : dist 1 (y, x0 ) ≤ 10 , t ∈ [0, 12 ]} has bounded 2 geometry; see below. Clearly this change of hypothesis does not make any substantial difference in the sequel. Proof. We follow the lines of the proof of Theorem 26.2. By scaling, we can take r0 = 1. Choose x ∈ M with dist1 (x, x0 ) < A. Define the reduced volume V˜ (τ ) by means of curves starting at (x, 1). An effective lower bound on V˜ (1) would imply that the solution is not κ-collapsed at (x, 1), on a scale less than 1, for an appropriate κ > 0. 1 We first note that the geometry of the region {(y, t) : dist 1 (y, x0 ) ≤ 10 , t ∈ [0, 12 ]} is 2 uniformly bounded. To see this, the upper sectional curvature bound implies that Ric ≤ 1, 1 so the distance distortion estimate of Section 27 implies that B 1 (x0 , 10 ) ⊂ B0 (x0 , 1). In 2 1 1 and particular, | Rm |(y, t) ≤ n on the region. By Remark 27.5, if dist 1 (y, x0 ) ≤ 10 2 1 1 1 t ∈ [0, 2 ] then B0 (y, 1000 ) ⊂ Bt (y, 100 ). The Bishop-Gromov inequality gives a lower bound 1 1 for the time-zero volume of B0 (y, 1000 ), of the form vol0 (B0 (y, 1000 )) ≥ C1 (n, A). The Ricci 1 flow equation then gives a lower bound for the time-t volume of B0 (y, 1000 ), of the form 1 1 1 volt (B0 (y, 1000 )) ≥ C2 (n, A). Thus the time-t volume of Bt (y, 100 ) satisfies vol(Bt (y, 100 )) ≥ C2 (n, A). This, along with the uniform sectional curvature bound, implies that the region has uniformly bounded geometry.
If we have an effective upper bound on miny l(y, 12 ), where y ranges over points that 1 , then we obtain a lower bound on V˜ (1). Thus it suffices to obtain satisfy dist 1 (y, x0 ) ≤ 10 2 an effective upper bound on miny l(y, 21 ) or, equivalently, on miny L(y, 12 ) (as defined using 1 L-geodesics from (x, 1)) for y satisfying dist 1 (y, x0) ≤ 10 . Applying the maximum principle 2 to (24.1) gave an upper bound on inf M L. The idea is to spatially localize this estimate near x0 , by means of a radial function φ. 1 1 Let φ = φ(u) be a smooth function that equals 1 on (−∞, 20 ), equals infinity on ( 10 , ∞) 1 1 and is increasing on ( 20 , 10 ), with
2(φ′ )2 /φ − φ′′ ≥ (2A + 100n)φ′ − C(A)φ
(28.4)
for some constant C(A) < ∞. To satisfy (28.4), it suffices to take φ(u) =
for u near
1 . 10
1
1
e(2A+100n)( 10 −u) −1
We claim that L + 2n + 1 ≥ 1 for t ≥ 12 . To see this, from the end of Section 24, n (28.5) R(·, τ ) ≥ − . 2(1 − τ )
Then for τ ∈ [0, 12 ], (28.6)
L(q, τ ) ≥ −
Z
0
τ
√
n τ dτ ≥ − n 2(1 − τ )
Z
0
τ
√
τ dτ = −
2n 3/2 τ . 3
54
BRUCE KLEINER AND JOHN LOTT
Hence (28.7)
L(q, τ ) = 2
√ 4n n τ L(q, τ ) ≥ − τ 2 ≥ − , 3 3
which proves the claim. Now put h(y, t) = φ(d(y, t) − A(2t − 1)) (L(y, 1 − t) + 2n + 1),
(28.8)
where d(y, t) = distt (y, x0 ). It follows from the above claim that h(y, t) ≥ 0 if t ≥ 21 . Also, min h(y, 1) ≤ h(x, 1) = φ(dist1 (x, x0 ) − A) · (2n + 1) = 2n + 1.
(28.9)
y
1 As φ is infinite on ( 10 , ∞) and L(·, 21 ) + 2n + 1 ≥ 1, the minimum of h(·, 12 ) is achieved at 1 . some y satisfying d(y, 21 ) ≤ 10
The calculations in I.8 give
h ≥ −(2n + C(A))h
(28.10)
at a minimum point of h, where = ∂t − △. Then dtd hmin (t) ≥ −(2n + C(A)) hmin (t), so C(A) C(A) 1 ≤ en+ 2 hmin (1) ≤ (2n + 1) en+ 2 . (28.11) hmin 2 It follows that C(A) 1 L(y, ) + 2n + 1 ≤ (2n + 1) en+ 2 . (28.12) min 1 1 2 y : d(y, 2 )≤ 10 This implies the theorem.
29. I.9. Perelman’s differential Harnack inequality This section is concerned with a localized version of the W-functional. It is mainly used in I.10. Let g(·) be a Ricci flow solution on a manifold M, defined for t ∈ (a, b). Put = ∂t − △. For f1 , f2 ∈ Cc∞ ((a, b) × M), we have Z b Z d (29.1) f1 (t, x)f2 (t, x) dV dt 0 = a dt M Z bZ Z bZ = ((∂t − △)f1 ) f2 dV + f1 (∂t + △ − R)f2 dV a M a M Z bZ Z bZ = (f1 ) f2 dV − f1 ∗ f2 dV, a
∗
M
a
M ∗
where = − ∂t − △ + R. In this sense, is the formal adjoint to . Now suppose that the Ricci flow is defined for t ∈ [0, T ). Suppose that
(29.2)
satisfies ∗ u = 0. Put (29.3)
u = (4π(T − t))−
n 2
e−f
v = [(T − t)(2△f − |∇f |2 + R) + f − n] u.
NOTES ON PERELMAN’S PAPERS
If M is compact then using (5.10), W(gij , f, T − t) =
(29.4)
Z
55
v dV.
M
Proposition 29.5. (cf. Proposition I.9.1) (29.6) ∗ v = − 2(T − t) Rij + ∇i ∇j f −
gij 2 u. 2(T − t)
Proof. We note that the right-hand side of I.(9.1) should be multiplied by u. To prove the proposition, we first claim that d△ = 2 Rij ∇i ∇j . (29.7) dt To see this, for f1 , f2 ∈ Cc∞ (M), we have Z Z (29.8) f1 △f2 dV = − hdf1 , df2 i dV. M
M
Differentiating with respect to t gives Z Z Z Z d△ f2 dV − f1 △f2 R dV = − 2 Ric(df1 , df2) dV + hdf1 , df2 i R dV, (29.9) f1 dt M M M M so d△ (29.10) f2 − R △f2 = 2 ∇i (Rij ∇j f2 ) − ∇i (R ∇i f2 ). dt Then (29.7) follows from the traced second Bianchi identity. Next, one can check that ∗ u = 0 is equivalent to n 1 + |∇f |2 − R. (29.11) (∂t + △) f = 2 T −t Then one obtains (29.12) u−1 ∗ v = − (∂t + △) (T − t) (2△f − |∇f |2 + R) + f − 2h∇ (T − t) (2△f − |∇f |2 + R) + f , u−1 ∇ui
= 2△f − |∇f |2 + R − (T − t) (∂t + △) (2△f − |∇f |2 + R)
− (∂t + △) f + 2 (T − t) h∇(2△f − |∇f |2 + R), ∇f i + 2 |∇f |2.
Now (29.13) (∂t + △) (2△f − |∇f |2 + R) = 2(∂t △)f + 2△ (∂t + △) f
− (∂t + △) |∇f |2 + (∂t + △) R
= 4 Rij ∇i ∇j f + 2 △ (|∇f |2 − R) − 2 Ric(df, df )
− 2h∇ft , ∇f i − △|∇f |2 + △R + 2 | Ric |2 + △R
= 4 Rij ∇i ∇j f + 2 △ |∇f |2 − 2 Ric(df, df )
− 2h∇(− △f + |∇f |2 − R), ∇f i − △|∇f |2 + 2 | Ric |2 .
56
BRUCE KLEINER AND JOHN LOTT
Hence the term in u−1 ∗ v proportionate to (T − t)−1 is n 1 (29.14) − . 2 T −t The term proportionate to (T − t)0 is
(29.15)
2△f − |∇f |2 + R − |∇f |2 + R + 2|∇f |2 = 2(△f + R).
The term proportionate to (T − t) is (T − t) times (29.16)
− 4 Rij ∇i ∇j f − 2 △ |∇f |2 + 2 Ric(df, df ) +
2h∇(− △f + |∇f |2 − R), ∇f i + △|∇f |2 − 2 | Ric |2 +
2 h∇(2△f − |∇f |2 + R), ∇f i =
− 4 Rij ∇i ∇j f − △ |∇f |2 + 2 Ric(df, df ) + 2h∇△f, ∇f i − 2 | Ric |2 = − 4 Rij ∇i ∇j f − 2 | Hess(f )|2 − 2 | Ric |2 .
Putting this together gives (29.17)
v = − 2 (T − t) Rij + ∇i ∇j f − ∗
This proves the proposition.
As a consequence of Proposition 29.5, Z d d (29.18) W(gij , f, T − t) = v dV dt dt M Z = 2 (T − t)
2 1 gij u. 2(T − t)
Z
= (∂t + △ − R)v dV M 2 1 gij u dV. Rij + ∇i ∇j f − 2(T − t) M
In this sense, Proposition 29.5 is a local version of the monotonicity of W.
Corollary 29.19. (cf. Corollary I.9.2) If M is closed, or whenever the maximum principle holds, then max v/u is nondecreasing in t. Proof. We note that the statement of Corollary I.9.2 should have max v/u instead of min v/u. To prove the corollary, we have v∗ u − u∗ v 2 D vE v − ∇u, ∇ . (29.20) (∂t + △) = u u2 u u As ∗ u = 0 and ∗ v ≤ 0, the corollary now follows from the maximum principle.
We now assume that the Ricci flow solution is defined on the closed interval [0, T ]. Corollary 29.21. (cf. Corollary I.9.3) Under the same assumptions, if the solution is defined for t ∈ [0, T ] and u tends to a δ-function as t → T then v ≤ 0 for all t < T . Proof. Suppose that h is a positive solution of h = 0. Then Z Z Z d ∗ (29.22) hv dV = ((h) v − h v) dV = − h ∗ v dV ≥ 0. dt M M M
NOTES ON PERELMAN’S PAPERS
57
R As t → T , the computation of M hv approaches the flat-space calculation, which one finds to be zero; see [50] for details. (Strictly speaking, the paper [50] deals with the case when M R is closed. It is indicated that the proof should extend to the noncompact setting.) Thus h(t0 )v(t0 ) dV is nonpositive for all t0 < T . As h(t0 ) can be taken to be an arbitrary M positive function, and then flowed forward to a positive solution of h = 0, it follows that v(t0 ) ≤ 0 for all t0 < T . The next result compares the function f used in the W-functional and the function l used in the reduced volume. Corollary 29.23. (cf. Corollary I.9.5) Under the assumptions of the previous corollary, let p ∈ M be the point where the limit δ-function is concentrated. Then f (q, t) ≤ l(q, T − t), where l is the reduced distance defined using curves starting from (p, T ). Proof. Equation (24.6) implies that ∗ (4πτ )−n/2 e−l ≤ 0. (This corrects the statement at ∗ −n/2 −f the top of page 23 of I.) From this and the fact that (4πτ ) e = 0, the argument of the proof of Corollary 29.19 gives that max ef −l is nondecreasing in t, so max(f − l) is nondecreasing in t. As t → T one obtains the flat-space result, namely that f − l vanishes. Thus f (t) ≤ l(T − t) for all t ∈ [0, T ). Remark 29.24. To give an alternative proof of Corollary 29.23, putting τ = T − t, Corollary I.9.4 of [51] says that for any smooth curve γ, 2 d 1 1 f (γ(τ ), τ ) ≤ f (γ(τ ), τ ), (29.25) R(γ(τ ), τ ) + γ(t) ˙ − dτ 2 2τ or 2 1 1/2 d 1/2 τ f (γ(τ ), τ ) ≤ τ R(γ(τ ), τ ) + γ(t) ˙ . (29.26) dτ 2 Take γ to be a curve emanating from (p, T ). For small τ , (29.27)
f (γ(τ ), τ ) ∼ d(p, γ(τ ))2 /4τ = O(τ 0 ).
Then integration gives τ 1/2 f ≤
1 2
L, or f ≤ l.
30. The statement of the pseudolocality theorem The next theorem says that, in a localized sense, if the initial data of a Ricci flow solution has a lower bound on the scalar curvature and satisfies an isoperimetric inequality close to that of Euclidean space then there is a sectional curvature bound in a forward region. The result is not used in the sequel. Theorem 30.1. (cf. Theorem I.10.1) For every α > 0 there exist δ, ǫ > 0 with the following property. Suppose that we have a smooth pointed Ricci flow solution (M, (x0 , 0), g(·)) defined for t ∈ [0, (ǫr0 )2 ], such that each time slice is complete. Suppose that for any x ∈ B0 (x0 , r0 ) and Ω ⊂ B0 (x0 , r0 ), we have R(x, 0) ≥ −r0−2 and vol(∂Ω)n ≥ (1 − δ) cn vol(Ω)n−1 , where cn is the Euclidean isoperimetric constant. Then | Rm |(x, t) < αt−1 + (ǫr0 )−2 whenever 0 < t ≤ (ǫr0 )2 and d(x, t) = distt (x, x0 ) ≤ ǫr0 .
58
BRUCE KLEINER AND JOHN LOTT
The sectional curvature bound | Rm |(x, t) < αt−1 + (ǫr0 )−2 necessarily blows up as t → 0, as nothing was assumed about the sectional curvature at t = 0. We first sketch the idea of the proof of Theorem 30.1. It is an argument by contradiction. One takes a Ricci flow solution that satisfies the assumptions and picks a point (x, t) where the desired curvature bound does not hold. One can assume, roughly speaking, that (x, t) is the first point in the given solution where the bound does not hold. (This will give the curvature bound needed for taking a limit in a sequence of counterexamples.) One now considers the solution u to the conjugate heat equation, starting as a δ-function at (x, t), and the corresponding function v. We know that v ≤ 0. The first goal is to get a negative upper bound for the integral of v over an appropriate ball B at a time e t near t; see Section 33. The argument to get such a bound is by contradiction. If there were R not such a bound then one could consider a rescaled sequence of counterexamples with B v dV → 0, and try to take a limit. If one has the R injectivity radius bounds needed to take a limit then one obtains a limit solution with B v dV = 0, which implies that the limit solution is a gradient shrinking soliton, which violates curvature assumptions. If one doesn’tRhave the injectivity radius bounds then one can do a further rescaling to see that in fact B v dV → −∞ for some subsequence, which is a contradiction. R If M is compact then M v dV is monotonically nondecreasing in t. As (29.6) is a localized version of this statement, whether M is compact or noncompact we can use a cutoff R function Rh and equation (29.6) to get a negative upper bound on M hv dV at time t = 0. Finally, M v dV is the expression that appears in the logarithmic Sobolev inequality. If the isoperimetric constant is sufficiently close to the Euclidean value cn then one concludes R that M hv dV must be bounded below by a constant close to zero, which contradicts the R negative upper bound on M hv dV . 31. Claim 1 of I.10.1. A point selection argument
In Theorem 30.1, we can assume that r0 = 1 and α < | Rm(x, t)| ≥ αt−1 }.
1 . 100n
Fix α and put Mα = {(x, t) :
The next lemma says that if we have a point (x, t) where the conclusion of Theorem 30.1 −1 does not hold then there is another point (x, t) with | Rm(x, t)| large (relative to t ) so that any other such point (x′ , t′ ) either has t′ > t or is much farther from x0 than x is. Lemma 31.1. (cf. Claim 1 of I.10.1) For any A > 0, if g(·) is a Ricci flow solution for 1 t ∈ [0, ǫ2 ], with Aǫ < 100n , and | Rm |(x, t) ≥ αt−1 + ǫ−2 for some (x, t) satisfying t ∈ (0, ǫ2 ] and d(x, t) ≤ ǫ, then one can find (x, t) ∈ Mα with t ∈ (0, ǫ2 ] and d(x, t) < (2A + 1)ǫ, such that | Rm(x′ , t′ )| ≤ 4 | Rm(x, t)|
(31.2) whenever (31.3)
(x′ , t′ ) ∈ Mα ,
t′ ∈ (0, t],
1
d(x′ , t′ ) ≤ d(x, t) + A| Rm |− 2 (x, t).
Proof. The proof is by a point selection argument as in Appendix H. By assumption, there is a point (x, t) satisfying t ∈ (0, ǫ2 ], d(x, t) ≤ ǫ and | Rm(x, t)| ≥ αt−1 + ǫ−2 . Clearly (x, t) ∈ Mα . Define points (xk , tk ) inductively as follows. First, (x1 , t1 ) = (x, t). Next,
NOTES ON PERELMAN’S PAPERS
59
suppose that (xk , tk ) is constructed but cannot be taken for (x, t). Then there is some point 1 (xk+1 , tk+1 ) ∈ Mα such that 0 < tk+1 ≤ tk , d(xk+1 , tk+1) ≤ d(xk , tk ) + A| Rm |− 2 (xk , tk ) and | Rm |(xk+1 , tk+1 ) > 4| Rm |(xk , tk ). Continuing in this way, the point (xk , tk ) constructed has | Rm |(xk , tk ) ≥ 4k−1 | Rm |(x1 , t1 ) ≥ 4k−1 ǫ−2 . Then (31.4)
1
1
d(xk , tk ) ≤ d(x1 , t1 ) + A| Rm |− 2 (x1 , t1 ) + . . . + A| Rm |− 2 (xk−1 , tk−1 ) 1
≤ ǫ + 2A| Rm |− 2 (x1 , t1 ) ≤ (2A + 1)ǫ.
As the solution is smooth, the induction process must terminate after a finite number of steps and the last value (xk , tk ) can be taken for (x, t). 32. Claim 2 of I.10.1. Getting parabolic regions In Lemma 31.1, we know that (31.2) is satisfied under the condition (31.3). The spacetime region described in (31.3) is not a product region, due to the fact that d(x, t) is timedependent. The next goal is to obtain the estimate (31.2) on a product region in spacetime; this will be necessary when taking limits of Ricci flow solutions. To get the estimate on a product region, one needs to bound how fast distances are changing with respect to t. Lemma 32.1. (cf. Claim 2 of I.10.1) For the point (x, t) constructed in Lemma 31.1, | Rm(x′ , t′ )| ≤ 4 | Rm(x, t)|
(32.2) holds whenever (32.3)
t −
1 αQ−1 ≤ t′ ≤ t, 2
distt (x′ , x) ≤
1 1 AQ− 2 , 10
where Q = | Rm(x, t)|. Proof. We first claim that if (x′ , t′ ) satisfies t− 21 αQ−1 ≤ t′ ≤ t and d(x′ , t′ ) ≤ d(x, t)+AQ−1/2 then | Rm |(x′ , t′ ) ≤ 4Q. To see this, if (x′ , t′ ) ∈ Mα then it is true by Lemma 31.1. If −1 (x′ , t′ ) ∈ / Mα then | Rm |(x′ , t′ ) < α(t′ )−1 . As (x, t) ∈ Mα , we know that Q ≥ αt . Then −1 t′ ≥ t − 21 αQ−1 ≥ 12 t and so | Rm |(x′ , t′ ) < 2 αt ≤ 2Q.
Thus we have a uniform curvature bound on the time-t′ distance ball B(x0 , d(x, t) + AQ−1/2 ), provided that t − 12 αQ−1 ≤ t′ ≤ t. We now claim that the time-t ball 1 B(x0 , d(x, t) + 10 AQ−1/2 ) lies in the time-t′ distance ball B(x0 , d(x, t) + AQ−1/2 ). To see 1 this, applying Lemma 27.8 with r0 = 100 AQ−1/2 and the above curvature bound, if x′ is in 1 AQ−1/2 ) then the time-t ball B(x0 , d(x, t) + 10 (32.4) 2 1 1 ′ −1 −1/2 −1 1/2 ′ distt′ (x0 , x ) − distt (x0 , x ) ≤ αQ · 2(n − 1) · 4Q( AQ ) + 100 A Q . 2 3 100
Assuming that A is sufficiently large (we’ll take A → ∞ later) and using the fact that 1 1 α < 100n , it follows that d(x′ , t′ ) ≤ d(x′ , t) + 12 AQ−1/2 ≤ d(x, t) + AQ− 2 , which is what we want to show. We note that the argument also shows that is indeed self-consistent to use the curvature bounds in the application of Lemma 27.8.
60
BRUCE KLEINER AND JOHN LOTT
Now suppose that (x′ , t′ ) satisfies (32.3). By the triangle inequality, x′ lies in the time-t 1 distance ball B(x0 , d(x, t) + 10 AQ−1/2 ). Then x′ is in the time-t′ distance ball B(x0 , d(x, t) + AQ−1/2 ) and so | Rm(x′ , t′ )| ≤ 4Q, which proves the lemma. 33. Claim 3 of I.10.1. An upper bound on the integral of v We first make some remarks about the fundamental solution to the backward heat equation. Let (M, (x, b), g(·)) be a smooth one-parameter family of complete pointed Riemannian manifolds, parametrized by t ∈ (a, b]. The fundamental solution u of the backward heat equation is a positive solution of ∗ u = 0 on M ×(a, b) such that u(·, t) converges to δx in the distributional sense, as t → b− . It is constructed as follows (cf. [26, Section 3]). Let {Di }∞ i=1 be an exhaustion of M by an increasing sequence of smooth compact codimension-zero submanifolds-with-boundary containing x in the interior. Let u(i) be the unique solution of ∗ u(i) = 0 on Di × (a, b) with limt→b− u(i) (x, t) = δx (x), as constructed using Dirichlet boundary conditions on Di . If Di ⊂ Dj then u(i) ≤ u(j) on Di . Then the fundamental solution is defined to be the limit u = limi→∞ u(i) , with smooth convergence on compact subsets of M × (a, b). The function Ru is independent of the choice of exhaustion sequence R ∞ {Di }i=1 . For any t ∈ (a, b), we have M u(x, t) dV (x) ≤ 1. If M u(x, t) dV (x) = 1 for all t then we say that (M, (x, b), g(·)) is stochastically complete for ∗ . This will be the case if one has bounded curvature on compact time intervals, but need not be the case in general. Lemma 33.1. Let {(Mk , (xk , b), gk (·))}∞ k=1 be a sequence of manifolds as above, each defined on the time interval (a, b]. Suppose that limk→∞ (Mk , (xk , b), gk (·)) = (M∞ , (x∞ , b), g∞ (·)) in the pointed smooth topology, and that (M∞ , (x∞ , b), g∞ (·)) is stochastically complete for ∗ . Then after passing to a subsequence, the fundamental solutions {uk }∞ k=1 converge smoothly on compact subsets of M∞ × (a, b) to the fundamental solution u∞ . (Of course, we use the pointed diffeomorphisms inherent in the statement of pointed convergence in order to compare the uk ’s with u∞ .) Proof. From the uniform upper L1 -bound on {uk (·, t)}∞ k=1 and parabolic regularity, after ∞ passing to a subsequence we can assume that {uk }k=1 converges smoothly on compact subsets of M∞ × (a, b) to some function U. From the construction of u∞ , it follows easily that u∞ ≤ U. For any t ∈ (a, b), we have (33.2) Z
M∞
Z
lim inf (uk (x, t) − u∞ (x, t)) dvol(x) k Z ≤ lim inf (uk (x, t) − u∞ (x, t)) dvol(x) ≤ 0,
(U(x, t) − u∞ (x, t)) dvol(x) =
M∞
k
so U = u∞ .
M∞
Starting the proof of Theorem 30.1, we suppose that the theorem is not true. Then there are sequences ǫk → 0 and δk → 0, and pointed Ricci flow solutions (Mk , (x0,k , 0), gk (·)) which satisfy the hypotheses of the theorem but for which there is a point (xk , tk ) with 0 < tk ≤ ǫ2k , −2 d(xk , tk ) ≤ ǫk and | Rm |(xk , tk ) ≥ αt−1 k + ǫk . Given the flow (Mk , gk (·)), we reduce ǫk as
NOTES ON PERELMAN’S PAPERS
61
much as possible so that there is still such a point (xk , tk ). Then −2 | Rm |(x, t) < αt−1 k + 2ǫk
(33.3)
1 whenever 0 < t ≤ ǫ2k and d(x, t) ≤ ǫk . Put Ak = 100nǫ . Construct points (xk , tk ) as k n in Lemma 31.1. Consider fundamental solutions uk = (4π(tk − t))− 2 e−fk of ∗ uk = 0 satisfying limt→t− u(x, t) = δxk (x). Construct the corresponding functions vk from (29.3). k
Lemma 33.4. (cf. Claim 3 of I.10.1) There Ris some β > 0 so that for all sufficiently large k, there is some e tk ∈ [tk − 21 αQ−1 k , tk ] with Bk vk dVk ≤ −β, where Qk = | Rm |(xk , tk ) p tk centered at xk . and Bk is the time-e tk ball of radius tk − e
Proof. Suppose that the claim is notR true. After passing to a subsequence, we can assume that for any choice of e tk , lim inf k→∞ Bk vk dVk ≥ 0.
Consider the pointed solution (Mk , (xk , tk ), gk (·)) parabolically rescaled by Qk . Suppose first that there is a subsequence so that the injectivity radii of the scaled metrics at (xk , tk ) are bounded away from zero. Since Ak → ∞, we can use Lemma 32.1 and Appendix E to take a subsequence that converges to a complete Ricci flow solution (M∞ , (x∞ , t∞ ), g∞ (·)) on a time interval (t∞ − 12 α, t∞ ], with | Rm | ≤ 4 and | Rm |(x∞ , t∞ ) = 1. Consider the fundamental solution u∞ of ∗ on M∞ with limt→t−∞ u∞ (x∞ , t) = δx∞ (x∞ ). As before, let uk be the fundamental solution of ∗ on Mk with limt→t− uk (xk , t) = δxk (xk ). In view of the k pointed convergence of the rescalings of (Mk , (xk , tk ), gk (·)) to (M∞ , (x∞ , t∞ ), g∞ (·)), Lemma 33.1 implies that after passing to a further subsequence we can ensure that limk→∞ uk = u∞ , with smooth convergence on compact subsets of M∞ ×(t∞ − 21 α, t∞ ). (The curvature bounds on (M∞ , (x∞ , t∞ ), g∞ (·)) ensure that it is stochastically complete for ∗ .) From Corollary 29.21, v∞ ≤ 0. Note that we are applying Corollary 29.21 on M∞ × (t∞ − 21 α, t∞ ), where we have the curvature bounds needed to use the maximum principle. p t∞ ball of radius t∞ − e t∞ centered at x∞ . Given e t∞ ∈ (t∞ − 12 α, t∞ ), let B∞ be the time-e In view of the smooth Rconvergence limk→∞ uk = u∞ on compact subsets of M∞ × (t∞ − 1 α, t∞ ), it follows that B∞ v∞ dV∞ = 0 at time e t∞ , so v∞ vanishes on B∞ at time e t∞ . Let 2 tR∞ ) a nonnegative nonzero function h be a solution to h = 0 on M∞ × [e t∞ , t∞ ) with h(·, e supported in B∞ . As in the proof of Corollary 29.21, M∞ hv∞ dV∞ is nondecreasing in t R and vanishes for t = e t∞ and t → t∞ . Thus M∞ hv∞ dV∞ vanishes for all t ∈ [e t∞ , t∞ ). However, for t ∈ (e t∞ , t∞ ), h is strictly positive and v∞ is nonpositive. Thus v∞ vanishes on M∞ for all t ∈ (e t∞ , t∞ ), and so 1 (33.5) Ric(g∞ ) + Hess f∞ − g∞ = 0. 2(t − t)
on this interval. We know that | Rm | ≤ 4 on M∞ × (t∞ − 21 α, t∞ ]. From the evolution equation, 1 dg∞ = − 2 Ric(g∞ ) = 2 Hess f∞ − g∞ . (33.6) dt t−t It follows that the supremal and infimal sectional curvatures of g∞ (·, t) go like (t∞ − t)−1 . Hence g∞ is flat, which contradicts the fact that | Rm |(x∞ , t∞ ) = 1.
62
BRUCE KLEINER AND JOHN LOTT
Suppose now that there is a subsequence so that the injectivity radii of the scaled metrics at (xk , tk ) tend to zero. Parabolically rescale (Mk , (xk , tk ), gk (·)) further so that the injectivity radius becomes one. After passing to a subsequence we will have convergence to a flat Ricci flow solution (−∞, 0] × L. The complete flat manifold L can be described as the total space of a flat orthogonal Rm -bundle over a flat compact manifold C. After separating variables, the fundamental solution u∞ on L will be Gaussian in the fiber directions and will decay exponentially fast to a constant in the base directions, i.e. |x|2
1 u(x, τ ) ∼ (4πτ )−m/2 e− 4τ vol(C) , where |x| is the fiber norm. With this for u∞ , one finds √ of radius τ , that v∞ = (m − n) 1 + 12 ln(4πτ ) u∞ . With Bτ the ball around a basepoint R the integral of u over Bτ has a positive limit as τ → ∞, and so limτ →∞ Bτ v∞ dV∞ = − ∞. R , t ] so that lim v dVk = −∞, which is a Then there are times e tk ∈ [tk − 21 αQ−1 k k→∞ k Bk k contradiction.
34. Theorem I.10.1. Proof of the pseudolocality theorem Continuing with the proof of Theorem 30.1, we now use Lemma 33.4 to get a contradiction to a log Sobolev inequality. For simplicity of notation, we drop the subscript k and deal with a particular (Mk , (xk , tk ), gk (·)) for k large. Define a smooth function φ on R which is one on (−∞, 1], decreasing on [1, 2] and zero on [2, ∞], with φ′′ ≥ −10φ′ and (φ′ )2 ≤ 10φ. To construct φ we can take the function which is 1 on (−∞, 1], 1 − 2(x − 1)2 on [1, 3/2], 2(x − 2)2 on [3/2, 2] and 0 on [2, ∞), and smooth it slightly. √ e t) = d(y, t) + 200n t. We claim that if 10Aǫ ≤ d(y, e t) ≤ 20Aǫ then dt (y, t) − Put d(y, 100n 2 e t) ≤ 20Aǫ and A is △d(y, t) + √ ≥ 0. To see this, recalling that t ∈ [0, ǫ ], if 10Aǫ ≤ d(y, t
sufficiently large then 9Aǫ ≤ √ d(y, t) ≤ 21Aǫ. We apply Lemma 27.18 with the parameter r0 of Lemma 27.18 equal to t. As r0 ≤ ǫ, we have y ∈ / B(x0 , r0 ). From (33.3), on B(x0 , r0 ) −1 −2 we have | Rm |(·, t) ≤ α t + 2 ǫ . Then from Lemma 27.18, at (y, t) we have
(34.1)
2 dt − △d ≥ − (n − 1) (α t−1 + 2 ǫ−2 )t1/2 + t−1/2 3 2 4 −2 = − (n − 1) 1 + α + ǫ t t−1/2 . 3 3
√ It follows that dt − △d + 100n ≥ 0. t e . Then h = Now put h(y, t) = φ d(y,t) 10Aǫ e
1 10Aǫ
dt − △d +
100n √ t
φ′ −
1 φ′′ , (10Aǫ)2
where
√ the arguments of φ′ and φ′′ are d(y,t) . Where φ′ 6= 0, we have dt − △d + 100n ≥ 0. The 10Aǫ t −n −f (x,t) ∗ fundamental solution u(x, t) = (4π(t − t)) 2 e of is positive for t ∈ [0, t) and we R have M u dV ≤ 1 for all t. (Recall that we are not assuming stochastic completeness.)
NOTES ON PERELMAN’S PAPERS
63
Then (34.2) Z
hu dV M
t
Z
Z
1 = ((h)u − h u) dV = (h)u dV ≤ − (10Aǫ)2 M M Z Z 10 10 10 φu dV ≤ u dV ≤ . ≤ 2 2 (10Aǫ) M (10Aǫ) M (10Aǫ)2 ∗
Z
φ′′ u dV
M
Hence Z
(34.3)
M
hu dV
t=0
≥
Z
M
hu dV
t=t
−
t ≥ 1 − A−2 . 2 (Aǫ)
Similarly, using Proposition 29.5 and Corollary 29.21, (34.4) Z Z Z ∗ − hv dV = − ((h)v − h v) dV ≤ − (h)v dV M M M t Z Z Z 1 10 10 ′′ ≤ φ v dV ≤ − φv dV = − hv dV. (10Aǫ)2 M (10Aǫ)2 M (10Aǫ)2 M
p t ∈ [t/2, t]. p Then t − e t ≤ 2−1/2 ǫ and Consider the time e t of Lemma 33.4. As (x, t) ∈ Mα , e so for large A, h will be one on the ball B at time e t of radius t − e t centered at x. Then at time e t, −
(34.5)
Z
M
hv dV ≥ −
Z
B
v dV ≥ β.
Thus (34.6)
−
Z
M
hv dV
t=0
−
≥ βe
e t (Aǫ)2
−
≥ βe
t (Aǫ)2
≥ β 1−
t (Aǫ)2
≥ β(1 − A−2 ).
Working at time 0, put u e = hu and fe = f − log h. In what follows we implicitly integrate over supp(h). We have (34.7)
−2
β(1 − A ) ≤ −
Z
hv dV = M
Z
M
[(−2△f + |∇f |2 − R)t − f + n]hu dV.
We claim that Z Z −f |∇h|2 2 2 e he−f dV. (34.8) −2△f + |∇f | he dV = − |∇f | + 2 h M M
64
BRUCE KLEINER AND JOHN LOTT
This follows from Z Z −f 2 (34.9) −2△f + |∇f | he dV = 2h∇f, ∇(he−f )i + |∇f |2 he−f dV M ZM ∇h 2 = 2h∇f, − ∇f i + |∇f | he−f dV h ZM ∇h − ∇f i he−f dV = h∇f, 2 h ZM ∇h ∇h e he−f dV = h∇fe + , − ∇fi h h ZM 2 |∇h| 2 e + = − |∇f| he−f dV. 2 h M Then
Z
(34.10)
[(−2△f + |∇f |2 − R)t − f + n]hu dV = Z ZM 2 e e u dV + [t(|∇h|2 /h − Rh) − h log h]u dV. [−t|∇f | − f + n]e M
M
Next, Then
|∇h|2 h
≤
Z
(34.11) Also, (34.12)
10 (10Aǫ)2
−
Z
M
t M
and − Rh ≤ 1 (from the assumed lower bound on R at time zero).
|∇h|2 10 2 − Rh u dV ≤ ǫ + 1 ≤ A−2 + ǫ2 . h (10Aǫ)2
uh log h dV = −
Z
B(x0 ,20Aǫ)−B(x0 ,10Aǫ)
≤ 1 − Putting h(y) = φ (34.13)
d(y) 5Aǫ
Z
uh log h dV ≤
Z
u dV
M −B(x0 ,10Aǫ)
u dV. B(x0 ,10Aǫ)
, a result similar to (34.3) shows that Z Z u dV ≥ hu dV ≥ 1 − cA−2 B(x0 ,10Aǫ)
M
for an appropriate constant c. Putting this together gives Z −2 e 2 − fe + n u (34.14) β(1 − A ) ≤ −t|∇f| e dV + (1 + c)A−2 + ǫ2 . M
n n b b = (2t) 2 u e and define fb by u b = (2π)− 2 e−f . From (34.3) and (34.14), if we Put b g = 2t1 g, u R restore the subscript k then limk→∞ M u bk dVbk = 1 and for large k, Z 1 1 2 b b (34.15) β ≤ − |∇fk | − fk + n u bk dVbk . 2 2 Mk
NOTES ON PERELMAN’S PAPERS
If we normalize u bk by putting Uk = for large k, we also have
1 β ≤ 2
(34.16)
R
Mk
u bk , u bk dVbk
65 n
and define Fk by Uk = (2π)− 2 e− Fk , then
1 2 − |∇Fk | − Fk + n Uk dVbk . 2 Mk
Z
On the other hand, the logarithmic Sobolev inequality for Rn [8, I.(8)] says that Z 1 2 (34.17) − |∇F | − F + n U dV ≤ 0, 2 Rn R provided that the compactly-supported function U = (2π)−n/2 e−F satisfies Rn U dV = 1. As was mentioned to us by Peter Topping, one can get a sharper inequality by applying (34.17) to the rescaled function Uc (x) = cn U(cx) and optimizing with respect to c. The result is Z R 2 (34.18) |∇F |2 U dV ≥ n e1 − n Rn F U dV . Rn
Given this inequality on Rn , one can use a symmetrization argument to prove the same inequality for a compactly-supported function on any complete Riemannian manifold, provided that the Euclidean isoperimetric inequality holds for domains in the support of U. See, for example, [48, Proposition 4.1] which gives the symmetrization argument for (34.17), attributing it to Perelman. Again using the inequality for Rn , if instead we have vol(∂Ω)n ≥ (1 − δk ) cn vol(Ω)n−1 for domains Ω ⊂ supp(Uk ) then the symmetrization argument gives Z R 2 1− 2 F U dVb (34.19) |∇Fk |2 Uk dVbk ≥ (1 − δk ) n n e n Mk k k k . Mk
Equations (34.16) and (34.19) imply that Z R 2 bk 2 2 β n 1− n F U d V k k Mk (1 − δk ) n e − 1 − 1− Fk Uk dVbk ≤ − . (34.20) 2 n Mk 2
However, (34.21)
2 x n lim inf (1 − δk ) e − 1 − x = 0.
k→∞ x∈R
This is a contradiction.
35. I.10.2. The volumes of future balls The next result gives a lower bound on the volumes of future balls. Corollary 35.1. (cf. Corollary √ I.10.2)n Under the assumptions of Theorem 30.1, for 0 < t ≤ (ǫr0 )2 we have vol(Bt (x, t)) ≥ ct 2 for x ∈ B0 (x0 , ǫr0 ), where c = c(n) is a universal constant.
66
BRUCE KLEINER AND JOHN LOTT
Proof. (Sketch) If the corollary were not true then taking√a sequence of counterexamples, we can center ourselves around the collapsing balls B(x, t) to obtain functions f as in Section 34. As in the proof of Theorem 13.3, the volume condition with the fact that R R along 1 −n/2 −f 2 (2π) e dV → 1 means that f → −∞, which implies that − |∇f | − f + n udV → M M 2 ∞. This contradicts the logarithmic Sobolev inequality. 36. I.10.4. κ-noncollapsing at future times The next result gives κ-noncollapsing at future times. Corollary 36.1. (cf. Corollary I.10.4) There are δ, ǫ > 0 such that for any A > 0 there exists κ = κ(A) > 0 with the following property. Suppose that we have a Ricci flow solution g(·) defined for t ∈ [0, (ǫr0 )2 ] which has bounded | Rm | and complete time slices. Suppose that for any x ∈ B(x0 , r0 ) and Ω ⊂ B(x0 , r0 ), we have R(x, 0) ≥ −r0−2 and vol(∂Ω)n ≥ (1 − δ) cn vol(Ω)n−1 , where cn is the Euclidean isoperimetric constant. If (x, t) satisfies A−1 (ǫr0 )2 √ ≤ t ≤ (ǫr0 )2 and distt (x, x0 ) ≤ Ar0 then g(·) is not κ-collapsed at (x, t) on scales less than t. Proof. Using Theorem 30.1 and Corollary 35.1, we can apply Theorem 28.2 starting at time A−1 (ǫr0 )2 . 37. I.10.5. Diffeomorphism finiteness In this section we prove the diffeomorphism finiteness of Riemannian manifolds with local isoperimetric inequalities, a lower bound on scalar curvature and an upper bound on volume. Theorem 37.1. Given n ∈ Z+ , there is a δ > 0 with the following property. For any r0 , V > 0, there are finitely many diffeomorphism types of compact n-dimensional Riemannian manifolds (M, g0 ) satisfying 1. R ≥ −r0−2 . 2. vol(M, g0 ) ≤ V . 3. Any domain Ω ⊂ M contained in a metric r0 -ball satisfies vol(∂Ω)n ≥ (1−δ)cn vol(Ω)n−1 , where cn is the Euclidean isoperimetric constant. Proof. Choose α > 0. Let δ and ǫ be the parameters of Theorem 30.1. Consider Ricci flow g(·) starting from (M, g0 ). Let T > 0 be the maximal number so that a smooth flow exists for t ∈ [0, T ). If T < ∞ then limt→T − supx∈M | Rm(x, t)| = ∞. It follows from Theorem 30.1 that T > (ǫr0 )2 . Put b g = g((ǫr0 )2 ). Theorem 30.1 gives a uniform double-sided sectional curvature bound on (M, b g ). Corollary 35.1 gives a uniform lower bound on the volumes of (ǫr0 )-balls in (M, b g ). Let {xi }N b). i=1 be a maximal (2ǫr0 )-separated net in (M, g
From the lower bound R ≥ −r0−2 on (M, g0 ) and the maximum principle, we have R(x, t) ≥ −r0−2 for t ∈ [0, (ǫr0 )2 ]. Then the Ricci flow equation gives a uniform upper bound on vol(M, b g ). This implies a uniform upper bound on N or, equivalently, a uniform upper bound on diam(M, gb). The theorem now follows from the diffeomorphism finiteness of n-dimensional Riemannian manifolds with double-sided sectional curvature bounds, upper bounds on diameter and lower bounds on volume.
NOTES ON PERELMAN’S PAPERS
67
38. I.11.1. κ-solutions Definition 38.1. Given κ > 0, a κ-solution is a Ricci flow solution (M, g(·)) that is defined on a time interval of the form (−∞, C) (or (−∞, C]) such that • The curvature | Rm | is bounded on each compact time interval [t1 , t2 ] ⊂ (−∞, C) (or (−∞, C]), and each time slice (M, g(t)) is complete. • The curvature operator is nonnegative and the scalar curvature is everywhere positive. • The Ricci flow is κ-noncollapsed at all scales.
By abuse of terminology, we may sometimes write that “(M, g(·)) is a κ-solution” if it is a κ-solution for some κ > 0. From Appendix F, Rt ≥ 0 for an ancient solution. This implies the essential equivalence of the notions of κ-noncollapsing in Definitions 13.14 and 26.1 , when restricted to ancient solutions. Namely, if a solution is κ-collapsed in the sense of Definition 26.1 then it is automatically κ-collapsed in the sense of Definition 13.14. Conversely, if a time-t0 slice of an ancient solution is collapsed in the sense of Definition 13.14 then the fact that Rt ≥ 0, together with bounds on distance distortion, implies that it is collapsed in the sense of Definition 26.1 (possibly for a different value of κ). The relevance of κ-solutions is that a blowup limit of a finite-time singularity on a compact manifold will be a κ-solution. For examples of κ-solutions, if n ≥ 3 then there is a κ-solution on the cylinder R×S n−1(r), where the radius satisfies r 2 (t) = r02 − 2(n−2)t. There is also a κ-solution on the Z2 -quotient R ×Z2 S n−1 (r), where the generator of Z2 acts by reflection on R and by the antipodal map on S n−1 . On the other hand, the quotient solution on S 1 × S n−1 (r) is not κ-noncollapsed for any κ > 0, as can be seen by looking at large negative time. Bryant’s gradient steady soliton is a three-dimensional κ-solution given by g(t) = φ∗t g0 , where g0 = dr 2 + µ(r) dΘ2 is a certain rotationally symmetric metric on R3 . It has sectional curvatures that go like r −1 , and µ(r) ∼ r. The gradient function f satisfies Rij + ∇i ∇j f = 0, with f (r) ∼ −2r. Then for r and r − 2t large, φt (r, Θ) ∼ (r − 2t, Θ). In particular, if R0 ∈ C ∞ (R3 ) is the scalar curvature function of g0 then R(t, r, Θ) ∼ R0 (r − 2t, Θ). To check the conclusion of Corollary 47.2 in this case, given a point (r0 , Θ) ∈ R3 at time 0, the scalar curvature goes like r0−1 . Multiplying the soliton metric by r0−1 and sending t → r0 t gives the asymptotic metric (38.2)
√ r − 2r0 t dΘ2 . d(r/ r0 )2 + r0
√ Putting u = (r − r0 )/ r0 , the rescaled metric is approximately u 2 (38.3) du + 1 + √ − 2t dΘ2 . r0 Given ǫ > 0, this will be ǫ-biLipschitz close to the evolving cylinder du2 + (1 − 2t) dΘ2 √ provided that |u| ≤ ǫ r0 , i.e. |r − r0 | ≤ ǫr0 . To have an ǫ-neck, we want this to hold
68
BRUCE KLEINER AND JOHN LOTT
whenever |r − r0 |2 ≤ (ǫr0−1 )−1 . This will be the case if r0 ≥ ǫ−3 . Thus Mǫ is approximately
(38.4)
{(r, Θ) ∈ R3 : r ≤ ǫ−3 }
and Q = R(x0 , 0) ∼ ǫ3 . Then diam(Mǫ ) ∼ ǫ−3 and at the origin 0 ∈ Mǫ , R(0, 0) ∼ ǫ0 . It follows that for the value of κ corresponding to this solution, C(ǫ, κ) must grow at least as fast as ǫ−3 as ǫ → 0. 39. I.11.2. Asymptotic solitons This section shows that every κ-solution has a gradient shrinking soliton buried inside of it, in an asymptotic sense as t → −∞. Such a soliton will be called an asymptotic soliton.
Heuristically, the existence of an asymptotic soliton is a consequence of the compactness results and the monotonicity of the reduced volume. Taking an appropriate sequence of spacetime points going backward in time, one constructs a limiting rescaled solution. As the limit reduced volume is constant in time, the monotonicity formula implies that this limit solution is a gradient shrinking soliton. This is the basic idea but the rigorous argument is a bit more subtle. Pick an arbitrary point (p, t0 ) in the κ-solution (M, g(·)). Define the reduced volume Ve (τ ) and the reduced length l(q, τ ) as in Section 15, by means of curves starting from (p, t0 ), with τ = t0 − t. From Section 24, for each τ > 0 there is some q(τ ) ∈ M such that l(q(τ ), τ ) ≤ n2 . (Note that l ≥ 0 from the curvature assumption.)
Proposition 39.1. (cf. Proposition I.11.2) There is a sequence τ i → ∞ so that if we consider the solution g(·) on the time interval [t0 − τ i , t0 − 12 τ i ] and parabolically rescale it at the point (q(τ i ), t0 − τ i ) by the factor τi −1 then as i → ∞, the rescaled solutions converge to a nonflat gradient shrinking soliton (restricted to [−1, − 21 ]). Proof. Equation (25.5) implies that |∇l1/2 |2 ≤
C , 4τ
r
and so
C distt0 −τ (q, q(τ )). 4τ We apply this estimate initially at some fixed time τ = τ , to obtain r r !2 C n distt0 −τ (q, q(τ )) + . (39.3) l(q, τ ) ≤ 4τ 2 (39.2)
|l1/2 (q, τ ) − l1/2 (q(τ ), τ )| ≤
From (18.13), (18.14) and (25.5), R |∇l|2 l (1 + C)l ∂τ l = − − ≥ − . 2 2 2τ 2τ This implies that for τ ∈ 12 τ , τ , r r !2 1+C 2 τ C n (39.5) l(q, τ ) ≤ distt0 −τ (q, q(τ )) + . τ 4τ 2 (39.4)
Also from (25.5), we have τ R ≤ Cl. Then we can plug in the previous bound on l to get 1 an upper bound on τ R for τ ∈ 2 τ , τ . The upshot is that for any ǫ > 0, one can find
NOTES ON PERELMAN’S PAPERS
69
δ > 0 so that both l(q, τ ) and τ R(q, t0 − τ ) do not exceed δ −1 whenever 12 τ ≤ τ ≤ τ and dist2t0 −τ (q, q(τ )) ≤ ǫ−1 τ .
Varying τ , as the rescaled solutions (with basepoints at (q(τ ), t0 − τ )) are uniformly noncollapsing and have uniform curvature bounds on balls, Appendix E implies that we can take a sequence τ i → ∞ to get a pointed limit (M , q, g(·)) that is a complete Ricci flow solution (in the backward parameter τ ) for 12 < τ < 1. We may assume that we have locally Lipschitz convergence of l to a limit function l. We define the reduced volume V˜ (τ ) for the limit solution using the limit function l. We 1 claim that for any τ ∈ 2 , 1 , if we put τi = τ τ i then the number V˜ (τ ) for the limit solution is the limit of numbers V˜ (τi ) for the original solution. One wishes to apply dominated R −n/2 −n/2 convergence to the integrals M e−l(q,τi ) τi dvol(q, t0 −τi ). (Note that τi dvol(q, t0 −τi ) = −n/2 −n/2 −n/2 τi dvol(q, t0 −τi ) and τ i dvol(q, t0 −τi ) is the volume form for the rescaled metric τ −1 τ i g(t0 − τi ).) However, to do so one needs uniform lower bounds on l(q, τ ′ ) for the original solution in terms of dt0 −τ ′ (q, q(τ ′ )), for τ ′ ∈ (−∞, 0). By an argument of Perelman, written in detail in [67], one does indeed have a lower bound of the form dt0 −τ ′ (q, q(τ ′ ))2 . τ′ The nonnegative curvature gives polynomial volume growth for distance balls, so using (39.6) R −n/2 one can apply dominated convergence to the integrals M e−l(q,τi ) τi dvol(q, t0 − τi ). Thus ˜ ˜ limi→∞ V (τi ) = V (τ ). (39.6)
l(q, τ ′ ) ≥ − l(q(τ ′ )) − 1 + C(n)
As (39.5) gives a uniform upper bound on l on an appropriate ball around q(τi ), and there is a lower volume bound on the ball, it follows that as i → ∞, V˜ (τi ) is uniformly bounded away from zero. From this argument and the monotonicity of V˜ , V˜ (τ ) is a positive constant c as a function of τ , namely the limit of the reduced volume of the original solution as real time goes to −∞. As the original solution is nonflat, the constant c is strictly less than the n limit of the reduced volume of the original solution as real time goes to zero, which is (4π) 2 . Next, we will apply (24.6) and (24.8). As (24.6) holds distributionally for each rescaled solution, it follows that it holds distributionally for l. In particular, the nonpositivity implies that the left-hand side of (24.6), when computed for the limit solution, is actually a nonpositive measure. If the left-hand side of (24.6) (for the limit solution) were not ˜ strictly zero then using (24.7) we would conclude that ddτV is somewhere negative, which is a contradiction. (We use the fact that (39.6) passes to the limit to give a similar lower bound on l.) Thus we must have equality in (24.6) for the limit solution. This implies equality in (21.2), which implies equality in (24.8). Writing (24.8) as l l − n −l (39.7) (4△ − R) e− 2 = e 2, τ elliptic theory gives smoothness of l. In I.11.2 it is said that equality in (24.8) implies equality in (23.9), which implies that one has a gradient shrinking soliton. There is a problem with this argument, as the use of (23.9) implicitly assumes that the solution is defined for all τ ≥ 0, which we do not know. (The function l is only defined by a limiting procedure, and not in terms of L-geodesics on some
70
BRUCE KLEINER AND JOHN LOTT
Ricci flow solution.) However, one can instead use Proposition 29.5, with f = l. Equality in (24.8) implies that v = 0, so (29.6) directly gives the gradient shrinking soliton equation. (The problem with the argument using (23.9), and its resolution using (29.6), were pointed out by the UCSB group.) If the gradient shrinking soliton g(·) is flat then, as it will be κ-noncollapsed at all scales, g n it must be Rn . From the soliton equation, ∂i ∂j l = 2τij and △l = 2τ . Putting this into the equality (24.8) gives |∇l|2 = τl . It follows that the level sets of l are distance spheres. Then 2 (24.6) implies that with an appropriate choice of origin, l = |x| . The reduced volume V˜ (τ ) 4τ n for the limit solution is now computed to be (4π) 2 , which is a contradiction. Therefore the gradient shrinking soliton is not flat. We remark that the gradient soliton constructed here does not, a priori, have bounded curvature on compact time intervals, i.e. it may not be a κ-solution. In the 2 and 3dimensional cases one can prove this using additional reasoning. See Section 43 where it is shown that 2-dimensional κ-solutions are round spheres, and Section 46 where the 3-dimensional case is discussed. 40. I.11.3. Two dimensional κ-solutions The next result is a classification of two-dimensional κ-solutions. It is important when doing dimensional reduction. Corollary 40.1. (cf. Corollary I.11.3) The only oriented two-dimensional κ-solution is the shrinking round 2-sphere. Proof. First, the only nonflat oriented nonnegatively curved gradient shrinking 2-D soliton is the round S 2 . The reference [30] given in I.11.3 for this fact does not actually cover it, as the reference only deals with compact solitons. A proof using Proposition 39.1 to rule out the noncompact case appears in [68]. Given this, the limit solution in Proposition 39.1 is a shrinking round 2-sphere. Thus the rescalings τ −1 i g(t0 − τ i ) converge to a round 2-sphere as i → ∞. However, by [30] the Ricci flow makes an almost-round 2-sphere become more round. Thus any given time slice of the original κ-solution must be a round 2-sphere. Remark 40.2. One can employ a somewhat different line of reasoning to prove Corollary 40.1; see Section 43. 41. I.11.4. Asymptotic scalar curvature and asymptotic volume ratio In this section we first show that the asymptotic scalar curvature ratio of a κ-solution is infinite. We then show that the asymptotic volume ratio vanishes. The proofs are somewhat rearranged from those in I.11.4. They are logically independent of Section 40, i.e. also cover the case n = 2. We will use results from Appendices F and G, in particular (F.14).
NOTES ON PERELMAN’S PAPERS
71
Definition 41.1. If M is a complete connected Riemannian manifold then its asymptotic scalar curvature ratio is R = lim supx→∞ R(x) d(x, p)2 . It is independent of the choice of basepoint p. Theorem 41.2. Let (M, g(·)) be a noncompact κ-solution. Then the asymptotic scalar curvature ratio R is infinite for each time slice. Proof. Suppose that M is n-dimensional, with n ≥ 2. Pick p ∈ M and consider a time-t0 slice (M, g(t0 )). We deal with the cases R ∈ (0, ∞) and R = 0 separately and show that they lead to contradictions. Case 1: 0 < R < ∞. We choose a sequence xk ∈ M such that dt0 (xk , p) → ∞ and R(xk , t0 )d2 (xk , p) → R. Consider the rescaled pointed solution (M, xk , gk (t)) with gk (t) = R(xk , t0 )g(t0 + R(xkt ,t0 ) ) and t ∈ (−∞, 0]. We have Rk (xk , 0) = 1, and for all b > 0, for sufficiently large k, we have Rk (x, t) ≤ Rk (x, 0) ≤ d22R for all x such that dk (x, p) > b. k (x,p) √ Fix numbers b, B > 0 so that b < R < B. The κ-noncollapsing assumption gives a uniform positive lower bound on the injectivity radius of gk (0) at xk , and so by Appendix E we may extract a pointed limit solution (M∞ , x∞ , g∞ (·)), defined on a time interval (−∞, 0] from the sequence (Mk , xk , gk (·)) where Mk = {x ∈ M | b < dk (x, p) < B}. Note that g∞ has nonnegative curvature operator and the time slice (M∞ , g∞ (0)) is locally isometric to an annular portion of a nonflat metric cone, since (Mk , p, gk (0)) Gromov-Hausdorff converges to the Tits cone CT (M, g(t0 )). (We use the word “locally” because the annulus in CT (M, g(t0 )) need not be geodesically convex in CT (M, g(t0 )), so we are only saying that the distance functions in small balls match up.) When n = 2 this contradicts the fact that R∞ (x∞ , 0) = 1. When n ≥ 3, we will derive a contradiction from Hamilton’s curvature evolution equation (41.3)
Rmt = ∆ Rm +Q(Rm).
Let dv : CT (M, g(t0 )) → R be the distance function from the vertex and let ρ : M∞ → R be the pullback of dv under the inclusion of the annulus M∞ in CT (M, g(t0 )). Lemma 41.4. The metric cone structure on (M∞ , g∞ (0)) is smooth, i.e. ρ is a smooth function. Proof. Consider a unit speed geodesic segment γ in the Tits cone CT (M, g(t0 )), such that γ is disjoint from the vertex v ∈ CT (M, g(t0 )). Note that since CT (M, g(t0 )) is a Euclidean cone over the Tits boundary ∂T (M, g(t0 )), the geodesic γ lies in the cone over a geodesic segment γˆ ⊂ ∂T (M, g(t0 )). Thus γ lies in a 2-dimensional locally convex flat subspace of CT (M, g(t0 )). Also, as in a flat 2-dimensional cone, the second derivative of the composite function d2v ◦ γ is identically 2.
Since ρ is obtained from dv by composition with a locally isometric embedding (M∞ , g∞ (0)) → CT (M, g(t0 )), the composition of ρ2 with any unit speed geodesic segment in M∞ also has second derivative identically equal to 2. Since ρ2 is Lipschitz, Rademacher’s theorem implies that ρ2 is differentiable almost everywhere. Let y ∈ M∞ be a point of differentiability of ρ2 . If the injectivity radius of g∞ (0)
72
BRUCE KLEINER AND JOHN LOTT
at y is > a, then the function hy given by the composition expy
ρ2
Ty M∞ ⊃ B(0, a) −→ M∞ −→ R
has the property that its second radial derivative is identically 2, and it is differentiable at the origin 0 ∈ Ty M∞ . Therefore hy is a second order polynomial with Hessian identically 2, and is smooth. As the injectivity radius is a continuous function, this implies that ρ2 is smooth everywhere in M∞ . p Since ρ2 is strictly positive on M∞ , it follows that ρ = ρ2 is smooth as well.
By the lemma, we may choose a smooth local orthonormal frame e1 , . . . , en for (M∞ , g∞ (0)) near x∞ such that e1 points radially outward (with respect to the cone structure), and e2 , e3 span a 2-plane at x∞ with strictly positive curvature; such a 2-plane exists because R∞ (x∞ , 1) = 1. Put P = e1 ∧ e2 . In terms of the curvature operator, the fact that Rm∞ (e1 , e2 , e2 , e1 ) = 0 is equivalent to hP, Rm∞ P i = 0. As the curvature operator is nonnegative, it follows that Rm∞ P = 0. (In fact, this is true for any metric cone.) Differentiating gives (41.5)
(∇ei Rm∞ ) P + Rm∞ (∇ei P ) = 0
and (41.6)
(△ Rm∞ ) P + 2
X i
(∇ei Rm∞ ) ∇ei P + Rm∞ (△P ) = 0.
Taking the inner product of (41.6) with P gives X (41.7) 0 = hP, (△ Rm∞ ) P i + 2 hP, (∇ei Rm∞ ) ∇ei P i i
X = hP, (△ Rm∞ ) P i + 2 h∇ei P, (∇ei Rm∞ ) P i. i
Then (41.5) gives (41.8)
hP, (△ Rm∞ )P i = 2
X i
h∇ei P, Rm∞ (∇ei P )i.
As the sphere of distance r from the vertex in a metric cone has principal curvatures r1 , we have ∇e3 e1 = − 1r e3 . Then
1 (e2 ∧ e3 ) + e1 ∧ ∇e3 e2 . r This shows that ∇e3 P has a nonradial component 1r e2 ∧e3 . Thus (∆ Rm∞ )(e1 , e2 , e2 , e1 ) > 0. The zeroth order quadratic term Q(Rm) appearing in (41.3) is nonnegative when Rm is nonnegative, so we conclude that ∂t Rm∞ (e1 , e2 , e2 , e1 ) > 0 at t = 0. This means that Rm∞ (−ǫ)(e1 , e2 , e2 , e1 ) < 0 for ǫ > 0 sufficiently small, which is impossible. (41.9)
∇e3 (e1 ∧ e2 ) = (∇e3 e1 ) ∧ e2 + e1 ∧ ∇e3 e2 =
Case 2: R = 0. Let us take any sequence xk ∈ M with dt0 (xk , p) → ∞. Set rk = dt0 (xk , p), put (41.10)
gk (t) = rk−2 g(t0 + rk2 t)
NOTES ON PERELMAN’S PAPERS
73
for t ∈ (−∞, 0], and let dk (·, ·) be the distance function associated to gk (0). For any 0 < b < B, put (41.11)
Mk (b, B) = {x ∈ M | 0 < b < dk (x, p) < B}.
Since R = 0, we get that supx∈Mk (b,B) | Rmk (x, 0)| → 0 as k → ∞. Invoking the κnoncollapsed assumption as in the previous case, we may assume that (M, p, gk (0)) GromovHausdorff converges to a metric cone (M∞ , p∞ , g∞ ) (the Tits cone CT (M, g(t0 ))) which is flat and smooth away from the vertex p∞ , and the convergence is smooth away from p∞ . The “unit sphere” in CT (M, g(t0 )) defines a compact smooth hypersurface S∞ in (M∞ − {p∞ }, g∞ (0)) whose principal curvatures are identically 1. If n ≥ 3 then S∞ must be a quotient of the standard (n − 1)-sphere by the free action of a finite group of isometries. We have a sequence Sk ⊂ Mk of approximating smooth hypersurfaces whose principal curvatures (with respect to gk (0)) go to 1 as k → ∞. In view of the convergence to (M∞ , p∞ , g∞ ), for sufficiently large k, the inward principal curvatures of Sk with respect to gk (0) are close to 1. As M has nonnegative curvature, Sk is diffeomorphic to a sphere [28, Theorem A]. Thus S∞ is isometric to the standard (n − 1)-sphere, and so CT (M, g(t0 )) is isometric to n-dimensional Euclidean space. Then (M, g(t0 )) is isometric to Rn , which contradicts the definition of a κ-solution. In the case n = 2 we know that S∞ is diffeomorphic to a circle but we do not know a priori that it has length 2π. To handle the case n = 2, we use the fact that gk (t) is a Ricci flow solution, to extract a limiting smooth incomplete time-independent Ricci flow solution (M∞ \p∞ , g∞ (t)) for t ∈ [−1, 0]. Note that this solution is unpointed. In view of the convergence to the limiting solution, for sufficiently large k, the inward principal curvatures of Sk with respect to gk (t) are close to 1 for all t ∈ [−1, 0]. This implies that Sk bounds a domain Bk ⊂ M whose diameter with respect to gk (t) is uniformly bounded above, say by 10 (see Appendix G). Applying the Harnack inequality (F.14) with yk ∈ Sk (at time 0) and x ∈ Bk (at time −1), we see that supx∈Bk | Rmk (x, −1)| → 0 as k → ∞. Thus (Bk , p, gk (−1)) Gromov-Hausdorff converges to a flat manifold (B∞ , p¯∞ , g∞ (−1)) with convex boundary. As all of the principal curvatures of ∂B∞ are 1, B∞ must be isometric to a Euclidean unit ball. This implies that S∞ is isometric to the standard S 1 of length 2π, and we obtain a contradiction as before. Definition 41.12. If M is a complete n-dimensional Riemannian manifold with nonnegative Ricci curvature then its asymptotic volume ratio is V = limr→∞ r −n vol(B(p, r)). It is independent of the choice of basepoint x0 . Proposition 41.13. (cf. Proposition I.11.4) Let (M, g(·)) be a noncompact κ-solution. Then the asymptotic volume ratio V vanishes for each time slice (M, g(t0)). Moreover, there is a sequence of points xk ∈ M going to infinity such that the pointed sequence {(M, (xk , t0 ), g(·))}∞ k=1 converges, modulo rescaling by R(xk , t0 ), to a κ-solution which isometrically splits off an R-factor. Proof. Consider the time-t0 slice. Suppose that V > 0. As R = ∞, there are sequences sk xk ∈ M and sk > 0 such that dt0 (xk , p) → ∞, dt (x → 0, R(xk , t0 )s2k → ∞, and k ,p) 0 R(x, t0 ) ≤ 2R(xk , t0 ) for all x ∈ Bt0 (xk , sk ) [33, Lemma 22.2]. Consider the rescaled pointed
74
BRUCE KLEINER AND JOHN LOTT
solution (M, xk , gk (t)) with gk (t) = R(xk , t0 ) g(t0 + R(xtk ,t0 ) ) and t ∈ (−∞, 0]. As Rt ≥ 0, we have Rk (x, t) ≤ 2 whenever t ≤ 0 and dk (x, xk ) ≤ R(xk , t0 )1/2 sk , where dk is the distance function for gk (0) and Rk (·, ·) is the scalar curvature of gk (·). The κ-noncollapsing assumption gives a uniform positive lower bound on the injectivity radius of gk (0) at xk , so by Appendix E we may extract a complete pointed limit solution (M∞ , x∞ , g∞ (t)), t ∈ (−∞, 0], of a subsequence of the sequence of pointed Ricci flows. By relative volume comparison, (M∞ , x∞ , g∞ (0)) has positive asymptotic volume ratio. By Appendix G, the Riemannian manifold (M∞ , x∞ , g∞ (0)) is isometric to an Alexandrov space which splits off a line, which means that it is a Riemannian product R × N. This implies a product structure for earlier times; see Appendix A. Now when n = 2, we have a contradiction, since R(x∞ , 0) = 1 but (M∞ , g∞ (0)) is a product surface, and must therefore be flat. When n > 2 we obtain a κ-solution on an (n − 1)-manifold with positive asymptotic volume ratio at time zero, and by induction this is impossible. 42. In a κ-solution, the curvature and the normalized volume control each other In this section we show that, roughly speaking, in a κ-solution the curvature and the normalized volume control each other. Corollary 42.1. 1. If B(x0 , r0 ) is a ball in a time slice of a κ-solution, then the normalized volume r0−n vol(B(x0 , r0 )) is controlled (i.e. bounded away from zero) ⇐⇒ the normalized scalar curvature r02 R(x0 ) is controlled (i.e. bounded above). 2. If B(x0 , r0 ) is a ball in a time slice of a κ-solution, then the normalized volume vol(B(x0 , r0 )) is almost maximal ⇐⇒ the normalized scalar curvature r02 R(x0 ) is almost zero. r0−n
3. (Precompactness) If (Mk , (xk , tk ), gk (·)) is a sequence of pointed κ-solutions (without the assumption that R(xk , tk ) = 1) and for some r > 0, the r-balls B(xk , r) ⊂ (Mk , gk (tk )) have controlled normalized volume, then a subsequence converges to an ancient solution (M∞ , (x∞ , 0), g∞(·)) which has nonnegative curvature operator, and is κ-noncollapsed (though a priori the curvature may be unbounded on a given time slice). 4. There is a constant η = η(n, κ) such that for every n-dimensional κ-solution (M, g(·)), 3 and all x ∈ M, we have |∇R|(x, t) ≤ ηR 2 (x, t) and |Rt |(x, t) ≤ ηR2 (x, t). More generally, there are scale invariant bounds on all derivatives of the curvature tensor, that only depend on n and κ. That is, for each ρ, k, l < ∞ there is a constant C = C(n, ρ, k, l, κ) < ∞ such l 1 ∂k l that ∂tk ∇ Rm (y, t) ≤ C R(x, t)(k+ 2 +1) for any y ∈ Bt (x, ρR(x, t)− 2 ).
5. There is a function α : [0, ∞) → [0, ∞) depending only on κ such that lims→∞ α(s) = ∞, and for every κ-solution (M, g(·)) and x, y ∈ M, we have R(y)d2(x, y) ≥ α(R(x)d2 (x, y)).
Proof. Assertion 1, =⇒. Suppose we have a sequence of κ-solutions (Mk , gk (·)), and sequences tk ∈ (−∞, 0], xk ∈ Mk , rk > 0, such that at time tk , the normalized volume of B(xk , rk ) is ≥ c > 0, and R(xk , tk )rk2 → ∞. By Appendix H, for each k, we can find yk ∈ B(xk , 5rk ), r¯k ≤ rk , such that R(yk , tk )¯ rk2 ≥ R(xk , tk )rk2 , and R(z, tk ) ≤ 2R(yk , tk ) for
NOTES ON PERELMAN’S PAPERS
75
all z ∈ B(yk , r¯k ). Note that by relative volume comparison, whenever rek ≤ rk we have (42.2)
vol(B(yk , r¯k )) vol(B(yk , 10rk )) vol(B(xk , rk )) c vol(B(yk , rek )) ≥ ≥ ≥ ≥ . rekn r¯kn (10rk )n (10rk )n 10n
Rescaling the sequence of pointed solutions (Mk , (yk , tk ), gk (·)) by R(yk , tk ), we get a sequence satisfying the hypotheses of Appendix E (we use here the fact that Rt ≥ 0 for an ancient solution), so it accumulates on a limit flow (M∞ , (y∞ , 0), g∞(·)) which is a κ-solution. By (42.2), the asymptotic volume ratio of (M, g∞ (0)) is ≥ 10cn > 0. This contradicts Proposition 41.13. Assertion 3. By relative volume comparison, it follows that every r-ball in (Mk , gk (tk )) has normalized volume bounded below by a (k-independent) function of its distance to xk . By 1, this implies that the curvature of (Mk , gk (tk )) is bounded by a k-independent function of the distance to xk , and hence we can apply Appendix E to extract a smoothly converging subsequence. Assertion 1, ⇐=. Suppose we have a sequence (Mk , gk (·)) of κ-solutions, and sequences xk ∈ Mk , rk > 0, such that R(xk , tk )rk2 < c for all k, but rk−n vol(B(xk , rk )) → 0. For large k, we can choose r¯k ∈ (0, rk ) such that r¯k−n vol(B(xk , r¯k )) = 21 cn where cn is the volume of the unit Euclidean n-ball. By relative volume comparison, rr¯kk → 0. Applying 3, we see that the pointed sequence (Mk , (xk , tk ), gk (·)), rescaled by the factor r¯k−2 , accumulates on a pointed ancient solution (M∞ , (x∞ , 0), g∞(·)), such that the ball B(x∞ , 1) ⊂ (M∞ , g∞ ) has normalized volume 12 cn at t = 0. Suppose the ball B(x∞ , 1) ⊂ (M∞ , g∞ (0)) were flat. Then by the Harnack inequality (F.14) (applied to the approximators) we would have R∞ (x, t) = 0 for all x ∈ M∞ , t ≤ 0, i.e. (M∞ , g∞ (t)) would be a time-independent flat manifold. It cannot be Rn since vol(B(x∞ , 1)) = 12 cn . But flat manifolds other than Euclidean space have zero asymptotic volume ratio (as follows from the Bieberbach theorem that if N = Rn /Γ is a flat manifold and Γ is nontrivial then there is a Γ-invariant affine subspace A ⊂ Rn of dimension at least 1 on which Γ acts cocompactly). This contradicts the assumption that the sequence (Mk , gk (·)) is κ-noncollapsed. Thus B(x∞ , 1) ⊂ (M∞ , g∞ (0)) is not flat, which means, by the Harnack inequality, that the scalar curvature of g∞ (0) is strictly positive everywhere. Therefore, with respect to gk , we have 2 2 rk rk 2 2 (42.3) lim inf R(xk , tk )rk = lim inf (R(xk , tk ))¯ rk ) ≥ const. lim inf = ∞, k→∞ k→∞ k→∞ r¯k r¯k which is a contradiction. Assertion 2, =⇒. Apply 1, the precompactness criterion, and the fact that a nonnegativelycurved manifold whose balls have normalized volume cn must be flat. Assertion 2, ⇐=. Apply 1, the precompactness criterion, and the Harnack inequality (F.14) (to the approximators). Assertion 4. This follows by rescaling g so that R(x, t) = 1, and applying 1 and 3.
76
BRUCE KLEINER AND JOHN LOTT
Assertion 5. The quantity R(z)d2 (u, v) is scale invariant. If the assertion failed then we would have sequences (Mk , gk (·)), xk , yk ∈ Mk , such that R(yk ) = 1 and d(xk , yk ) remains bounded, but the curvature at xk blows up. This contradicts 1 and 3. 43. An alternate proof of Corollary 40.1 using Proposition 41.13 and Corollary 42.1 In this section we give an alternate proof of Corollary 40.1. It uses Proposition 41.13 and Corollary 42.1 To clarify the chain of logical dependence, we remark that this section is concerned with 2-dimensional κ-solutions, and does not use anything from Sections 39 or 40. It does use Proposition 41.13. However, we avoid circularity here because the proof of Proposition 41.13 given in Section 41, unlike the proof in [51], does not use Corollary 40.1. Lemma 43.1. There is a constant v = v(κ) > 0 such that if (M, g(·)) is a 2-dimensional κ-solution (a priori either compact or noncompact), x, y ∈ M and r = d(x, y) then (43.2)
vol(Bt (x, r)) ≥ vr 2 .
Proof. If the lemma were not true then there would be a sequence (Mk , gk (·)) of 2-dimensional κ-solutions, and sequences xk , yk ∈ Mk , tk ∈ R such that rk−2 vol(Btk (xk , rk )) → 0, where rk = d(xk , yk ). Let zk be the midpoint of a shortest segment from xk to yk in the tk -time slice (Mk , gk (tk )). For large k, choose r¯k ∈ (0, rk /2) such that π (43.3) r¯k−2 vol(Btk (zk , r¯k )) = , 2 2 i.e. half the area of the unit disk in R . As π = r¯k−2 vol(Btk (zk , r¯k )) ≤ r¯k−2 vol(Btk (xk , rk )) = (¯ rk /rk )−2 rk−2 vol(Btk (xk , rk )), (43.4) 2 it follows that limk→∞ rr¯kk = 0. Then by part 3 of Corollary 42.1, the sequence of pointed Ricci flows (Mk , (zk , tk ), gk (·)), when rescaled by r¯k−2 , accumulates on a complete Ricci flow (M∞ , (z∞ , 0), g∞ (·)). The segments from zk to xk and yk accumulate on a line in (M∞ , g∞ (0)), and hence (M∞ , g∞ (0)) splits off a line. By (43.3), (M∞ , g∞ (0)) cannot be isometric to R2 , and hence must be a cylinder. Considering the approximating Ricci flows, we get a contradiction to the κ-noncollapsing assumption. Lemma 43.1 implies that the asymptotic volume ratio of any noncompact 2-dimensional κ-solution is at least v > 0. By Proposition 41.13 we therefore conclude that every 2dimensional κ-solution is compact. (This was implicitly assumed in the proof of Corollary I.11.3 in [51], as its reference [30] is about compact surfaces.) Consider the family F of 2-dimensional κ-solutions (M, (x, 0), g(·)) with diam(M, g(0)) = 1. By Lemma 43.1, there is uniform lower bound on the volume of the t = 0 time slices of κ-solutions in F . Thus F is compact in the smooth topology by part 3 of Corollary 42.1 (the precompactness leads to compactness in view of the diameter bound). This implies (recall that R > 0) that there is a constant K ≥ 1 such that every time slice of every 2-dimensional κ-solution has K-pinched curvature.
NOTES ON PERELMAN’S PAPERS
77
Hamilton has shown that volume-normalized Ricci flow on compact surfaces with positively pinched initial data converges exponentially fast to a constant curvature metric [30]. His argument shows that there is a small ǫ > 0, depending continuously on the initial data, so that when the volume of the (unnormalized) solution has gone down by a factor of at least ǫ−1 , the pinching is at most the square root of the initial pinching. By the compactness of the family F , this ǫ can be chosen uniformly when we take the initial data to be the t = 0 time slice of a κ-solution in F .
Now let K0 be the worst pinching of a 2-dimensional κ-solution, and let (M, g(·)) be a κ-solution where the curvature pinching of (M, g(0)) is K0 . Choosing t < 0 such that ǫ vol(M, g(t)) = vol(M, g(0)), the previous paragraph implies the curvature pinching of (M, g(t)) is at least K02 . This would contradict the fact that K0 is the upper bound on the pinching for all κ-solutions, unless K0 = 1.
44. I.11.5. A volume bound In this section we give a consequence of Proposition 41.13 concerning the volumes of metric balls in Ricci flow solutions with nonnegative curvature operator. Corollary 44.1. (cf. Corollary I.11.5) For every ǫ > 0, there is an A < ∞ with the following property. Suppose that we have a sequence of (not necessarily complete) Ricci flow solutions gk (·) with nonnegative curvature operator, defined on Mk × [tk , 0], such that 1. For each k, the time-zero ball B(xk , rk ) has compact closure in Mk . 2. For all (x, t) ∈ B(xk , rk ) × [tk , 0], 21 R(x, t) ≤ R(xk , 0) = Qk . 3. limk→∞ tk Qk = −∞. 4. limk→∞ rk2 Qk = ∞. −
1
−
1
Then for large k, vol(B(xk , AQk 2 )) ≤ ǫ(AQk 2 )n at time zero.
Proof. Given ǫ > 0, suppose that the corollary is not true. Then there is a sequence of such −1 −1 Ricci flow solutions with vol(B(xk , Ak Qk 2 )) > ǫ(Ak Qk 2 )n at time zero, where Ak → ∞. −
1
−
n
By Bishop-Gromov, vol(B(xk , Qk 2 )) > ǫQk 2 at time zero, so we can parabolically rescale by Qk and take a convergent subsequence. The limit (M∞ , g∞ (·)) will be a nonflat complete ancient solution with nonnegative curvature operator, bounded curvature and V(0) > 0. By Proposition 41.13, it cannot be κ-noncollapsed for any κ. Thus for each κ > 0, there are a point (xκ , tκ ) ∈ M∞ × (−∞, 0] and a radius rκ so that | Rm(xκ , tκ )| ≤ rκ−2 on the time-tκ ball B(xκ , rκ ), but vol(B(xκ , rκ )) < κ rκn . From the Bishop-Gromov inequality, V(tκ ) < κ for the limit solution. R ) = U R dV ≥ 0 We claim that V(t) is nonincreasing in t. To see this, we have dvol(U dt for any domain U ⊂ M∞ . Also, as R ≤ 2 on M∞ × (−∞, 0], Corollary 27.16 gives that distances on M∞ decrease at most linearly in t, which implies the claim. Thus V(0) = 0 for the limit solution, which is a contradiction.
78
BRUCE KLEINER AND JOHN LOTT
45. I.11.6. Curvature bounds for Ricci flow solutions with nonnegative curvature operator, assuming a lower volume bound In this section we show that for a Ricci flow solution with nonnegative curvature operator, a lower bound on the volume of a ball implies an earlier upper curvature bound on a slightly smaller ball. This will be used in Section 54. Corollary 45.1. (cf. Corollary I.11.6) For every w > 0, there are B = B(w) < ∞, C = C(w) < ∞ and τ0 = τ0 (w) > 0 with the following properties. (a) Take t0 ∈ [−r02 , 0). Suppose that we have a (not necessarily complete) Ricci flow solution (M, g(·)), defined for t ∈ [t0 , 0], so that at time zero the metric ball B(x0 , r0 ) has compact closure. Suppose that for each t ∈ [t0 , 0], g(t) has nonnegative curvature operator and vol(Bt (x0 , r0 )) ≥ wr0n . Then (45.2)
R(x, t) ≤ Cr0−2 + B(t − t0 )−1
whenever distt (x, x0 ) ≤ 41 r0 . (b) Suppose that we have a (not necessarily complete) Ricci flow solution (M, g(·)), defined for t ∈ [−τ0 r02 , 0], so that at time zero the metric ball B(x0 , r0 ) has compact closure. Suppose that for each t ∈ [−τ0 r02 , 0], g(t) has nonnegative curvature operator. If we assume a timezero volume bound vol(B0 (x0 , r0 )) ≥ wr0n then (45.3)
R(x, t) ≤ Cr0−2 + B(t + τ0 r02 )−1
whenever t ∈ [−τ0 r02 , 0] and distt (x, x0 ) ≤ 14 r0 . Remark 45.4. The statement in [51, Corollary 11.6(a)] does not have any constraint on t0 . In our proof we seem to need that −t0 ≤ cr02 for some arbitrary but fixed constant c < ∞. (The statement R(x, t) > C + B(t − t0 )−1 in [51, Proof of Corollary 11.6(a)] is the issue.) For simplicity we take −t0 ≤ r02 . This point does not affect the proof of Corollary 45.1(b), which is what ends up getting used. Proof. For part (a), we can assume that r0 = 1. Given B, C > 0, suppose that g(·) is a Ricci flow solution for t ∈ [t0 , 0] that satisfies the hypotheses of the corollary, with R(x, t) > C + B(t − t0 )−1 for some (x, t) satisfying distt (x, x0 ) ≤ 14 . Following the notation b put A b = λC 21 of the proof of Theorem 30.1, except changing the A of Theorem 30.1 to A, 1 and α = min(λ2 C 2 , B), where we will take λ to be a sufficiently small number that only depends on n. Put (45.5) Clearly (x, t) ∈ Mα .
Mα = {(x′ , t′ ) : R(x′ , t′ ) ≥ α(t′ − t0 )−1 }.
We first go through the analog of the proof of Lemma 31.1. We claim that there is some (x, t) ∈ Mα , with t ∈ (t0 , 0] and distt (x, x0 ) ≤ 13 , such that R(x′ , t′ ) ≤ 2Q = 2R(x, t) b − 12 . Put (x1 , t1 ) = whenever (x′ , t′ ) ∈ Mα , t′ ∈ (t0 , t] and distt′ (x′ , x0 ) ≤ distt (x, x0 ) + AQ (x, t). Inductively, if we cannot take (xk , tk ) for (x, t) then there is some (xk+1 , tk+1) ∈ Mα with tk+1 ∈ (t0 , tk ], R(xk+1 , tk+1 ) > 2R(xk , tk ) and disttk+1 (xk+1 , x0 ) ≤ disttk (xk , x0 ) +
NOTES ON PERELMAN’S PAPERS
79
−1 b AR(x k , tk ) 2 . As the process must terminate, we end up with (x, t) satisfying 1 1 b R(x, t)− 12 ≤ 1 p + A (45.6) distt (x, x0 ) ≤ 4 3 1 − 1/2
if λ is sufficiently small.
Next, we go through the analog of the proof of Lemma 32.1. As in the proof of Lemma 32.1, b − 12 . R(x′ , t′ ) ≤ 2R(x, t) whenever t − 21 αQ−1 ≤ t′ ≤ t and distt′ (x′ , x0 ) ≤ distt (x, x0 ) + AQ 1 b −1/2 We claim that the time-t ball B(x0 , distt (x, x0 ) + 10 AQ ) is contained in the time-t′ ball b − 12 ). To see this, we apply Lemma 27.8 with r0 = 1 Q−1/2 to give B(x0 , distt (x, x0 ) + AQ 2 (45.7)
b −1/2 . distt (x0 , x) − distt (x0 , x) ≤ const.(n) αQ−1/2 ≤ λ const.(n)AQ
If λ is sufficiently small then the claim follows. The argument also shows that it is consistent to use the curvature bound when applying Lemma 27.8. Hence R(x′ , t′ ) ≤ 2R(x, t) whenever t − 21 αQ−1 ≤ t′ ≤ t and distt (x′ , x0 ) ≤ distt (x, x0 ) + 1 b −1/2 AQ . It follows that R(x′ , t′ ) ≤ 2R(x, t) whenever t− 12 αQ−1 ≤ t′ ≤ t and distt (x′ , x) ≤ 10 1 b −1/2 AQ . This shows that there is an A′ = A′ (B, C), which goes to infinity as B, C → ∞, 10
so that R(x′ , t′ ) ≤ 2R(x, t) whenever t − A′ Q−1 ≤ t′ ≤ t and distt (x′ , x) ≤ A′ Q−1/2 .
Now suppose that Corollary 45.1(a) is not true. Fixing w > 0, for any sequences {Bk }∞ k=1 and {Ck }∞ going to infinity and for each k, there is a Ricci flow solution g (·) which satisfies k k=1 the hypotheses of the corollary but for which R(xk , tk ) ≥ Ck + Bk (tk − t0,k )−1 for some point 1
(xk , tk ) satisfying disttk (xk , x0,k ) ≤ 41 . We can assume that λ2 Ck2 ≥ Bk . From the preceding discussion, there is a sequence A′k → ∞ and points (xk , tk ) with disttk (xk , x0,k ) ≤ 13 so that −1/2 ≤ t′k ≤ tk and disttk (x′k , xk ) ≤ A′k Qk , R(x′k , t′k ) ≤ 2R(xk , tk ) whenever tk − A′k Q−1 k where (45.8)
Qk = R(xk , tk ) ≥ Bk (tk − t0,k )−1 ≥ Bk .
By Corollary 44.1, for any ǫ > 0 there is some A = A(ǫ) < ∞ so that for large k, p p (45.9) vol(B(xk , A/ Qk )) ≤ ǫ(A/ Qk )n
at time zero. By the Bishop-Gromov inequality, vol(B(xk , 1)) ≤ ǫ for large k, since Qk → ∞. If we took ǫ sufficiently small from the beginning then we would get a contradiction to the fact that n n 2 2 2 ≥ vol(B(x0,k , 1)) ≥ w. (45.10) vol(B(xk , 1)) ≥ vol B x0,k , 3 3 3
For part (b), the idea is to choose the parameter τ0 sufficiently small so that we will still have the estimate vol(B(x0 , r0 )) ≥ 5−n w r0n for the time-t ball B(x0 , r0 ) when t ∈ [−τ0 r02 , 0], and so we can apply part (a) with w replaced by w5 . The value of τ0 will emerge from the proof. More precisely, putting r0 = 1 and with a given τ0 , let τ be the largest number in [0, τ0 ] so that the time-t ball B(x0 , 1) satisfies vol(B(x0 , 1)) ≥ 5−n w whenever t ∈ [−τ, 0]. If τ < τ0 then at time −τ , we have vol(B(x0 , 1)) = 5−n w. The conclusion of part (a) holds in the sense that (45.11)
R(x, t) ≤ C(5−n w) + B(5−n w)(t + τ )−1
80
BRUCE KLEINER AND JOHN LOTT
whenever t ∈ [−τ, 0] and distt (x, x0 ) ≤ 41 . Lemma 27.8, along with (45.11), implies that √ √ the time-(−τ ) ball B(x0 , 41 ) contains the time-0 ball B(x0 , 14 − 10(n − 1)(τ C + 2 Bτ )). From the nonnegative curvature, the time-(−τ ) volume of the first ball is at least as large as the time-0 volume of the second ball. Then 1 (45.12) 5−n w = vol(B(x0 , 1)) ≥ vol(B(x0 , )) 4 √ √ 1 ≥ vol(B(x0 , − 10(n − 1)(τ C + 2 Bτ ))) 4 √ √ 1 ≥ ( − 10(n − 1)(τ C + 2 Bτ ))n vol(B(x0 , 1)) 4 √ √ 1 ≥ ( − 10(n − 1)(τ C + 2 Bτ ))n w, 4 where the balls on the top at time-(−τ ), and the other balls are at time-0. √ line of√(45.12) are 1 1 Thus 4 − 10(n − 1)(τ C + 2 √Bτ ) ≤ 5 . This contradicts our assumption that τ < τ0 √ provided that 41 − 10(n − 1)(τ0 C + 2 Bτ0 ) = 15 . Finally, we give a version of Corollary 45.1(b) where instead of assuming a nonnegative curvature operator, we assume that the curvature operator in the time-dependent ball of radius r0 around x0 is bounded below by − r0−2 . Corollary 45.13. (cf. end of Section I.11.6) For every w > 0, there are B = B(w) < ∞, C = C(w) < ∞ and τ0 = τ0 (w) > 0 with the following property. Suppose that we have a (not necessarily complete) Ricci flow solution (M, g(·)), defined for t ∈ [−τ0 r02 , 0], so that at time zero the metric ball B(x0 , r0 ) has compact closure. Suppose that for each t ∈ [−τ0 r02 , 0], the curvature operator in the time-t ball B(x0 , r0 ) is bounded below by − r0−2 . If we assume a time-zero volume bound vol(B0 (x0 , r0 )) ≥ wr0n then (45.14)
R(x, t) ≤ Cr0−2 + B(t + τ0 r02 )−1
whenever t ∈ [−τ0 r02 , 0] and distt (x, x0 ) ≤ 41 r0 .
Proof. The blowup argument goes through as before. The only real difference is that the −2 be at least e− const. τ r0 times the volume of the volume of the time-(−τ ) ball B(x0√, 41 ) will √ time-0 ball B(x0 , 14 − 10(n − 1)(τ C + 2 Bτ )). 46. I.11.7. Compactness of the space of three-dimensional κ-solutions In this section we prove a compactness result for the space of three-dimensional κsolutions. The three-dimensionality assumption is used to show that the limit solution has bounded curvature. If a three-dimensional κ-solution M is compact then it is diffeomorphic to a quotient of S 3 or R × S 2 , as it has nonnegative curvature and is nonflat. If its asymptotic soliton (see Section 39) is also closed then M is a quotient of the round S 3 or R × S 2 . There are κ-solutions on S 3 and RP 3 with noncompact asymptotic soliton; see [52, Section 1.4]. They are not isometric to the round metric; this corrects the statement in the first paragraph of [51, Section 11.7].
NOTES ON PERELMAN’S PAPERS
81
Theorem 46.1. (cf. Theorem I.11.7) Given κ > 0, the set of oriented three-dimensional κ-solutions is compact modulo scaling. That is, from any sequence of such solutions and points (xk , 0), after appropriate dilations we can extract a smoothly converging subsequence that satisfies the same conditions. Proof. If (Mk , (xk , 0), gk (·)) is a sequence of such κ-solutions with R(xk , 0) = 1 then parts 1 and 3 of Corollary 42.1 imply there is a subsequence that converges to an ancient solution (M∞ , (x∞ , 0), g∞(·)) which has nonnegative curvature operator and is κ-noncollapsed. The remaining issue is to show that it has bounded curvature. Note that Rt ≥ 0 since g∞ (·) is a limit of a sequence of Ricci flows satisfying Rt ≥ 0. Hence it is enough to show that (M∞ , g(0)) has bounded scalar curvature. If not, there is a sequence of points yi going to infinity in M∞ such that R(yi , 0) → ∞ 1 and R(y, 0) ≤ 2R(yi , 0) for y ∈ B(yi , Ai R(yi, 0)− 2 ), where Ai → ∞; compare [33, Lemma 22.2]. Using the κ-noncollapsing, a subsequence of the rescalings (M∞ , yi, R(yi, 0)g∞ ) will converge to a limit manifold N∞ . As in the proof of Proposition 41.13 from Appendix G, N∞ will split off a line. By Corollary 40.1 or Section 43, N∞ must be the standard solution on R × S 2 . Thus (M, g(0)) contains a sequence Di of neck regions, with their cross-sectional radii tending to zero as i → ∞.
Note that M has to be 1-ended. Otherwise, it would contain a line, and would therefore have to split off a line isometrically [18, Theorem 8.17]. But then M, the product of a line and a surface, could not have neck regions with cross-sections tending to zero.
From theStheory of nonnegatively curved manifolds [18, Chapter 8.5], there is an exhaustion M = t≥0 Ct by nonempty totally convex compact sets Ct so that (t1 ≤ t2 ) ⇒ (Ct1 ⊂ Ct2 ), and (46.2)
Ct1 = {q ∈ Ct2 : dist(q, ∂Ct2 ) ≥ t2 − t1 }.
Now consider a neck region D which is close to a cylinder. Note by triangle comparison – or simply because the distance function in D is close to that of a product metric – any minimizing geodesic segment γ ⊂ D of length large compared to cross-sectional radius of D must be nearly orthogonal to the cross-section. It follows from this and (46.2) that if t > 0 and ∂Ct contains a point p ∈ D such that d(p, ∂D) is large compared to the cross-section of D, then ∂Ct ∩ D is an approximate 2-sphere cross-section of D. Fix such a neck region D0 and let Ct0 be the corresponding convex set. As M has one end, ∂Ct0 has only one connected component, namely the approximate 2-sphere cross-section. For all t > t0 , there is a distance-nonincreasing retraction r : Ct → Ct0 which maps Ct − Ct0 onto ∂Ct0 [60]. Let D be a neck region with a very small cross-section and let Ct be a convex set so that ∂Ct intersects D in an approximate 2-sphere cross-section. Then ∂Ct consists entirely of this approximate cross-section. The restriction of r to ∂Ct is distance-nonincreasing, but will map the 2-sphere ∂Ct onto the 2-sphere ∂Ct0 . This is a contradiction. Remark 46.3. The statement of [51, Theorem 11.7] is about noncompact κ-solutions but the proof works whether the solutions are compact or noncompact.
82
BRUCE KLEINER AND JOHN LOTT
Remark 46.4. One may wonder where we have used the fact that we have a Ricci flow solution, i.e. whether the curvature is bounded for any κ-noncollapsed Riemannian 3manifold with nonnegative sectional curvature. Following the above argument, we could again split off a line in a rescaling around high-curvature points. However, we would not necessarily know that the ensuing nonnegatively-curved surface is compact. (A priori, it could be a smoothed-out cone, for example.) In the case of a Ricci flow, the compactness comes from Corollary 40.1 or Section 43. Corollary 46.5. Let (M, g(·)) be a 3-dimensional κ-solution. Then any asymptotic soliton constructed as in Section 39 is also a κ-solution. 47. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - weak version The next corollary says that outside of a compact region, any oriented noncompact threedimensional κ-solution looks necklike (after rescaling). In this section we give a simple argument to prove the corollary, except for a diameter bound on the compact region. In the next section we give an argument that also proves the diameter bound. More information on three-dimensional κ-solutions is in Section 59. Definition 47.1. Fix ǫ > 0. Let (M, g(·)) be an oriented three-dimensional κ-solution. We say that a point x0 ∈ M is the center of an ǫ-neck if the solution g(·) in the set {(x, t) : −(ǫQ)−1 < t ≤ 0, dist0 (x, x0 )2 < (ǫQ)−1 }, where Q = R(x0 , 0), is, after scaling with the factor Q, ǫ-close in some fixed smooth topology to the corresponding subset of the evolving round cylinder (having scalar curvature one at time zero). (See Definition 58.1 below for a more precise statement.) We let Mǫ denote the points in M that are not centers of ǫ-necks. Corollary 47.2. (cf. Corollary I.11.8) For any ǫ > 0, there exists C = C(ǫ, κ) > 0 such that if (M, g(·)) is an oriented noncompact three-dimensional κ-solution then 1 1. Mǫ is compact with diam(Mǫ ) ≤ CQ− 2 and 2. C −1 Q ≤ R(x, 0) ≤ CQ whenever x ∈ Mǫ , where Q = R(x0 , 0) for some x0 ∈ ∂Mǫ . Proof. We prove here the claims of Corollary 47.2, except for the diameter bound. In the next section we give another argument which also proves the diameter bound. We claim first that Mǫ is compact. Suppose not. Then there is a sequence of points xk ∈ Mǫ going to infinity. Fix a basepoint x0 ∈ M. Then R(x0 ) dist20 (x0 , xk ) → ∞. By part 5 of Corollary 42.1, R(xk ) dist20 (x0 , xk ) → ∞. Rescaling around (xk , 0) to make its scalar curvature one, we can use Theorem 46.1 to extract a convergent subsequence (M∞ , x∞ ). As in the proof of Proposition 41.13, we can say that (M∞ , x∞ ) splits off a line. Hence for large k, xk is the center of an ǫ-neck, which is a contradiction. Next we claim that for any ǫ, there exists C = C(ǫ, κ) > 0 such that if gij (t) is a κ-solution then for any point x ∈ Mǫ , there is a point x0 ∈ ∂Mǫ such that dist0 (x, x0 ) ≤ CQ−1/2 and C −1 Q ≤ R(x, 0) ≤ CQ, where Q = R(x0 , 0).
NOTES ON PERELMAN’S PAPERS
83
If not then there is a sequence {Mi }∞ i=1 of κ-solutions along with points xi ∈ Mi,ǫ such that for each yi ∈ ∂Mi,ǫ , we have 1. dist20 (xi , yi) R(yi , 0) ≥ i or 2. R(yi , 0) ≥ i R(xi , 0) or 3. R(xi , 0) ≥ i R(yi , 0).
Rescale the metric on Mi so that R(xi , 0) = 1. From Theorem 46.1, a subsequence of the pointed spaces (Mi , xi ) will converge smoothly to a κ-solution (M∞ , x∞ ). Also, x∞ ∈ M∞,ǫ . Taking a subsequence, we can assume that 1. occurs for each i, or 2. occurs for each i, or 3. occurs for each i. If M∞ 6= M∞,ǫ , choose y∞ ∈ ∂M∞,ǫ . Then y∞ is the limit of a subsequence of points yi ∈ ∂Mi,ǫ .
If 1. occurs for each i then dist20 (x∞ , y∞ ) R(y∞ , 0) = ∞, which is impossible. If 2. occurs for each i then R(y∞ , 0) = ∞, which is impossible. If 3. occurs for each i then R(y∞ , 0) = 0. It follows from (F.14) that M∞ is flat, which is impossible, as R(x∞ , 0) = 1. Hence M∞ = M∞,ǫ , i.e. no point in the noncompact ancient solution M∞ is the center of an ǫ-neck. This contradicts the previous conclusion that M∞,ǫ is compact. 48. I.11.8. Necklike behavior at infinity of a three-dimensional κ-solution - strong version The following corollary is an application of the compactness result Theorem 46.1. it is a refinement of [51, Cor. I.11.8]. Corollary 48.1. For all κ > 0, there exists an ǫ0 > 0 such that for all 0 < ǫ < ǫ0 there exists an α = α(ǫ, κ) with the property that for any κ-solution (M, g(·)), and at any time t, precisely one of the following holds (Mǫ denotes the set of points which are not centers of ǫ-necks at time t): A. (M, g(·)) is round cylindrical flow, and so every point at every time is the center of an ǫ-neck for all ǫ > 0. B. M is noncompact, Mǫ 6= ∅, and for all x, y ∈ Mǫ , we have R(x)d2 (x, y) < α.
C. M is compact, and there is a pair of points x, y ∈ Mǫ such that R(x)d2 (x, y) > α, (48.2)
1
1
Mǫ ⊂ B(x, αR(x)− 2 ) ∪ B(y, αR(y)− 2 ),
and there is a minimizing geodesic xy such that every z ∈ M − Mǫ satisfies R(z)d2 (z, xy) < α. D. M is compact and there exists a point x ∈ Mǫ such that R(x)d2 (x, z) < α for all z ∈ M. Lemma 48.3. For all ǫ > 0, κ > 0, there exists α = α(ǫ, κ) with the following property. Suppose (M, g(·)) is any κ-solution, x, y, z ∈ M, and at time t we have x, y ∈ Mǫ and R(x)d2 (x, y) > α. Then at time t either R(x)d2 (z, x) < α or R(y)d2 (z, y) < α or (R(z)d2 (z, xy) < α and z ∈ / Mǫ ). Proof. Pick ǫ > 0, κ > 0, and suppose no such α exists. Then there is a sequence αk → ∞, a sequence of κ-solutions (Mk , gk (·)), and sequences xk , yk , zk ∈ Mk , tk ∈ R violating the
84
BRUCE KLEINER AND JOHN LOTT
αk -version of the statement for all k. In particular, xk , yk ∈ (Mk )ǫ and (48.4) R(xk , tk )d2tk (xk , yk ) → ∞, R(xk , tk )d2tk (zk , xk ) → ∞, and R(yk , tk )d2tk (zk , yk ) → ∞. Let zk′ ∈ xk yk be a point in xk yk nearest zk in (Mk , gk (tk )).
We first show that R(xk , tk )d2tk (zk′ , xk ) → ∞. If not, we may pass to a subsequence on which R(xk , tk )d2tk (zk′ , xk ) remains bounded. Applying Theorem 46.1, we may pass to a subsequence and rescale by R(xk , tk ), to make the sequence (Mk , (xk , tk ), gk (·)) converge to a κ-solution (M∞ , (x∞ , 0), g∞(·)), the segments xk yk ⊂ (Mk , gk (tk )) converge to a ray x∞ ξ ⊂ ′ η. Recall that the comparison (M∞ , g∞ (0)), and the segments zk′ zk converge to a ray z∞ e z ′ (u, v) tends to the Tits angle ∂T (ξ, η) as u ∈ z ′ ξ, v ∈ z ′ η tend to infinity. Since angle ∠ ∞ ∞ ∞ ′ ξ d(zk , zk′ ) = d(zk , xk yk ) we must have ∂T (ξ, η) ≥ π2 . Now consider a sequence uk ∈ z∞ tending to infinity. By Theorem 46.1, part 5 of Corollary 42.1, and the remarks about Alexandrov spaces in Appendix G, if we rescale (M∞ , (uk , 0), g∞(·)) by R(uk , 0), we get round cylindrical flow as a limit. When k is sufficiently large, we may find an almost ′ η, and whose crossproduct region D ⊂ (M∞ , g∞ (·)) containing uk which is disjoint from z∞ ′ ξ transversely at a single point. section Σ × {0} ⊂ Σ × (−1, 1) ≃ D intersects the ray z∞ ′ ξ ∪ z ′ η from each other; hence M This implies that Σ × {0} separates the two ends of z∞ ∞ is ∞ two-ended, and (M∞ , g∞ (·)) is round cylindrical flow. This contradicts the assumption that xk is not the center of an ǫ-neck. Hence R(xk , tk )d2tk (zk′ , xk ) → ∞, and similar reasoning shows that R(yk , tk )d2tk (zk′ , yk ) → ∞.
By part 5 of Corollary 42.1, we therefore have R(zk′ , tk )d2tk (zk′ , xk ) → ∞ and R(zk′ , tk )d2tk (zk′ , yk ) → ∞. Rescaling the sequence (Mk , (zk′ , tk ), gk (·)) by R(zk′ , tk ), we get convergence to round cylindrical flow (since any limit flow contains a line), and zk′ zk subconverges to a segment orthogonal to the R-factor, which implies that R(zk′ , tk )d2tk (zk , zk′ ) is bounded and zk is the center of an ǫ-neck for large k. This contradicts our assumption that the αk -version of the lemma is violated for each k.
Proof of Corollary 48.1. Let (M, g(·)) be a κ-solution, and ǫ > 0. Case 1: Every x ∈ (M, g(t)) is the center of an ǫ-neck. In this case, if ǫ > 0 is sufficiently small, M fibers over a 1-manifold with fiber S 2 . If the 1-manifold is homeomorphic to R, then M has two ends, which implies that the flow (M, g(·)) is an evolving round cylinder. ˜ , g˜(t)) would split off a If the base of the fibration were a circle, then the universal cover (M line, which would imply that the universal covering flow would be a round cylindrical flow; but this would violate the κ-noncollapsed assumption at very negative times. Thus A holds in this case. Case 2: There exist x, y ∈Mǫ such that R(x)d2 (x, y) > α. By Lemma 48.3 and Corollary − 21 − 12 42.1 part 5, for all z ∈ M − B(x, αR(x) ) ∪ B(y, αR(y) ) , we have R(z)d2 (z, xy) < α and z ∈ / Mǫ . This implies (again by Corollary 42.1 part 5) that there exists a γ = γ(ǫ, κ) such that for every z ∈ M there is a z ′ ∈ xy for which R(z ′ )d2 (z ′ , z) < γ, which means that M must be compact, and C holds.
NOTES ON PERELMAN’S PAPERS
85
Case 3: Mǫ 6= ∅, and for all x, y ∈ Mǫ , we have R(x)d2 (x, y) < α. If M is noncompact then we are in case B and are done, so assume that M is compact. Pick x ∈ Mǫ , and suppose z ∈ M maximizes R(x)d2 (·, x). If R(x)d2 (z, x) ≥ α, then z is the center of an ǫ-neck, and we may look at the cross-section Σ of the neck region. If Σ separates M, then when ǫ > 0 is sufficiently small, we get a contradiction to the assumption that z maximizes R(x)d2 (x, ·). Hence Σ cannot separate M, and there is a loop passing through x which ˜ , g˜(·)) intersects Σ transversely at one point. It follows that the universal covering flow (M is cylindrical flow, a contradiction. Hence R(x)d2 (x, z) < α for all z ∈ M, so D holds. 49. More properties of κ-solutions In this section we prove some additional properties of κ-solutions. In particular, Corollary 49.2 implies that if z lies in a geodesic segment η in a κ-solution M and if the endpoints of 1 η are sufficiently far from z (relative to R(z)− 2 ) then z ∈ / Mǫ . The results of this section will be used in the proof of Theorem 52.7. Proposition 49.1. For all κ > 0, α > 0, θ > 0, there exists a β(κ, α, θ) < ∞ such that if (M, g(t)) is a time slice of a κ-solution, x, y1 , y2 ∈ M, R(x)d2 (x, yi ) > β for i = 1, 2, and e x (y1 , y2) ≥ θ, then (a) x is the center of an α-neck, and (b) ∠ e x (y1 , y2 ) ≥ π − α. ∠
Proof. The proof of this is similar to the first part of the proof of Lemma 48.3. Note that when α is small, then after enlarging β if necessary, the neck region around x will separate y1 from y2 ; this implies (b). Corollary 49.2. For all κ > 0, ǫ > 0, there exists a ρ = ρ(κ, ǫ) such that if (M, g(t)) is a time slice of a κ-solution, η ⊂ (M, g(t)) is a minimizing geodesic segment with endpoints y1 , y2 , z ∈ M, z ′ ∈ η is a point in η nearest z, and R(z ′ )d2 (z ′ , yi) > ρ for i = 1, 2, then z, z ′ are centers of ǫ-necks, and max(R(z)d2 (z, z ′ ), R(z ′ )d2 (z, z ′ )) < 4π 2 . Proof. Pick ǫ′ > 0. Under the assumptions, if (49.3)
min(R(z ′ )d2 (z ′ , y1 ), R(z ′ )d2 (z ′ , y2 ))
is sufficiently large, we can apply the preceding proposition to the triple z ′ , y1, y2 , to conclude that z ′ is the center of an ǫ′ -neck. Since the shortest segment from z to z ′ is orthogonal to η, when ǫ′ is small enough the segment zz ′ will lie close to an S 2 cross-section in the approximating round cylinder, which gives R(z ′ )d2 (z, z ′ ) . 2π 2 . 50. I.11.9. Getting a uniform value of κ Proposition 50.1. There is a κ0 > 0 so that if (M, g(·)) is an oriented three-dimensional κ-solution, for some κ > 0, then it is a κ0 -solution or it is a quotient of the round shrinking S 3. Proof. Let (M, g(·)) be a κ-solution. Suppose that for some κ′ > 0, the solution is κ′ collapsed at some scale. After rescaling, we can assume that there is a point (x0 , 0) so that | Rm(x, t)| ≤ 1 for all (x, t) satisfying dist0 (x, x0 ) < 1 and t ∈ [−1, 0], with vol(B0 (x0 , 1)) < κ′ . Let Ve (t) denote the reduced volume as a function of t ∈ (−∞, 0], as defined using curves
86
BRUCE KLEINER AND JOHN LOTT
from (x0 , 0). It is nondecreasing in t. As in the proof of Theorem 26.2, there is an estimate Ve (−κ′ ) ≤ 3(κ′ )3/2 . Take a sequence of times ti → −∞. For each ti , choose qi ∈ M so that l(qi , ti ) ≤ 23 . From the proof of Proposition 39.1, for all ǫ > 0 there is a δ > 0 such that l(q, t) does not exceed δ −1 whenever t ∈ [ti , ti /2] and dist2ti (q, qi ) ≤ ǫ−1 ti . Given the monotonicity of Ve and the obtain an upper bound on the volume p upper bound on l(q, t), we 3/2 δ−1 of the time-ti ball B(qi , ti /ǫ) of the form const. ti e (κ′ )3/2 . On the other hand, from Proposition 39.1, a subsequence of the rescalings of the ancient solution around (qi , ti ) converges to a nonflat gradient shrinking soliton. If the gradient shrinking soliton is compact then it must be a quotient of the round shrinking S 3 [35]. Otherwise, Corollary 51.22 says that if the gradient shrinking soliton is noncompact then it must bep an evolving cylinder or its Z2 -quotient. Fixing ǫ, this gives a lower bound on vol(Bti (qi , ti /ǫ)) in terms of the noncollapsing constants of the evolving cylinder and its Z2 -quotient. Hence there is a universal constant κ0 so that if κ′ < κ0 then we obtain a contradiction to the assumption of κ′ -collapsing.
Remark 50.2. The hypotheses of Corollary 51.22 assume a global upper bound on the sectional curvature of any time slice, which in the n-dimensional case is not a priori true for the asymptotic soliton of Proposition 39.1. However, in our 3-dimensional case, the argument of Theorem 46.1 shows that there is such an upper bound. 51. II.1.2. Three-dimensional noncompact κ-noncollapsed gradient shrinkers are standard In this section we show that any complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking soliton with bounded nonnegative curvature is either the evolving round cylinder R × S 2 or its Z2 -quotient.
The basic example of √ a gradient shrinking soliton is the metric on R × S 2 which gives the 2-sphere a radius of −2t at time t ∈ (−∞, 0). With coordinates (s, θ) on R × S 2 , the 2 function f is given by f (t, s, θ) = − s4t .
Lemma 51.1. (cf. Lemma of II.1.2) There is no complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking soliton with bounded positive sectional curvature. Proof. The idea of the proof is to show that the soliton has the qualitative features of a shrinking cylinder, and then to get a contradiction to the assumption of positive sectional curvature. Applying ∇i to the gradient shrinker equation ∇i ∇j f + Rij +
(51.2)
1 gij = 0 2t
gives (51.3) As ∇i Rij =
(51.4)
1 2
△∇j f + ∇i Rij = 0.
∇j R and △∇j f = ∇j △f + Rjk ∇k f = ∇j −R − ∇i R = 2 Rij ∇j f.
n 2t
+ Rjk ∇k f , we obtain
NOTES ON PERELMAN’S PAPERS
87
Fix a basepoint x0 ∈ M and consider a normalized minimal geodesic γ : [0, s] → M in the time −1 slice with γ(0) = x0 . Put X(s) = dγ . As in the proof of Lemma 27.8, ds Rs Ric(X, X) ds ≤ const. for some constant independent of s. If {Yi }3i=1 are orthonormal 0 parallel vector fields along γ then 2 Z s Z s 3 Z s X 2 | Ric(X, Y1 )| ds ≤ s | Ric(X, Y1 )| ds ≤ s | Ric(X, Yi )|2 ds. (51.5) 0
0
P3
i=1
0
Thinking of Ric as a self-adjoint linear operator on T M, i=1 | Ric(X, Yi)|2 = hX, Ric2 Xi. In terms of a pointwise orthonormal frame {ei } of eigenvectors of Ric, with eigenvalues λi , P3 write X = i=1 Xi ei . Then (51.6)
Hence (51.7)
hX, Ric2 Xi = Z
3 X i=1
λ2i Xi2 ≤ (
s 0
| Ric(X, Y1 )| ds
2
3 X i=1
3 X λi ) ( λi Xi2 ) = R · Ric(X, X).
≤ (sup R) s M
i=1
Z
s
Ric(X, X) ds ≤ const. s.
0 2
(γ(s)) Multiplying (51.2) by X i X j and summing gives d fds + Ric(X, X) − 12 = 0. Then 2 Z s 1 1 df (γ(s)) df (γ(s)) + s − Ric(X, X) ds ≥ s − const. (51.8) = ds ds 2 2 s=s s=0 0 This implies that there is a compact subset of M outside of which f has no critical points.
If Y is a unit vector field perpendicular to X then multiplying (51.2) by X i Y j and d (Y · f )(γ(s)) + Ric(X, Y ) = 0. Then summing gives ds Z s (51.9) (Y · f )(γ(s)) = (Y · f )(γ(0)) − Ric(X, Y ) ds 0
and
(51.10)
√ |(Y · f )(γ(s))| ≤ const.( s + 1).
For large s, |(Y · f )(γ(s))| is small compared to (X · f )(γ(s)). This means that as one approaches infinity, the gradient of f becomes more and more parallel to the gradient of the distance function from x0 , where by the latter we mean the vectors X that are tangent to minimal geodesics. The gradient flow of f is given by the equation dx = (∇f )(x). (51.11) du Then along a flowline, equation (51.4) implies that dR(x) dx (51.12) = ∇R, = 2 Ric(∇f, ∇f ). du du
In particular, outside of a compact set, R is strictly increasing along the flowlines. Put R = lim supx→∞ R. Take points xα tending toward infinity, with R(xα ) → R. Putting
88
BRUCE KLEINER AND JOHN LOTT
p α rα = → 0 and R(xα ) rα2 → ∞. Then the argument dist−1 (x0 , xα ), we have dist−1r(x 0 ,xα ) of the proof of Proposition 41.13 shows that any convergent subsequence of the rescalings around (xα , −1) splits off a line. Hence the limit is a shrinking round cylinder with scalar curvature R at time −1. Because our original solution exists up to time zero, we must have R ≤ 1. Now equation (C.13) says that the Ricci flow is given by g(−t) = −tφ∗t g(−1), where φt is the flow generated by ∇f . It follows that inf x∈M R(x, t) = Ct−1 for some C > 0. That is, the curvature blows up uniformly as t → 0. Comparing this with the singularity time of the shrinking round cylinder implies that R = 1. Performing a similar argument with any sequence of xα ’s tending toward infinity, with the property that R(xα ) has a limit, shows that limx→∞ R(x) = 1. Let N denote a (connected component of a) level surface of f . At a point of N, choose an orthonormal frame {e1 , e2 , e3 } with e3 = X normal to N. From the Gauss-Codazzi equation, (51.13)
RN = 2 K N (e1 , e2 ) = 2(K M (e1 , e2 ) + det(S)),
where S is the shape operator. As R = 2(K M (e1 , e2 ) + K M (e1 , e3 ) + K M (e2 , e3 )) and Ric(X, X) = K M (e1 , e3 ) + K M (e2 , e3 ), we obtain (51.14)
RN = R − 2 Ric(X, X) + 2 det(S).
|T N The shape operator is given by S = Hessf . From (51.2), Hess f = 12 − Ric. We can |∇f | r1 0 c1 diagonalize Ric to write Ric = 0 r2 c2 , where r3 = Ric(X, X). Then TN c1 c2 r3 1 1 1 = (51.15) det Hess f − r1 − r2 = (1 − r1 − r2 )2 − (r1 − r2 )2 2 2 4 TN 1 1 ≤ (1 − r1 − r2 )2 = (1 − R + Ric(X, X))2 . 4 4 This shows that the scalar curvature of N is bounded above by
(51.16)
R − 2 Ric(X, X) +
(1 − R + Ric(X, X))2 . 2|∇f |2
If |∇f | is large then 1 − R + Ric(X, X) < 2 |∇f |2 . As 1 − R + Ric(X, X) is positive when the distance from x to x0 is large enough, (51.17) and so (51.18) Hence (51.19)
(1 − R + Ric(X, X))2 < 2(1 − R + Ric(X, X))|∇f |2
≤ 2(1 − R + Ric(X, X))|∇f |2 + 2|∇f |2 Ric(X, X)
(1 − R + Ric(X, X))2 < 1 − R + 2 Ric(X, X). 2|∇f |2 R − 2 Ric(X, X) +
(1 − R + Ric(X, X))2 < 1. 2|∇f |2
NOTES ON PERELMAN’S PAPERS
89
This shows that RN < 1 if N is sufficiently far from x0 . If Y is a unit vector that is tangential to N then from (51.2), 1 (51.20) ∇Y ∇Y f = − Ric(Y, Y ). 2 If {Y, Z, W } is an orthonormal basis then
1 R. 2 Hence ∇Y ∇Y f ≥ 12 (1 − R), which is positive if N is sufficiently far from x0 . Thus N is convex and so the area of the level set increases as the level increases. On the other hand, we can take points xα on the level sets going to infinity, apply the previous splitting argument and use the fact that grad f becomes almost parallel to grad d(·, x0 ). Within one of the approximate cylinders coming from the splitting argument, there is a projection π to its base S 2 . As a tangent plane of N is almost perpendicular to Ker(dπ), the restriction of π to N is an almost-isometry from N to S 2 . By the monotonicity of area(N), we conclude that area(N) ≤ 8π if N is sufficiently far from R x0 . However as N is a topologically a 2-sphere, the Gauss-Bonnet theorem says that N RN dA = 8π. This contradicts the facts that RN < 1 and area(N) ≤ 8π. (51.21) Ric(Y, Y ) = K M (Y, Z)+K M (Y, W ) ≤ K M (Y, Z)+K M (Y, W )+K M (Z, W ) =
Corollary 51.22. The only complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking solitons with bounded nonnegative sectional curvature are the round evolving R × S 2 and its Z2 -quotient R ×Z2 S 2 . Proof. Let (M, g(·)) be a complete oriented 3-dimensional noncompact κ-noncollapsed gradient shrinking soliton with bounded nonnegative sectional curvature. By Lemma 51.1, M cannot have positive sectional curvature. From Theorem A.7, M must locally split off an R-factor. Then the universal cover splits off an R-factor and so, by Corollary 40.1 or Section 43, must be the standard R × S 2 . From the κ-noncollapsing, M must be R × S 2 or R ×Z2 S 2 . 52. I.12.1. Canonical neighborhood theorem In this section we show that a high-curvature region of a three-dimensional Ricci flow is modeled by part of a κ-solution. We first define the notion of Φ-almost nonnegative curvature. Definition 52.1. (cf. I.12) Let Φ ∈ C ∞ (R) be a positive nondecreasing function such that for positive s, Φ(s) is a decreasing function which tends to zero as s → ∞. A Ricci flow s solution is said to have Φ-almost nonnegative curvature if for all (x, t), we have (52.2)
Rm(x, t) ≥ − Φ(R(x, t)).
Remark 52.3. Note that Φ-almost nonnegative curvature implies that the scalar curvature is uniformly bounded below by − 6 Φ(0). The formulation of the pinching condition in [51, Section 12] is that there is a decreasing function φ, tending to zero at infinity, so that Rm(x, t) ≥ − φ(R(x, t)) R(x, t) for each (x, t). This formulation has a problem when R(x, t) < 0, if one takes φ to be defined on all of R. The condition in Definition 52.1 is what
90
BRUCE KLEINER AND JOHN LOTT
comes out of the three-dimensional Hamilton-Ivey pinching result (applied to the rescaled metric g˜(t) = g(t) ) if we assume normalized initial conditions; see Appendix B. t We note that since the sectional curvatures have to add up to R, the lower bound (52.2) implies a double-sided bound on the sectional curvatures. Namely, R n(n − 1) (52.4) −Φ(R) ≤ Rm ≤ + − 1 Φ(R). 2 2 The main use of the pinching condition is to show that blowup limits have nonnegative sectional curvature. Lemma 52.5. Let {(Mk , pk , gk )}∞ k=1 be a sequence of complete pointed Riemannian manifolds with Φ-almost nonnegative curvature. Given a sequence Qk → ∞, put g k = Qk gk and suppose that there is a pointed smooth limit (M∞ , p∞ , g ∞ ) = limk→∞ (Mk , pk , g k ). Then M∞ has nonnegative sectional curvature. Proof. First, the Φ-almost nonnegative curvature condition implies that the scalar curvature of Mk is bounded below uniformly in k. For m ∈ M∞ , let mk ∈ (Mk , g¯k ) be a sequence of approximants to m. Then limk→∞ Rm(mk ) = Rm∞ (m), where Rm(mk ) = Q−1 k Rm(mk ). There are two possibilities : either the numbers R(mk ) are uniformly bounded above or they are not. If they are uniformly bounded above then (52.4) implies that Rm(mk ) is uniformly bounded above and below, so Rm∞ (m) = 0. Suppose on the other hand that a subsequence of the numbers R(mk ) tends to infinity. We pass to this subsequence. Now R∞ (m) = limk→∞ R(mk ) exists by assumption and is nonnegative. Applying (52.2) gives that Rm∞ (m), the limit of Rm(mk ) (52.6) Q−1 , k Rm(mk ) = R(mk ) R(mk ) is nonnegative. We now prove the first version of the “canonical neighborhood” theorem. Theorem 52.7. (cf. Theorem I.12.1) Given ǫ, κ, σ > 0 and a function Φ as above, one can find r0 > 0 with the following property. Let g(·) be a Ricci flow solution on a threemanifold M, defined for 0 ≤ t ≤ T with T ≥ 1. We suppose that for each t, g(t) is complete, and the sectional curvature is bounded on compact time intervals. Suppose that the Ricci flow has Φ-almost nonnegative curvature and is κ-noncollapsed on scales less than σ. Then for any point (x0 , t0 ) with t0 ≥ 1 and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) : dist2t0 (x, x0 ) < (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution. Remark 52.8. Our statement of Theorem 52.7 differs slightly from that in [51, Theorem 12.1]. First, we allow M to be noncompact, provided that there is bounded sectional curvature on compact time intervals. This generalization will be useful for later work. More importantly, the statement in [51, Theorem 12.1] has noncollapsing at scales less than r0 , whereas we require noncollapsing at scales less than σ. See Remark 52.19 for further comment. In the phrase “t0 ≥ 1” there is an implied scale which comes from the Φ-almost nonnegativity assumption, and similarly for the statement “scales less than σ”.
NOTES ON PERELMAN’S PAPERS
91
Proof. We give a proof which differs in some points from the proof in [51] but which has the same ingredients. We first outline the argument. Suppose that the theorem is false. Then for some ǫ, κ, σ > 0, we have a sequence of such Φ-nonnegatively curved 3-dimensional Ricci flows (Mk , gk (·)) defined on intervals [0, Tk ], and sequences rk → 0, xˆk ∈ Mk , tˆk ≥ 1 such that Mk is κ-noncollapsed on scales < σ and Qk = R(ˆ xk , tˆk ) ≥ rk−2 , but the Qk -rescaled solution in B(ǫQk )−1/2 (ˆ xk ) × [tˆk − (ǫQk )−1 , tˆk ] is not ǫ-close to the corresponding subset of any κ-solution. We note that if the statement were false for ǫ then it would also be false for any smaller ǫ. Because of this, somewhat paradoxically, we will begin the argument with a given ǫ but will allow ourselves to make ǫ small enough later so that the argument works. To be clear, we will eventually get a contradiction using a fixed (small) value of ǫ, but as the proof goes along we will impose some upper bounds on this value in order for the proof to work. (If we tried to list all of the constraints at the beginning of the argument then they would look unmotivated.) The goal is to get a contradiction based on the “bad” points (b xk , b tk ). In a sense, the method of proof of Theorem 52.7 is an induction on the curvature scale. For example, if we were to make the additional assumption in the theorem that R(x, t) ≤ R(x0 , t0 ) for all x ∈ M and t ≤ t0 then the theorem would be very easy to prove. We would just take a convergent subsequence of the rescaled solutions, based at (ˆ xk , tˆk ), to get a κ-solution; this would give a contradiction. This simple argument can be considered to be the first step in a proof by induction on curvature scale. In the proof of Theorem 52.7 one effectively proves the result at a given curvature scale inductively by assuming that the result is true at higher curvature scales. The actual proof consists of four steps. Step 1 consists of replacing the sequence (b xk , b tk ) by another sequence of “bad” points (xk , tk ) which have the property that points near (xk , tk ) with distinctly higher scalar curvature are “good” points. It then suffices to get a contradiction based on the existence of the sequence (xk , tk ). In steps 2-4 one uses the points (xk , tk ) to build up a κ-solution, whose existence then contradicts the “badness” of the points (xk , tk ). More precisely, let (Mk , (xk , tk ), g¯k (·)) be the result of rescaling gk (·) by R(xk , tk ). We will show that the sequence of pointed flows (Mk , (xk , tk ), g¯k (·)) accumulates on a κ-solution (M∞ , (x∞ , t0 ), g ∞ (·)), thereby obtaining a contradiction. In step 2 one takes a pointed limit of the manifolds (Mk , xk , g¯k (tk )) in order to construct what will become the final time slice of the κ-solution, (M∞ , x∞ , g ∞ (t0 )). In order to take this limit, it is necessary to show that the manifolds (Mk , xk , g¯k (tk )) have uniformly bounded curvature on distance balls of a fixed radius. If this were not true then for some radius, a subsequence of the manifolds (Mk , xk , g¯k (tk )) would have curvatures that asymptotically blowup on the ball of that radius. One shows that geometrically, the curvature blowup is due to the asymptotic formation of a cone-like point at the blowup radius. Doing a further rescaling at this cone-like point, one obtains a Ricci flow solution that ends on a part of a nonflat metric cone. This gives a contradiction as in the case 0 < R < ∞ of Theorem 41.2.
Thus one can construct the pointed limit (M∞ , x∞ , g ∞ (t0 )). The goal now is to show that (M∞ , x∞ , g ∞ (t0 )) is the final time slice of a κ-solution (M∞ , (x∞ , t0 ), g ∞ (·)). In step 3
92
BRUCE KLEINER AND JOHN LOTT
one shows that (M∞ , x∞ , g ∞ (t0 )) extends backward to a Ricci flow solution on some time interval [t0 − ∆, t0 ], and that the time slices have bounded nonnegative curvature. In step 4 one shows that the Ricci flow solution can be extended all the way to time (−∞, t0 ], thereby constructing a κ-solution Step 1: Adjusting the choice of basepoints. We first modify the points (b xk , b tk ) slightly in time to points (xk , tk ) so that in the given Ricci flow solution, there are no other “bad” points with much larger scalar curvature in a earlier time interval whose length is large compared to R(xk , tk )−1 . (The phrase “nearly the smallest curvature Q” in [51, Proof of Theorem 12.1] should read “nearly the largest curvature Q”. This is clear from the sentence in parentheses that follows.) The proof of the next lemma is by pointpicking, as in Appendix H. Lemma 52.9. We can find Hk → ∞, xk ∈ Mk , and tk ≥ 12 such that Qk = R(xk , tk ) → ∞, and for all k the conclusion of Theorem 52.7 fails at (xk , tk ), but holds for any (y, t) ∈ Mk × [tk − Hk Q−1 k , tk ] for which R(y, t) ≥ 2R(xk , tk ). 1 Proof. Choose Hk → ∞ such that Hk (R(b xk , b tk ))−1 ≤ 10 for all k. For each k, initially set ˆ (xk , tk ) = (ˆ xk , tk ). Put Qk = R(xk , tk ) and look for a point in Mk × [tk − Hk Q−1 k , tk ] at which Theorem 52.7 fails, and the scalar curvature is at least 2R(xk , tk ). If such a point exists, replace (xk , tk ) by this point; otherwise do nothing. Repeat this until the second alternative occurs. This process must terminate with a new choice of (xk , tk ) satisfying the lemma.
Hereafter we use this modified sequence (xk , tk ). Let (Mk , (xk , tk ), g¯k (·)) be the result of ¯ k to denote its scalar curvature; in particular, rescaling gk (·) by Qk = R(xk , tk ). We use R ¯ Rk (xk , tk ) = 1. Note that the rescaled time interval of Lemma 52.9 has duration Hk → ∞; this is what we want in order to try to extract an ancient solution. ¯ k is uniformly bounded on the ρ-balls Step 2: For every ρ < ∞, the scalar curvature R B(xk , ρ) ⊂ (Mk , g¯k (tk )) (the argument for this is essentially equivalent to [51, Pf. of Claim 2 of Theorem I.12.1]). Before proceeding, we need some bounds which come from our choice of basepoints, and the derivative bounds inherited (by approximation) from κ-solutions. Lemma 52.10. There is a constant C = C(κ) so that for any (x, t) in a Ricci flow solution, if R(x, t) > 0 and the solution in Bt (x, (ǫR(x, t))−1/2 ) × [t − (ǫR(x, t))−1 , t] is ǫ-close to a corresponding subset of a κ-solution then |∇R−1/2 |(x, t) ≤ C and |∂t R−1 |(x, t) ≤ C. Proof. This follows from the compactness in Theorem 46.1.
Note that the same value of C in Lemma 52.10 also works for smaller ǫ. Lemma 52.11. (cf. Claim 1 of I.12.1) For each (x, t) with tk − −1
1 2
Hk Q−1 ≤ t ≤ tk , k −1/2
we have Rk (x, t) ≤ 4Qk whenever t − c Qk ≤ t ≤ t and distt (x, x) ≤ c Qk Qk = Qk + |Rk (x, t)| and c = c(κ) > 0 is a small constant.
, where
NOTES ON PERELMAN’S PAPERS
93
Proof. If Rk (x, t) ≤ 2Qk then there is nothing to show. If Rk (x, t) > 2Qk , consider a spacetime curve γ that goes linearly from (x, t) to (x, t), and then goes from (x, t) to (x, t) along a minimizing geodesic. If there is a point on γ with curvature 2Qk , let p be the nearest such point to (x, t). If not, put p = (x, t). From the conclusion of Lemma 52.9, we can apply Lemma 52.10 along γ from (x, t) to p. The claim follows from integrating the ensuing derivative bounds along γ. Lemma 52.12. In terms of the rescaled solution g k (·), for each (x, t) with tk − 21 Hk ≤ t ≤ tk , ek whenever t − c Q e−1 ≤ t ≤ t and distt (x, x) ≤ c Q e−1/2 , where we have Rk (x, t) ≤ 4Q k k ek = 1 + |Rk (x, t)|. Q Proof. This is just the rescaled version of Lemma 52.11.
For all ρ ≥ 0, put
(52.13)
¯ k (x, tk ) | k ≥ 1, x ∈ B(xk , ρ) ⊂ (Mk , g¯k (tk ))}, D(ρ) = sup{R
and let ρ0 be the supremum of the ρ’s for which D(ρ) < ∞. Note that ρ0 > 0, in view of Lemma 52.12 (taking (¯ x, t¯) = (xk , tk )). Suppose that ρ0 < ∞. After passing to a subsequence if necessary, we can find a sequence yk ∈ Mk with disttk (xk , yk ) → ρ0 and ¯ k , tk ) → ∞. Let ηk ⊂ (Mk , g¯k (tk )) be a minimizing geodesic segment from xk to yk . Let R(y ¯ k , tk ) = 2, and let γk be the subsegment zk ∈ ηk be the point on ηk closest to yk at which R(z of ηk running from yk to zk . By Lemma 52.12 the length of γk is bounded away from zero independent of k. Due to the Φ-pinching (see (52.4)), for all ρ < ρ0 , we have a uniform bound on | Rm | on the balls B(xk , ρ) ⊂ (Mk , g¯k (tk )). The injectivity radius is also controlled in B(xk , ρ), in view of the curvature bounds and the κ-noncollapsing. Therefore after passing to a subsequence, we can assume that the pointed sequence (B(xk , ρ0 ), g¯k (tk ), xk ) converges in the pointed Gromov-Hausdorff topology (i.e. for all ρ < ρ0 we have the usual GromovHausdorff convergence) to a pointed C 1 -Riemannian manifold (Z, g¯∞ , x∞ ), the segments ηk converge to a segment (missing an endpoint) η∞ ⊂ Z emanating from x∞ , and γk converges to γ∞ ⊂ η∞ . Let Z¯ denote the completion of (Z, g¯∞), and y∞ ∈ Z¯ the limit point of η∞ . Note that by Lemma 52.9 and part 4 of Corollary 42.1, the Riemannian structure near γ∞ may be chosen to be many times differentiable. (Alternatively, this follows from ¯∞ Lemma 52.12 and the Shi estimates of Appendix D.) In particular the scalar curvature R is defined, differentiable, and satisfies the bound in Lemma 52.10 near γ∞ . Lemma 52.14. 1. There is a function c : (0, ∞) → R depending only on κ, with limt→0 c(t) = ¯ ∞ (w) d(y∞ , w)2 > c(ǫ). ∞, such that if w ∈ γ∞ then R
2. There is a function ǫ′ : (0, ∞) → R ∪ {∞} depending only on κ, with limt→0 ǫ′ (t) = 0, ¯ ∞ (w)¯ such that if w ∈ γ∞ and d(y∞ , w) is sufficiently small then the pointed manifold (Z, w, R g∞ ) ′ 2 is 2ǫ (ǫ)-close to a round cylinder in the C topology. Proof. It follows from Lemma 52.9 that for all w ∈ γ∞ , the pointed Riemannian manifold ¯ ∞(w)¯ (Z, w, R g∞ ) is 2ǫ-close to (a time slice of) a pointed κ-solution. From the definition of pointed closeness, there is an embedded region around w, large on the scale defined by ¯ ∞ (w), which is close to the corresponding subset of a pointed κ-solution. This gives a lower R bound on the distance ρ0 − d(w, x∞ ) to the point of curvature blowup, thereby proving part 1 of the lemma.
94
BRUCE KLEINER AND JOHN LOTT
¯ ∞(w)¯ We know that (Z, w, R g∞ ) is 2ǫ-close to a pointed κ-solution (N, ⋆, h(t)) in the 2 ¯ ∞ (w) tends to ∞ as d(w, x∞) → ρ0 , for pointed C -topology. By Lemma 52.10, we know R 2 ¯ w ∈ γ∞ . In particular, R∞ (w)d (w, x∞ ) → ∞. From part 1 above, we can choose ǫ small ¯ ∞ (w)d2(w, y∞ ) large enough to apply Proposition 49.1. Hence the enough in order to make R pointed manifold (N, ⋆, R(⋆)h(t)) is ǫ′′ (ǫ)-close to a round cylinder, where ǫ′′ : (0, ∞) → R is a function with limt→0 ǫ′′ (t) = 0. The lemma follows. Note that the function ǫ′ in Lemma 52.14 is independent of the particular manifold Z that arises in our proof. From part 2 of Lemma 52.14, if ǫ is small and w ∈ γ∞ is sufficiently close to y∞ then (Z, w, R∞ (w)g ∞ ) is C 2 -close to a round cylinder. The cross-section of the cylinder has diam1 1 eter approximately π(R∞ (w)/2)− 2 . If we form the union of the balls B(w, 2π(R∞ (w)/2)− 2 ), as w ranges over such points in γ∞ , then we obtain a connected Riemannian manifold W . ¯ which is locally complete, and geodesic By adding in the point y∞ , we get a metric space W near y∞ . As the original manifolds Mk had Φ-almost nonnegative curvature, it follows from Lemma 52.5 that W is nonnegatively curved. Furthermore, y∞ cannot be an interior point ¯ , since such a geodesic would have to pass through a cylindrical of any geodesic segment in W region near y∞ twice. The usual proof of the Toponogov triangle comparison inequality now ¯. applies near y∞ since minimizers remain in the smooth nonnegatively curved part of W Then W has nonnegative curvature in the Alexandrov sense. ¯ , y∞ ) converge to the tangent cone Cy∞ W ¯ . As W is This implies that blowups of (W 2 ¯ three-dimensional, so is Cy∞ W [12, Corollary 7.11]. It will be C -smooth away from the ¯ such that d(z, y∞ ) = 1. vertex and nowhere flat, by part 2 of Lemma 52.14. Pick z ∈ Cy∞ W 1 ¯ Then the ball B(z, 2 ) ⊂ Cy∞ W is the Gromov-Hausdorff limit of a sequence of rescaled balls B(b zk , rbk ) ⊂ (Mk , g¯k (tk )) where rbk → 0, whose center points (b zk , tk ) satisfy the conclusions of Theorem 52.7. Applying Lemma 52.12 and Appendix B, we get the curvature bounds needed to extract a limiting Ricci flow solution whose time zero slice is isometric to B(z, 12 ). Now we can apply the reasoning from the 0 < R < ∞ case of Theorem 41.2 to get a contradiction. This completes step 2. Step 3: The sequence of pointed flows (Mk , g¯k (·), (xk , tk )) accumulates on a pointed Ricci flow (M∞ , g¯∞ (·), (x∞ , t0 )) which is defined on a time interval [t′ , t0 ] with t′ < t0 . By step 2, we know that the scalar curvature of (Mk , g¯k (tk )) at y ∈ Mk is bounded by a function of the distance from y to xk . Lemma 52.12 extends this curvature control to a backward parabolic neighborhood centered at y whose radius depends only on d(y, xk ). Thus we can conclude, using Φ-pinching (52.4) and Shi’s estimates (Appendix D), that all derivatives of the curvature (Mk , g¯k (tk )) are controlled as a function of the distance from xk , which means that the sequence of pointed manifolds (Mk , g¯k (tk ), xk ) accumulates to a smooth manifold (M∞ , g¯∞ ). From Lemma 52.5, M∞ has nonnegative sectional curvature. We claim that M∞ has bounded curvature. If not then there is a sequence of points qk ∈ M∞ so that limk→∞ R(qk ) = 1 ∞ and R(q) ≤ 2R(qk ) for q ∈ B(yk , Ak R(qk )− 2 ), where Ak → ∞; compare [33, Lemma 22.2]. Lemma 52.9 implies that for large k, a rescaled neighborhood of (M∞ , qk ) is ǫ-close to the corresponding subset of a time slice of a κ-solution. As in the proof of Theorem 46.1, we
NOTES ON PERELMAN’S PAPERS
95
obtain a sequence of neck-like regions in M∞ with smaller and smaller cross-sections, which contradicts the existence of the Sharafutdinov retraction. By Lemma 52.12 again, we now get curvature control on (Mk , g¯k (·)) for a time interval [tk − ∆, tk ] for some ∆ > 0, and hence we can extract a subsequence which converges to a pointed Ricci flow (M∞ , (x∞ , t0 ), g¯∞ (·)) defined for t ∈ [t0 − ∆, t0 ], which has nonnegative curvature and bounded curvature on compact time intervals. Step 4: Getting an ancient solution. Let (t′ , t0 ] be the maximal time interval on which we can extract a limiting solution (M∞ , g¯∞ (·)) with bounded curvature on compact time intervals. Suppose that t′ > −∞. By Lemma 52.12 the maximum of the scalar curvature on the time slice (M∞ , g¯∞ (t)) must tend to infinity as t → t′ . From the trace Harnack R inequality, Rt + t−t ′ ≥ 0, and so ′
¯ ∞ (x, t) ≤ Q t0 − t , R t − t′ where Q is the maximum of the scalar curvature on (M∞ , g¯∞ (t0 )). Combining this with Corollary 27.16, we get r d t0 − t′ (52.16) dt (x, y) ≥ const. Q . dt t − t′ Since the right hand side is integrable on (t′ , t0 ], and using the fact that distances are nonincreasing in time (since Rm ≥ 0), it follows that there is a constant C such that (52.15)
(52.17)
for all x, y ∈ M∞ , t ∈ (t′ , t0 ].
|dt (x, y) − dt0 (x, y)| < C
If M∞ is compact then by (52.17) the diameter of (M∞ , g¯∞ (t)) is bounded independent of t ∈ (t′ , t0 ]. Since the minimum of the scalar curvature is increasing in time, it is also bounded independent of t. Now the argument in Step 2 shows that the curvature is bounded everywhere independent of t. (We can apply the argument of Step 2 to the time-t slice because the main ingredient was Lemma 52.9, which holds for rescaled time t.) We may therefore assume M∞ is noncompact. To be consistent with the notation of I.12.1, we now relabel the basepoint (x∞ , t0 ) as (x0 , t0 ). Since nonnegatively curved manifolds are asymptotically conical (see Appendix G), there is a constant D such that if y ∈ M∞ , and dt0 (y, x0 ) > D, then there is a point x ∈ M∞ such that 3 (52.18) dt0 (x, y) = dt0 (y, x0) and dt0 (x, x0 ) ≥ dt0 (y, x0); 2 ′ by (52.17) the same conditions hold at all times t ∈ (t , t0 ], up to error C. If for some such y, ¯ ∞ (y, t)¯ and some t ∈ (t′ , t0 ] the scalar curvature were large, then (M∞ , (y, t), R g∞(t)) would be 2ǫ-close to a κ-solution (N, h(·), (z, t0 )). When ǫ is small we could use Proposition 49.1 1 to see that y lies in a neck region U in (M∞ , g¯∞ (t)) of diameter ≈ R(y, t)− 2 ≪ 1. We claim that U separates x0 from x in the sense that x0 and x belong to disjoint components of M∞ − U, where x0 , y, and x satisfy (52.18). To see this, we recall that if M∞ has more than one end then it splits isometrically, in which case the claim is clear. If M∞ has one end then we consider an exhaustion of M∞ by totally convex compact sets Ct as
96
BRUCE KLEINER AND JOHN LOTT
in Section 46. One of the sets Ct will have boundary consisting of an approximate 2-sphere cross-section in the neck region U, giving the separation. e y (x0 , x) > π (For another argument, suppose that U does not separate x0 from x. Since ∠ 2 (from Proposition 49.1), the segments yx0 and yx must exit U by different ends. If x0 and x can be joined by a curve avoiding U then there is a nonzero element of H1 (M∞ , Z). The corresponding infinite cyclic cover of M∞ will then isometrically split off a line and its quotient M∞ will be compact, which is a contradiction. We thank one of the referees for this argument.) Obviously, at time t0 the set U still separates x0 from x. Since g¯∞ has nonnegative curvature, we have diamt0 (U) ≤ diamt (U) ≪ 1. Since (M∞ , g¯∞ (t0 )) has bounded geometry, there cannot be topologically separating subsets of arbitrarily small diameter. Thus there must be a uniform upper bound on R(y, t) and the curvature of (M∞ , g¯∞ ) is uniformly bounded (in space and time) outside a set of uniformly bounded diameter. Repeating the reasoning from Step 2, we get uniform bounds everywhere. This contradicts our assumption that the curvature blows up as t → t′ .
It remains to show that the ancient solution is a κ-solution. The only remaining point is to show that it is κ-noncollapsed at all scales. This follows from the fact that the original Ricci flow solutions (Mk , gk (·)) were κ-noncollapsed on scales less than the fixed number σ.
Remark 52.19. As mentioned in Remark 52.8, the statement of [51, Theorem 12.1] instead assumes noncollapsing at all scales less than r0 . Bing Wang pointed out that with this assumption, after constructing the ancient solution in Step 4 of the proof, one only gets that it is κ-collapsed at all scales less than one. Hence it may not be a κ-solution. The literal statement of [51, Theorem 12.1] is not used in the rest of [51, 52], but rather its method of proof. Because of this, the change of hypotheses does not seem to lead to any problems. The method of proof of Theorem 52.7 is used in two different ways. The first way is to construct the Ricci flow with surgery on a fixed finite time interval, as in Section 77. In this case the noncollapsing at a given scale σ comes from Theorem 26.2, and its extension when surgeries are allowed. The second way is to analyze the large-time behavior of the Ricci flow, as in the next few sections. 53. I.12.2. Later scalar curvature bounds on bigger balls from curvature and volume bounds The next theorem roughly says that if one has a sectional curvature bound on a ball, for a certain time interval, and a lower bound on the volume of the ball at the initial time, then one obtains an upper scalar curvature bound on a larger ball at the final time. We first write out the corrected version of the theorem (see II.6.2). Theorem 53.1. For any A < ∞, there exist K = K(A) < ∞ and ρ = ρ(A) > 0 with the following property. Suppose in dimension three we have a Ricci flow solution with Φalmost nonnegative curvature. Given x0 ∈ M and r0 > 0, suppose that r02 Φ(r0−2 ) < ρ, the solution is defined for 0 ≤ t ≤ r02 and it has | Rm |(x, t) ≤ 3r12 for all (x, t) satisfying 0
NOTES ON PERELMAN’S PAPERS
97
dist0 (x, x0 ) < r0 . Suppose in addition that the volume of the metric ball B(x0 , r0 ) at time zero is at least A−1 r03 . Then R(x, r02 ) ≤ Kr0−2 whenever distr02 (x, x0 ) < Ar0 . Remark 53.2. The added restriction that r02 Φ(r0−2 ) < ρ (see II.6.2) imposes an upper bound on r0 . This is necessary, as otherwise the conclusion would imply that neck pinches cannot occur. There is an apparent gap in the proof of [51, Theorem 12.2], in the sentence (There is a little subtlety...). We instead follow the proof of II.6.3(b,c) (see Proposition 84.1(b,c)), which proves the same statement in the presence of surgeries. The volume assumption in the theorem is used to guarantee noncollapsing, by means of Theorem 28.2. The reason for the “3” in the hypothesis | Rm |(x, t) ≤ 3r12 comes from 0 Remark 28.3. Proof. The proof is in two steps. In the first step one shows that if R(x, r02 ) is large then a parabolic neighborhood of (x, r02 ) is close to the corresponding subset of a κ-solution. In the second part one uses this to prove the theorem. The first step is the following lemma. Lemma 53.3. For any ǫ > 0 there exists K = K(A, ǫ) < ∞ so that for any r0 , whenever we have a solution as in the statement of the theorem and distr02 (x, x0 ) < Ar0 then (a) R(x, r02 ) < Kr0−2 or (b) The solution in {(x′ , t′ ) : distt (x′ , x) < (ǫQ)−1 , t − (ǫQ)−1 ≤ t′ ≤ t} is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution. Here t = r02 and Q = R(x, t). Remark 53.4. One can think of this lemma as a localized analog of Theorem 52.7, where “localized” refers to the fact that both the hypotheses and the conclusion involve the point x0 . Proof. To prove the lemma, suppose that there is a sequence of such pointed solutions (Mk , x0,k , gk (·)), along with points xˆk ∈ Mk , so that distr02 (ˆ xk , x0,k ) < Ar0 and r02 R(ˆ xk , r02 ) → ∞, but (ˆ xk , r02 ) does not satisfy conclusion (b) of the lemma. As in the proof of Theorem 52.7, we will allow ourselves to make ǫ smaller during the course of the proof. We first show that there is a sequence Dk → ∞ and modified points (xk , tk ) with 43 r02 ≤ tk ≤ r02 , disttk (xk , x0,k ) < (A + 1)r0 and Qk = R(xk , tk ) → ∞, so that any point (x′k , t′k ) −1/2 ′ ′ ′ with R(x′k , t′k ) > 2Qk , tk − Dk2 Q−1 k ≤ tk ≤ tk and disttk (xk , x0,k ) < disttk (xk , x0,k ) + Dk Qk satisfies conclusion (b) of the lemma, but (xk , tk ) does not satisfy conclusion (b) of the lemma. (Of course, in saying “(x′k , t′k ) satisfies conclusion (b)” or “(xk , tk ) does not satisfy conclusion (b)”, we mean that the (x, t) in conclusion (b) is replaced by (x′k , t′k ) or (xk , tk ), respectively.) r R(ˆ x ,r 2 )1/2
k 0 . Start The construction of (xk , tk ) is by a pointpicking argument. Put Dk = 0 10 2 ′ ′ ′ ′ with (xk , tk ) = (ˆ xk , r0 ) and look if there is a point (xk , tk ) with R(xk , tk ) > 2R(xk , tk ), 2 −1 tk − Dk R(xk , tk ) ≤ t′k ≤ tk and distt′k (x′k , x0,k ) < disttk (xk , x0,k ) + Dk R(xk , tk )−1/2 , but
98
BRUCE KLEINER AND JOHN LOTT
which does not have a neighborhood that is ǫ-close to the corresponding subset of a κsolution. If there is such a point, we replace (xk , tk ) by (x′k , t′k ) and repeat the process. The process must terminate after a finite number of steps to give a point (xk , tk ) with the desired property. −1/2
involves the metric (Note that the condition distt′k (x′k , x0,k ) < disttk (xk , x0,k ) + Dk Qk at time t′k . In order to construct an ancient solution, one of the issues will be to replace this by a condition that only involves the metric at time tk , i.e. that involves a parabolic neighborhood around (xk , tk ).) Let g k (·) denote the rescaling of the solution gk (·) by Qk . We normalize the time interval of the rescaled solution by fixing a number t∞ and saying that for all k, the time-tk slice of (Mk , gk ) corresponds to the time-t∞ slice of (Mk , g k ). Then the scalar curvature Rk of g k satisfies Rk (xk , t∞ ) = 1. By the argument of Step 2 of the proof of Theorem 52.7, a subsequence of the pointed spaces (Mk , xk , g k (t∞ )) will smoothly converge to a nonnegatively-curved pointed space (M∞ , x∞ , g ∞ ). By the pointpicking, if m ∈ M∞ has R(m) ≥ 3 then a parabolic neighborhood of m is ǫ-close to the corresponding region in a κ-solution. It follows, as in Step 3 of the proof of Theorem 52.7, that the sectional curvature of M∞ will be bounded above by some C < ∞. Using Lemma 52.12, the metric on M∞ is the time-t∞ slice of a nonnegativelycurved Ricci flow solution defined on some time interval [t∞ −c, t∞ ], with c > 0, and one has convergence of a subsequence g k (t) → g ∞ (t) for t ∈ [t∞ −c, t∞ ]. As Rt ≥ 0, the scalar curvature on this time interval will be uniformly bounded above by 6C and so from the Φ-almost nonnegative curvature (see (52.4)), the sectional curvature will be uniformly bounded above on the time interval. Hence we can apply Lemma 27.8 to get a uniform additive bound on the length distortion between times t∞ − c and t∞ (see Step 4 of the proof of Theorem 52.7). More precisely, in applying Lemma 27.8, we use the curvature bound coming from the hypotheses of the theorem near x0 , and the just-derived upper curvature bound near xk . ′ It follows that for a given A′ > 0, for large k, if t′k ∈ [tk − cQ−1 k /2, tk ] and disttk (xk , xk ) < −1/2 −1/2 A′ Qk then distt′k (x′k , x0,k ) < disttk (xk , x0,k )+Dk Qk . In particular, if a point (x′k , t′k ) lies ′ ′ −1/2 in the parabolic neighborhood given by t′k ∈ [tk − cQ−1 , k /2, tk ] and disttk (xk , xk ) < A Qk and has R(x′k , t′k ) > 2Qk , then it has a neighborhood that is ǫ-close to the corresponding subset of a κ-solution.
As in Step 4 of the proof of Theorem 52.7, we now extend (M∞ , g∞ , x∞ ) backward to an ancient solution g ∞ (·), defined for t ∈ (−∞, t∞ ]. To do so, we use the fact that if the solution is defined backward to a time-t slice then the length distortion bound, along with the pointpicking, implies that a point m in a time-t slice with R∞ (m) > 3 has a neighborhood that is ǫ-close to the corresponding subset of a κ-solution. The ancient solution is κ-noncollapsed at all scales since the original solution was κ-noncollapsed at some scale, by Theorem 28.2. Then we obtain smooth convergence of parabolic regions of the points (xk , tk ) to the κ-solution, which is a contradiction to the choice of the (xk , tk )’s. We now know that regions of high scalar curvature are modeled by corresponding regions in κ-solutions. To continue with the proof of the theorem, fix A large. Suppose that the
NOTES ON PERELMAN’S PAPERS
99
theorem is not true. Then there are 1. Numbers ρk → 0, −2 2 2. Numbers r0,k with r0,k Φ(r0,k ) ≤ ρk , 2 3. Solutions (Mk , gk (·)) defined for 0 ≤ t ≤ r0,k , 4. Points x0,k ∈ Mk and 5. Points xk ∈ Mk so that −2 2 a. | Rm |(x, t) ≤ r0,k for all (x, t) ∈ Mk × [0, r0,k ] satisfying dist0 (x, x0,k ) < r0,k , 3 b. The volume of the metric ball B(x0,k , r0,k ) at time zero is at least A−1 r0,k and 2 (xk , x0,k ) < Ar0,k , but c. distr0,k 2 2 d. r0,k R(xk , r0,k ) → ∞.
We now apply Step 2 of the proof of Theorem 52.7 to obtain a contradiction. That is, we −2 2 take a subsequence of {(Mk , x0,k , r0,k gk (r0,k ))}∞ k=1 that converges on a maximal ball. The only difference is that in Theorem 52.7, the nonnegative curvature of W came from the Φ-almost nonnegative curvature assumption on the original manifolds Mk along with the fact (with the notation of the proof of Theorem 52.7) that the numbers Qk = R(xk , tk ), which we used to rescale, go to infinity. In the present case the rescaled scalar curvatures 2 2 r0,k R(x0,k , r0,k ) at the basepoints x0,k stay bounded. However, if a point y ∈ W is a limit of points x ek ∈ Mk then the equations 2 2 Rm(e xk , r0,k ) ≥ − Φ(R(e xk , r0,k ))
(53.5)
in the form (53.6)
2 r0,k
2 Rm(e xk , r0,k )
−2 2 2 Φ r0,k R(e xk , r0,k ) · r0,k 2 2 r0,k R(e xk , r0,k ) ≥ − 2 R(e xk , r0,k )
2 2 pass to the limit to give Rm(y) ≥ 0 (using that y ∈ W , so r0,k Rm(e xk , r0,k ) → R(y) > 0). This is enough to carry out the argument.
54. I.12.3. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds The main result of this section says that if one has a lower bound on volume and sectional curvature on a ball at a certain time then one obtains an upper scalar curvature bound on a smaller ball at an earlier time. We first prove a result in Riemannian geometry saying that under certain hypotheses, metric balls have subballs of a controlled size with almost-Euclidean volume. Lemma 54.1. Given w ′ > 0 and n ∈ Z+ , there is a number c = c(w ′ , n) > 0 with the following property. Let B be a radius-r ball with compact closure in an n-dimensional Riemannian manifold. Suppose that the sectional curvatures of B are bounded below by − r −2 . Suppose that vol(B) ≥ w ′r n . Then there is a subball B ′ ⊂ B of radius r ′ ≥ cr so that vol(B ′ ) ≥ 21 ωn (r ′ )n , where ωn is the volume of the unit ball in Rn . Proof. Suppose that the lemma is not true. Rescale so that r = 1. Then there is a sequence of Riemannian manifolds {Mi }∞ i=1 with balls B(xi , 1) ⊂ Mi having compact closure
100
BRUCE KLEINER AND JOHN LOTT
so that Rm
B(xi ,1)
≥ − 1 and vol(B(xi , 1)) ≥ w ′, but with the property that all balls
B(x′i , r ′) ⊂ B(xi , 1) with r ′ ≥ i−1 satisfy vol(B(x′i , r ′ )) < 12 ωn (r ′ )n . After taking a subsequence, we can assume that limi→∞ (B(xi , 1), xi ) = (X, x∞ ) in the pointed GromovHausdorff topology. From [12, Theorem 10.8], the Riemannian volume forms dvolMi converge weakly to the three-dimensional Hausdorff measure µ of X. From [12, Corollary 6.7 and Section 9], for any ǫ > 0, there are small balls B(x′∞ , r ′ ) ⊂ X with compact closure in X such that µ(B(x′∞ , r ′ )) ≥ (1 − ǫ) ωn (r ′)n . This gives a contradiction. Theorem 54.2. (cf. Theorem I.12.3) For any w > 0 there exist τ = τ (w) > 0, K = K(w) < ∞ and ρ = ρ(w) > 0 with the following property. Suppose that g(·) is a Ricci flow on a closed three-manifold M, defined for t ∈ [0, T ), with Φ-almost nonnegative curvature. Let (x0 , t0 ) be a spacetime point and let r0 > 0 be a radius with t0 ≥ 4τ r02 and r02 Φ(r0−2 ) < ρ. Suppose that volt0 (B(x0 , r0 )) ≥ wr03 and the time-t0 sectional curvatures on B(x0 , r0 ) are bounded below by − r0−2 . Then R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) ≤ 1 r . 4 0 Proof. Let τ0 (w), B(w) and C(w) be the constants of Corollary 45.13. Put τ (w) = 12 τ0 (w) and K(w) = C(w) + 2 τB(w) . The function ρ(w) will be specified in the course of the proof. 0 (w) Suppose that the theorem is not true. Take a counterexample with a point (x0 , t0 ) and a radius r0 > 0 such that the time-t0 ball B(x0 , r0 ) satisfies the assumptions of the theorem, but the conclusion of the theorem fails. We claim there is a counterexample coming from a point (b x0 , b t0 ) and a radius rb0 > 0, with the additional property that for any (x′0 , t′0 ) and r0′ ′ having t0 ∈ [b t0 − 2τ rb02 , b t0 ] and r0′ ≤ 21 rb0 , if volt′0 (B(x′0 , r0′ )) ≥ w(r0′ )3 and the time-t′0 sectional curvatures on B(x′0 , r0′ ) are bounded below by − (r0′ )−2 then R(x, t) ≤ K(r0′ )−2 whenever t ∈ [t′0 − τ (r0′ )2 , t′0 ] and distt (x, x′0 ) ≤ 41 r0′ . This follows from a pointpicking argument suppose that it is not true for the original x0 , t0 , r0 . Then there are (x′0 , t′0 ) and r0′ with t′0 ∈ [t0 − 2τ r02 , t0 ] and r0′ ≤ 12 r0 , for which the assumptions of the theorem hold but the conclusion does not. If the triple (x′0 , t′0 , r0′ ) satisfies the claim then we stop, and otherwise we iterate the procedure. The iteration must terminate, which provides the desired triple (b x0 , b t0 , rb0 ). Note that b t0 > t0 − 4τ r02 ≥ 0. We relabel (b x0 , b t0 , rb0 ) as (x0 , t0 , r0 ). For simplicity, let us assume that the time-t0 sectional curvatures on B(x0 , r0 ) are strictly greater than − r0−2 ; the general case will follow from continuity. Let τ ′ > 0 be the largest number such that Rm(x, t) ≥ − r0−2 whenever t ∈ [t0 − τ ′ r02 , t0 ] and distt (x, x0 ) ≤ r0 . If τ ′ ≥ 2τ = τ0 (w) then Corollary 45.13 implies that R(x, t) ≤ Cr0−2 + B(t − t0 + 2τ r02 )−1 whenever t ∈ [t0 − 2τ r02 , t0 ] and distt (x, x0 ) ≤ 14 r0 . In particular, R(x, t) ≤ K whenever t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) ≤ 41 r0 , which contradicts our assumption that the conclusion of the theorem fails. Now suppose that τ ′ < 2τ . Put t′ = t0 − τ ′ r02 . From estimates on the length and volume distortion under the Ricci flow, we know that there are numbers α = α(w) > 0 and w ′ = w ′ (w) > 0 so that the time-t′ ball B(x0 , αr0) has volume at least w ′ (αr0 )3 . From Lemma 54.1, there is a subball B(x′ , r ′ ) ⊂ B(x0 , αr0 ) with r ′ ≥ cαr0 and vol(B(x′ , r ′ )) ≥ 12 ω3 (r ′ )3 . From the preceding pointpicking argument, we have the estimate R(x, t) ≤ K(r ′ )−2 whenever t ∈ [t′ − τ (r ′ )2 , t′ ] and distt (x, x′ ) ≤ 41 r ′ . From the Φ-almost nonnegative curvature, we have
NOTES ON PERELMAN’S PAPERS
101
a bound | Rm |(x, t) ≤ const. K (r ′ )−2 + const. Φ(K(r ′ )−2 ) at such a point (see (52.4)). If ρ(w) is taken sufficiently small then we can ensure that r0 is small enough, and hence r ′ is small enough, to make K (r ′ )−2 + Φ(K(r ′ )−2 ) ≤ 2 K (r ′)−2 . Then we can apply Theorem 53.1 to a time interval ending at time t′ , after a redefinition of its constants, to obtain a bound of the form R(x, t′ ) ≤ K ′ (r ′ )−2 whenever distt (x, x′ ) ≤ 10r0 , where K ′ is related to the constant K of Theorem 53.1. (We also obtain a similar estimate at times slightly less than t′ .) Thus at such a point, Rm(x, t′ ) ≥ − Φ(K ′ (r ′ )−2 ). If we choose ρ(w) to be sufficiently small to force r ′ to be sufficiently small to force − Φ(K ′ (r ′ )−2 ) > − r0−2 then we have Rm > −r0−2 on Bt′ (x0 , r0 ) ⊂ Bt′ (x′ , 10r0 ), which contradicts the assumed maximality of τ ′ . We note that in the application of Theorem 53.1 at the end of the proof, we must take into account the extra hypothesis, in the notation of Theorem 53.1, that r02 Φ(r0−2 ) < ρ (see Remark 53.2). This will be satisfied if the r0 in Theorem 53.1 is small enough, which is ensured by taking the ρ of Theorem 54.2 small enough. 55. I.12.4. Small balls with strongly negative curvature are volume-collapsed In this section we show that under certain hypotheses, if the infimal sectional curvature on an r-ball is exactly − r −2 then the volume of the ball is small compared to r 3 . Corollary 55.1. (cf. Corollary I.12.4) For any w > 0, one can find ρ > 0 with the following property. Suppose that g(·) is a Φ-almost nonnegatively curved Ricci flow solution on a closed three-manifold M, defined for t ∈ [0, T ) with T ≥ 1. If B(x0 , r0 ) is a metric ball at time t0 ≥ 1 with r0 < ρ and if inf x∈B(x0 ,r0 ) Rm(x, t0 ) = −r0−2 then vol(B(x0 , r0 )) ≤ wr03. Proof. Fix w > 0. The number ρ will be specified in the course of the proof. Suppose that the corollary is not true, i.e. there is a Ricci flow solution as in the statement of the corollary along with a metric ball B(x0 , r0 ) at a time t0 ≥ 1 so that inf x∈B(x0 ,r0 ) Rm(x, t0 ) = −r0−2 and vol(B(x0 , r0 )) > wr0n . The idea is to use Theorem 54.2, along with the Φ-almost nonnegative curvature, to get a double-sided sectional curvature bound on a smaller ball at an earlier time. Then one goes forward in time using Theorem 53.1, along with the Φ-almost nonnegative curvature, to get a lower sectional curvature bound on the original ball, thereby obtaining a contradiction. 1
Looking at the hypotheses of Theorem 54.2, if we require r0 < (4τ )− 2 then 4τ r02 < 1 ≤ t0 . From Theorem 54.2, R(x, t) ≤ Kr0−2 whenever t ∈ [t0 − τ r02 , t0 ] and distt (x, x0 ) ≤ 14 r0 , provided that r0 is small enough that r02 Φ(r0−2 ) is less than the ρ of Theorem 54.2. If in addition r0 is sufficiently small then it follows that | Rm(x, t)| ≤ const. Φ(Kr0−2 ) ≤ r0−2 .
From the Bishop-Gromov inequality and the bounds on length and volume distortion under Ricci flow, there is a small number c so that we are ensured that | Rm(x, t)| ≤ (cr0 )−2 for all (x, t) satisfying distt0 −(cr0 )2 (x, x0 ) < cr0 and t ∈ [t0 − (cr0 )2 , t0 ], and in addition the volume of B(x0 , cr0 ) at time t0 − (cr0 )2 is at least c(cr0 )3 . Choosing the constant A of Theorem 53.1 appropriately in terms of c, we can apply Theorem 53.1 to the ball B(x0 , cr0 ) and the time interval [t0 − (cr0 )2 , t0 ] to conclude that at time t0 , R(·, t0 ) ≤ B(x0 ,r0 )
102
BRUCE KLEINER AND JOHN LOTT
K(A) (cr0 )−2 , where K(A) is as in the statement of Theorem 53.1. From the Φ-almost nonnegative curvature condition, (55.2) Rm ≥ − Φ K(A) (cr0 )−2 . B(x0 ,r0 )
If r0 is sufficiently small then we contradict the assumption that inf Rm(x, t0 )
−
B(x0 ,r0 )
1 . r02
=
56. I.13.1. Thick-thin decomposition for nonsingular flows The main result of this section says that if g(·) is a Ricci flow solution on a closed oriented three-dimensional manifold M that exists for t ∈ [0, ∞) then for large t, (M, g(t)) has a thick-thin decomposition. A fuller description is in Sections 87-92. We assume that at time zero, the sectional curvatures are bounded below by −1. This can always be achieved by rescaling the initial metric. Then we have the Φ-almost nonnegative curvature result of (B.8). If the metric g(t) has nonnegative sectional curvature then it must be flat, as we are assuming that the Ricci flow exists for all time. Let us assume that g(t) is not flat, so it has some negative sectional curvature. Given x ∈ M, consider the time-t ball Bt (x, r). Clearly if r is sufficiently small then Rm > − r −2 , while if r is sufficiently large (maybe Bt (x,r) > − r −2 . Let rb(x, t) > 0 greater than the diameter of M) then it is not true that Rm Bt (x,r) −2 be the unique number such that inf Rm = − rb . Let Mthin (w, t) be the set of points x ∈ M for which (56.1)
Bt (x,b r)
vol(Bt (x, rb(x, t))) < w rb(x, t)3 .
Put Mthick (w, t) = M − Mthin (w, t).
As the statement of (B.8) is invariant under parabolic rescaling (although we must take t ≥ t0 for (B.8) to apply), if t ≥ t0 and we are interested in the Ricci flow at time t then we can apply Theorem 53.1, Theorem 54.2 and Corollary 55.1 to the rescaled flow g(t′ ) = t−1 g(tt′ ).√ From Corollary 55.1, for any w > 0 we can find ρb = ρb(w) > 0 so that if rb(x, t) < ρb t then x ∈ Mthin (w, t), provided that t is sufficiently large (depending on w). Equivalently, if t is sufficiently large (depending on w) and x ∈ Mthick (w, t) then √ rb(x, t) ≥ ρb t.
Theorem 56.2. (cf. I.13.1) There are numbers T = T (w) > 0, ρ = ρ(w) > 0 √ and −1 K = K(w) < ∞ so that if t ≥ T and x ∈ Mthick (w, t) then | Rm | ≤ Kt on Bt (x, ρ t), √ √ 3 1 and vol(Bt (x, ρ t)) ≥ 10 w ρ t .
Proof. The method of proof is the same as in Corollary 55.1. By assumption, Rm ≥ Bt (x,b r(x,t)) √ − rb(x, t)−2 and vol(Bt (x, rb(x, t))) ≥ w rb(x, t)3 . As rb(x, t) ≥ ρb t, for any c ∈ (0, 1) we have
NOTES ON PERELMAN’S PAPERS
Rm
√ Bt (x,cb ρ t)
(56.3)
103
≥ − (cb ρ)−2 t−1 . By the Bishop-Gromov inequality,
√ vol(Bt (x, cb ρ t)) ≥ ≥
R
√ cρ b t r b(x,t)
0
R1 0
sinh2 (u) du 2
sinh (u) du
3 1 w (cb ρ)3 t 2 . 10
w rb(x, t)3 ≥
3
R1 0
1
3
2
sinh (u) du
w (cb ρ )3 t 2
w Considering Theorem 54.2 with its w replaced by 10 , if c = c(w) is taken sufficiently small √ 2 (to ensure t ≥ 4τ (cb ρ t) ) and t is larger than a certain w-dependent constant (to ensure √ √ Φ((cb ρ t)−2 ) √ < ρ) then we can apply Theorem 54.2 with r = cb ρ t to obtain R(x′ , t′ ) ≤ 0 −2 (cb ρ t) √ K ′ (w)c−2 ρb−2 t−1 whenever t′ ∈ [t − τ c2 ρb2 t, t] and distt′ (x′ , x) ≤ 14 cb ρ t. From the Φ-almost nonnegative curvature (see (52.4)),
(56.4)
| Rm |(x′ , t′ ) ≤ const. K ′ c−2 ρb−2 t−1 + const. Φ(K ′ c−2 ρb−2 t−1 ),
which is bounded above by 2 const. K ′ c−2 ρb−2 t−1 if t is larger than a certain w-dependent constant. Then from length and volume distortion √ estimates for the Ricci flow, we obtain a √ lower volume bound vol(Bt′ (x, c′ ρb t)) ≥ w ′ (c′ ρb t)3 on a smaller ball of controlled radius, √ for ′ ′ ′′ some c = c√ (w). Using Theorem 53.1, we finally obtain an upper bound R ≤ K (w)(cb ρ t)−2 nonnegative curvature, an upper bound of the on Bt (x, cb ρ t) and hence, by the Φ-almost √ form | Rm | ≤ K(w) t−1 on Bt (x, cb ρ t), provided that t ≥ T for an appropriate T = T (w). ρ, the theorem follows. Taking ρ = cb Remark 56.5. The use of Theorem 53.1 in the √ proof of Theorem 56.2 also gives an upper bound | Rm |(x, t) ≤ K(A, w) t−1 on Bt (x, Aρ t) if x ∈ Mthick (w, t), for any A > 0. We now take w sufficiently small. Then for large t, Mthick (w, t) has a boundary consisting of tori that are incompressible in M and the interior of Mthick (w, t) admits a complete Riemannian metric with constant sectional curvature − 41 and finite volume; see Sections 90 and 91. In addition, Mthin (w, t) is a graph manifold; see Section 92. 57. Overview of Ricci Flow with Surgery on Three-Manifolds [52] The paper [52] is concerned with the Ricci flow on compact oriented 3-manifolds. The main difference with respect to [51] is that singularity formation is allowed, so the paper deals with a “Ricci flow with surgery”. The main part of the paper is concerned with setting up the surgery procedure and showing that it is well-defined, in the sense that surgery times do not accumulate. In addition, the long-time behavior of a Ricci flow with surgery is analyzed. The paper can be divided into three main parts. Sections II.1-II.3 contain preparatory material about ancient solutions, the so-called standard solution and the geometry at the first singular time. Sections II.4-II.5 set up the surgery procedure and prove that it is well-defined. Sections II.6-II.8 analyze the long-time behavior.
104
BRUCE KLEINER AND JOHN LOTT
57.1. II.1-II.3. Section II.1 continues the analysis of three-dimensional κ-solutions from I.11. From I.11, any κ-solution contains an “asymptotic soliton”, a gradient shrinking soliton that arises as a rescaled limit of the κ-solution as t → −∞. It is shown that any such gradient shrinking soliton must be a shrinking round cylinder R × S 2 , its Z2 -quotient R ×Z2 S 2 or a finite quotient of the round shrinking S 3 . Using this, one obtains a finer description of the κ-solutions. In particular, any compact κ-solution must be isometric to a finite quotient of the round shrinking S 3 , or diffeomorphic to S 3 or RP 3 . It is shown that there is a universal number κ0 > 0 so that any κ-solution is a finite quotient of the round shrinking S 3 or is a κ0 -solution. This implies universal derivative bounds on the scalar curvature of a κ-solution. Section II.2 defines and analyzes the Ricci flow of the so-called standard solution. This is a Ricci flow on R3 whose initial metric is a capped-off half cylinder. The surgery procedure will amount to gluing in a truncated copy of the time-zero slice of the standard solution. Hence one needs to understand the Ricci flow on the standard solution itself. It is shown that the Ricci flow of the standard solution exists on a maximal time interval [0, 1), and the solution goes singular everywhere as t → 1.
The geometry of the solution at the first singular time T (assuming that there is one) is considered in II.3. Put Ω = {x ∈ M : lim supt→T − | Rm(x, t)| < ∞}. Then Ω is an open subset of M, and x ∈ M − Ω if and only if limt→T − R(x, t) = ∞. If Ω = ∅ then for t slightly less than T , the manifold (M, g(t)) consists of nothing but high-scalar-curvature regions. Using Theorem I.12.1, one shows that M is diffeomorphic to S 1 × S 2 , RP 3 #RP 3 or a finite isometric quotient of S 3 . If Ω 6= ∅ then there is a well-defined limit metric g on Ω, with scalar curvature function R. The set Ω could a priori have an infinite number of connected components, for example if an infinite number of distinct 2-spheres simultaneously shrink to points at time T . For small ρ > 0, put Ωρ = {x ∈ Ω : R(x) ≤ ρ−2 }, a compact subset of M. The connected components of Ω can be divided into those that intersect Ωρ and those that do not. If a connected component does not intersect Ωρ then it is a “capped ǫ-horn” (consisting of a hornlike end capped off by a ball or a copy of RP 3 − B 3 ) or a “double ǫ-horn” (with two hornlike ends). If a connected component of Ω does intersect Ωρ then it has a finite number of ends, each being an ǫ-horn. Topologically, the surgery procedure of II.4 will amount to taking each connected component of Ω that intersects Ωρ , truncating each of its ǫ-horns and gluing a 3-ball onto each truncated horn. The connected components of Ω that do not intersect Ωρ are thrown away. Call the new manifold M ′ . At a time t slightly less than T , the region M − Ωρ consists of high-scalar-curvature regions. Using the characterization of such regions in I.12.1, one shows that M can be reconstructed from M ′ by taking the connected sum of its connected components, along possibly with a finite number of S 1 × S 2 and RP 3 factors. 57.2. II.4-II.5. Section II.4 defines the surgery procedure. A Ricci flow with surgery consists of a sequence of smooth 3-dimensional Ricci flows on adjacent time intervals with the property that for any two adjacent intervals, there is a a compact 3-dimensional submanifoldwith-boundary that is common to the final slice of the first time interval and the initial slice of the second time interval.
NOTES ON PERELMAN’S PAPERS
105
There are two a priori assumptions on a Ricci flow with surgery, the pinching assumption and the canonical neighborhood assumption. The pinching assumption is a form of Hamilton-Ivey pinching. The canonical neighborhood assumption says that every spacetime point (x, t) with R(x, t) ≥ r(t)−2 has a neighborhood which, after rescaling, is ǫ-close to one of the neighborhoods that occur in a κ-solution or in a time slice of the standard solution. Here ǫ is a small but universal constant and r(·) is a decreasing function, which is to be specified. One wishes to define a Ricci flow with surgery starting from any compact oriented 3manifold, say with a normalized initial metric. There are various parameters that will enter into the definition : the above canonical neighborhood scale r(·), a nonincreasing function δ(·) that decays to zero, the truncation scale ρ(t) = δ(t)r(t) and the surgery scale h. In order to show that one can construct the Ricci flow with surgery, it turns out that one wants to perform the surgery only on necks with a radius that is very small compared to the canonical neighborhood scale; this is the role of the parameter δ(·). Suppose that the Ricci flow with surgery is defined at times less than T , with the a priori assumptions satisfied, and goes singular at time T . Define the open subset Ω ⊂ M as before and construct the compact subset Ωρ ⊂ M using ρ = ρ(T ). Any connected component N of Ω that intersects Ωρ has a finite number of ends, each of which is an ǫ-horn. This means that each point in the horn is in the 1center of2an ǫ-neck, i.e. has a neighborhood that, after 1 rescaling, is ǫ-close to a cylinder − ǫ , ǫ × S . In II.4.3 it is shown that as one goes down the end of the horn, there is a self-improvement phenomenon; for any δ > 0, one can find h < δρ so that if a point x in the horn has R(x) ≥ h−2 then it is actually in the center of a δ-neck. With δ = δ(T ), let h be the corresponding number. One then cuts off the ǫ-horn at a 2-sphere in the center of such a δ-neck and glues in a copy of a rescaled truncated standard solution. One does this for each ǫ-horn in N and each connected component N that intersects Ωρ , and throws away the connected components of Ω that do not intersect Ωρ . One lets the new manifold evolve under the Ricci flow. If one encounters another singularity then one again performs surgery. Based on an estimate on the volume change under a surgery, one concludes that a finite number of surgeries occur in any finite time interval. (However, one is not able to conclude from volume arguments that there is a finite number of surgeries altogether.) The preceding discussion was predicated on the condition that the a priori assumptions hold for all times. For the Ricci flow before the first surgery time, the pinching condition follows from the Hamilton-Ivey result. One shows that surgery can be performed so that it does not make the pinching any worse. Then the pinching condition will hold up to the second surgery time, etc. The main issue is to show that one can choose the parameters r(·) and δ(·) so that one knows a priori that the canonical neighborhood assumption, with parameter r(·), will hold for the Ricci flow with surgery. (For any singularity time T , one needs to know that the canonical neighborhood assumption holds for t ∈ [0, T ) in order to do the surgery at time T .)
106
BRUCE KLEINER AND JOHN LOTT
As a preliminary step, in Lemma II.4.5 it is shown that after one glues in a standard solution, the result will still look similar to a standard solution, for as long of a time interval as one could expect, unless the entire region gets removed by some exterior surgery. The result of II.5 is that the time-dependent parameters r(·) and δ(·) can be chosen so as to ensure that the a priori assumptions hold. The normalization of the initial metric implies that there is a time interval [0, C], for a universal constant C, on which the Ricci flow is smooth and has explicitly bounded curvature. On this time interval the canonical neighborhood assumption holds vacuously, if r(·) [0,C] is sufficiently small. To handle later times, the strategy is to divide [ǫ, ∞) into a countable sequence of finite time intervals and proceed by induction. In II.5 the intervals {[2j−1 ǫ, 2j ǫ]}∞ j=1 are used, although the precise choice of intervals is immaterial. We recall from I.12.1 that in the case of smooth flows, the proof of the canonical neighborhood assumption used the fact that one has κ-noncollapsing. It is not immediate that the method of proof of I.12.1 extends to a Ricci flow with surgery. (It is exactly for this reason that one takes δ(·) to be a time-dependent function which can be forced to be very small.) Hence one needs to prove κ-noncollapsing and the canonical neighborhoood assumption together. The main proposition of II.5 says that there are decreasing sequences rj , κj and δ j so that if δ(·) is a function with δ(·) [2j−1 ǫ,2j ǫ] ≤ δ j for each j > 0 then any Ricci flow with surgery, defined with the parameters r(·) and δ(·), is κj -noncollapsed on the time interval [2j−1 ǫ, 2j ǫ] at scales less than ǫ and satisfies the canonical neighborhood assumption there. Here we take r(·) [2j−1 ǫ,2j ǫ) = rj .
The proof of the proposition is by induction. Suppose that it is true for 1 ≤ j ≤ i. In the induction step, besides defining the parameters ri+1 , κi+1 and δ i+1 , one redefines δ i . As one only redefines δ in the previous interval, there is no circularity.
The first step of the proof, Lemma II.5.2, consists of showing that there is some κ > 0 so that for any r, one can find δ = δ(r) > 0 with the following property. Suppose that g(·) is a Ricci flow with surgery defined on [0, T ), with T ∈ [2i ǫ, 2i+1 ǫ], that satisfies the proposition on [0, 2iǫ]. Suppose that it also satisfies the canonical neighborhood assumption with parameter r on [2i ǫ, T ), and is constructed using a function δ(·) that satisfies δ(t) ≤ δ on [2i−1 ǫ, T ). Then it is κ-noncollapsed at all scales less than ǫ. The proof of this lemma is along the lines of the κ-noncollapsing result of I.7, with some important modifications. One again considers the L-length of curves γ(τ ) starting from the point at which one wishes to prove the noncollapsing. One wants to find a spacetime point (x, t), with t ∈ [2i−1 ǫ, 2i ǫ], at which one has an explicit upper bound on l. In I.7, the analogous statement came from a differential inequality for l. In order to use this differential equality in the present case, one needs to know that any curve γ(τ ) that is competitive to be a minimizer for L(x, t) will avoid the surgery regions. Choosing δ small enough, one can ensure that the surgeries in the time interval [2i−1 ǫ, T ) are done on very long thin necks. Using Lemma II.4.5, one shows that a curve γ(τ ) passing near such a surgery region obtains a large value of L, thereby making it noncompetitive as a minimizer for L(x, t). (This is the underlying reason that the surgery parameter δ(·) is chosen in a time-dependent way.) One
NOTES ON PERELMAN’S PAPERS
107
also chooses the point (x, t) so that there is a small parabolic neighborhood around it with a bound on its geometry. One can then run the argument of I.7 to prove κ-noncollapsing in the time interval [2i ǫ, T ). The proof of the main proposition of II.5 is now by contradiction. Suppose that it is not α true. Then for some sequences r α → 0 and δ → 0, for each α there is a counterexample α to the proposition with ri+1 ≤ r α and δi , δi+1 ≤ δ . That is, there is some spacetime point in the interval [2i ǫ, 2i+1 ǫ] at which the canonical neighborhood assumption fails. Take a first such point (xα , tα ). By Lemma II.5.2, one has κ-noncollapsing up to the time of this first counterexample. Using this noncollapsing, one can consider taking rescaled limits. If there are no surgeries in an appropriate-sized backward spacetime region around (xα , tα ) then one can extract a convergent subsequence as α → ∞ and construct, as in the proof of Theorem I.12.1, a limit κ-solution, thereby giving a contradiction. If there are nearby interfering surgeries then one argues, using Lemma II.4.5, that the point (xα , tα ) is in fact in a canonical neighborhood, again giving a contradiction. Having constructed the Ricci flow with surgery, if the initial manifold is simply-connected then according to [24, 25, 53], there is a finite extinction time. One then concludes that the Poincar´e Conjecture holds. 57.3. II.6-II.8. Sections II.6 and II.8 analyze the large-time behavior of a Ricci flow with surgery. Section II.6 establishes back-and-forth curvature estimates. Proposition II.6.3 is an analog of Theorem I.12.2 and Proposition II.6.4 is an analog of Theorem I.12.3. The proofs are along the lines of the proofs of Theorems I.12.2 and I.12.3, but are complicated by the possible presence of surgeries. The thick-thin decomposition for large-time slices is considered in Section II.7. Using monotonicity arguments of Hamilton, it is shown that as t → ∞ the metric on the w-thick part M + (w, t) becomes closer and closer to having constant negative sectional curvature. Using a hyperbolic rigidity argument of Hamilton, it is stated that the hyperbolic pieces stabilize in the sense that there is a finite collection {(Hi , xi )}ki=1 of pointed finite-volume 3-manifolds of constant sectional curvature − 14 so that for large t, the metric gb(t) = 1t g(t) S on the w-thick part M + (w, t) approaches the metric on the w-thick part of ki=1 Hi . It is stated that the cuspidal tori (if any) of the hyperbolic pieces are incompressible in M. To show this (following Hamilton), if there is a compressing 3-disk then one takes a minimal such 3-disk, say of area A(t), and shows from a differential inequality for A(·) that for large t the function A(t) is negative, which is a contradiction. Theorem II.7.4, a statement in Riemannian geometry, characterizes the thin part M − (w, t), for small w and large t, as a graph manifold. The main hypothesis of the theorem is that for each point x, there is a radius ρ = ρ(x) so that the ball B(x, ρ) has volume at most wρ3 and sectional curvatures bounded below by − ρ−2 . In this sense the manifold is locally volume collapsed with respect to a lower sectional curvature bound. Section II.8 contains an alternative proof of the incompressibility of cuspidal tori, using the functional λ1 (g) = λ1 (−4△+R). (At the beginning of Section 93, we give a simpler argument 2 using the functional Rmin (g) vol(M, g) 3 .) More generally, the functional λ1 (g) is used to
108
BRUCE KLEINER AND JOHN LOTT
define a topological invariant that determines the nature of the geometric decomposition. First, the manifold M admits a Riemannian metric g with λ1 (g) > 0 if and only if it admits a Riemannian metric with positive scalar curvature, which in turn is equivalent to saying that M is diffeomorphic to a connected sum of S 1 × S 2 ’s and round quotients of S 3 . If M 2 does not admit a Riemannian metric with λ1 > 0, let λ be the supremum of λ1 (g)·vol(M, g) 3 over all Riemannian metrics g on M. If λ = 0 then M is a graph manifold. If λ < 0 then the geometric decomposition of M contains a nonempty hyperbolic piece, with total volume 3 2 − 32 λ 2 . The proofs of these statements use the monotonicity of λ(g(t)) · vol(M, g(t)) 3 , when it is nonpositive, under a smooth Ricci flow. The main work is to show that in the case 2 of a Ricci flow with surgery, one can choose δ(·) so that λ(g(t)) · vol(M, g(t)) 3 is arbitrarily close to being nondecreasing in t.
58. II. Notation and terminology B(x, t, r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P (x, t, r, ∆t) denotes a parabolic neighborhood, that is the set of all points (x′ , t′ ) with x′ ∈ B(x, t, r) and t′ ∈ [t, t + ∆t] or t′ ∈ [t + ∆, t], depending on the sign of ∆t. Definition 58.1. We say that a Riemannian manifold (M1 , g1 ) has distance ≤ ǫ in the C N topologyPto another Riemannian manifold (M2 , g2 ) if there is a diffeomorphism φ : M2 → M1 1 so that |I| ≤ N |I|! k ∇I (φ∗ g1 − g2 ) k∞ ≤ ǫ. An open set U in a Riemannian 3-manifold M is an ǫ-neck if modulo rescaling, it has distance less than ǫ, in the C [1/ǫ]+1-topology, to the product of the round 2-sphere of scalar curvature 1 (and therefore Gaussian curvature 21 ) with an interval I of length greater than 2ǫ−1 . If a point x ∈ M and a neighborhood U of x are specified then we will understand that “distance” refers to the pointed topology, where the basepoint in S 2 × I projects to the center of I.
We make a similar definition of ǫ-closeness in the spacetime case, where ∇I now includes time derivatives. A subset of the form U ×[a, b] ⊂ M ×[a, b], where U ⊂ M is open, sitting in the spacetime of a Ricci flow is a strong ǫ-neck if after parabolic rescaling and time shifting, it has distance less than ǫ to the product Ricci flow defined on the time interval [−1, 0] which, at its final time, is isometric to the product of a round 2-sphere of scalar curvature 1 with an interval of length greater than 2ǫ−1 . (Evidently, the time-0 slice of the product has 3-dimensional scalar curvature equal to 1.) Our definition of an ǫ-neck differs in an insubstantial way from that on p. 1 of II. In the definition of [52], a ball B(x, t, ǫ−1 r) is called an ǫ-neck if, after rescaling the metric with a factor r −2 , it is ǫ-close, i.e. has distance less than ǫ, to the corresponding subset of the standard neck S 2 × I... (italicized words added by us). (The issue is that a large metric ball in the cylinder R × S 2 does not have a smooth boundary.) Clearly after a slight change of the constants, an ǫ-neck in our sense is contained in an ǫ-neck in the sense of [52], and vice versa. An important fact is that the notion of (x, t) being contained in an ǫ-neck is an open condition with respect to the pointed C [1/ǫ]+1-topology on Ricci flow solutions.
NOTES ON PERELMAN’S PAPERS
109
With an ǫ-approximation f : S 2 × I → U being understood, a cross-sectional sphere in U will mean the image of S 2 × {λ} under f , for some λ ∈ (−ǫ−1 , ǫ−1 ). Any curve γ in U that intersects both f (S 2 ×{ǫ−1 }) and f (S 2 ×{−ǫ−1 }) must intersect each cross-sectional sphere. If γ is a minimizing geodesic and ǫ is small enough then γ will intersect each cross-sectional sphere exactly once. There is a typo in the definition of a strong ǫ-neck in [52] : the parabolic neighborhood should be P (x, t, ǫ−1 r, −r 2 ), i.e. it should go backward in time rather than forward. We note that the time interval involved in the definition of strong ǫ-neck, i.e. 1 after rescaling, is different than the rescaled time interval ǫ−1 in Theorem 52.7. In the next definition, I is an open interval and B 3 is an open ball. Definition 58.2. A metric on S 2 × I such that each point is contained in some ǫ-neck is called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, or tends to infinity on both ends, respectively. A metric on B 3 or RP 3 − B 3 , such that each point outside some compact subset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively. An example of an ǫ-tube is S 2 × (−ǫ−1 , ǫ−1 ) with the product metric. For a relevant example of an ǫ-horn, consider the metric 1 r 2 dθ2 (58.3) g = dr 2 + 8 ln 1r on (0, R) × S 2 , where dθ2 is the metric on S 2 with R = 1. From [6], q the metric g models a rotationally symmetric neckpinch. Rescaling around r0 , we put s = 8 ln r10 rr0 − 1 and find ! 1 8 ln r0 1 1 (58.4) 2+ s + O(s2) dθ2 . g = ds2 + 1 + q 1 2 1 r0 ln r0 8 ln r0
− 41 then the region with s ∈ (− ǫ−1 , ǫ−1 ) will be ǫFor small r0 , if we take ǫ ∼ ln r10 biLipschitz close to the standard cylinder. Note that as r0 → 0, the constant ǫ improves; this is related to Lemma 71.1.
An ǫ-cap is the result of capping off an ǫ-tube by a 3-ball or RP 3 − B 3 with an arbitrary metric. A capped ǫ-horn is the result of capping off an ǫ-horn by a 3-ball or RP 3 − B 3 with an arbitrary metric. Remark 58.5. Throughout the rest of these notes, ǫ denotes a small positive constant that is meant to be universal. The precise value of ǫ is unspecified. If the statement of a lemma or theorem invokes ǫ then the statement is meant to be true uniformly with respect to the other variables, provided ǫ is sufficiently small. When going through the proofs one is allowed to make ǫ small enough so that the arguments work, but one is only allowed to make a finite number of such reductions.
110
BRUCE KLEINER AND JOHN LOTT
Lemma 58.6. Let U be an ǫ-neck in an ǫ-tube (or horn) and let S be a cross-sectional sphere in U. Then S separates the two ends of the tube (or horn). Proof. Let W denote the tube (or horn). As any point m ∈ W lies in some ǫ-neck, there is a unique lowest eigenvalue of the Ricci operator Ric ∈ End(Tm W ) at m. Let ξm ⊂ Tm W be the corresponding eigenspace. As m varies, the ξm ’s form a smooth line field ξ on W , to which S is transverse. Suppose that S does not separate the two ends of W . Then S represents a trivial element of H2 (S 2 × I) ∼ = π2 (S 2 × I) and there is an embedded 3-disk D ⊂ W for which ∂D = S. This contradicts the fact that the line field ξ is transverse to S and extends over D.
59. II.1. Three-dimensional κ-solutions This section is concerned with properties of three-dimensional oriented κ-solutions. For brevity, in the rest of these notes we will generally omit the phrases “three-dimensional” and “oriented”. If (M, g(·)) is a κ-solution then its topology is easy to describe. By definition, (M, g(t)) has nonnegative sectional curvature. If it does not have strictly positive curvature then the universal cover splits off a line (see Theorem A.7), from which it follows (using Corollary 40.1 and the κ-noncollapsing) that (M, g(·)) is a standard shrinking cylinder R × S 2 or its Z2 quotient R ×Z2 S 2 . If (M, g(t)) has strictly positive curvature and M is compact then it is diffeomorphic to a spherical space form [35]. If (M, g(t)) has strictly positive curvature and M is noncompact then it is diffeomorphic to R3 [19]. The lemmas in this section give more precise geometric information. Recall that Mǫ consists of the points in a κ-solution which are not the center of an ǫ-neck. Lemma 59.1. If (M, g(t)) is a time slice of a noncompact κ-solution and Mǫ 6= ∅ then there is a compact submanifold-with-boundary X ⊂ M so that Mǫ ⊂ X, X is diffeomorphic to B 3 or RP 3 − B 3 , and M − int(X) is diffeomorphic to [0, ∞) × S 2 . Proof. If (M, g(t)) does not have positive sectional curvature and Mǫ 6= ∅ then M must be isometric to R×Z2 S 2 , in which case the lemma is easily seen to be true with X diffeomorphic to RP 3 − B 3 . Suppose that (M, g(t)) has positive sectional curvature. Choose x ∈ Mǫ . Let γ : [0, ∞) → M be a ray with γ(0) = x. As Mǫ is compact, there is some a > 0 so that if t > a then γ(t) ∈ / Mǫ . We can cover (a, ∞) by open intervals Vj so that γ is a geodesic Vj
segment in an ǫ-neck of rescaled length approximately 2ǫ−1 . Then we can find a cover of (a, ∞) by linearly ordered open intervals Ui , refining the previous cover, so that 1 −1 ǫ . 1. The rescaled length of γ is approximately 10 Ui 2. Choosing some xi ∈ Ui ∩ Ui+1 , the rescaled length (with rescaling at xi ) of γ is Ui ∩Ui+1 1 −1 approximately 40 ǫ and γ lies in an ǫ-neck Wi centered at xi . Ui ∩Ui+1
NOTES ON PERELMAN’S PAPERS
111
Let φi be projection on the first factor in the assumed diffeomorphism Wi ∼ = (−ǫ−1 , ǫ−1 ) × −1 S . If ǫ is sufficiently small then the composition φi ◦ γ Ui : Ui → (−ǫ , ǫ−1 ) is a diffeo )) ⊂ Wi and pi = (φi ◦ γ )−1 ◦ φi : morphism onto its image. Put Ni = φ−1 (Im(φ ◦ γ i i Ui Ui Ni → Ui ⊂ (a, ∞). Then pi is ǫ-close to being a Riemannian submersion and on overlaps Ni ∩ Ni+1 , the maps pi and pi+1 are C K -close. Choosing an appropriate partition of unity P {bi } subordinate to the Ui ’s, if ǫ is small then the function f = i bi pi is a submersion from S 2 −1 (t, ∞). i Ni to (a, ∞). The fiber is seen to be S . Given t ∈ (a, ∞), put X = M − f −1 2 Then M − int(X) = f ([t, ∞)) is diffeomorphic to [0, ∞) × S . 2
Recall from Section 46 that we have an exhaustion of M by certain convex compact subsets. As M is one-ended, the subsets have connected boundary. As in Section 46, if the boundary of such a subset intersects an ǫ-neck then the intersection will be a nearly cross-sectional 2-sphere in the ǫ-neck. Hence with an appropriate choice of t, the set X will be isotopic to one of our convex subsets and so diffeomorphic to a closed 3-ball. Lemma 59.2. If (M, g(t)) is a time slice of a κ-solution with Mǫ = ∅ then the Ricci flow is the evolving round cylinder R × S 2 .
Proof. By assumption, each point (x, t) lies in an ǫ-neck. If ǫ is sufficiently small then piecing the necks together, we conclude that M must be diffeomorphic to S 1 × S 2 or R × S 2 ; see f is R × S 2 . the proof of Lemma 59.1 for a similar argument. Then the universal cover M As it has nonnegative sectional curvature and two ends, Toponogov’s theorem implies that f, e f (M g (t)) splits off an R-factor. Using the strong maximum principle, the Ricci flow on M f, e splits off an R-factor; see Theorem A.7. Using Corollary 40.1, it follows that (M g (t)) is 2 the evolving round cylinder R × S . From the κ-noncollapsing, the quotient M cannot be S 1 × S 2. A κ-solution has an asymptotic soliton (Section 39) that is either compact or noncompact. If the asymptotic soliton of a compact κ-solution (M, g(·)) is also compact then it must be a shrinking quotient of the round S 3 [35], so the same is true of M. Lemma 59.3. If a κ-solution (M, g(·)) is compact and has a noncompact asymptotic soliton then M is diffeomorphic to S 3 or RP 3 . Proof. We use Corollary 48.1 in Section 48. First, we claim that the time slices of the type-D κ-solutions of Corollary 48.1 have a universal upper bound on maxM R · diam(M)2 . To see this,√we can rescale at the point x ∈ Mǫ by R(x), after which the diameter is bounded above by 2 α. We then use Theorem 46.1 to get an upper bound on the rescaled scalar curvature, which proves the claim. Given an upper bound on maxM R · diam(M)2 , the asymptotic soliton cannot be noncompact. Thus we are in case C of Corollary 48.1. Take a sequence ti → −∞ and choose points xi , yi ∈ Mǫ (ti ) as in Corollary 48.1.C. Rescale by R(xi , ti ) and take a subsequence that converges to a pointed Ricci flow solution (M∞ , (x∞ , t∞ )). The limit M∞ cannot be compact, as otherwise we would have a uniform upper bound on R · diam2 for (M, g(ti )), which would contradict the existence of the noncompact asymptotic soliton. Thus M∞ is a noncompact 1 κ-solution. We can find compact sets Xi ⊂ M containing B(xi , αR(xi , ti )− 2 ) so that {Xi }
112
BRUCE KLEINER AND JOHN LOTT
converges to a set X∞ ⊂ M∞ as in Lemma 59.1. Taking a further subsequence, we find 1 similar compact sets Yi ⊂ M containing B(yi , αR(yi, ti )− 2 ) so that {Yi} converges to a set Y∞ ⊂ M∞ as in Lemma 59.1. In particular, for large i, Xi and Yi are each diffeomorphic to either B 3 or RP 3 − B 3 . Considering a minimizing geodesic segment xi yi as in the statement of Corollary 48.1.C, we can use an argument as in the proof of Lemma 59.1 to construct a submersion from M − (Xi ∪ Yi ) to an interval, with fiber S 2 . Hence M is diffeomorphic to the result of gluing Xi and Yi along a 2-sphere. As M has finite fundamental group, no more than one of Xi and Yi can be diffeomorphic to RP 3 − B 3 . Thus M is diffeomorphic to S 3 or RP 3 . From Lemma 50.1, every ancient solution which is a κ-solution for some κ is either a κ0 -solution or a metric quotient of the round S 3 . Lemma 59.4. There is a universal constant η such that at each point of every ancient solution that is a κ-solution for some κ, we have estimates (59.5)
3
|∇R| < ηR 2 ,
|Rt | < ηR2 .
Proof. This is obviously true for metric quotients of the round S 3 . For κ0 -solutions it follows from the compactness result in Theorem 46.1, after rescaling the scalar curvature at the given point to be 1. It is sometimes useful to rewrite (59.5) as a pair of estimates on the spacetime derivatives of the quantity R−1 at points where R 6= 0: 1 η |(R−1 )t | < η. (59.6) |∇(R− 2 )| < , 2 Lemma 59.7. For every sufficiently small ǫ > 0 one can find C1 = C1 (ǫ) and C2 = C2 (ǫ) such that for each point (x, t) in every κ-solution there is a radius r ∈ [R(x, t)−1/2 , C1 R(x, t)−1/2 ] and a neighborhood B, B(x, t, r) ⊂ B ⊂ B(x, t, 2r), which falls into one of the four categories: (a) B is a strong ǫ-neck (more precisely, B is the slice of a strong ǫ-neck at its maximal time, and an appropriate parabolic neighborhood of B satisfies the condition to be a strong ǫ-neck), or (b) B is an ǫ-cap, or (c) B is a closed manifold, diffeomorphic to S 3 or RP 3 , or (d) B is a closed manifold of constant positive sectional curvature. Furthermore: • The scalar curvature in B at time t is between C2−1 R(x, t) and C2 R(x, t). 3 • The volume of B in cases (a), (b) and (c) is greater than C2−1 R(x, t)− 2 . • In case (b), there is an ǫ-neck U ⊂ B with compact complement in B (i.e. the end of B is entirely contained in the ǫ-neck) such that the distance from x to U is at least 10000R(x, t)−1/2 . • In case (c) the sectional curvature in B at time t is greater than C2−1 R(x, t). Remark 59.8. The statement of the lemma is slightly stronger than the corresponding statement in II.1.5, in that we have r ≥ R(x, t)−1/2 as opposed to r > 0.
NOTES ON PERELMAN’S PAPERS
113
Proof. We may assume that we are talking about a κ0 -solution, as if M is a metric quotient of a round sphere then it falls into category (d) for any r > π(R(xi , ti )/6)−1/2 (since then M = B(xi , ti , r) = B(xi , ti , 2r)). Fix a small ǫ and suppose that the claim is not true. Then there is a sequence of κ0 solutions Mi that together provide a counterexample. That is, there is a sequence Ci → ∞ and a sequence of points (xi , ti ) ∈ Mi ×(−∞, 0] so that for any r ∈ [R(xi , ti )−1/2 , Ci R(xi , ti )−1/2 ] one cannot find a B between B(xi , ti , r) and B(xi , ti , 2r) falling into one of the four categories and satisfying the subsidiary conditions with parameter C2 = Ci . Rescale the metric by R(xi , ti ) and take a convergent subsequence of (Mi , (xi , ti )) to obtain a limit κ0 -solution (M∞ , (x∞ , t∞ )). Then for any r > 1, one cannot find a B∞ between B(x∞ , t∞ , r) and B(x∞ , t∞ , 2r) falling into one of the four categories and satisfying the subsidiary conditions for any parameter C2 . If M∞ is compact then for any r greater than the diameter of the time-t∞ slice of M∞ , B(x∞ , t∞ , r) = M∞ = B(x∞ , t∞ , 2r) falls into category (c) or (d). For the subsidiary conditions, M∞ clearly has a lower volume bound, a positive lower scalar curvature bound and an upper scalar curvature bound. As a compact κ0 -solution has positive sectional curvature, M∞ also has a lower sectional curvature bound. This is a contradiction. If M∞ is noncompact then Lemma 59.1 (or more precisely its proof) and Lemma 59.2 imply that for some r > 1, there will be a B between B(x∞ , t∞ , r) and B(x∞ , t∞ , 2r) falling into category (a) or (b). In case (b), by choosing the parameter r sufficiently large, the existence of the ǫ-neck U with the desired properties follows from the proof of Lemma 59.1. For the other subsidiary conditions, B clearly has a lower volume bound, a positive lower scalar curvature bound and an upper scalar curvature bound. This contradiction completes the proof of the lemma. 60. II.2. Standard solutions The next few sections are concerned with the properties of special Ricci flow solutions on M = R3 . We fix a smooth rotationally symmetric metric g0 which is the result of gluing a hemispherical-type cap to a half-infinite cylinder of scalar curvature 1. Among other properties, g0 is complete and has nonnegative curvature operator. We also assume that g0 has scalar curvature bounded below by 1. Remark 60.1. In Section 72 we will further specialize the initial metric g0 of the standard solution, for technical convenience in doing surgeries. Definition 60.2. A Ricci flow (R3 , g(·)) defined on a time interval [0, a) is a standard solution if it has initial condition g0 , the curvature | Rm | is bounded on compact time intervals [0, a′ ] ⊂ [0, a), and it cannot be extended to a Ricci flow with the same properties on a strictly longer time interval. It will turn out that every standard solution is defined on the time interval [0, 1). To motivate the next few sections, let us mention that the surgery procedure will amount to gluing in a truncated copy of (R3 , g0 ). The metric on this added region will then evolve as part of the Ricci flow that takes up after the surgery is performed. We will need to
114
BRUCE KLEINER AND JOHN LOTT
understand the behavior of the Ricci flow after performing a surgery. Near the added region, this will be modeled by a standard solution. Hence one first needs to understand the Ricci flow of a standard solution. The main results of II.2 concerning the Ricci flow on a standard solution (Sections 61-64) are used in II.4.5 (Lemma 74.1) to show that, roughly speaking, the part of the manifold added by surgery acquires a large scalar curvature soon after the surgery time. This is used crucially in II.5 (Sections 79 and 80) to adapt the noncollapsing argument of I.7 to Ricci flows with surgery. Our order of presentation of the material in II.2 is somewhat different than that of [52]. In Sections 61-63, we cover Claims 2, 4 and 5 of II.2. These are what’s needed in the sequel. The other results of II.2, Claims 1 and 3, are concerned with proving the uniqueness of the standard solution. Although it may seem intuitively obvious that there should be a unique and rotationally-symmetric standard solution, the argument is not routine since the manifold is noncompact. In fact, the uniqueness is not really needed for the sequel. (For example, the method of proof of Lemma 74.1 produces a standard solution in a limiting argument and it is enough to know certain properties of this standard solution.) Because of this we will talk about a standard solution rather than the standard solution. Consequently, we present the material so that we do not logically need the uniqueness of the standard solution. Having uniqueness does not shorten the subsequent arguments any. Of course, one can ask independently whether the standard solution is unique. In Section 65 we show that a standard solution is rotationally symmetric. In Section 66 we sketch the argument for uniqueness. Papers concerning the uniqueness of the standard solution are [21, 41]. We end this section by collecting some basic facts about standard solutions. Lemma 60.3. Let (R3 , g(·)) be a standard solution. Then (1) The curvature operator of g is nonnegative. (2) All derivatives of curvature are bounded for small time, independent of the standard solution. (3) The scalar curvature satisfies limt→a− supx∈R3 R(x, t) = ∞.
(4) (R3 , g(·)) is κ-noncollapsed at scales below 1 on any time interval contained in [0, 2], where κ depends only on the choice of the initial condition g0 . (5) (R3 , g(·)) satisfies the conclusion of Theorem 52.7, in the sense that for any ∆t > 0, there is an r0 > 0 so that for any point (x0 , t0 ) with t0 ≥ ∆t and Q = R(x0 , t0 ) ≥ r0−2 , the solution in {(x, t) : dist2t0 (x, x0 ) < (ǫQ)−1 , t0 − (ǫQ)−1 ≤ t ≤ t0 } is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution. Moreover, any Ricci flow which satisfies all of the conditions of Definition 60.2 except maximality of the time interval can be extended to a standard solution. In particular, using short-time existence [62, Theorem 1.1], there is at least one standard solution. Proof. (1) follows from [63, Theorem 4.14].
NOTES ON PERELMAN’S PAPERS
115
(2) follows from Appendix D. (3) In view of (1), this is equivalent to saying that limt→a− supx∈R3 | Rm |(x, t) = ∞. The argument for this last assertion is in [22, Chapter 6.7.2]. The proof in [22, Chapter 6.7.2] is for the compact case but using the derivative estimates of Appendix D, the same argument works in the present case. (4) See Theorem 26.2. (5) See Theorem 52.7. The final assertion of the lemma follows from the method of proof of (3).
61. Claim 2 of II.2. The blow-up time for a standard solution is ≤ 1 Lemma 61.1. (cf. Claim 2 of II.2) Let [0, TS ) be the maximal time interval such that the curvature of all standard solutions is uniformly bounded for every compact subinterval [0, a] ⊂ [0, TS ). Then on the time interval [0, TS ), the family of standard solution converges uniformly at (spatial) infinity to the standard Ricci flow on the round infinite cylinder S 2 ×R of scalar curvature one. In particular, TS is at most 1. ∞ Proof. Let {Mi}∞ i=1 be a sequence of standard solutions, and let {xi }i=1 be a sequence tending to infinity in the time-zero slice M.
By (2) of Lemma 60.3, the gradient estimates in Appendix D, and Appendix E, every subsequence of {Mi, (xi , 0)}∞ i=1 has a subsequence which converges in the pointed smooth topology on the time interval [0, TS ). Therefore, it suffices to show that if {Mi, (xi , 0)}∞ i=1 converges to some pointed Ricci flow (M∞ , (x∞ , 0)) then M∞ is round cylindrical flow.
Since gi (0) = g0 for all i, the sequence of pointed time-zero slices {(M, xi , gi(0))}∞ i=1 converges in the pointed smooth topology to the round cylinder, i.e. (M∞ , g∞ (0)) is a round cylinder of scalar curvature 1. Each time slice (M∞ , g∞ (t)) is biLipschitz equivalent to (M∞ , g∞ (0)). In particular, it has two ends. As it also has nonnegative sectional curvature, Toponogov’s theorem implies that (M∞ , g∞ (t)) splits off an R-factor. Using the strong maximum principle, the Ricci flow M∞ splits off an R-factor; see Theorem A.7. Then using the uniqueness of the Ricci flow on the round S 2 , it follows that M∞ is a standard shrinking cylinder, which proves the lemma. In particular, TS ≤ 1.
62. Claim 4 of II.2. The blow-up time of a standard solution is 1 Lemma 62.1. (cf. Claim 4 of II.2) Let TS be as in Lemma 61.1. Then TS = 1. In particular, every standard solution survives until time 1. Proof. First, there is an α > 0 so that TS > α [62, Theorem 1.1]. In what follows we will apply Theorem 52.7. The hypothesis of Theorem 52.7 says that the flow should exist on a time interval of duration at least one, but by rescaling we can apply Theorem 52.7 just as well with the alternative hypothesis that the flow exists on a time interval of duration at least α.
116
BRUCE KLEINER AND JOHN LOTT
Suppose that TS < 1. Then there is a sequence of standard solutions {Mi }∞ i=1 , times ti → TS and points (xi , ti ) ∈ Mi so that limi→∞ R(xi , ti ) = ∞.
We first argue that no subsequence of the points xi can go to infinity (with respect to the time-zero slice). Suppose, after relabeling the subsequence, that {xi }∞ i=1 goes to infinity. From Lemma 61.1, for any fixed t′ < TS the pointed solutions (M, (xi , 0), gi(·)), defined for t ∈ [0, t′ ], approach that of the shrinking cylinder on the same time interval. Lemma 60.3 and the characterization of high-curvature regions from Theorem 52.7 implies a uniform bound on high-curvature regions of the time derivative of R, of the form (59.5). Then taking t′ sufficiently close to TS , we get a contradiction. We conclude that outside of a compact region the curvature stays uniformly bounded as t → TS ; compare with the proof of Lemma 52.11. (Alternatively, one could apply Theorem 30.1 to compact approximants, as is done in II.2.) Thus we may assume that the sequence {xi }∞ i=1 stays in a compact region of the timezero slice. By Theorem 52.7, there is a sequence ǫi → 0 so that after rescaling the pointed solution (M, (xi , ti )) by R(xi , ti ), the result is ǫi -close to the corresponding subset of an ancient solution. By Proposition 41.13, the ancient solutions have vanishing asymptotic volume ratio. Hence for every β > 0, there is some L < ∞ so that in the original unscaled 1
1
3
solution, for large i we have vol B(xi , ti , L R(xi , ti )− 2 ) ≤ β L R(xi , ti )− 2 . Applying the Bishop-Gromov inequality to the time-ti slices, we conclude that for any D > 0, limi→∞ D−3 vol(B(xi , ti , D)) = 0. However, this contradicts the previously-shown fact that the solution extends smoothly to time TS < 1 outside of a compact set. Thus TS = 1.
Lemma 62.2. The infimal scalar curvature on the time-t slice tends to infinity as t → 1− uniformly for all standard solutions. Proof. Suppose the lemma failed and let {(Mi, (xi , ti ))}∞ i=1 be a sequence of pointed standard solutions, with {R(xi , ti )}∞ uniformly bounded and lim i→∞ ti = 1. i=1
Suppose first that after passing to a subsequence, the points xi go to infinity in the timezero slice. From Lemma 61.1, for any t′ ∈ [0, 1) we have limi→∞ R−1 (xi , t′ ) = 1 − t′ . −1 Combining this with the derivative estimate ∂R∂t ≤ η at high curvature regions gives a contradiction; compare with the proof of Lemma 52.11. Thus the points xi stay in a compact region. We can now use the bounded-curvature-at-bounded-distance argument in Step 2 of the proof of Theorem 52.7 to extract a convergent subsequence of {(Mi, (xi , 0))}∞ i=1 with a limit Ricci flow solution (M∞ , (x∞ , 0)) that exists on the time interval [0, 1]. (In this case, the nonnegative curvature of the blowup region W comes from the fact that a standard solution has nonnegative curvature.) As in Step 3 of the proof of Theorem 52.7, M∞ will have bounded curvature for t ∈ [0, 1]. Note that M∞ is a standard solution. This contradicts Lemma 61.1. 63. Claim 5 of II.2. Canonical neighborhood property for standard solutions Let p be the center of the hemispherical region in the time-zero slice.
NOTES ON PERELMAN’S PAPERS
117
Lemma 63.1. (cf. Claim 5 of II.2) Given ǫ > 0 sufficiently small, there are constants η = η(ǫ), C1 = C1 (ǫ) and C2 = C2 (ǫ) so that every standard solution M satisfies the conclusions of Lemmas 59.4 and 59.7, except that the ǫ-neck neighborhood need not be strong. (Here the constants do not depend on the standard solution.) More precisely, any point (x, t) is covered by one of the following cases : 1.The time t lies in ( 34 , 1) and (x, t) has an ǫ-cap neighborhood or a strong ǫ-neck neighborhood as in Lemma 59.7. 2. x ∈ B(p, 0, ǫ−1 ), t ∈ [0, 43 ] and (x, t) has an ǫ-cap neighborhood as in Lemma 59.7.
3. x ∈ / B(p, 0, ǫ−1 ), t ∈ [0, 34 ] and there is an ǫ-neck B(x, t, ǫ−1 r) such that the solution in P (x, t, ǫ−1 r, −t) is, after scaling with the factor r −2 , ǫ-close to the appropriate piece of the evolving round infinite cylinder. Moreover, we have an estimate Rmin (t) ≥ const. (1 − t)−1 , where the constant does not depend on the standard solution. Proof. We first show that the conclusion of Lemma 59.7 is satisfied. In view of Lemma 62.2, there is a δ > 0 so that if t ∈ (1 − δ, 1) then we can apply Theorem 52.7 and Lemma 59.7 to a point (x, t) to see that the conclusions of Lemma 59.7 are satisfied in this case. If t ∈ [0, 1 − δ] and x is sufficiently far from p (i.e. dist0 (x, p) ≥ D for an appropriate D) then Lemma 61.1 implies that (x, t) has a strong ǫ-neck neighborhood or there is an ǫ-neck B(x, t, ǫ−1 r) such that the solution in P (x, t, ǫ−1 r, −t) is, after scaling with the factor r −2 , ǫ-close to the appropriate piece of the evolving round infinite cylinder. (To elaborate a bit on the last possibility, the issue here is that there is no backward extension of the solution to t < 0. Because of this, if t > 0 is close to 0 then the backward neighborhood P (x, t, ǫ−1 r, −t) will not exist for rescaled time one, as required to have a strong ǫ-neck neighborhood. Since inf x∈M R(x, 0) = 1, we know from (B.2) that R(x, t) ≥ 1 . Then if t > 35 , the time from the initial slice to (x, t), after rescaled by the scalar 1− 2 t 3
curvature, is bounded below by t
1 1− 23 t
> 1. In particular, if t ≥
3 4
then r 2 t is at least one
and we are ensured that the backward neighborhood P (x, t, ǫ−1 r, −t) does contain a strong ǫ-neck neighborhood.) If t ∈ [0, 1 − δ] and dist0 (x, p) < D then, provided that D and ǫ are chosen appropriately, we can say that (x, t) has an ǫ-cap neighborhood. We now show that the conclusion of Lemma 59.4 is satisfied. If t ∈ [1 − δ, 1) then the conclusion follows from Theorem 52.7 and Lemma 59.4. If δ ′ > 0 is sufficiently small and t ∈ [0, δ ′ ] then the conclusion follows from Appendix D. If t ∈ [ 21 δ ′ , 1 − 12 δ] then we have an upper scalar curvature bound from Lemma 62.1. From Hamilton-Ivey pinching (see Appendix B), this implies a double-sided sectional curvature bound. The conclusion of Lemma 59.4, when t ∈ [ 12 δ ′ , 1 − 21 δ], now follows from the Shi estimates of Appendix D. −1 The last statement of the lemma follows from the estimate ∂R∂t ≤ const., which holds for t near 1 (see Lemmas 59.4, 60.3(5) and 62.2) and then can be extended to all t ∈ [0, 1) (see Lemma 61). From Lemma 62.2, limt→1 R−1 (x, t) = 0 for every x. Thus
118
BRUCE KLEINER AND JOHN LOTT
R−1 (x, t) ≤ const. (1 − t) for any (x, t). Equivalently,
R(x, t) ≥ const. (1 − t)−1 .
(63.2)
64. Compactness of the space of standard solutions Lemma 64.1. The family ST of pointed standard solutions {(M, (p, 0))} is compact with respect to pointed smooth convergence. Proof. This follows immediately from Appendix E and the fact that the constant TS from Lemma 61.1 is equal to 1, by Lemma 62.1. 65. Claim 1 of II.2. Rotational symmetry of standard solutions Consider a standard solution (M, g(·)). Since the time-zero metric g0 is rotationally symmetric, it is clear by separation of variables that there is a rotationally symmetric solution for some time interval [0, T ). In this section we show that every standard solution is rotationally symmetric for each t ∈ [0, 1). Of course this would follow from the uniqueness of the standard solution; see [21, 41]. But the direct argument given here is the first step toward a uniqueness proof as in [41]. Lemma 65.1. (cf. Claim 1 of II.2) Any Ricci flow solution in the space ST is rotationally symmetric for all t ∈ [0, 1). Proof. We first describe an evolution equation for vector fields which turns out Pto send m Killing vector fields to Killing vector fields. Suppose that a vector field u = m u ∂m evolves by m k m i um t = u ; k + R iu .
(65.2)
Then (65.3) m k ∂t (um;i ) = um t ;i + (∂t Γ ki ) u = (um; kk + Rmk uk ); i + (∂t Γmki ) uk = um; kki + Rmk;iuk + Rmk uk;i + (∂t Γmki ) uk = um; kik − Rmlki ul ; k − Rk lki um; l + Rmk;iuk + Rmk uk;i + (∂t Γmki) uk
= um; ki k − Rmlki ul ; k − Rli um; l + Rmk;i uk + Rmk uk;i + (∂t Γmki ) uk
= um; ik k − (Rmlkiul ); k − Rmlki ul ; k − Rki um; k + Rmk;iuk + Rmk uk;i + (∂t Γmki) uk
= um; ik k − Rmlki; k ul − 2Rmlki ul ; k − Rik um; k + Rmk;iuk + Rmk uk;i + (∂t Γmki) uk .
Contracting the second Bianchi identity gives (65.4)
Rmlki; k = Ril;m − Rim;l .
NOTES ON PERELMAN’S PAPERS
119
Also, (65.5)
∂t Γmki = ∂t (g ml Γlki) = 2Rml Γlki − g ml (Rlk,i + Rli,k − Rik,l ) = −Rmk;i − Rmi;k + Rik;m .
Substituting (65.4) and (65.5) in (65.3) gives (65.6) Then (65.7)
∂t (um;i ) = um; ik k − 2Rmlki ul ; k − Rik um; k + Rmk uk;i . ∂t (uj;i) = ∂t (gjm um;i) = −2Rjm um;i + gjm ∂t (um;i ) = uj; ik k − 2Rjlki ul ; k − Rik uj ;k − Rjk uk;i
= uj; ik k + 2Rikjl ul ; k − Rik uj ;k − Rkj uk;i.
Equivalently, writing vij = uj;i gives
∂t vij = vij;kk + 2Ri kj l vkl − Ri k vkj − Rkj vik .
Then putting Lij = vij + vji gives
∂t Lij = Lij;kk + 2Ri kj l Lkl − Ri k Lkj − Rkj Lik . For any λ ∈ R, we have
(65.8) ∂t (e2λt Lij Lij ) = 2λ (e2λt Lij Lij ) + (e2λt Lij Lij );k k − 2 e2λt Lij;k Lij;k + Q(Rm, eλt L),
where Q(Rm, L) is an algebraic expression that is linear in the curvature tensor Rm and quadratic in L. Putting Mij = eλt Lij gives (65.9)
∂t (Mij M ij ) = 2λ Mij M ij + (Mij M ij );k k − 2 Mij;k M ij;k + Q(Rm, M).
Suppose that we have a Ricci flow solution g(t), t ∈ [0, T ], with g(0) = g0 . Let u(0) be a rotational Killing vector field for g0 . Let u∞ (0) be its restriction to (any) S 2 , which we will think of as the 2-sphere at spatial infinity. Solve (65.2) for t ∈ [0, T ] with u(t) bounded at spatial infinity for each t; due to the asymptotics coming from Lemma 61.1 (which is independent of the rotational symmetry question), there is no problem in doing so. Arguing as in the proof of Lemma 61.1, one can show that for any t ∈ [0, T ], at spatial infinity u(t) converges to u∞ (0). Construct Mij (t) from u(t). As u(0) is a Killing vector field, Mij (0) = 0. For any t ∈ [0, T ], at spatial infinity the tensor Mij (t) converges smoothly to zero. Suppose that λ is sufficiently negative, relative to the L∞ -norm of the sectional curvature on the time interval [0, T ]. We can apply the maximum principle to (65.9) to conclude that Mij (t) = 0 for all t ∈ [0, T ]. Thus u(t) is a Killing vector field for all t ∈ [0, T ]. To finish the argument, as Ben Chow pointed out, any Killing vector field u satisfies
(65.10)
um; kk + Rmi ui = 0.
To see this, we use the Killing field equation to write (65.11)
0 = um;k k + uk;m k = um;k k + uk;m k − uk; km = um;k k + uk;mk − uk;km = um; kk − Rkimk ui = um;kk + Rmi ui .
Then from (65.2), um = 0 and the Killing vector fields are not changing at all. This t implies that g(t) is rotationally symmetric for all t ∈ [0, T ].
120
BRUCE KLEINER AND JOHN LOTT
66. Claim 3 of II.2. Uniqueness of the standard solution In this section, which is not needed for the sequel, we outline an argument for the uniqueness of the standard solution. We do this for the convenience of the reader. Papers on the uniqueness issue are [21, 41]. Our argument is somewhat different than that of [52, Proof of Claim 3 of Section 2], which seems to have some unjustified statements. c ≡ In general, suppose that we have two Ricci flow solutions M ≡ (M, g(·)) and M (M, b g (·)) with bounded curvature on each compact time interval and the same initial condition. We want to show that they coincide. As the set of times for which g(t) = b g (t) is closed, it suffices to show that it is relatively open. Thus it is enough to show that g and b g agree on [0, T ) for some small T .
We will carry out the Deturck trick in this noncompact setting, using a time-dependent background metric as in [5, Section 2]. The idea is to define a 1-parameter family of metrics {h(t)}t∈[0,T ) by h(t) = φ−1 (t)∗ g(t), where {φ(t)}t∈[0,T ) is a 1-parameter family of diffeomorphisms of M whose generator is the negative of the time-dependent vector field (66.1)
W i(t) = hjk Γ(h)ijk − Γ(b g )ijk ,
with φ0 = Id. More geometrically, as in [33, Section 6], we consider the solution of the harmonic heat flow equation ∂F = △F for maps F : M → M between the manifolds ∂t (M, g(t)) and (M, b g (t)), with F (0) = Id.
We now specialize to the case when (M, g(·)) and (M, b g (·)) come from standard solutions. The technical issue, which we do not address here, is to show that a solution to the harmonic heat flow will exist for some time interval [0, T ) with uniformly bounded derivatives; see [21]. One is allowed to use the asymptotics of Section 61 here and from Section 65, one can also assume that all of the metrics are rotationally invariant. In the rest of this section we assume the existence of such a solution F . By further reducing the time interval if necessary, we may assume that F (t) is a diffeomorphism of M for each t ∈ [0, T ). Then h(t) = F −1 (t)∗ g(t). Clearly h(0) = g(0) = b g (0).
By Section 61, g and b g have the same spatial asymptotics, namely that of the shrinking cylinder. We claim that this is also true for h. That is, we claim that (M, h(·)) converges smoothly to the shrinking cylinder solution on [0, T ). It suffices to show that F converges smoothly to the identity on [0, T ). Suppose not. Let {xi }∞ i=1 be a sequence of points in the time-zero slice so that no subsequence of the pointed spacetime maps (F, (xi , 0)) converges to the identity. Using the derivative bounds, we can extract a subsequence that converges to some Fe : [0, T ) × R × S 2 → R × S 2 in the pointed smooth topology. However, Fe will satisfy the harmonic heat flow equation from the shrinking cylinder R × S 2 to itself, with Fe(0) being the identity, and will have bounded derivatives. The uniqueness of Fe follows by standard methods. Hence Fe(t) is the identity for all t ∈ [0, T ), which is a contradiction. By construction, the family of metrics {h(t)}t∈[0,T ) satisfies the equation
(66.2)
dhij = − 2 Rij (h) + ∇(h)i Wj + ∇(h)j Wi . dt
NOTES ON PERELMAN’S PAPERS
121
In local coordinates, the right-hand side of (66.2) is a polynomial in hij , hij , hij,k and hij,kl . The leading term in (66.2) is dhij = hkl ∂k ∂l hij + . . . . dt A particular solution of (66.2) is h(t) = b g (t), since if we had g = gb then we would have W = 0 and φt = Id. (66.3)
Put w(t) = h(t) − b g (t). We claim that w satisfies an equation of the form
dw = − ∇(b g )∗ ∇(b g )w + P w + Qw, dt where P is a first-order operator and Q is a zeroth-order operator. To obtain the leading derivative terms in (66.4), using (66.3) we write
(66.4)
(66.5)
dwij = hkl ∂k ∂l hij − b g kl ∂k ∂l b gij + . . . dt = b g kl ∂k ∂l wij + hkl − b g kl ∂k ∂l hij + . . . = gbkl ∂k ∂l wij − gbka wab hbl ∂k ∂l hij + . . .
= gbkl ∂k ∂l wij − gbka hbl ∂k ∂l hij wab + . . .
A similar procedure can be carried out for the lower order terms, leading to (66.4). By construction the operators P and Q have smooth coefficients which, when expressed in terms of orthonormal frames, will be bounded on M. In fact, as h and gb have the same spatial asymptotics, it follows from [5, Proposition 4] that the operator on the right-hand side of (66.4) converges at spatial infinity to the Lichnerowicz Laplacian △L(b g ). By assumption, w(0) = 0. We now claim that w(t) = 0 for all t ∈ [0, T ). Let K ⊂ M be a codimension-zero compact submanifold-with-boundary. For any λ ∈ R, we have (66.6) Z 1 −2λt 2λt d 2 e e |w(t)| dvolbg(t) = dt 2 K Z dw R 2 |w| + hw, i dvolbg(t) = −λ − 2 dt K Z R 2 2 abcdi |w| − |∇(b g )w| + wab P ∇(b g )i wcd + hw, Qwi dvolbg(t) ± −λ − 2 K Z hw, ∇n wi dvol∂bg(t) = ∂K Z R 1 abcdi 1 abcdi 2 2 2 cd −λ − |w| − |∇(b g )i w − P wab | + |P wab | + hw, Qwi dvolbg(t) ± 2 2 4 K Z hw, ∇n wi dvol∂bg(t) . ∂K
Choose (66.7)
λ > sup v6=0
1 |P abcdi vab |2 4
+ hv, Qvi . hv, vi
122
BRUCE KLEINER AND JOHN LOTT
On any subinterval [0, T ′ ] ⊂ [0, T ), since w converges to zeroRat infinity and (M, b g (t)) is standard at infinity, by choosing K appropriately we can make ∂K hw, ∇n wi dvol∂bg(t) small. It follows that there is an exhaustion {Ki }∞ i=1 of M so that Z 1 −2λt 1 2 2λt d e |w(t)| dvolbg(t) ≤ (66.8) e dt 2 i Ki for t ∈ [0, T ′ ]. Then (66.9)
Z
2
Ki
|w(t)| dvolgb(t)
e2λt − 1 ≤ λi
for all t ∈ [0, T ′]. Taking i → ∞ gives w(t) = 0.
c ∈ ST then Thus h = b g . From (66.1), W = 0 and so h = g. This shows that if M, M c M = M. 67. II.3. Structure at the first singularity time
This section is concerned with the structure of the Ricci flow solution at the first singular time, in the case when the solution does go singular. Let M be a connected closed oriented 3-manifold. Let g(·) be a Ricci flow on M defined on a maximal time interval [0, T ) with T < ∞. One knows that limt→T − maxx∈M | Rm |(x, t) = ∞.
From Theorem 26.2 and Theorem 52.7, given ǫ > 0 there are numbers r = r(ǫ) > 0 and κ = κ(ǫ) > 0 so that for any point (x, t) with Q = R(x, t) ≥ r −2 , the solution 1 in P (x, t, (ǫQ)− 2 , (ǫQ)−1 ) is (after rescaling by the factor Q) ǫ-close to the corresponding subset of a κ-solution. By Lemma 59.4, the estimate (59.5) holds at (x, t), provided ǫ is sufficiently small. In addition, there is a neighborhood B of (x, t) as described in Lemma 59.7. In particular, B is a strong ǫ-neck, an ǫ-cap or a closed manifold with positive sectional curvature. If M has positive sectional curvature at some time t then it is diffeomorphic to a finite quotient of the round S 3 and shrinks to a point at time T [35]. The topology of M satisfies the conclusion of the geometrization conjecture and M goes extinct in a finite time. Therefore for the remainder of this section we will assume that the sectional curvature does not become everywhere positive. We now look at the behavior of the Ricci flow as one approaches the singular time T . Definition 67.1. Define a subset Ω of M by (67.2)
Ω = {x ∈ M : sup | Rm |(x, t) < ∞}. t∈[0,T )
Suppose that x ∈ M − Ω, so there is a sequence of times {ti } in [0, T ) with limi→∞ ti = T and limi→∞ | Rm |(x, ti ) = ∞. As minM R(·, t) in nondecreasing in t, the largest sectional curvature at (x, ti ) goes to infinity as i → ∞. Then by the Φ-almost nonnegative sectional curvature result of Appendix B, limi→∞ R(x, ti ) = ∞. From the time-derivative estimate of (59.5), limt→T − R(x, t) = ∞. Thus x ∈ M − Ω if and only if limt→T − R(x, t) = ∞.
NOTES ON PERELMAN’S PAPERS
123
Lemma 67.3. Ω is open in M. Proof. Given x ∈ Ω, using the time-derivative estimate in (59.6) gives a bound of the form |R(x, t)| ≤ C for t ∈ [0, T ). Then using the spatial-derivative estimate in (59.6) gives a number rb > 0 so that so that |R(·, t)| ≤ 2C on B(x, t, rb), for each t ∈ [0, T ). The Φ-almost nonnegative sectional curvature implies a bound of the form | Rm(·, t)| ≤ C ′ on B(x, t, rb), for each t ∈ [0, T ). Then the length-distortion estimate of Section 27 implies that we can pick a neighborhood N of x so that | Rm | ≤ C ′ on N × [0, T ). Thus N ⊂ Ω. Lemma 67.4. Any connected component C of Ω is noncompact. Proof. Since M is connected, if C were compact then it would be all of M. This contradicts the assumption that there is a singularity at time T . We remark that a priori, the structure of M −Ω can be quite complicated. For example, it is not ruled out that an accumulating collection of 2-spheres in M can simultaneously shrink 2 2 to points. That is, M −Ω could have a subset of the form ({0} ∪{ 1i }∞ i=1 ) ×S ⊂ (−1, 1)×S , the picture being that Ω contains a sequence of smaller and smaller adjacent double horns. One could even imagine a Cantor set’s worth of 2-spheres simultaneously shrinking, although conceivably there may be additional arguments to rule out both of these cases. Lemma 67.5. If Ω = ∅ then M is diffeomorphic to S 3 , RP 3, S 1 × S 2 or RP 3 #RP 3 . Proof. The time-derivative estimate in (59.6) implies that for t slightly less than T , we have R(x, t) ≥ r −2 for all x ∈ M. Thus at that time, every x ∈ M has a neighborhood that is in an ǫ-neck or an ǫ-cap, as described in Lemma 59.7. (Recall that we have already excluded the positively-curved case of the lemma.) As in the proof of Lemma 59.1, by splicing together the projection maps associated with neck regions, one obtains an open subset U ⊂ M and a 2-sphere fibration U → N where the fibers are nearly totally geodesic, and the complement of U is contained in a union of ǫ-caps. It follows that U is connected. If there are any ǫ-caps then there must be exactly two of them U1 , U2 , and they may be chosen to intersect U in connected open sets Vi = Ui ∩ U which are isotopic to product regions in both U and in the Ui ’s. The caps being diffeomorphic to B 3 or RP 3 − B 3 , it follows that M is diffeomorphic to S 3 , RP 3 or RP 3 #RP 3 if U 6= M; otherwise M is diffeomorphic to an S 2 bundle over a circle, and the orientability assumption implies that this bundle is diffeomorphic to S 1 × S 2 . In the rest of this section we assume that Ω 6= ∅. From the local derivative estimates of Appendix D, there is a smooth Riemannian metric g = limt→T − g(t) on Ω. Let R denote Ω its scalar curvature. Thus the scalar curvature function extends to a continuous function on the subset (M × [0, T )) ∪ (Ω × {T }) ⊂ M × [0, T ]. Lemma 67.6. (Ω, g) has finite volume. Proof. From the lower scalar curvature bound of (B.2) and the formula dtd vol(M, g(t)) = R 3 − M R dvolM , we obtain an estimate of the form vol(M, g(t)) ≤ const. + const. t 2 , for t < T . The lemma follows.
124
BRUCE KLEINER AND JOHN LOTT
Lemma 67.7. There is a open neighborhood V of (M − Ω) × {T } in M × [0, T ] such that R−1 extends to a continuous function on V which vanishes on (M − Ω) × {T }. Proof. As observed above Lemma 67.3, x ∈ M − Ω if and only if limt→T − R−1 (x, t) = 0. The lemma follows by applying (59.6) to suitable spacetime paths. Definition 67.8. For ρ < 2r , put Ωρ = {x ∈ Ω : R(x) ≤ ρ−2 }. Lemma 67.9. The function R : Ω → R is proper; equivalently, if {xi } ⊂ Ω is a sequence which leaves every compact subset of Ω, then limi→∞ R(xi ) = ∞. In particular, Ωρ is a compact subset of M for every ρ < r. Proof. Suppose {xi } ⊂ Ω is a sequence such that {R(xi )} is bounded. After passing to a subsequence, we may assume that {xi } converges to some point x∞ ∈ M. But R−1 is well-defined and continuous on V , and vanishes on (M − Ω) × {T }, so we must have x∞ ∈ Ω. We now consider the connected components of Ω according to whether they intersect Ωρ or not. First, let C be a connected component of Ω that does not intersect Ωρ . Given x ∈ C, there is a neighborhood Bx of x which is ǫ-close to a region as described in Lemma 59.7. From Lemma 67.4, the neighborhood Bx cannot be of type (c) or (d) in the terminology of Lemma 59.7. We now introduce some terminology. If a manifold Z is diffeomorphic to R3 or RP 3 − B 3 then any embedded 2-sphere Σ ⊂ Z separates Z into two connected subsets, one of which has compact closure and the other contains the end of Z. We refer to the first component as the compact side and the other component as the noncompact side. An open subset R of a Riemannian manifold is a good cylinder if: • It is ǫ-close, modulo rescaling, to a segment of a round cylinder of scalar curvature 1. • The diameter of R is approximately 100 times its cross-section. • Every point in R, lies in an ǫ-neck in the ambient Riemannian manifold. From Lemma 59.7, every ǫ-cap neighborhood Bx contains a good cylinder lying in the ǫ-neck at the end of Bx . Lemma 67.10. Suppose that for all x ∈ C, the neighborhood Bx can be taken to be a strong ǫ-neck as in case (a) of Lemma 59.7. Then C is a double ǫ-horn. Proof. Each point x has an ǫ-neck neighborhood. We can glue these ǫ-necks together to form a submersion from C to a 1-manifold, with fiber S 2 ; cf. the proof of Lemma 59.1. (We can do the gluing by successively adding on good cylinders, where the intersections of successive cylinders have diameter approximately 10 times the diameter of the cross-sections.) In view of Lemma 67.7, it follows in this case that C is a double ǫ-horn.
NOTES ON PERELMAN’S PAPERS
125
Lemma 67.11. Suppose that there is some x ∈ C whose neighborhood Bx is an ǫ-cap as in case (a) of Lemma 59.7. Then C is a capped ǫ-horn. Proof. Put p1 = x. Let R be a good cylinder in the ǫ-neck at the end of Bp1 . Now glue on successive good cylinders to R, as in the proof of the preceding lemma, going away from p1 . Case 1 : Suppose this gluing process can be continued indefinitely. Then taking the union of Bp1 with the good cylinders, we obtain an open subset W of C which is diffeomorphic to R3 or RP 3 − B 3 . We claim that W is a closed subset of Ω. If not then there is a sequence ∞ {xk }∞ k=1 ⊂ W converging to some x∞ ∈ Ω − W . This implies that {R(xk )}k=1 remains bounded. In view of the overlap condition between successive good cylinders, a subsequence of {xk }∞ k=1 lies in an infinite number of mutually disjoint good cylinders, whose volumes have a positive lower bound (because of the upper scalar curvature bound at the points xk ). This contradicts Lemma 67.6. Thus W is open and closed in Ω. Hence W = C and we are done. Case 2 : Now suppose that the gluing process cannot be continued beyond some good cylinder R1 . Then there must be a point p2 ∈ R1 such that Bp2 is an ǫ-cap. Also, note that the union W1 of Bp1 with the good cylinders is diffeomorphic to R3 or RP 3 − B 3 , and that R1 has compact complement in W1 . Let Σ ⊂ R1 be a cross-sectional 2-sphere passing through p2 . We first claim that if V is the compact side of Σ in W1 , then V coincides with the compact side V ′ of Σ in Bp2 . To see this, note that V and V ′ are both connected open sets disjoint from Σ, with topological frontiers ∂V = ∂V ′ = Σ. Then V − V ′ = V ∩ (C − (V ′ ∪ Σ)) and we obtain two open decompositions (67.12)
V = (V ∩ V ′ ) ⊔ (V − V ′ ),
V ′ = (V ∩ V ′ ) ⊔ (V ′ − V ).
If V ∩ V ′ = ∅, then V ∪ V ′ is a union of two compact manifolds with the same boundary Σ, and disjoint interiors. Hence it is an open and closed subset of the connected component C, which contradicts Lemma 67.4. Thus V ∩V ′ is nonempty. By (67.12) and the connectedness of V and V ′ , we get V ⊂ V ′ and V ′ ⊂ V , so V = V ′ as claimed.
Next, we claim that if R2 ⊂ Bp2 is a good cylinder with compact complement in Bp2 , then R2 is disjoint from W1 . To see this, note that R2 is disjoint from Σ because p2 ∈ Σ and the diameter of Σ is close to π(R(p2 )/6)−1/2 , whereas by Lemma 59.7 there is an ǫ-neck U ⊂ Bp2 with compact complement in Bp2 , at distance at least 9000R(p2 )−1/2 from p2 . Thus R2 must lie in the noncompact side of Σ in Bp2 , and hence is disjoint from V . As the good cylinder R1 ∋ p2 lies within B(p2 , 1000R(p2 )−1/2 ) ⊂ Ω, it follows that R2 is also disjoint from R1 , so R2 is disjoint from W1 = V ∪ R1 ⊂ Bp2 .
We continue adding good cylinders to R2 as long as we can. If we come to another cap point p3 then we jump to its cap Bp3 and continue the process. When so doing, we encounter successive cap points p1 , p2 , . . . with associated caps Bp1 ⊂ Bp2 ⊂ . . . and disjoint
good cylinders R1 , R2 , . . .. Since the ratio
supBp inf Bp
k
k
R
R
has an a priori bound by Lemma 59.7,
in view of the disjoint good cylinders in Bpk we get vol(Bpk ) ≥ const. k R(p1 )−3/2 . Then
126
BRUCE KLEINER AND JOHN LOTT
Lemma 67.6 gives an upper bound on k. Hence we encounter a finite number of cap points. Arguing as in Case 1, we conclude that C is a capped ǫ-horn. We note that there could be an infinite number of connected components of Ω that do not intersect Ωρ . Now suppose that C is a connected component of Ω that intersects Ωρ . As C is noncompact, there must be some point x ∈ C that is not in Ωρ . Again, any such x has a neighborhood B as in Lemma 59.7. If one of the boundary components of B intersects Ω2ρ then we terminate the process in that direction. For the directions of the boundary components of B that do not intersect Ω2ρ , we perform the above algorithm of looking for an adjacent ǫ-neck, etc. The only difference from before is that in at least one direction any such sequence of overlapping ǫ-necks will be finite, as it must eventually intersect Ω2ρ . (In the other direction it may terminate in Ω2ρ , in an ǫ-cap, or not terminate at all.) Once a cross-sectional 2-sphere intersects Ω2ρ , if ǫ is small then the entire 2-sphere lies in Ωρ . Thus any connected component of C − (C ∩ Ωρ ) is contained in an ǫ-tube with both boundary components in Ωρ , an ǫ-cap with boundary in Ωρ or an ǫ-horn with boundary in Ωρ . We note that Ωρ need not have a nice boundary. There is an a priori ρ-dependent lower bound for the volume of any such connected component of C − (C ∩ Ωρ ), in view of the fact that it contains ǫ-necks that adjoin Ωρ . From Lemma 67.6, there is a finite number of connected components of Ω that intersect Ωρ . Any such connected component has a finite number of ends, each being an ǫ-horn. Note that the ǫ-horns can be made disjoint, each with a quantitative lower volume bound. The surgery procedure, which will be described in detail in Section 73, is performed as follows. First, one throws away all connected components of Ω that do not intersect Ωρ . For each connected component Ωj of Ω that intersects Ωρ and for each ǫ-horn of Ωj , take a cross-sectional sphere that lies far in the ǫ-horn. Let X be what’s left after cutting the ǫ-horns at these 2-spheres and removing the tips. The (possibly-disconnected) postsurgery manifold M ′ is the result of capping off ∂X by 3-balls. We now discuss how to reconstruct the original manifold M from M ′ . Lemma 67.13. M is the result of taking connected sums of components of M ′ and possibly taking additional connected sums with a finite number of S 1 × S 2 ’s and RP 3 ’s. Proof. At a time shortly before T , each point of M − X has a neighborhood as in S Lemma ′ 59.7. The components of M − X are ǫ-tubes and ǫ-caps. Writing M = X ∪ B 3 and M = X ∪ (M − X), one builds M from M ′ as follows. If the boundary of an ǫ-tube of M − X lies in two disjoint components of X then it gives rise to a connected sum of two components of M ′ . If the boundary of an ǫ-tube lies in a single connected component of X then it gives rise to the connected sum of the corresponding component of M ′ with a new copy of S 1 × S 2 . If an ǫ-cap in M − X is a 3-ball it does not have any effect on M ′ . If an ǫ-cap is RP 3 − B 3 then it gives rise to the connected sum of the corresponding component of M ′ with a new copy of RP 3 . The lemma follows. Remark 67.14. We do not assume that the diameter of (M, g(t)) stays bounded as t → T ; it is an open question whether this is the case.
NOTES ON PERELMAN’S PAPERS
127
68. Ricci flow with surgery: the general setting In this section we introduce some notation and terminology in order to treat Ricci flows with surgery. The principal purpose of sections II.4 and II.5 is to show that one can prescribe the surgery procedure in such a way that Ricci flow with surgery is well-defined for all time. This involves showing that • One can give a sufficiently precise description of the formation of singularities so that one can envisage defining a geometric surgery. In the case of the formation of the first singularity, such a description was given in Section 67. • The sequence of surgery times cannot accumulate.
The argument in Section 67 strongly uses both the κ-noncollapsing result of Theorem 26.2 and the characterization of the geometry in a spacetime region around a point (x0 , t0 ) with large scalar curvature, as given in Theorem 52.7. The proofs of both of these results use the smoothness of the solution at times before t0 . If surgeries occur before t0 then one must have strong control on the scales at which the surgeries occur, in order to extend the arguments of Theorems 26.2 and 52.7. This forces one to consider time-dependent scales. Section II.4 introduces Ricci flow with surgery, in varying degrees of generality. Our treatment of this material follows Perelman’s. We have added some terminology to help formalize the surgery process. There is some arbitrariness in this formalization, but the version given below seems adequate. For later use, we now summarize the relevant notation that we introduce. More precise definitions will be given below. We will avoid using new notation as much as possible. • M is a Ricci flow with surgery. • Mt is the time-t slice of M.
• Mreg is the set of regular points of M.
• If T is a singular time then MT− is the limit of time slices Mt as t → T − (called Ω in II.4.1) and MT+ is the outgoing time slice (for example, the result of performing surgery on + Ω). If T is a nonsingular time then M− T = MT = MT . The basic notion of a Ricci flow with surgery is simply a sequence of Ricci flows which “fit together” in the sense that the final (possibly singular) time slice of each flow is isometric, modulo surgery, to the initial time slice of the next one. Definition 68.1. A Ricci flow with surgery is given by + • A collection of Ricci flows {(Mk ×[t− k , tk ), gk (·))}1≤k≤N , where N ≤ ∞, Mk is a compact + − (possibly empty) manifold, tk = tk+1 for all 1 ≤ k < N, and the flow gk goes singular at t+ k for each k < N. We allow t+ to be ∞. N
• A collection of limits {(Ωk , g¯k )}1≤k≤N , in the sense of Section 67, at the respective final times t+ k that are singular if k < N. (Recall that Ωk is an open subset of Mk .)
− • A collection of isometric embeddings {ψk : Xk+ → Xk+1 }1≤k
128
BRUCE KLEINER AND JOHN LOTT
Xk± ’s are the subsets which survive the transition from one flow to the next, and the ψk ’s give the identifications between them. − + We will say that t is a singular time if t = t+ k = tk+1 for some 1 ≤ k < N, or t = tN and the metric goes singular at time t+ N.
A Ricci flow with surgery does not necessarily have to have any real surgeries, i.e. it could be a smooth nonsingular flow. Our definition allows Ricci flows with surgery that are more general than those appearing in the argument for geometrization, where the transitions/surgeries have a very special form. Before turning to these more special flows in Section 73, we first discuss some basic features of Ricci flow with surgery. It will be convenient to associate a (non-manifold) spacetime M to the Ricci flow with surgery. This is constructed by taking the disjoint union of the smooth manifolds-withboundary + + − + (68.2) Mk × [t− k , tk ) ∪ Ωk × {tk } ⊂ Mk × [tk , tk ]
for 1 ≤ k ≤ N and making identifications using the ψk ’s as gluing maps. We denote the quotient space by M and the quotient map by π. We will sometimes also use M to refer to the whole Ricci flow with surgery structure, rather than just the associated spacetime. The time-t slice Mt of M is the image of the time-t slices of the constituent Ricci flows under the quotient map. + − + If t = t+ k is a singular time then we put Mt = π(Ωk × {tk }); if in addition t 6= tN then + − − we put M+ t = π(Mk+1 × {tk+1 }). If t is not a singular time then we put Mt = Mt = Mt . − We refer to M+ t and Mt as the forward and backward time slices, respectively.
− Let us summarize the structure of M near a singular time t = t+ k = tk+1 . The backward + time slice M− t is a copy of Ωk . The forward time slice Mt is a copy of Mk+1 . The time slice Mt is the result of gluing Ωk and Mk+1 using ψk . Thus it is the disjoint union of Ωk − Xk+ , − − . If s > 0 is small then in going from Mt−s to Mt+s , the and Xk+ ∼ Mk+1 − Xk+1 = Xk+1 − . topological change is that we remove Mk − Xk+ from Mk and add Mk+1 − Xk+1
We let M(t,t′ ) = ∪t¯∈(t,t′ ) Mt¯ denote the time slab between t and t′ , i.e. the union of the time slices between t and t′ . The closed time slab M[t,t′ ] is defined to be the closure of − M(t,t′ ) in M, so M[t,t′ ] = M+ t ∪ M(t,t′ ) ∪ Mt′ . We (ab)use the notation (x, t) to denote a point x ∈ M lying in the time t slice Mt , even though M may no longer be a product.
The spacetime M has three types of points: + 1. The 4-manifold points, which include all points at nonsingular times in (t− 1 , tN ) and all + + − − points in π(Interior(Xk ) × {tk }) (or π(Interior(Xk ) × {tk })) for 1 ≤ k < N, + 2. The boundary points of M, which are the images in M of M1 × {t− 1 }, ΩN × {tN }, + + − − (Ωk − Xk ) × {tk } for 1 ≤ k < N, and (Mk − Xk ) × {tk } for 1 < k ≤ N, and 3. The “splitting” points, which are the images in M of ∂Xk+ × {t+ k } for 1 ≤ k < N.
Here the classification of points is according to the smooth structure, not the topology. In fact, M is a topological manifold-with-boundary. We say that (x, t) is regular if it is either a 4-manifold point, or it lies in the initial time slice Mt−1 or final time slice Mt+ . Let Mreg N denote the set of regular points. It has a natural smooth structure since the gluing maps ψk , being isometries between smooth Riemannian manifolds, are smooth maps.
NOTES ON PERELMAN’S PAPERS
129
Note that the Ricci flows on the Mk ’s define a Riemannian metric g on the “horizontal” subbundle of the tangent bundle of Mreg . It follows from the definition of the Ricci flow that g is actually smooth on Mreg .
We metrize each time slice Mt , and the forward and backward time slices M± t , by infimizing the path length of piecewise smooth paths. We allow our distance functions to be infinite, since the infimum will be infinite when points lie in different components. If (x, t) ∈ Mt and r > 0 then we let B(x, t, r) denote the corresponding metric ball. Similarly, ± B ± (x, t, r) denotes the ball in M± t centered at (x, t) ∈ Mt . A ball B(x, t, r) ⊂ Mt is proper if the distance function d(x,t) : B(x, t, r) → [0, r) is a proper function; a proper ball “avoids singularities”, except possibly at its frontier. Proper balls B ± (x, t, r) ⊂ M± t are defined likewise. An admissible curve in M is a path γ : [c, d] → M, with γ(t) ∈ Mt for all t ∈ [c, d], such + that for each k, the part of γ landing in M[t− t+ ] lifts to a smooth map into Mk × [t− k , tk ) ∪ k k Ωk × {t+ ˙ to denote the “horizontal” part of the velocity of an admissible k }. We will use γ curve γ. If t < t0 , a point (x, t) ∈ M is accessible from (x0 , t0 ) ∈ M if there is an admissible curve running from (x, t) to (x0 , t0 ). An admissible curve γ : [c, d] → M is static if its lifts to the product spaces have constant first component. That is, the points in the image of a static curve are “the same”, modulo the passage of time and identifications taking place at surgery times. A barely admissible curve is an admissible curve γ : [c, d] → M such that − the image is not contained in M+ c ∪ Mreg ∪ Md . If γ : [c, d] → M is barely admissible then − + there is a surgery time t = tk = tk+1 ∈ (c, d) such that γ(t) lies in (68.3)
− − π(∂Xk+ × {t+ k }) = π(∂Xk+1 × {tk+1 }).
If (x, t) ∈ M+ t , r > 0, and ∆t > 0 then we define the forward parabolic region P (x, t, r, ∆t) to be the union of (the images of) the static admissible curves γ : [t, t′ ] → M starting in B + (x, t, r), where t′ ≤ t + ∆t. That is, we take the union of all the maximal extensions of all static curves, up to time t + ∆t, starting from the initial time slice B + (x, t, r). When ∆t < 0, the parabolic region P (x, t, r, ∆t) is defined similarly using static admissible curves ending in B − (x, t, r). If Y ⊂ Mt , and t ∈ [c, d] then we say that Y is unscathed in [c, d] if every point (x, t) ∈ Y lies on a static curve defined on the time interval [c, d]. If, for instance, d = t then this will force Y ⊂ M− t . The term “unscathed” is intended to capture the idea that the set is unaffected by singularities and surgery. (Sometimes Perelman uses the phrase “the solution is defined in P (x, t, r, ∆t)” as synonymous with “the solution is unscathed in P (x, t, r, ∆t)”, for example in the definition of canonical neighborhood in II.4.1.) We may use the notation Y × [c, d] for the set of points lying on static curves γ : [c, d] → M which pass through Y , when Y is unscathed on [c, d]. Note that if Y is open and unscathed on [c, d] then we can think of the Ricci flow on Y × [c, d] as an ordinary (i.e. surgery-free) Ricci flow.
The definitions of ǫ-neck, ǫ-cap, ǫ-tube and (capped/double) ǫ-horn from Section 58 do not require modification for a Ricci flow with surgery, since they are just special types of Riemannian manifolds; they will turn up as subsets of forward or backward time slices of a Ricci flow with surgery. A strong ǫ-neck is a subset of the form U × [c, d] ⊂ M, where
130
BRUCE KLEINER AND JOHN LOTT
U ⊂ M− d is an open set that is unscathed on the interval [c, d], which is a strong ǫ-neck in the sense of Section 58. 69. II.4.1. A priori assumptions This section introduces the notion of canonical neighborhood. The following definition captures the geometric structure that emerges by combining Theorem 52.7 and its extension to Ricci flows with surgery (Section 77) with the geometric description of κ-solutions. The idea is that blowups either yield κ-solutions, whose structure is well understood from Section 59, or there are surgeries nearby in the recent past, in which case the local geometry resembles that of the standard solution. Both alternatives produce canonical neighborhoods. Definition 69.1. (Canonical neighborhoods, cf. Definition in II.4.1) Let ǫ > 0 be small enough so that Lemmas 59.7 and 63.1 hold. Let C1 be the maximum of 30ǫ−1 and the C1 (ǫ)’s of Lemmas 59.7 and 63.1. Let C2 be the maximum of the C2 (ǫ)’s of Lemmas 59.7 and 63.1. Let r : [a, b] → (0, ∞) be a positive nonincreasing function. A Ricci flow with surgery M defined on the time interval [a, b] satisfies the r-canonical neighborhood assumption if every −2 (x, t) ∈ M± has a canonical neighborhood in the t with scalar curvature R(x, t) ≥ r(t) corresponding (forward/backward) time slice, as in Lemma 59.7. More precisely, there is an 1 1 ± rˆ ∈ (R(x, t)− 2 , C1 R(x, t)− 2 ) and an open set U ⊂ M± ˆ) ⊂ U ⊂ B ± (x, t, 2ˆ r) t with B (x, t, r that falls into one of the following categories : (a) U × [t − ∆t, t] ⊂ M is a strong ǫ-neck for some ∆t > 0. (Note that after parabolic rescaling the scalar curvature at (x, t) becomes 1, so the scale factor must be ≈ R(x, t), which implies that ∆t ≈ R(x, t)−1 .) (b) U is an ǫ-cap which, after rescaling, is ǫ-close to the corresponding piece of a κ0 solution or a time slice of a standard solution (cf. Section 60). (c) U is a closed manifold diffeomorphic to S 3 or RP 3 . (d) U is ǫ-close to a closed manifold of constant positive sectional curvature. Moreover, the scalar curvature in U lies between C2−1 R(x, t) and C2 R(x, t). In cases (a), 3 (b), and (c), the volume of U is greater than C2−1 R(x, t)− 2 . In case (c), the infimal sectional curvature of U is greater than C2−1 R(x, t). Finally, we require that (69.2)
3 2
|∇R(x, t)| < ηR(x, t) ,
∂R < ηR(x, t)2 , (x, t) ∂t
where η is the constant from (59.5). Here the time dervative ∂R (x, t) should be interpreted ∂t as a one-sided derivative when the point (x, t) is added or removed during surgery at time t. Remark 69.3. Note that the smaller of the two balls in B ± (x, t, rˆ) ⊂ U ⊂ B ± (x, t, 2ˆ r) is closed, in order to make it easier to check the openness of the canonical neighborhood condition. The requirement that C1 be at least 30ǫ−1 will be used in the proof of Lemma 73.7; see Remark 73.8.
NOTES ON PERELMAN’S PAPERS
131
Remark 69.4. For convenience, in case (b) we have added the extra condition that U is ǫ-close to the corresponding piece of a κ0 -solution or a time slice of a standard solution. One does not need this extra condition, but it is consistent to add it. We remark that when − surgery is performed according to the recipe of Section 73, if a point (p, t) lies in M+ t − Mt (i.e. it is “added” by surgery) then it will sit in an ǫ-cap, because M+ t will resemble a standard solution from Section 60 near (p, t). Points lying somewhat further out on the capped neck will belong to a strong ǫ-neck which extends backward in time prior to the surgery. The next condition, which will ultimately be guaranteed by the Hamilton-Ivey curvature pinching result and careful surgery, is also essential in blowup arguments `a la Section 52. Definition 69.5. (Φ-pinching) Let Φ ∈ C ∞ (R) be a positive nondecreasing function such that for positive s, Φ(s) is a decreasing function which tends to zero as s → ∞. The Ricci s flow with surgery M satisfies the Φ-pinching assumption if for all (x, t) ∈ M, one has Rm(x, t) ≥ −Φ(R(x, t)). We remark that the notion of Φ-pinching here is somewhat different from Perelman’s φpinching. The purpose of this definition is to distill out the properties of the Hamilton-Ivey pinching condition which are needed in the rest of the proof. Definition 69.6. A Ricci flow with surgery satisfies the a priori assumptions if it satisfies the Φ-pinching and r-canonical neighborhood assumptions on the time interval of the flow. Note that the a priori assumptions depend on ǫ, the function r(t) of Definition 69.1 and the function Φ of Definition 69.5. 70. II.4.2. Curvature bounds from the a priori assumptions In this section we state some technical lemmas about Ricci flows with surgery that satisfy the a priori assumptions of the previous section. The first one is the surgery analog of Lemma 52.11. Lemma 70.1. (cf. Claim 1 of II.4.2) Given (x0 , t0 ) ∈ M put Q = |R(x0 , t0 )| + r(t0 )−2 . 1 Then R(x, t) ≤ 8Q for all (x, t) ∈ P (x0 , t0 , 12 η −1 Q− 2 , − 81 η −1 Q−1 ), where η is the constant from (69.2). Proof. The lemma follows from the estimates (69.2). One integrates these derivative bounds 1 along a subinterval of a path that goes in B(x0 , t0 , 12 η −1 Q− 2 ) and then backward in time along a static path. See the proof of Lemma 52.11. We also use the fact that if t′ ≤ t0 and R(x′ , t′ ) ≥ Q then the inequalities (69.2) are valid at (x′ , t′ ), since r(·) is nonincreasing. The next lemma expresses the main consequence of Claim 2 of II.4.2. Lemma 70.2. (cf. Claim 2 of II.4.2) If ǫ is small enough then the following holds. Suppose that M is a Ricci flow with surgery that satisfies the Φ-pinching assumption. Then for any A < ∞ and rb > 0 there exist ξ = ξ(A) > 0 and K = K(A, rb) < ∞ with the following property. Suppose that M also satisfies the r-canonical neighborhood assumption for some function r(·). Then for any time t0 , if (x0 , t0 ) is a point so that Q = R(x0 , t0 ) > 0 satisfies
132
BRUCE KLEINER AND JOHN LOTT 1
Φ(Q) Q
< ξ and (x, t0 ) is a point so that distt0 (x0 , x) ≤ AQ− 2 then R(x, t0 ) ≤ KQ, where K = K(A, r(t0 )). Proof. The proof is similar to the proof of Step 2 of Theorem 52.7. (The canonical neighborhood assumption replaces Step 1 of the proof of Theorem 52.7.) Assuming that the lemma fails, one obtains a piece of a nonflat metric cone as a blowup limit. Using the canonical neighborhood assumption, one concludes that the corresponding points in M have a neighborhood of type (a), i.e. a strong ǫ-neck, since the neighborhoods of type (b), (c) and (d) of Definition 69.1 are not close to a piece of metric cone. A strong ǫ-neck, has the time interval needed to apply the strong maximum principle as in Step 2 of the proof of Theorem 52.7, in order to get a contradiction. 71. II.4.3. δ-necks in ǫ-horns In this section we show that an ǫ-horn has a self-improving property as one goes down the horn. For any δ > 0, if the scalar curvature at a point is sufficiently large then the point actually lies in a δ-neck. In the statement of the next lemma we will write Ω synonymously with the M− T of Section 68. Lemma 71.1. (cf. Lemma II.4.3) Given the pinching function Φ, a number Tb ∈ (0, ∞), a positive nonincreasing function r : [0, Tb] → R and a number δ ∈ 0, 21 , there is a nonincreasing function h : [0, Tb] → R with 0 < h(t) < δ 2 r(t) so that the following property is satisfied. Let M be a Ricci flow with surgery defined on [0, T ), with T < Tb, which satisfies the a priori assumptions (Definition 69.6) and which goes singular at time T . Let (Ω, g) denote the time-T limit, in the sense of Section 67. Put ρ = δ r(T ) and (71.2)
Ωρ = {(x, t) ∈ Ω | R(x, T ) ≤ ρ−2 }.
Suppose that (x, T ) lies in an ǫ-horn H ⊂ Ω whose boundary is contained in Ωρ . Suppose 1 also that R(x, T ) ≥ h−2 (T ). Then the parabolic region P (x, T, δ −1 R(x, T )− 2 , −R(x, T )−1 ) is contained in a strong δ-neck. (As usual, ǫ is a fixed constant that is small enough so that the result holds uniformly with respect to the other variables.) Proof. Fix δ ∈ (0, 1). Suppose that the claim is not true. Then there is a sequence of Ricci flows with surgery Mα and points (xα , T α ) ∈ Mα with T α < Tb such that 1. Mα satisfies the Φ-pinching and r-canonical neighborhood assumptions, 2. Mα goes singular at time T α , 3. (xα , T α ) belongs to an ǫ-horn Hα ⊂ Ωα whose boundary is contained in Ωαρ , and 4. R(xα , T α ) → ∞, but 1 5. For each α, P (xα , T α , δ −1 R(xα , T α )− 2 , −R(xα , T α )−1 ) is not contained in a strong δ-neck.
Recall that when ǫ is small enough, any cross-sectional 2-sphere sitting in an ǫ-neck V ⊂ Hα separates the ends of Hα ; see Section 58. We may find a properly embedded minimizing geodesic γ α ⊂ Hα which joins the two ends of Hα . As γ α must intersect a
NOTES ON PERELMAN’S PAPERS
133 1
cross-sectional 2-sphere containing (xα , T α ), it must pass within distance ≤ 10R(xα , T α )− 2 of (xα , T α ), when ǫ is small. Let y α be the endpoint of γ α contained in Ωαρ and let ybα be the first point, moving along γ α from the noncompact end of Hα toward y α, where −1 R(b y α , T α ) = ρ−2 . As the gradient bound |∇R 2 | ≤ 21 η is valid along γ α starting from ybα and going out the noncompact end (since such points on γ α have scalar curvature greater 1 than r(T α )−2 ), we have distT α (xα , y α ) ≥ distT α (xα , ybα) ≥ η2 ρ − R(xα , T α )− 2 . Let Lα
denote the time-T α distance from xα to the other end of Hα . Since R goes to infinity as one 1 exits the end, Lemma 70.2 implies that limα→∞ R(xα , T α) 2 Lα = ∞. From the existence of 1 γ α , whose length in either direction from xα is large compared to R(xα , T α )− 2 , it is clear that for large α, the canonical neighborhood of (xα , T α) must be of type (a) or (b) in the terminology of Definition 69.1. By Lemmas 67.9 and 70.2, we also know that for any fixed 1 σ < ∞, for large α the ball B(xα , T α, σR(xα , T α )− 2 ) has compact closure in the time-T α slice of Mα .
By Lemma 70.2, after rescaling the metric on the time-T α slice by R(xα , T α ) we have uniform curvature bounds on distance balls. We also have a uniform lower bound on the injectivity radius at (xα , T α) of the rescaled solution, in view of its canonical neighborhood. Hence after passing to a subsequence, we may take a pointed smooth complete limit (M ∞ , x∞ , g∞ ) of the time-T α slices, where the derivative bounds needed to take a smooth limit come from the canonical neighborhood assumption. By the Φ-pinching assumption, M ∞ will have nonnegative curvature.
After passing to a subsequence, we can also assume that the γ α ’s converge to a minimizing geodesic γ in M ∞ that passes within distance 10 from x∞ . The rescaled length of γ α from 1 xα to y α is bounded below by η2 R(xα , T α) 2 ρ − 1 , which tends to infinity as α → ∞. We have shown that the rescaled length of γ α from xα to the other end of Hα also tends to infinity as α → ∞. It follows that γ is bi-infinite. Thus by Toponogov’s theorem, M ∞ splits off an R-factor. Then for large α, the canonical neighborhood of (xα , T α ) must be an ǫ-neck, and M ∞ = R × S 2 for some positively curved metric on S 2 . In particular, M ∞ has scalar curvature uniformly bounded above. Any point x b ∈ M ∞ is a limit of points (b xα , T α ) ∈ Mα . As R∞ (b x) > 0 and R(xα , T α ) → α α α α ∞, it follows that R(b x , T ) → ∞. Then for large α, (b x , T ) is in a canonical neighborhood ∞ which, in view of the R-factor in M , must be a strong ǫ-neck. From the upper bound on the scalar curvature of M ∞ , along with the time interval involved in the definition of a strong ǫ-neck, it follows that we can parabolically rescale the pointed flows (Mα , xα , T α ) by R(xα , T α ), shift time and extract a smooth pointed limiting Ricci flow (M∞ , x∞ , 0) which is defined on a time interval (ξ, 0], for some ξ < 0. In view of the strong ǫ-necks around the points (b xα , T α ), if we take ξ close to zero then we are ensured that the Ricci flow (M∞ , x∞ , 0) has positive scalar curvature R∞ . Given (b x, t) ∈ M∞ , as R∞ (b x, t) > 0 and R(xα , T α ) → ∞, the Φ-pinching implies that the time-t slice M∞ b. Thus M∞ has nonnegative curvature. The timet has nonnegative curvature at x ∞ 0 slice M0 splits off an R-factor, which means that the same will be true of all time slices; cf. the proof of Lemma 61.1. Hence M∞ is a product Ricci flow.
134
BRUCE KLEINER AND JOHN LOTT
Let ξ be the minimal negative number so that after parabolically rescaling the pointed flows (Mα , xα , T α ) by R(xα , T α ), we can extract a limit Ricci flow (M∞ , (x∞ , 0), g∞ (·)) which is the product of R with a positively curved Ricci flow on S 2 , and is defined on the time interval (ξ, 0]. We claim that ξ = −∞. Suppose not, i.e. ξ > −∞. Given (b x, t) ∈ M∞ , as R∞ (b x, t) > 0 and R(xα , T α ) → ∞, it follows that (x, t) is a limit of points α α α (b x , t ) ∈ M that lie in canonical neighborhoods. In view of the R-factor in M∞ , for large α these canonical neighborhoods must be strong ǫ-necks. This implies in particular that −2 ∂R∞ R∞ (b x , t) > 0, so ∂R∂t∞ (b x, t) > 0. Then there is a uniform upper bound Q for the ∂t 1 ∞ scalar curvature on M . Extending backward from a time-(ξ + 100Q ) slice, we can construct ′ ′ a limit Ricci flow that exists on some time interval (ξ , 0] with ξ < ξ. As before, using the strong ǫ-neck condition and the Φ-pinching, if ξ ′ is sufficiently close to ξ then we are ensured that the Ricci flow on (ξ ′ , 0] is the product of R with a positively curved Ricci flow on S 2 . This is a contradiction. Thus we obtain an ancient solution M∞ with the property that each point (x, t) lies in a strong ǫ-neck. Removing the R-factor gives an ancient solution on S 2 . In view of the fact that each time slice is ǫ-close to the round S 2 , up to rescaling, it follows that the ancient solution on S 2 must be the standard shrinking solution (see Sections 40 and 43). Then M∞ is the standard shrinking solution on R × S 2 . Hence for an infinite number 1 of α, P (xα , T α , δ −1 R(xα , T α )− 2 , −R(xα , T α )−1 ) is in fact in a strong δ-neck, which is a contradiction. Remark 71.3. If a given h makes Lemma 71.1 work for a given function r then one can check that logically, h also works for any r ′ with r ′ ≥ r. Because of this, we may assume that h only depends on min r = r(T ) and is monotonically nondecreasing as a function of r(T ). Similarly, if a given h makes Lemma 71.1 work for a given value of δ then h also works for any δ ′ with δ ′ ≥ δ. Thus we may assume that h is monotonically nondecreasing as a function of δ. 72. Surgery and the pinching condition This section describes how one can take a δ-neck satisfying the time-t Hamilton-Ivey pinching condition, and perform surgery so as to obtain a new manifold which also satisfies the time-t pinching condition, and which is δ ′ -close to the standard solution modulo rescaling. Here δ ′ is a nonexplicit function of δ but satisifes the important property that δ ′ (δ) → 0 as δ → 0.
The main geometric idea which handles the delicate part of the surgery procedure is contained in the following lemma. It says that one can “round off” the boundary of an approximate round half-cylinder so as to simultaneously increase the scalar curvature and the minimum of sectional curvature at each point. As the statement of the following lemma involves the curvature operator, we state our conventions. If M has constant sectional curvature k then the curvature operator acts on 2-forms as multiplication by 2k. This is consistent with the usual Ricci flow literature, e.g. [22]. Recall that ǫ is our global parameter, which is taken sufficiently small.
NOTES ON PERELMAN’S PAPERS
135
Lemma 72.1. Let gcyl denote the round cylindrical metric of scalar curvature 1 on R × S 2 . Let z denote the coordinate in the R-direction. Given A > 0, suppose that f : (−A, 0] → R is a smooth function such that • f (k) (0) = 0 for all k ≥ 0. • On (−A, 0),
(72.2)
f (z) < 0 ,
f ′ (z) > 0 ,
f ′′ (z) < 0.
• kf kC 2 < ǫ.
• For every z ∈ (−A, 0),
(72.3)
max (|f (z)|, |f ′ (z)|) ≤ ǫ |f ′′ (z)| .
Then if h0 is a smooth metric on (−A, 0] × S 2 with kh0 − gcyl kC 2 < ǫ and we set h1 = e h0 , it follows that for all p ∈ (−A, 0) × S 2 we have Rh1 (p) > Rh0 (p) − f ′′ (z(p)). Also, if λ1 (p) denotes the lowest eigenvalue of the curvature operator at p then λh1 1 (p) > λh1 0 (p) − f ′′ (z(p)). 2f (z)
Proof. We will use the variational characterization of λ1 (p) : (72.4)
ωij Rijkl ω kl ω6=0 ωij ω ij
λ1 (p) = inf
where ω ∈ Λ2 (Tp M). We will also use the following formulas about curvature quantities for conformally related metrics in dimension 3 : (72.5) Rh1 = e−2f Rh0 − 4 △f − 2 |∇f |2
and (72.6) ij ij j j j i j −2f i i j i 2 i i j e e e e R kl (h1 ) = e R kl (h0 ) − f k δ l + f l δ k + f k δ l − f l δ k − |∇f | δ k δ l − δ l δ k ,
where feij = f;ij − f;i f;j . That is, fe = Hess(f ) − df ⊗ df . The right-hand sides of these expressions are computed using the metric h0 .
To motivate the proof, let us first consider the linearization of these expressions around h0 . Keeping only the linear terms in f gives to leading order, (72.7) and (72.8)
Rh1 ∼ Rh0 −2f Rh0 − 4 △f Rijkl (h1 ) ∼ Rijkl (h0 ) − 2f Rijkl (h0 ) − f; ik δ jl + f; il δ jk + f; jk δ il − f; jl δ i k .
From the assumptions, Rh0 ∼ 1 and f < 0 on (−A, 0) × S 2 . As h0 is close to gcyl , we have △f ∼ f ′′ (z), so −2f Rh0 − 4 △f ≥ − f ′′ (z). Similarly, in the case of gcyl a minimizer ω in (72.4) is of the form ω = X ∧ ∂z , where X is a unit vector in the S 2 -direction. As h0 is close to gcyl , a minimizing ω for h0 will be close to something of the form X ∧ ∂z . Then (72.9)
λh1 1 ∼ λh1 0 − 2f λh1 0 − 2 f ′′ (z) ≥ λh1 0 + 2f (z) |λh1 0 | − 2 f ′′ (z). g
As h0 is close to gcyl , λh1 0 is close to λ1cyl = 0. Then we can use (72.3) to say that 2f (z)|λh1 0 | − 2 f ′′ (z) ≥ − f ′′ (z).
136
BRUCE KLEINER AND JOHN LOTT
The remaining issue is to show that the increase in R and λ1 coming from the linear approximation is still approximately valid in the nonlinear case, provided that ǫ is sufficiently small. For this, we have to show that the increase from the linear approximation dominates the error terms that we have neglected. To deal with the scalar curvature first, from (72.5) we have (72.10) Rh1 = e−2f Rh0 − 4 △gcyl f + 4e−2f (△gcyl f − △f ) − 2 e−2f |∇f |2 ≥ Rh0 − 4 f ′′ (z) + 4e−2f (△gcyl f − △f ) − 2 e−2f |∇f |2 .
Next, there is an estimate of the form (72.11)
|△gcyl f − △f | ≤ const. k h0 − gcyl kC 2 (|f (z)| + |f ′ (z)| + |f ′′ (z)|) ≤ const. ǫ (|f (z)| + |f ′ (z)| + |f ′′(z)|) .
As e−2f ≤ e2ǫ , if ǫ is small then −2f e (72.12) (△gcyl f − △f ) ≤ const. ǫ (|f (z)| + |f ′(z)| + |f ′′ (z)|) Similarly,
e−2f |∇f |2 ≤ const. |f ′(z)|2 ≤ const. ǫ |f ′ (z)|.
(72.13)
When combined with (72.3), if ǫ is taken sufficiently small then (72.14)
− 4 f ′′ (z) + 4e−2f (△gcyl f − △f ) − 2 e−2f |∇f |2 ≥ − f ′′ (z).
This shows the desired estimate for Rh1 (p).
To estimate λh1 1 we use (72.4) and (72.6) to write (72.15)
λh1 1 (p) = e−2f (z)
Comparing with (72.16)
! ωij Rijkl (h0 ) ω kl − 4 ωij feik ω kj − 2 |∇f |2 (z) . inf ω6=0 ωij ω ij ωij Rijkl (h0 ) ω kl ω6=0 ωij ω ij
λh1 0 (p) = inf
gives (72.17)
λh1 0 (p) ≤ e2f (z) λh1 1 (p) +
where ω is a minimizer in (72.15), or (72.18)
4 ωij feik ω kj + 2 |∇f |2 (z), ωij ω ij
λh1 1 (p) ≥ e−2f (z) λh1 0 (p) − 4 e−2f (z)
ωij feik ω kj − 2 e−2f (z) |∇f |2(z). ωij ω ij
Using the variational formula (72.4), one can show that |λh1 0 (p)| ≤ const. ǫ. From eigenvalue perturbation theory [55, Chapter 12], ω will be of the form X ∧ ∂z + O(ǫ) for some unit vector X tangential to S 2 . Then we get an estimate (72.19)
λh1 1 (p) ≥ λh1 0 (p) − 2f ′′ (z) − const. ǫ (|f (z)| + |f ′ (z)| + |f ′′ (z)|) .
From (72.3), if ǫ is taken sufficiently small then λh1 1 (p) − λh1 0 (p) ≥ − f ′′ (z).
NOTES ON PERELMAN’S PAPERS
137
Recall that the initial condition S0 for the standard solution is an O(3)-symmetric metric g0 on R3 with nonnegative curvature operator, whose end is isometric to a round halfcylinder of scalar curvature 1. To facilitate the surgery procedure, we will assume that some metric ball around the O(3)-fixed point has constant positive curvature. Outside of this ball we use radial coordinates (z, θ) ∈ (−B, ∞) × S 2 , with g0 = e2F (z) gcyl . Here gcyl is the round cylindrical metric of scalar curvature one and F ∈ C ∞ (−B, ∞).
Lemma 72.20. Given A > 0, we can choose B > A and F ∈ C ∞ (−B, ∞) so that 1. F ≡ 0 on [0, ∞) × S 2 . 2. The restriction of F to (−A, 0] × S 2 satisfies the hypotheses of Lemma 72.1. 3. The metric e2F (z) gcyl on (−B, ∞) × S 2 has nonnegative sectional curvature and extends smoothly to a metric on R3 by adding a ball of constant positive curvature at {−B} × S 2 . Proof. For a metric of the forme2F (z) gcyl , one computes that the sectional curvatures are − e− 2F F ′′ and e− 2F 12 − (F ′)2 . In particular, the conditions for positive sectional curvature are F ′′ < 0 and |F ′ | < √12 . The 3-sphere of constant sectional curvature k 2 , with two points removed, has a metric given by √ ! √ 1 2 + √ z − log 1 + e 2z . (72.21) Fk (z) = log k 2
(Shifting z gives other metrics of constant curvature k 2 . We have normalized so that the z = 0 slice is the slice of maximal area.) Note that the derivative √
√ 1 e 2z √ D(z) = √ − 2 2 1 + e 2z
(72.22) is independent of k.
Given A > 0, we take F to be 0 on [0, ∞) and of the form c1 ec2 /z on (−A, 0]. We can take the constant c1 > 0 sufficiently small and the constant c2 < ∞ sufficiently large so that the hypotheses of Lemma 72.1 are satisfied. It remains to smoothly cap off ([−A, ∞) × S 2 , e2F (z) gcyl ) with something of positive sectional curvature. With our given choice of F , we have F ′ (−A) ∈ (0, ǫ). As limz→−∞ D(z) = √1 , (−A,0]
2
′ we can choose B > A so that D(−B) > F ′ (−A). As F ′′ (−A) < 0 and D (−B) < 0, we e ′ < 0 and which e : (−B, ∞) → 0, √1 which has D can extend F ′ to a smooth function D 2 coincides with D on a small interval (−B, −B + δ). Putting Z z e (72.23) F (z) = F (0) + D(w) dw, 0
∞
we obtain F ∈ C (B, ∞) which coincides with Fk on (−B, −B + δ), for some k > 0. Then we can glue on a round metric ball of constant curvature k 2 to {−B}×S 2 , in order to obtain the desired metric. In the statement of the next lemma we continue with the metric constructed in Lemma 72.20.
138
BRUCE KLEINER AND JOHN LOTT
Lemma 72.24. There exists δ ′ = δ ′ (δ) with limδ→0 δ ′ (δ) = 0 and a constant δ0 > 0 such that the following holds. Suppose that δ < δ0 , x ∈ {0} × S 2 and h0 is a Riemannian metric on (−A, 1δ ) × S 2 with R(x) > 0 such that: • h0 satisfies the time-t Hamilton-Ivey pinching condition of Definition B.5. 1
• R(x)h0 is δ-close to gcyl in the C [ δ ]+1-topology.
Then there is a smooth metric h on R3 = D3 ∪ (−B, 1δ ) × S 2 such that • h satisfies the time-t pinching condition. • The restriction of h to [0, 1δ ) × S 2 is h0 .
• The restriction of R(x)h to (−B, −A)×S 2 is g0 , the initial metric of a standard solution.
• The restriction of R(x)h to D3 has constant curvature k 2 .
1 • R(x)h is δ ′ -close to e2F gcyl in the C [ δ′ ]+1 -topology on −B, 1δ × S 2 .
Proof. Put (72.25)
A U1 = (−B, − ) × S 2 , 2
1 U2 = (−A, ) × S 2 δ
and let {α1 , α2 } be a C ∞ partition of unity subordinate to the open cover {U1 , U2 } of (−B, 1δ ) × S 2 . We set (72.26)
h = α1 R(x)−1 g0 + α2 e2F h0
on −B, 1δ × S 2 and cap it off with a 3-ball of constant curvature k, as in Lemma 72.20.
Given δ ′ , we claim that if δ is sufficiently small then the conclusion of the lemma holds. The only of the lemma that is not obvious is the pinching condition. Note that on part A 1 2 − 2 , δ × S the metric h agrees with e2F h0 and hence, when δ is sufficiently small, the pinching condition will hold on − A2 , 1δ × S 2 by Lemmas 72.1 and B.6. On the other hand, when δ is sufficiently small, the restrictions of the metrics g0 = e2F gcyl and R(x)e2F h0 to (−A, − A2 ) × S 2 will be very close and will have strictly positive curvature. (The positive curvature for e2F h0 also follows from Lemma 72.1; if δ is small enough then λh1 0 will be close to zero, while −f ′′ (z) is strictly positive for z ∈ (−A, − A2 ).) Thus h will have positive curvature on (−B, − A2 ) × S 2 and the pinching condition will hold there. We have now fixed the initial condition g0 for a standard solution, along with the procedure to meld g0 to an approximate cylinder. 73. II.4.4. Performing surgery and continuing flows This section discusses the surgery procedure and shows how to prolong a Ricci flow with surgery, provided that the a priori assumptions hold.
NOTES ON PERELMAN’S PAPERS
139
Definition 73.1. (Ricci flow with cutoff) Suppose that a ≥ 0 and let M be a Ricci flow with surgery defined on [a, b] that satisfies the a priori assumptions of Definition 69.6. Let δ : [a, b] → (0, δ0 ) be a nonincreasing function, where δ0 is the parameter of Lemma 72.24. Then M is a Ricci flow with (r, δ)-cutoff if at each singular time t, the forward time slice − M+ t is obtained from the backward time slice Ω = Mt by applying the following procedure: A. Discard each component of Ω that does not intersect
(73.2)
Ωρ = {(x, t) ∈ Ω | R(x, t) ≤ ρ−2 },
where ρ = δ(t)r(t). B. In each ǫ-horn Hij of each of the remaining components Ωi , find a point (xij , t) such that R(xij , t) = h−2 , where h = h(t) is as in Lemma 71.1. 1
C. Find a strong δ-neck Uij × [t − h2 , t] containing P (xij , t, δ −1 R(xij , t)− 2 , −R(xij , t)−1 ); this is guaranteed to exist by Lemma 71.1. S D. For each ij, let Sij ⊂ Uij be a cross-sectional 2-sphere containing (xij , t). Cut i Ωi along the Sij ’s and throw away the tips of the horns Hij , to obtain a compact manifoldwith-boundary X having a spherical boundary component for each ij. E. Glue caps onto X, using Lemma 72.24, to obtain the closed manifold M+ t . For concreteness, we take the parameter A of Lemma 72.24 to be 10. The neighborhood of a boundary component of X is parametrized as [−A, δ −1 ) × S 2 , with Sij = {−A} × S 2 . The metric on [0, δ −1 ) × S 2 is unaltered by the surgery procedure. The corresponding region in the new manifold M+ t , minus a metric ball of constant curvature, is parametrized by (−B, δ −1 ) × S 2 . Put Sij′ = {0} × S 2 ⊂ M− t . We will consider the part added by surgery + on Hij to be the 3-disk in Mt bounded by Sij′ . In terms of Definition 68.1, if t = t+ k then S ′ + − + − the subset Xk of Ωk = Mt has boundary ij Sij . The added part Mt − Xk+1 is a union of 3-balls. Remark 73.3. Our definition of surgery differs slightly from that in [52]. The paper [52] has two extra steps involving throwing away certain components of the postsurgery manifold. We omit these steps in order to simplify the definition of surgery, but there is no real loss either way. First, in the setup of [52, Section 4.4], any component of M+ t that is ǫ-close to a metric 3 quotient of the round S is thrown away. The motivation of [52] was to not have to include these in the list of canonical neighborhoods. Such components are topologically standard. We do include such manifolds in the list of canonical neighborhoods and do not throw them away in the surgery procedure. Second, when considering the long-time behavior of Ricci flow in [52, Section 7], any component of M+ t which admits a metric of nonnegative scalar curvature is thrown away. The motivation for this extra step is that any such component admits a metric that is either flat or has finite extinction time. In either case one concludes that the component is a graph manifold and, for the purposes of the geometrization conjecture, is standard. (Recall the definition of graph manifolds from Appendix I.) Again, we do not throw away such components.
140
BRUCE KLEINER AND JOHN LOTT
Note that the definition of Ricci flow with (r, δ)-cutoff also depends on the function r(t) through the a priori assumption. We now state how the topology of the time slice changes when going backward through the singular time t. Recall that for t′ < t close to t, the time slices Mt′ are all diffeomorphic; we refer to this diffeomorphism type as the presurgery manifold, and the forward time slice M+ t as the postsurgery manifold. Lemma 73.4. The presurgery manifold may be obtained from the postsurgery manifold by applying the following operations finitely many times: • Replacing two connected components with their connected sum. • Taking connected sum of a connected component with S 1 × S 2 or RP 3 . • Taking the disjoint union with an additional S 1 × S 2 or an isometric quotient of the round S 3 . Proof. The proof is basically the same as that of Lemma 67.13. The only difference is that we must take into account the compact components of Ω that do not intersect Ωρ ; these are thrown away in Step A. (Such components did not occur in Lemma 67.13 because in Lemma 67.13 we were dealing with the first surgery for the Ricci flow on the initial connected manifold; see Lemma 67.4, which is valid for the first surgery time.) Any such component is diffeomorphic to S 1 × S 2 , RP 3 #RP 3 or a quotient of the round S 3 , in view of the canonical neighborhood assumption; see the proof of Lemma 67.5. − 3 Remark 73.5. When δ > 0 is sufficiently small, we will have vol(M+ t ) < vol(Mt ) − h(t) for each surgery time t ∈ (a, b). This is because each component that is discarded in step D contains at least “half” of the δ-neck Uij , which has volume at least const. δ −1 h(t)3 , while the cap added has volume at most const. h(t)3 .
Remark 73.6. For a Ricci flow with surgery whose original manifold is nonaspherical and irreducible, one wants to know that the Ricci flow goes extinct within a finite time [24, 25, 53]. Consider the effect of a first surgery, say at time t. Among the connected components of the postsurgery manifold M+ t , one will be diffeomorphic to the presurgery manifold and the others will be 3-spheres. Let Nt+ be a component of M+ t that is diffeomorphic to the presurgery manifold. By the nature of the surgery procedure, there is a function ξ defined on a small interval (t − α, t) so that limt′ →t ξ(t′ ) = 1 and for t′ ∈ (t − α, t), there is a homotopyequivalence from (Mt′ , g(t′)) to Nt+ that expands distances by at most ξ(t′ ). Following the subsequent evolution of Nt+ , there is a similar statement for the later singular times. This fact is needed in [24, 25, 53] in order to control the decay of a certain area functional as one goes through a surgery. We discuss how to continue Ricci flows after surgery. We recall that in Definition 68.1 of a Ricci flow with surgery defined on an interval [a, c], the final time slice Mc consists of a single manifold M− c = Ω that may or may not be singular. Lemma 73.7. (Prolongation of Ricci flows with cutoff ) Take the function Φ to be the timedependent pinching function associated to Definition B.5 in Appendix B. Suppose that r and δ are nonincreasing positive functions defined on [a, b]. Let M be a Ricci flow with (r, δ)-cutoff defined on an interval [a, c] ⊂ [a, b]. Provided sup δ is sufficiently small, either (1) M can be prolonged to a Ricci flow with (r, δ)-cutoff defined on [a, b], or
NOTES ON PERELMAN’S PAPERS
141
(2) There is an extension of M to a Ricci flow with surgery defined on an interval [a, T ] with T ∈ (c, b], where a. The restriction of the flow to any subinterval [a, T ′ ], T ′ < T , is a Ricci flow with (r, δ)-cutoff, but b. The r-canonical neighborhood assumption fails at some point (x, T ) ∈ M− T. In particular, the only obstacle to prolongation of Ricci flows with (r, δ)-cutoff is the potential breakdown of the r-canonical neighborhood assumption. Proof. Consider the time slice of Mc = M− c at time c. If it is singular then we perform steps + − A-E of Definition 73.1 to produce Mc ; otherwise we set M+ c = Mc . Since the surgery is done using Lemma 72.24, provided δ > 0 is sufficiently small, the forward time slice M+ c will satisfy the Φ-pinching assumption. We claim that the r-canonical neighborhood assumption holds in M+ c . More precisely, if + −1 − a point (x, c) ∈ Mc lies within a distance of 10ǫ h from the added part M+ c − Mc then −1 + it lies in an ǫ-cap, while if (x, c) lies at distance greater than 10ǫ h from Mc − M− c and −2 has scalar curvature greater than r(c) then it lies in a canonical neighborhood that was 1 present in the presurgery manifold M− c . (We are assuming that ǫ < 100 .) In view of Lemma 63.1, the only point to observe is that points at distance roughly 10ǫ−1 h lie in ǫ-necks, as they are unaltered by the surgery and they were in δ-necks before the surgery. This gives the ǫ-neck needed to define an ǫ-cap. We now prolong M by Ricci flow with initial condition M+ c . If the flow extends smoothly up to time b then we are done because either the canonical neighborhood assumption holds up to time b yielding (a), or it fails at some time in the interval (c, b], and we have (b). Otherwise, there is some time tsing ≤ b at which it goes singular. We add the singular limit Ω at time tsing to obtain a Ricci flow with surgery defined on [a, tsing ]. From Lemma 72.24, M satisfies the Hamilton-Ivey pinching condition of Definition B.5 on [a, tsing ]. As the function r is nonincreasing in t, it follows from Definition 69.1 that the set of times t ∈ [c, tsing ] for which the r-canonical neighborhood assumption holds is relatively open to the right (i.e. if the r-canonical neighborhood assumption holds at time t ∈ [c, tsing ) then it also holds within some interval [t, t′ )). Thus the set of times t ∈ [c, tsing ] for which the r-canonical neighborhood assumption holds is either an interval [c, T ), with T ≤ tsing , or [c, tsing ]. If the set of such times t is (c, T ) for some T ≤ tsing then the lemma holds. Otherwise, the r-canonical neighborhood assumption holds at tsing . In this case we repeat the construction with c replaced by tsing , and iterate if necessary. Either we will reach time b after a finite number of iterations, or we will reach a time T satisfying (2), or we will hit an infinite number of singular times before time b. However, the last possibility cannot occur. A singular time corresponds to a component going extinct or to a surgery. The number of components going extinct before time b can be bounded in terms of the number of surgeries before time b, so it suffices to show that the latter is finite. Each surgery removes a volume of at least h3 , but the lower bound on the scalar curvature during the flow, coming from the maximum principle, gives a finite upper bound on the total volume growth during the complement of the singular times. Remark 73.8. The condition C1 ≥ 30ǫ−1 in Definition 69.1 was in order to ensure that the ǫ-cap coming from a surgery satisfies the requirements to be a canonical neighborhood.
142
BRUCE KLEINER AND JOHN LOTT
74. II.4.5. Evolution of a surgery cap Let M be a Ricci flow with (r, δ)-cutoff. The next result says that provided δ is small, after a surgery at scale h there is a ball B of radius Ah ≫ h centered in the surgery cap whose evolution is close to that of a standard solution for an elapsed time close to h2 , unless another surgery occurs during which the entire ball is thrown away. Note that the elapsed time h2 corresponds, modulo parabolic rescaling, to the duration of the standard solution. Lemma 74.1. (cf. Lemma II.4.5) ˆ θ, rˆ) > 0 with the following For any A < ∞, θ ∈ (0, 1) and rˆ > 0, one can find δˆ = δ(A, property. Suppose that we have a Ricci flow with (r, δ)-cutoff defined on a time interval [a, b] ˆ with min r = r(b) ≥ rˆ. Suppose that there is a surgery time T0 ∈ (a, b), with δ(T0 ) ≤ δ. + Consider a given surgery at the surgery time and let (p, T0 ) ∈ MT0 be the center of the ˆ = h(δ(T0 ), ǫ, r(T0 ), Φ) be the surgery scale given by Lemma 71.1 and put surgery cap. Let h ˆ 2 ). Then one of the two following possibilities occurs : T1 = min(b, T0 + θh ˆ T1 − T0 ). The pointed solution there (with (1) The solution is unscathed on P (p, T0, Ah, respect to the basepoint (p, T0 )) is, modulo parabolic rescaling, A−1 -close to the pointed flow ˆ −2 ], where U0 is an open subset of the initial time slice S0 of a standard on U0 × [0, (T1 − T0 )h solution S and the basepoint is the center c of the cap in S0 . (2) Assertion (1) holds with T1 replaced by some t+ ∈ [T0 , T1 ), where t+ is a surgery time. ˆ becomes extinct at time t+ , i.e. P (p, T0, Ah, ˆ t+ − T0 ) ∩ Moreover, the entire ball B(p, T0 , Ah) + − Mt+ ⊂ Mt+ − Mt+ . Proof. We give a proof with the same ingredients as the proof in [52], but which is slightly rearranged. We first show the following result, which is almost the same as Lemma 74.1. ˆ θ, rˆ) > 0 with Lemma 74.2. For any A < ∞, θ ∈ (0, 1) and rˆ > 0, one can find δˆ = δ(A, the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff defined on a time interval [a, b] with min r = r(b) ≥ rˆ. Suppose that there is a surgery time T0 ∈ (a, b), with ˆ Consider a given surgery at the surgery time and let (p, T0 ) ∈ M+ be the center δ(T0 ) ≤ δ. T0 ˆ = h(δ(T0 ), ǫ, r(T0 ), Φ) be the surgery scale given by Lemma 71.1 and of the surgery cap. Let h ˆ 2 ). Suppose that the solution is unscathed on P (p, T0 , Ah, ˆ T1 − T0 ). put T1 = min(b, T0 + θh Then the pointed solution there (with respect to the basepoint (p, T0 )) is, modulo parabolic ˆ −2 ], where U0 is an open subset rescaling, A−1 -close to the pointed flow on U0 × [0, (T1 − T0 )h of the initial time slice S0 of a standard solution S and the basepoint is the center c of the cap in S0 . Proof. Fix θ and rˆ. Suppose that the lemma is not true. Then for some A > 0, there is a α α sequence {Mα , (pα , T0α )}∞ α=1 of pointed Ricci flows with (r , δ )-cutoff that together provide a counterexample. In particular, 1. limα→∞ δ α (T0α ) = 0. ˆ α , T α − T α ). 2. Mα is unscathed on P (pα, T0α , Ah 1 0 α α c 3. If (M , (ˆ p , 0)) is the pointed Ricci flow arising from (Mα , (pα , T0α )) by a time shift of ˆ α then P (ˆ ˆ α )−2 ) is not A−1 -close to a T0α and a parabolic rescaling by h pα , 0, A, (T1α − T0α )(h pointed subset of a standard solution.
NOTES ON PERELMAN’S PAPERS
143
ˆ α )−2 . (We do not exclude that T2 = 0.) Clearly T2 ≤ θ. Put T2 = lim inf α→∞ (T1α − T0α )(h ˆ α )−2 . Let T3 be After passing to a subsequence, we can assume that T2 = limα→∞ (T1α −T0α )(h the supremum of the set of times τ ∈ [0, T2 ] with the property that we can apply Appendix cα , (ˆ E, if we want, to take a convergent subsequence of the pointed solutions (M pα , 0)) on the time interval [0, τ ] to get a limit solution with bounded curvature. (In applying Appendix E, we use the case l > 0 of Appendix D to get bounds on the curvature derivatives near time 0. In particular, if T2 > 0 then T3 > 0.) From the nature of the surgery gluing in Lemma 72.24, since limα→∞ δ α (T0α ) = 0 we know that we can at least take a limit of the cα , (ˆ pointed solutions (M pα , 0)) on the time interval [0, 0], so T3 is well-defined. Sublemma 74.3. T3 = T2 .
Proof. Suppose not. Consider the interval [0, T3 ) (where we define [0, 0) to be {0}). Given α σ ∈ (0, T2 −T3 ], for any subsequence of {Mα , (ˆ pα , 0)}∞ pα , 0)}∞ α=1 (which we relabel as {M , (ˆ α=1 ) either 1. There is some λ > 0 and an infinite number of α for which the set B(ˆ pα , 0, λ) becomes scathed on [0, T3 + σ], or 2. For each λ > 0 the set P (ˆ pα , 0, λ, T3 + σ) is unscathed for large α, but for each Λ > 0 there is some λΛ > 0 such that lim supα→∞ supP (ˆpα,0,λΛ ,T3 +σ) | Rm | ≥ Λ.
c∞ , (ˆ By Appendix E, after passing to a subsequence, there is a complete limit solution (M p∞ , 0)) defined on the time interval [0, T3 ) with bounded curvature on compact time intervals. Rec∞ , (ˆ label the subsequence by α. By Lemma 60.3, (M p∞ , 0)) must be the same as the c∞ restriction of some standard solution to [0, T3 ). From Lemma 62.1, the curvature of M is uniformly bounded on [0, T3 ); therefore by the canonical neighborhood assumption and equation (69.2), we can choose σ ∈ (0, T2 − T3 ] and Λ′ > 0 so that for any λ > 0, we have lim supα→∞ supP (ˆpα ,0,λ,T3 +σ) | Rm | ≤ Λ′ . However, limα→∞ δ α (T0α ) = 0 and surgeries only occur near the centers of δ-necks. From the curvature bound on the time interval [0, T3 + σ] and the length distortion estimates of Lemma 27.8, for a given λ the balls B(ˆ pα , 0, λ) will stay within a uniformly bounded distance from pˆα on the time interval [0, T3 + σ]. Hence they cannot be scathed on [0, T3 + σ] for an infinite number of α, as the collar length of the δ-neck around the supposed surgery locus would be large enough to prohibit the cap point pˆα from being within a bounded distance from the surgery locus. This, along with the fact that lim supα→∞ supP (ˆpα ,0,λ,T3 +σ) | Rm | ≤ Λ′ for all λ > 0, gives a contradiction.
cα pα , 0)}∞ , Returning to the original sequence {Mα , (pα , T0α )}∞ α=1 and its rescaling {M , (ˆ α=1 we can now take a subsequence that converges on the time interval [0, T2 ), again necessarcαβ , (ˆ ily to a standard solution. Then there will be an infinite subsequence {M pαβ , 0)}∞ β=1 αβ αβ ˆ αβ −2 αβ αβ cα , (ˆ of {M pα , 0)}∞ , with lim (T − T )( h ) = T , so that P (ˆ p , 0, A, (T β→∞ 1 2 α=1 0 1 − αβ ˆ αβ −2 −1 T0 )(h ) ) is A -close to a pointed subset of a standard solution (by the canonical neighborhood assumption, equation (69.2) and Appendix D). This is a contradiction. ˆ T1 −T0 ) We now finish the proof of Lemma 74.1. If the solution is unscathed on P (p, T0 , Ah, then we can apply Lemma 74.2 to see that we are in case (1) of the conclusion of Lemma ˆ T1 − T0 ). 74.1. Suppose, on the other hand, that the solution is scathed on P (p, T0 , Ah,
144
BRUCE KLEINER AND JOHN LOTT
ˆ t − T0 ). We can Let t+ be the largest t so that the solution is unscathed on P (p, T0 , Ah, apply Lemma 74.2 to see that conclusion (1) of Lemma 74.1 holds with T1 replaced by t+ . As surgery is always performed near the middle of a δ-neck, if δˆ << A−1 then the final ˆ t+ − T0 ) cannot intersect a 2-sphere time slice in the parabolic neighborhood P (p, T0, Ah, where a surgery is going to be performed. The only other possibility is that the entire ball ˆ becomes extinct at time t+ . B(p, T0 , Ah)
75. II.4.6. Curves that penetrate the surgery region Let M be a Ricci flow with (r, δ)-cutoff. The next result, Corollary 75.1, says that if δ is sufficiently small then an admissible curve γ which comes close to a surgery cap at a surgery R 2 time will have a large value of γ (R(γ(t)) + |γ(t)| ˙ ) dt. Note that the latter quantity is not quite the same as L(γ), and is invariant under parabolic rescaling. Corollary 75.1 is used in the extension of Theorem 26.2 to Ricci flows with surgery. The idea is that if δ is small and L(x, t) isn’t too large then any L-minimizing sequence of admissible curves joining the basepoint (x0 , t0 ) to (x, t) must avoid surgery regions, and will therefore accumulate on a minimizing L-geodesic.
Corollary 75.1. (cf. Corollary II.4.6) For any l < ∞ and rˆ > 0, we can find A = A(l, rˆ) < ∞ and θ = θ(l, rˆ) with the following property. Suppose that we are in the situation ˆ θ, rˆ). As usual, h ˆ will be the surgery scale coming from of Lemma 74.1, with δ(T0 ) < δ(A, Lemma 71.1. Let γ : [T0 , Tγ ] → M be an admissible curve, with Tγ ∈ (T0 , T1 ]. Suppose that ˆ ˆ Tγ − T0 ), and either γ(T0 ) ∈ B(p, T0 , A2h ), γ([T0 , Tγ )) ⊂ P (p, T0 , Ah, ˆ 2, a. Tγ = T1 = T0 + θ(h) or ˆ × [T0 , Tγ ]. b. γ(Tγ ) ∈ ∂B(p, T0 , Ah) Then
(75.2)
Z
Tγ
T0
2 R(γ(t), t) + |γ(t)| ˙ dt > l.
ˆ θ, rˆ) so as to satisfy Proof. For the moment, fix A < ∞ and θ ∈ (0, 1). Choose δˆ = δ(A, Lemma 74.1. Let M, (p, T0 ), etc., be as in the hypotheses of Lemma 74.1. Let γ : [T0 , Tγ ] → M be a curve as in the hypotheses of the Corollary. From Lemma 74.1, we know that ˆ Tγ − T0 ) ⊂ M, there is a standard solution S such that the parabolic region P (p, T0 , Ah, ˆ −2 ) A−1 -close to a pointed flow with basepoint (p, T0 ), is (after parabolic rescaling by h ˆ −2 . U0 × [0, Tˆγ ] ⊂ S, the latter having basepoint (c, 0). Here U0 ⊂ S0 and Tˆγ = (Tγ − T0 )h −1 Then the image of γ, under the diffeomorphism implicit in the definition of A -closeness, gives rise to a smooth curve γ0 : [0, Tˆγ ] → U0 × [0, Tˆγ ] so that (if A is sufficiently large) :
NOTES ON PERELMAN’S PAPERS
145
3 γ0 (0) ∈ B(c, 0, A), 5 Z Tγ Z Tˆγ 1 |γ| ˙ 2 dt ≥ |γ˙ 0 |2 dt, 2 T0 0 Z Tˆγ Z Tγ 1 R(γ0 (t), t)dt, R(γ(t), t)dt ≥ 2 0 T0
(75.3) (75.4) (75.5) and (a) Tˆγ = θ, or
(b) γ0 (Tˆγ ) 6∈ P (c, 0, 45 A, Tˆγ ).
In case (a) we have, by Lemma 63.1, Z Z Z Tγ 1 θ 1 θ R(γ0 (t), t)dt ≥ const.(1 − t)−1 dt = const. log(1 − θ). (75.6) R(γ(t), t)dt ≥ 2 0 2 0 T0
If we choose θ sufficiently close to 1 then in this case, we can ensure that Z Tγ Z Tγ 2 (75.7) R(γ(t), t) + |γ(t)| ˙ dt ≥ R(γ(t), t) dt > l. T0
T0
In case (b), we may use the fact that the Ricci curvature of the standard solution is everywhere nonnegative, and hence the metric tensor is nonincreasing with time. So if π : S = S0 × [0, 1) → Sθ is projection to the time-θ slice and we put η = π ◦ γ0 then Z ˆ Z ˆ Z Tγ 2 1 Tγ 1 1 Tγ 2 2 2 ˆ (75.8) d(η(0), η(Tγ )) |γ˙0 (t)| dt ≥ |η(t)| ˙ dt ≥ |γ(t)| ˙ dt ≥ 2 0 2 0 2Tˆγ T0 2 1 ≥ d(η(0), η(Tˆγ )) . 2 With our given value of θ, in view of (b), if we take A large enough then we can ensure that 2 1 d(η(0), η(Tˆγ )) > l. This proves the lemma. 2
76. II.4.7. A technical estimate The next result is a technical result that will not be used in the sequel. Corollary 76.1. (cf. Corollary II.4.7) For any Q < ∞ and rˆ > 0, there is a θ = θ(Q, rˆ) ∈ (0, 1) with the following property. Suppose that we are in the situation of Lemma 74.1, with ˆ θ, rˆ) and A > ǫ−1 . If γ : [T0 , Tx ] → M is a static curve starting in B(p, T0 , Ah), ˆ δ(T0 ) < δ(A, and (76.2)
Q−1 R(γ(t)) ≤ R(γ(Tx )) ≤ Q(Tx − T0 )−1
ˆ 2. for all t ∈ [T0 , Tx ], then Tx ≤ T0 + θh
146
BRUCE KLEINER AND JOHN LOTT
Remark 76.3. The hypothesis (76.2) in the corollary means that in the scale of the scalar curvature R(γ(Tx )) at the endpoint γ(Tx ), the scalar curvature on γ is bounded and the elapsed time of γ is bounded. The conclusion says that given these bounds, the elapsed time is strictly less than that of the corresponding rescaled standard solution. ˆ 2 then by Lemma 63.1 and 74.1, Proof. If Tx > T0 + θh ˆ 2 )) ≥ const.(1 − θ)−1 h ˆ −2 . (76.4) R(γ(T0 + θh Thus by (76.2) we get (76.5) or
ˆ −2 ≤ R(γ(Tx )) ≤ Q(Tx − T0 )−1 , Q−1 const.(1 − θ)−1 h
ˆ 2, Tx − T0 ≤ const. Q2 (1 − θ)h ˆ 2 is less than θh ˆ 2 , which gives a If we choose θ close enough to 1 then const. Q2 (1 − θ)h contradiction. (76.6)
77. II.5. Statement of the the existence theorem for Ricci flow with surgery Our presentation of this material follows Perelman’s, except for some shuffling of the material. We will be using some terminology introduced in Section 68, as well as results about the L-function and noncollapsing from Sections 78 and 79. Definition 77.1. A compact Riemannian 3-manifold is normalized if | Rm | ≤ 1 everywhere, and the volume of every unit ball is at least half the volume of the Euclidean unit ball. We will use the fact that a smooth normalized Ricci flow, with bounded curvature on compact time intervals, satisfies the Hamilton-Ivey pinching condition of Definition B.5. The main result of the surgery procedure is Proposition 77.2 (cf. II.5.1), which implies that one can choose positive nonincreasing functions r : R+ → (0, ∞), δ : R+ → (0, ∞) such that the Ricci flow with (r, δ)-surgery flow starting with any normalized initial condition will be defined for all time. The actual statement is structured to facilitate a proof by induction: Proposition 77.2. (cf. Proposition II.5.1) There exist decreasing sequences 0 < rj < ǫ2 , κj > 0, 0 < δ¯j < ǫ2 for 1 ≤ j < ∞, such that for any normalized initial data and any nonincreasing function δ : [0, ∞) → (0, ∞) such that δ < δ¯j on [2j−1 ǫ, 2j ǫ], the Ricci flow with (r, δ)-cutoff is defined for all time and is κ-noncollapsed at scales below ǫ. Here, and in the rest of this section, r and κ will always denote functions defined on an interval [0, T ] ⊆ [0, ∞) with the property that r(t) = rj and κ(t) = κj for all t ∈ [0, T ] ∩ [2j−1 ǫ, 2j ǫ). By “κ-noncollapsed at scales below ǫ”, we mean that for each ρ < ǫ and all (x, t) ∈ M with t ≥ ρ2 , whenever P (x, t, ρ, −ρ2 ) is unscathed and | Rm | ≤ ρ−2 on P (x, t, ρ, −ρ2 ), then we also have vol(B(x, t, ρ)) ≥ κ(t)ρ3 .
Recall that ǫ is a “global” parameter which is assumed to be small, i.e. all statements involving ǫ (explicitly or otherwise) are true provided ǫ is sufficiently small. Proposition 77.2
NOTES ON PERELMAN’S PAPERS
147
does not impose any serious new constraints on ǫ. For example, instead of using the time intervals {[2j−1ǫ, 2j ǫ]}∞ j=1 , we could have taken any collection of adjoining time intervals starting at a small positive time. Also, we just need some fixed upper bound on rj and δ j . We will follow [52] and write these somewhat arbitrary constants in terms of the single global parameter ǫ. Note also that having normalized initial data sets a length scale for the Ricci flow. The phrase “the Ricci flow with (r, δ)-cutoff is defined for all time” allows for the possibility that the entire manifold goes extinct, i.e. that after some time we are talking about the flow on the empty set. In the rest of this section we give a sketch of the proof. The details are in the subsequent sections. Given positive nonincreasing functions r and δ, if one has a normalized initial condition (M, g(0)) then there will be a maximal time interval on which the Ricci flow with (r, δ)-cutoff is defined. This interval can be finite only if it is of the form [0, T ) for some T < ∞, and the Ricci flow with (r, δ)-cutoff on [0, T ) extends to a Ricci flow with surgery on [0, T ] for which the r-canonical neighborhood assumption fails at time T ; see Lemma 73.7. The main point here is that the r-canonical neighborhood assumption allows one to run the flow forward up to the singular time, and then perform surgery, while volume considerations rule out an accumulation of surgery times. Thus the crux of the proof is showing that the functions r and δ can be chosen so that the r-canonical neighborhood assumption will continue to hold, and the Ricci flow with surgery satisfies a noncollapsing condition. The strategy is to argue by induction on i that ri , δ¯i , and κi can be chosen (and δ¯i−1 can be adjusted) so that the statement of the proposition holds on the the finite time interval [0, 2i ǫ]. In the induction step, one establishes the canonical neighborhood assumption using an argument by contradiction similar to the proof of Theorem 52.7. (We recommend that the reader review this before proceeding). The main difference between the proof of Theorem 52.7 and that of Proposition 77.2 is that the non-collapsing assumption, the key ingredient that allows one to implement the blowup argument, is no longer available as a direct consequence of Theorem 26.2, due to the presence of surgeries. We now discuss the augmentations to the non-collapsing argument of Theorem 26.2 necessitated by surgery; this is treated in detail in sections 78 and 79. We first recall Theorem 26.2 and its proof: if a parabolic ball P (x0 , t0 , r0 , −r02 ) in Ricci flow (without surgery) is sufficiently collapsed then one uses the L-function with basepoint (x0 , t0 ), and the L-exponential map based at (x0 , t0 ), to get a contradiction. One considers the reduced volume of a suitably chosen time slice Mt . There is a positive lower bound on the reduced volume coming from the selection of a point where the reduced distance is at most 23 , which in turn comes from an application of the maximum principle to the L-function. On the other hand, there is an upper bound on the reduced volume, which the collapsing forces to be small, thereby giving the contradiction. The upper bound comes from the monotonicity of the weighted Jacobian of the L-exponential map. In fact, this upper bound works without significant modification in the presence of surgery, provided one considers only the reduced volume contributed by those points in the time t slice which may be joined to (x0 , t0 ) by minimizing L-geodesics lying in the regular part of spacetime (see Lemma 78.11).
148
BRUCE KLEINER AND JOHN LOTT
To salvage the lower bound on the reduced volume, the basic idea is that by making the surgery parameter δ small, one can force the L-length of any curve passing close to the surgery locus to be large (Lemma 79.3). This implies that if (x, t) is a point where L isn’t too large, then there will necessarily be an L-geodesic from (x0 , t0 ) to (x, t). To construct the minimizer, one takes a sequence of admissible curves from (x, t) to (x0 , t0 ) with L-length tending to the infimum, and argues that they must stay away from the surgeries; hence they remain in a compact part of spacetime, and subconverge to a minimizer. Therefore the calculations from Sections 15-26 will be valid near such a point (x, t). The maximum principle can then be applied as before to show that the minimum of the reduced length is ≤ 32 on each time slice (see Lemma 78.6). To be more precise, if one makes the surgery parameter δ(t′ ) small for a surgery at a given time t′ then one can force the L-length of any curve passing close to the time-t′ surgery locus to be large, provided that the endtime t0 of the curve is not too large compared to t′ . (If t0 is much larger than t′ then the curve may spend a long time in regions of negative scalar curvature after time t′ . The ensuing negative effect on L could overcome the positive effect of the small surgery parameter.) In the proof of Theorem 26.2, in order to show noncollapsing at time t0 , one went all the way back to a time slice near the initial time and found a point there where l was at most 32 . There would be a problem in using this method for Ricci flows with surgery - we would have to constantly redefine δ(t′ ) to handle the case of larger and larger t0 . The resolution is to not go back to a time slice near the initial time slice. Instead, in order to show κ-noncollapsing in the time slice [2i ǫ, 2i+1 ǫ], we will want to get a lower bound on the reduced volume for a time t-slice with t lying in the preceding time interval [2i−1 ǫ, 2i ǫ]. As we inductively have control over the geometry in the time slice [2i−1 ǫ, 2i ǫ], the argument works equally well.
Finally, as mentioned, after obtaining the a priori κ-noncollapsing estimate on the interval [2 ǫ, 2i+1 ǫ], one proves that the r-canonical neighborhood assumption holds at time T ∈ [2i ǫ, 2i+1 ǫ]. One difference here is that because of possible nearby surgeries, there are two ways to obtain the canonical neighborhood : either from closeness to a κ-solution, as in the proof of Theorem 52.7, or from closeness to a standard solution. i
78. The L-function of I.7 and Ricci flows with surgery In this section we examine several points which arise when one adapts the noncollapsing argument of Theorem 26.2 to Ricci flows with surgery. This material is implicit background for Lemma 79.12 and Proposition 84.1. We will use notation and terminology introduced in Section 68. Let M be a Ricci flow with surgery, and fix a point (x0 , t0 ) ∈ M. One may define the L-length of an admissible curve γ from (x0 , t0 ) to some (x, t), for t < t0 , using the formula Z t0 p t0 − t¯ R + |γ| ˙ 2 dt¯, (78.1) L(γ) = t
where γ˙ denotes the spatial part of the velocity of γ. One defines the L-function on M(−∞,t0 ) by setting L(x, t) to be the infimal L-length of the admissible curves from (x0 , t0 ) to (x, t) if such an admissible curve exists, and infinity otherwise. We note that if (x, t) is in a surgery
NOTES ON PERELMAN’S PAPERS
149
time slice M− t and is actually removed by the surgery then there will not be an admissible curve from (x0 , t0 ) to (x, t). If γ is an admissible curve lying in Mreg then the first variation formula applies. Hence an admissible curve in Mreg from (x0 , t0 ) to (x, t) whose L-length equals L(x, t) will satisfy the L-geodesic equation. If γ is a stable L-geodesic in Mreg then the proof of the monotonicity 3 along γ of the weighted Jacobian τ − 2 exp(−l(τ ))J(τ ) remains valid. Similarly, if U ⊂ M(−∞,t0 ) is an open set such that every (x, t) ∈ U is accessible from (x0 , t0 ) by a minimizing L-geodesic (i.e. an L-geodesic of L-length L(x, t)) contained in Mreg , then the arguments of Section 24 imply that the differential inequality (78.2)
¯ τ + ∆L ¯≤6 L
¯ = 2√τ L and l = holds in U, in the barrier sense, where τ = t0 − t, L
¯ L . 4τ
Lemma 78.3 (Existence of L-minimizers). Let M be a Ricci flow with surgery defined on [a, b]. Suppose that (x0 , t0 ) ∈ M lies in the backward time slice M− t0 .
(1) For each (x, t) ∈ M[a,t0 ) with L(x, t) < ∞, there exists an L-minimizing admissible path γ : [t, t0 ] → M from (x, t) to (x0 , t0 ) which satisfies the L-geodesic equation at every time t ∈ (t, t0 ) for which γ(t) ∈ Mreg .
(2) L is lower semicontinuous on M[a,t0 ) and continuous on Mreg ∩ M[a,t0 ) . (Note that M+ a ⊂ Mreg .)
(3) Every sequence (xj , tj ) ∈ M[a,t0 ) with lim supj L(xj , tj ) < ∞ has a convergent subsequence. Proof. (1) Let {γj : [t, t0 ] → M}∞ j=1 be a sequence of admissible curves from (x, t) to (x0 , t0 ) such that limj→∞ L(γj ) = L(x, t) < ∞. By restricting the sequence, we may assume that supj L(γj ) < 2L(x, t). We claim that there is a subsequence of the γj ’s that (a) converges uniformly to some γ∞ : [t, t0 ] → M,
and
(b) converges weakly to γ∞ in W 1,2 on any subinterval [t′ , t′′ ] ⊂ [t, t0 ) such that [t′ , t′′ ] is free of singular times. To see this, note that on any time interval [c, d] ⊂ [t, t0 ) which is free of singular times, one may apply the Schwarz inequality to the L-length, along with the fact that the metrics on the time slices Mt , t ∈ [c, d], are uniformly biLipschitz to each other, to conclude that the + γj ’s are uniformly H¨older-continuous on [c, d]. We know that γj (t′ ) lies in M− t′ ∩ Mt′ for each surgery time t′ ∈ (t, t0 ), and so one can use similar reasoning to get H¨older control on a short time interval of the form [t′′ , t′ ]. Using a change of variable as in (17.6), one obtains uniform H¨older control near t0 after reparametrizing with s. It follows that the γj ’s are equicontinuous and map into a compact part of spacetime, so Arzela-Ascoli applies; therefore, by passing to a subsequence we may assume that (a) holds. To show (b), we apply weak compactness to the sequence (78.4)
{γj |[t′ ,t′′ ] };
150
BRUCE KLEINER AND JOHN LOTT
this is justified by the fact that the paths γj |[t′ ,t′′ ] ’s remain in a part of M with bounded geometry. Thus we may assume that our sequence {γj } converges uniformly on [t, t0 ] and weakly on every subinterval [t′ , t′′ ] as in (b). By weak lower semicontinuity of L-length, it follows that the W 1,2 -path γ∞ has L-length ≤ L(x, t). Since any W 1,2 path may be approximated in W 1,2 by admissible curves with the same endpoints, it follows that γ∞ minimizes L-length among W 1,2 paths, and therefore it restricts to a smooth solution of the L-geodesic equation on each time interval [t′ , t′′ ] ⊂ [t, t0 ] such that (t′ , t′′ ) is free of singular times. Hence γ∞ is an L-minimizing admissible curve. (2) Pick (x, t) ∈ M[a,t0 ) . To verify lower semicontinuity at (x, t) we suppose the sequence {(xj , tj )} ⊂ M[a,t0 ) converges to (x, t) and lim inf j→∞ L(xj , tj ) < ∞. By (1) there is a sequence {γj } of L-minimizing admissible curves, where γj runs from (xj , tj ) to (x0 , t0 ). By the reasoning above, a subsequence of {γj } converges uniformly and weakly in W 1,2 to a W 1,2 curve γ∞ : [t, t0 ] → M going from (x, t) to (x0 , t0 ), with (78.5)
L(γ∞ ) ≤ lim inf L(γj ). j→∞
Therefore L(x, t) ≤ lim inf j→∞ L(xj , tj ), and we have established semicontinuity. If (x, t) ∈ Mreg , the opposite inequality obviously holds, so in this case (x, t) is a point of continuity.
(3) Because {L(xj , tj )} is uniformly bounded, any sequence {γj } of L-minimizing paths with γj (tj ) = (xj , tj ) will be equicontinuous, and hence by Arzela-Ascoli a subsequence converges uniformly. Therefore a subsequence of {(xj , tj )} converges. The fact that (78.2) can hold locally allows one to appeal – under appropriate conditions – to the maximum principle as in Section 24 to prove that min l ≤ 32 on every time slice. L L = 4τ . Recall that l = 2√ τ
Lemma 78.6. Suppose that M is a Ricci flow with surgery defined on [a, b]. Take t0 ∈ (a, b] and (x0 , t0 ) ∈ M− t0 . Suppose that for every t ∈ [a, t0 ), every admissible curve [t, t0 ] → M ending at (x0 , t0 ) which does not lie in Mreg ∪ M− t0 has reduced length strictly greater than 3 + . Then there is a point (x, a) ∈ M where l(x, a) ≤ 23 . a 2 Remark 78.7. In the lemma we consider the Ricci flow with surgery to begin at time a. − + Hence Mreg ∪ M− t0 = Ma ∪ Mreg ∪ Mt0 and so the hypothesis of the lemma is a statement about the reduced lengths of barely admissible curves, in the sense of Section 68. Proof. As in the case when there are no surgeries, the proof relies on the maximum principle and a continuity argument. Let β : M[a,t0 ) → R ∪ {∞} be the function (78.8)
¯ − 6τ = 4τ β = L
3 l− , 2
where as usual, τ (x, t) = t0 − t. Note that for each τ ∈ (0, t0 − a], the function β attains a minimum βmin (τ ) < ∞ on the slice Mt0 −τ , because by (2) of Lemma 78.3, it is continuous on the compact manifold M+ t0 −τ (as seen by changing the parameter a of Lemma 78.3 to t0 − τ ), and β ≡ ∞ on Mt0 −τ − M+ t0 −τ . Thus it suffices to show that βmin (t0 − a) ≤ 0.
NOTES ON PERELMAN’S PAPERS
151
From Lemma 24.3, βmin (τ ) < 0 for τ > 0 small. Let τ1 ∈ (0, t0 − a] be the supremum of the τ¯ ∈ (0, t0 − a] such that βmin < 0 on the interval (0, τ¯). We claim that
(a) βmin is continuous on (0, τ1 ) and (b) The upper right τ -derivative of βmin is nonpositive on (0, τ1 ). To see (a), pick τ ∈ (0, τ1 ), suppose that {τj } ⊂ (0, τ1 ) is a sequence converging to τ and choose (xj , t0 − τj ) ∈ M+ t0 −τj such that β(xj , t0 − τj ) = βmin (τj ) < 0. By Lemma 78.3 part (3), the sequence {(xj , t0 − τj )} subconverges to some (x, t0 − τ ) ∈ M+ t0 −τ for which β(x, t0 − τ ) ≤ lim inf j→∞ β(xj , t0 − τj ). Thus βmin is lower semicontinuous at τ . On the other hand, since βmin (τ ) < 0, the minimum of β on Mt0 −τ will be attained at a point − + (x, t0 − τ ) ∈ M+ t0 −τ lying in the interior of Mt0 −τ ∩ Mt0 −τ , as β > 0 elsewhere on Mt0 −τ (by Lemma 78.3 and the hypothesis on admissible curves). Therefore β is continuous at (x, t0 − τ ), which implies that βmin is upper semicontinuous at τ . This gives (a).
Part (b) of the claim follows from the fact that if τ ∈ (0, τ1 ) and the minimum of β on Mt0 −τ is attained at (x, t0 − τ ) then l(x, τ ) < 23 , so there is a neighborhood U of (x, t0 − τ ) such that the inequality ∂β (78.9) + ∆β ≤ 0 ∂τ holds in the barrier sense on U (by Lemma 78.3 and the hypothesis on admissible curves). d Hence the upper right derivative ds β(x, τ + s) is nonpositive, so the upper right τ s=0
derivative of βmin (τ ) is also nonpositive.
The claim implies that βmin is nonincreasing on (0, τ1 ), and so lim supτ →τ1− β(τ ) < 0. By parts (2) and (3) of Lemma 78.3, we have βmin(τ1 ) < 0, and the minimum is attained at some (x, t0 − τ1 ) ∈ Mreg . (Recall that Ma ⊂ Mreg .) This implies that τ1 = t0 − a, for otherwise βmin (τ ) would be strictly negative for τ ≥ τ1 close to τ1 , contradicting the definition of τ1 . The notion of local collapsing can be adapted to Ricci flows with surgery, as follows. Definition 78.10. Let M be a Ricci flow with surgery defined on [a, b]. Suppose that (x0 , t0 ) ∈ M and r > 0 are such that t0 − r 2 ≥ a, B(x0 , t0 , r) ⊂ M− t0 is a proper ball and the parabolic ball P (x0 , t0 , r, −r 2 ) is unscathed. Then M is κ-collapsed at (x0 , t0 ) at scale r if | Rm | ≤ r −2 on P (x0 , t0 , r, −r 2 ) and vol(B(x0 , t0 , r)) < κr3 ; otherwise it is κ-noncollapsed. We make use of the following variant of the noncollapsing argument from Section 26.
Lemma 78.11. (Local version of reduced volume comparison) There is a function κ′ : R+ → R+ , satisfying limκ→0 κ′ (κ) = 0, with the following property. Let M be a Ricci flow with surgery defined on √ [a, b]. Suppose that we are given t0 ∈ (a, b], (x0 , t0 ) ∈ Mt0 ∩ Mreg , t ∈ [a, t0 ) and r ∈ (0, t0 − t). Let Y be the set of points (x, t) ∈ Mt that are accessible from (x0 , t0 ) by means of minimizing L-geodesics which remain in Mreg . Assume in addition that M is κ-collapsed at (x0 , t0 ) at scale r, i.e. P (x0 , t0 , r, −r 2 )∩M[t0−r2 ,t0 ) ⊂ Mreg , | Rm | ≤ r −2
152
BRUCE KLEINER AND JOHN LOTT
on P (x0 , t0 , r, −r 2 ), and vol(B(x0 , t0 , r)) < κr3 . Then the reduced volume of Y is at most κ′ (κ). Proof. Let Yˆ ⊂ Tx0 Mt0 be the set of vectors v ∈ Tx0 Mt0 such that there is a minimizing L-geodesic γ : [t, t0 ] → Mreg running from (x0 , t0 ) to some point in Y , with p ˙ t¯) = −v. (78.12) lim t0 − t¯ γ( t¯→t0
The calculations from Sections 17-23 apply to L-geodesics sitting in Mreg . In particular, n the monotonicity of the weighted Jacobian τ − 2 exp(−l(τ ))J(τ ) holds. Now one repeats the proof of Theorem 26.2, working with the set Yˆ instead of the set of initial velocities of all minimizing L-geodesics. 79. Establishing noncollapsing in the presence of surgery The key result of this section, Lemma 79.12, gives conditions under which one can deduce noncollapsing on a time interval I2 , given a noncollapsing bound on a preceding interval I1 and lower bounds on r on I1 ∪ I2 . Definition 79.1. The L+ -length of an admissible curve γ is Z t0 √ 2 (79.2) L+ (γ, τ ) = t0 − t R+ (γ(t), t) + |γ(t)| ˙ dt, t0 −τ
where R+ (x, t) = max(R(x, t), 0).
Lemma 79.3. (Forcing L+ to be large, cf. Lemma II.5.3)
For all Λ < ∞, r¯ > 0 and rˆ > 0, there is a constant F0 = F0 (Λ, r¯, rˆ) with the following property. Suppose that • M is a Ricci flow with (r, δ)-cutoff defined on an interval containing [t, t0 ], where r([t, t0 ]) ⊂ [ˆ r , ǫ],
• r0 ≥ r¯, B(x0 , t0 , r0 ) is a proper ball which is unscathed on [t0 − r02 , t0 ], and | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ),
• γ : [t, t0 ] → M is an admissible curve ending at (x0 , t0 ) whose image is not contained in Mreg ∪ Mt0 , and • δ < F0 (Λ, r¯, rˆ) on [t, t0 ].
Then L+ (γ) > Λ.
Proof. The idea is that the hypotheses on γ imply that it must touch the part of the manifold added during surgery at some time t¯ ∈ [t, t0 ]. Then either γ has to move very fast at times close to t¯ or t0 , or it will stay in the surgery region while it develops large scalar curvature. In the first case L+ (γ) will be large because of the |γ| ˙ 2 term in the formula for L+ , and in the second case it will be large because of the R(γ) term.
NOTES ON PERELMAN’S PAPERS
153
q r¯ . Then since First, we can assume that F0 is small enough so that F0 < 100ǫ 2 (79.4) max h(t) ≤ max δ max r(t) ≤ F02 ǫ, [t,t0 ]
r¯ 100 −2
we have max[t,t0 ] h(t) <
≤
[t,t0 ]
[t,t0 ]
r0 . 100
Put ∆t = 10−10 r¯4 Λ . It always suffices to prove the lemma for a larger value of Λ, so without loss of generality we can assume that ∆t ≤ r¯2 ≤ r02 . Set (79.5)
1
A = A((∆t)− 2 Λ, rˆ),
1
θ = θ((∆t)− 2 Λ, rˆ)
where A(·, ·) and θ(·, ·) are the functions from Corollary 75.1. That is, we will eventually 1 be applying Corollary 75.1 with l = (∆t)− 2 Λ. We impose the additional constraint on F0 that ˆ + 2, θ, rˆ) (79.6) F0 ≤ δ(A on the interval [t, t0 ], where δˆ is the function from Lemma 74.1.
As γ is admissible but is not contained in Mreg ∪ Mt0 , it must pass through the boundary of a surgery cap at some time in the interval [t, t0 ) or it must start in the interior of a surgery cap at time t. By dropping an initial segment of γ if necessary, we may assume that γ(t) lies in a surgery cap. Let x denote the tip of the surgery cap. Note that (79.7)
P (x0 , t0 , r0 , −r02 ) ∩ P (x, t, Ah(t), θh2 (t)) = ∅
−2
4
since by Lemma 74.1 the scalar curvature on P (x, t, Ah(t), θh2 (t)) is at least h 2 > 102 r0−2 , while | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ). Therefore when going backward in time from (x0 , t0 ), γ must leave the parabolic region P (x0 , t0 , r0 , −r02 ) before it arrives at (x, t). If it exits at a time t˜ > t0 − ∆t then applying the Schwarz inequality we get (79.8) Z t0 2 Z t0 −1 Z t0 √ 1 2 2 −1/2 t0 − s |γ(s)| ˙ ds ≥ |γ(s)| ˙ ds (t0 − s) ds ≥ r0 (∆t)−1/2 > Λ, 100 ˜ ˜ ˜ t t t
1 where the factor of 100 comes from the length distortion estimate of Section 27, using the −2 fact that | Rm | ≤ r0 on P (x0 , t0 , r0 , −r02 ). So we can restrict to the case when γ exits P (x0 , t0 , r0 , −∆t) through the initial time slice at time t0 − ∆t. In particular, by (79.7), γ must exit the parabolic region P (x, t, Ah(t), θh2 (t)) by time t0 − ∆t.
By Lemma 74.1, the parabolic region P (x, t, Ah, θh2 ) is either unscathed, or it coincides (as a set) with the parabolic region P (x, t, Ah, s) for some s ∈ (0, θh2 ) and the entire final time slice P (x, t, Ah, s) ∩ Mt+s of P (x, t, Ah, s) is thrown away by a surgery at time t + s.
One possibility is that γ exits P (x, t, Ah, θh2 ) through the final time slice. If this is the case then P (x, t, Ah, θh2 ) must be unscathed (as otherwise the final face is removed by surgery at time t + s < t + θh2 and γ would have nowhere to go after this time), so γ lies in P (x, t, Ah, θh2 ) for the entire time interval [t, t + θh2 ]. The other possibility is that γ leaves P (x, t, Ah, θh2 ) before the final time slice of P (x, t, Ah, θh2 ), in which case it exits the ball B(x, t, Ah) by time t + θh2 .
154
BRUCE KLEINER AND JOHN LOTT
Corollary 75.1 applies to either of these two possibilities. Putting Tγ = sup{t¯ ∈ [t, t + θh2 ] | γ([t, t¯]) ⊂ P (x, t, Ah, θh2 )}
(79.9)
and using the fact that Tγ ≤ t0 − ∆t, we have (79.10) Z t0
Z
Tγ √ 2 t0 − s R+ (γ(s), s) + |γ(s)| ˙ ds ≥ t0 − s R+ (γ(s), s) + |γ(s)| ˙ ds ≥ t t Z Tγ 1/2 2 (∆t) R+ (γ(s), s) + |γ(s)| ˙ ds ≥ (∆t)1/2 l = Λ,
√
2
t
where the last inequality comes from Corollary 75.1 and the choice of A, θ, and δ in (79.5) and (79.6). This completes the proof. Lemma 79.11. If M is a Ricci flow with surgery, with normalized initial condition at time zero, then for all t ≥ 0, R(x, t) ≥ − 23 t+1 1 . 4
Proof. From the initial conditions, Rmin (0) ≥ −6. If the Ricci flow is smooth then (B.2) implies that Rmin (t) ≥ − 32 t+1 1 . If there is a surgery at time t0 then Rmin on M+ t0 equals 4
Rmin on M− t0 , as surgery is done in regions of high scalar curvature. The lemma follows by applying (B.2) on the time intervals between the singular times.
In the statement of the next lemma, one has successive time intervals [a, b) and [b, c). As a mnemonic we use the subscript − for quantities attached to the earlier interval [a, b), and + for those associated with [b, c). We will also assume that the global parameter ǫ is small enough that the Φ-pinching condition implies that whenever | Rm(x, t)| ≥ ǫ−2 , then R(x, t) > | Rm(x,t)| . (We remind the reader of the role of the parameter ǫ; see Remark 58.5.) 100 Lemma 79.12. (Noncollapsing estimate) (cf. Lemma II.5.2) Suppose ǫ ≥ r− ≥ r+ > 0, κ− > 0, E− > 0 and E < ∞. Then there are constants δ = δ(r− , r+ , κ− , E− , E) and κ+ = κ+ (r− , κ− , E− , E) with the following property. Suppose that • a < b < c,
b − a ≥ E− ,
c − a ≤ E,
• M is a Ricci flow with (r, δ)-cutoff with normalized initial condition defined on a time interval containing [a, c), • r ≥ r− on [a, b) and r ≥ r+ on [b, c), •r≤ǫ,
• M is κ− -noncollapsed at scales below ǫ on [a, b) and • δ ≤ δ¯ on [a, c),
Then M is κ+ -noncollapsed at scales below ǫ on [b, c).
Remark 79.13. The important point to notice here is that δ¯ is allowed to depend on the lower bound r+ on [b, c), but the noncollapsing constant κ+ does not depend on r+ .
NOTES ON PERELMAN’S PAPERS
155
p r+ Proof. In the proof, we can assume that 100 ≤ E− /3. If this were not the case then we p could prove the lemma with r+ replaced by 100 E− /3. Then the lemma would also hold for the original value of r+ . First, from Lemma 79.11, R ≥ −6 on M[a,c).
Suppose that r0 ∈ (0, ǫ), (x0 , t0 ) ∈ M[b,c), B(x0 , t0 , r0 ) is a proper ball unscathed on the interval [t0 − r02 , t0 ], and | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ). p r+ We first assume that r0 ≤ E− /3 and r0 ≥ 100 . We will consider L-length, L+ -length, etc in M[a,t0 ) with basepoint at (x0 , t0 ). Suppose that b t ∈ [a, t0 ). Then for any admissible curve γ : [b t, t0 ] → M[a,t0 ] ending at (x0 , t0 ), we have
(79.14)
L(γ) ≤ L+ (γ) ≤ L(γ) +
Z
c
a
√ 3 6 c − t dt ≤ L(γ) + 4E 2 3
and l(x, t) ≥
L+ − 4E 2 1
.
2E 2
1 3 r+ Assume that δ¯ ≤ F0 (4E 2 +4E 2 , 100 , r+ ) where F0 is the function from Lemma 79.3. Then by (79.14) and Lemma 79.3, we conclude that any admissible curve [b t, t0 ] → M[a,t0 ] ending at (x0 , t0 ) which does not lie in Mreg ∪ Mt0 has reduced length bounded below by 2 = 32 + 12 . By Lemma 78.6 there is an admissible curve γ : [a, t0 ] → M ending at (x0 , t0 ) such that √ √ (79.15) L(γ) = L(γ(a)) = 2 t0 − a l(γ(a)) ≤ 3 t0 − a,
so by (79.14) it follows that (79.16)
√ √ 3 3 L+ (γ) ≤ 3 t0 − a + 4E 2 ≤ 3 E + 4E 2 .
Set (79.17)
t1 = a +
b−a , 3
t2 = a +
2(b − a) 3
and (79.18)
√ 1 − 32 3 E− . ρ = 3 E + 4E 2 3
By construction, t2 ≤ t0 − r02 . Note that there is a t¯ ∈ [t1 , t2 ] such that R(γ(t¯)) ≤ ρ. Otherwise we would get (79.19) r r Z t2 Z t2 √ √ 3 1 1 1 L+ (γ) > E− ρ dt ≥ E− E− ρ = 3 E + 4E 2 , t0 − t R+ (γ(t)) dt ≥ 3 3 3 t1 t1 contradicting (79.16). Put x = γ(t). By Lemma 70.1, there is an estimate of the form (79.20)
R ≤ const. s−2
156
BRUCE KLEINER AND JOHN LOTT
−2 ). Appealing to Hamiltonon the parabolic ball Pˆ = P (x, t¯, s, −s2 ) with s−2 = const.(ρ+r− −2 Ivey curvature pinching as usual, we get that | Rm | ≤ const. s in Pˆ . If t − s2 < a then we shrink s (as little as possible) to ensure that Pˆ ⊂ M[a,b) . Provided that δ¯ is less than a small constant c1 = c1 (r− , E− , E), we can guarantee that Pˆ is unscathed, by forcing the curvature in a surgery cap to exceed our bound (79.20) on R. Put U = Mt¯− 1 s2 ∩ Pˆ . If s < ǫ 2 then the κ− -noncollapsing assumption on [a, b) gives a lower bound on vol(B(¯ x, t¯, s))s−3 . If s ≥ ǫ then the κ− -noncollapsing assumption gives a lower bound on vol(B(¯ x, t¯, ǫ/2))(ǫ/2)−3 . In either, case, we get a lower bound on vol(B(¯ x, t¯, s)) and hence a lower bound vol(U) ≥ v = v(r− , κ− , E− , E). Now every point in U can be joined to (x0 , t0 ) by a curve of L+ length at most Λ+ = Λ+ (r− , E− , E), by concatenating an admissible curve [t¯− 12 s2 , t¯] → M (of controlled L+ -length) with γ | . Shrinking δ¯ again, we can apply Lemmas 78.3 and [t¯,t0 ]
79.3 with (79.14) to ensure that every point in U can be joined to (x0 , t0 ) by a minimizing L-geodesic lying in Mreg ∪ Mt0 . Lemma 78.11 then implies that vol(B(x0 , t0 , r0 ))r0−3 ≥ κ1 = κ1 (r− , κ− , E− , E).
(79.21)
(We briefly recall the argument. We have a parabolic ball around (¯ x, t¯), of small but controlled size, on which we have uniform curvature bounds. The lower volume bound coming from the κ− -noncollapsing assumption on [a, b) means that we have bounded geometry on the parabolic ball. As we have a fixed upper bound on l(¯ x, t¯), we can estimate from below + the reduced volume of the accessible points Y ⊂ Mt¯− 1 s2 . Then we obtain a lower bound on vol(B(x0 , t0 , r0 ))r0−3 as in Theorem 26.2.)
2
p r+ . This completes the proof of the lemma when r0 ≤ E− /3 and r0 ≥ 100 p Now suppose that r0 > E− /3. Applying our noncollapsing estimate (79.21) to the ball p of radius E− /3 gives (79.22) 3 (E /3) 23 p (E− /3) 2 − −3 − 23 ≥ κ1 = κ2 , vol(B(x0 , t0 , r0 ))r0 ≥ vol(B(x0 , t0 , E− /3)(E− /3) r03 ǫ3 where κ2 = κ2 (r− , κ− , E− , E).
The next sublemma deals with the case when r0 < Sublemma 79.23. If r0 <
r+ 100
r+ . 100
then vol(B(x0 , t0 , r0 ))r0−3 ≥ κ3 = κ3 (r− , κ− , E− , E).
r+ Proof. Let s be the maximum of the numbers s¯ ∈ [r0 , 100 ] such that B(x0 , t0 , s¯) is unscathed 2 −2 −2 on [t0 − s¯ , t0 ], and | Rm | ≤ s¯ on P (x0 , t0 , s¯, −¯ s ). Then either
(a) Some point (x, t) on the frontier of P (x0 , t0 , s, −s2 ) lies in a surgery cap
or
(b) Some point (x, t) in the closure of P (x0 , t0 , s, −s2 ) has | Rm | = s−2 or
(c) s =
r+ . 100
NOTES ON PERELMAN’S PAPERS
157 −2
In case (a) the scalar curvature at (x, t) will satisfy R(x, t) ∈ ( h 2 , 10h−2 ), since (x, t) lies in the cap at the surgery time. Since | Rm(x, t)| ≤ s−2 we conclude that s ≤ const. h(t). If δ¯ is small then the pointed time slice (Mt , (x, t)) will be close, modulo scaling by h(t), to the initial condition of the standard solution with the basepoint somewhere in the cap. Using the fact that the time slices of P (x0 , t0 , s, −s2 ) have comparable metrics, along with the fact that r0 ≤ s ≤ const. h(t), we get a lower bound vol(B(x0 , t0 , r0 ))r0−3 ≥ const.
In case (b), we have a static curve γ : [t, t0 ] → M such that γ(t) = (x, t), γ(t0 ) ∈ −2 1 −2 B(x0 , t0 , s), and | Rm(x, t)| = s−2 ≥ 104 r+ . Hence by Φ-pinching, R(x, t) ≥ 100 s ≥ −2 100r+ (cf. the remark just before the statement of Lemma 79.12). With reference to the constant η of (69.2), put σ = 10−6 min 1, η1 . Let α : [0, ρb] → B(x0 , t0 , s) ⊂ M− t0 be a
minimizing geodesic from (x0 , t0 ) to γ(t0 ) ∈ B(x0 , t0 , s). We can find a point z along α with distt0 (z, γ(t0 )) ≤ σs and distt0 (z, x0 ) ≤ (1 − σ)s. Let γ¯ : [t, t0 ] → M be the static curve ending at z and put (¯ x, t) = γ¯ (t). In brief, we get (¯ x, t) by “pulling (x, t) slightly inward from the boundary”.
From the distance distortion estimate of Section 27, we can say that distt (¯ x, x) ≤ 106 σs. Then applying (69.2) along a minimizing time-t curve from x¯ to x, we conclude that (79.24) 1
1
1
|R− 2 (¯ x, t) − R− 2 (x, t)| ≤
1 1 η distt (¯ x, x) ≤ s, 2 2
1
−2 1 x, t) ≤ R− 2 (x, t) + 21 s ≤ 20s and hence R(¯ so R− 2 (¯ x, t) ≥ 400 s−2 ≥ 25 r+ . In particular, (¯ x, t) has a canonical neighborhood. We also know that R(¯ x, t) ≤ 6| Rm(¯ x, t)| ≤ 6s−2 . It follows from the definition of canonical neighborhoods (see Definition 69.1) that there is some universal constant so that vol(B(¯ x, t, 10−9 s)) ≥ const. s3 . (We recall that from Lemma 60.3, there is a κ > 0 such that a standard solution is κ-noncollapsed as scales < 1.) The distance distortion estimate ensures that B(¯ x, t, 10−9 s) ⊂ B(z, t0 , 10−6 s) ⊂ B(x0 , t0 , s). Then the standard volume distortion estimate implies that vol(B(x0 , t0 , s)) ≥ const. vol(B(¯ x, t, 10−9s)) ≥ const. s3 , again for some universal constant. Finally we use Bishop-Gromov volume comparison to get vol(B(x0 , t0 , r0 ))r0−3 ≥ const. vol(B(x0 , t0 , s))s−3 .
In case (c) we apply (79.21), replacing the r0 parameter there by s, and Bishop-Gromov volume comparison as in case (b). 80. Construction of the Ricci flow with surgery The proof is by induction on i. To start the induction process, we observe that the initial normalization | Rm | ≤ 1 at t = 0 implies that a smooth solution exists for some definite time [22, Corollary 7.7]. The curvature bound on this time interval, along with the volume assumption on the initial time balls, implies that the solution is κ-noncollapsed below scale 1 and satisfies the ρ-canonical neighborhood assumption vacuously for small ρ > 0. Now assume inductively that rj , κj , and δ¯j have been selected for 1 ≤ j ≤ i, thereby defining the functions r, κ, and δ¯ on [0, 2iǫ], such that for any nonincreasing function δ on ¯ [0, 2i ǫ] satisfying 0 < δ(t) ≤ δ(t), if one has normalized initial data then the Ricci flow with i (r, δ)-cutoff is defined on [0, 2 ǫ] and is κ(t)-noncollapsed at scales < ǫ.
158
BRUCE KLEINER AND JOHN LOTT
We will determine κi+1 using Lemma 79.12. So suppose it is not possible to choose ri+1 and δ¯i+1 (and, if necessary, make δ¯i smaller) so that if we put κi+1 = κ+ (ri , κi , 2i−1 ǫ, 3(2i−1 )ǫ) (where κ+ denotes the function from Lemma 79.12) then the inductive statement above holds with i replaced by i + 1. Then given sequences r α → 0 and δ¯α → 0, for each α there must be a counterexample, say (M α , g α (0)), to the statement with ri+1 = r α and δ¯i = δ¯i+1 = δ¯α . We assume that ˆ 1 − 1 , rα) (80.1) δ¯α < δ(α, α where δˆ is the quantity from Lemma 74.1; this will guarantee that for any A < ∞ and θ ∈ (0, 1) we may apply Lemma 74.1 with parameters A and θ for sufficiently large α. We also assume that ¯ i , r α , κi , 2i−1 ǫ, 3(2i−1 )ǫ) (80.2) δ¯α < δ(r ¯ i , r α , κi , 2i−1 ǫ, 3(2i−1 )ǫ) is from Lemma 79.12. By Lemma 73.7 each initial condition where δ(r (M α , g α (0)) will prolong to a Ricci flow with surgery Mα defined on a time interval [0, T α ] with T α ∈ (2iǫ, ∞], which restricts to a Ricci flow with (r, δ)-cutoff on any proper subinterval [0, τ ] of [0, T α ], but for which the r α -canonical neighborhood assumption fails at some point (¯ xα , T α ) lying in the backward time slice Mα− T α . (It is implicit in this statement that 1 α α α α R(¯ x , T ) ≥ (rα )2 .) Since (M , g (0)) violates the theorem, we must have T α ∈ (2i ǫ, 2i+1 ǫ]. By (80.2) and Lemma 79.12, it follows that Mα is κi+1 -noncollapsed at scales below ǫ on the interval [2i ǫ, T α ), where κi+1 = κ+ (ri , κi , 2i−1ǫ, 3(2i−1 )ǫ) and κ+ denotes the function from Lemma 79.12. cα , (¯ Let (M xα , 0)) be the pointed Ricci flow with surgery obtained from (Mα , (¯ xα , T α )) α α α by shifting time by T and parabolically rescaling by R(¯ x , T ). We also remove the part α α c cα to be diffeomorphic to of (M , (¯ x , 0)) after time zero and we take the time-zero slice M 0 α− MT α . In brief, the rest of the proof goes as follows. If surgeries occur further and further away from (¯ xα , 0) in spacetime as α → ∞, then the reasoning of Theorem 52.7 applies and we obtain a κ-solution as a limit. This would contradict the fact that (¯ xα , T α) does not have a canonical neighborhood. Thus there must be surgeries in a parabolic ball of a fixed size centered at (¯ xα , 0), for arbitrarily large α. Then one argues using Lemma 74.1 that the solution will be close to the (suitably rescaled and time-shifted) standard solution, which again leads to a canonical neighborhood and a contradiction. We now return to the proof. Recall that a metric ball B is proper if the distance function from the center is a proper function on B. If T is a surgery time for a Ricci flow with surgery then a metric ball in M− T need not be proper.
cα whose scalar curvature is strictly greater than Note that by continuity, every point in M α that of (¯ x , 0) has a neighborhood as in Definition 69.1, except that the error estimate is 2ǫ instead of ǫ. cα is proper for sufficiently large Sublemma 80.3. For all λ < ∞, the ball B(¯ xα , 0, λ) ⊂ M 0 α.
cα0 is Proof. As in Lemma 70.2, for each ρ < ∞ the scalar curvature on B(¯ xα , 0, ρ) ⊂ M uniformly bounded in terms of α. (In carrrying out the proof of Lemma 70.2, we now use the
NOTES ON PERELMAN’S PAPERS
159
aforementioned property of having canonical neighborhoods of quality 2ǫ.) Thus B(¯ xα , 0, ρ) cα (see Lemma 67.9), from which the sublemma follows. has compact closure in M 0
Let T1 ∈ [−∞, 0] be the infimum of the set of numbers τ ′ ∈ (−∞, 0] such that for all cα is proper, and unscathed on [τ ′ , 0] for sufficiently large λ < ∞, the ball B(¯ xα , 0, λ) ⊂ M 0 α.
cα , (¯ Lemma 80.4. After passing to a subsequence if necessary, the pointed flows (M xα , 0)) ∞ ∞ converge on the time interval (T1 , 0] to a Ricci flow (without surgery) (M , (¯ x , 0)) with a smooth complete nonnegatively-curved Riemannian metric on each time slice, and scalar curvature globally bounded above by some number Q < ∞. (We interpret (0, 0] to mean {0} rather than the empty set.) Proof. Suppose first that T1 < 0. Then the arguments of Theorem 52.7 apply in the time interval (T1 , 0], to give the Ricci flow (without surgery) (M∞ , (¯ x∞ , 0)). Since r α → 0, ∞ Hamilton-Ivey pinching implies that M will have nonnegative curvature. The fact that cα allows the canonical neighborhood assumption, with ǫ replaced by 2ǫ, holds for each M us to deduce that the scalar curvature of M∞ is globally bounded above by some number Q < ∞; compare with Section 46 and Step 4 of the proof of Theorem 52.7.
Now suppose that T1 = 0. The argument is similar to Steps 2 and 3 of the proof of Theorem 52.7. As in Step 2, or more precisely as in Lemma 70.2, for each ρ < ∞ the cα is uniformly bounded in terms of α. Given ρ < ∞ scalar curvature on B(¯ xα , 0, ρ) ⊂ M 0 cα , if the parabolic region of Lemma 70.1 (centered around and (xα , 0) ∈ B(¯ xα , 0, ρ) ⊂ M cα ) is unscathed then we can apply the 2ǫ-canonical neighborhood assumption (xα , 0) ∈ M cα , Lemma 70.1 and Appendix D to derive bounds on the curvature derivatives at on M cα that depend on ρ but are independent of α. If the parabolic region is scathed (xα , 0) ∈ M 0 then we can apply Lemma 74.1, along with our scalar curvature bound at (xα , 0), to again obtain uniform bounds on the curvature derivatives at (xα , 0). Hence there is a subsequence cα , x¯α )}∞ that converges to a smooth complete of the pointed Riemannian manifolds {(M 0 α=1 ∞ ∞ pointed Riemannian manifold (M0 , x ¯ ). As in the previous case, it will have bounded nonnegative sectional curvature. This proves the lemma. Alternatively, in the case T1 = 0 one can argue directly that if the parabolic region of Lemma 70.1 is scathed then (xα , T α ) has a canonical neighborhood; see the rest of the proof of Proposition 77.2.
If we can show that T1 = −∞ then (M∞ , (¯ x∞ , 0)) will be a κ-solution, which will α α contradict the assumption that (¯ x , T ) does not admit a canonical neighborhood. Suppose that T1 > −∞. We know that for all τ ′ ∈ (T1 , 0] and λ < ∞, the scalar curvature in P (¯ xα , 0, λ, τ ′ ) is bounded by Q + 1 when α is sufficiently large. By Lemma 70.1, there cα is unscathed on exists σ < T1 such that for all λ < ∞, if (for large α) the solution M P (¯ xα , 0, λ, tα ) for some tα > σ then (80.5)
R(x, t) < 8(Q + 2) for all (x, t) ∈ P (¯ xα , 0, λ, tα).
(In applying Lemma 70.1, we use the fact that in the unscaled variables, r(T α )−2 ≤ R(¯ xα , T α ) by assumption, along with the fact that r(·) is a nonincreasing function.)
160
BRUCE KLEINER AND JOHN LOTT
By the definition of T1 , and after passing to a subsequence if necessary, there exist λ < ∞ cα of static curves so that and a sequence γ α : [σ α , 0] → M α α 1. γ (0) ∈ B(¯ x , 0, λ) and 2. The point γ α (σ α ) is inserted during surgery at time σ α > σ. For each α, we may assume that σ α is the largest number having this property. Put ξ α = σ α + (hα (σ α ))2 . (In the notation of [52], (hα (σ α ))2 would be written as R(¯ x, t¯) h2 (T0 ). Note that we have no a priori control on hα (σ α ).) Then ξ α is the blowup time of the cα , γ α (σ α )). We rescaled and shifted standard solution that Lemma 74.1 compares with (M α claim that lim inf α→∞ ξ > 0. Otherwise, Lemma 74.1 would imply that after passing to cα , starting from time σ α , that are better and better a subsequence, there are regions of M approximated by rescaled and shifted standard solutions whose blowup times ξ α have a limit that is nonpositive, thereby contradicting (80.5). Lemma 74.1, along with the fact that R(¯ xα , 0) = 1, also gives a uniform upper bound on ξ α . cα to the time interval Now Lemma 74.1 implies that for large α, the restriction of M α [σ , 0] is well approximated by the restriction to [σ , 0] of a rescaled and shifted standard solution. Then Lemma 63.1 implies that (¯ xα , T α) has a canonical neighborhood. The canonical neighborhood may be either a strong ǫ-neck or an ǫ-cap. (Note a strong ǫ-neck may arise when an ǫ-neck around (¯ xα , T α ) extends smoothly backward in time to form a strong ǫ-neck that incorporates part of the Ricci flow solution that existed before the surgery time σ α .) α
This is a contradiction.
81. II.6. Double sided curvature bound in the thick part Having shown that for a suitable choice of the functions r and δ, the Ricci flow with (r, δ)cutoff exists for all time and for every normalized initial condition, one wants to understand its implications. The main results in II.6 are noncollapsing and curvature estimates which form the basis of the analysis of the large-time behavior given in II.7. Lemma 81.1. If M is a Ricci flow with (r, δ)-cutoff on a compact manifold and g(0) has positive scalar curvature then the solution goes extinct after a finite time, i.e. MT = ∅ for some T > 0. Proof. We apply (B.2). This formula is initially derived for smooth flows but because surgeries are performed in regions of high scalar curvature, it is also valid for a Ricci flow with surgery; cf. the proof of Lemma 79.11. It follows that the flow goes extinct by time 3 . 2Rmin (0) Lemma 81.2. If M is a Ricci flow with surgery that goes extinct after a finite time, then the initial (compact connected orientable) 3-manifold is diffeomorphic to a connected sum of S 1 × S 2 ’s and quotients of the round S 3 . Proof. This follows from Lemma 73.4.
NOTES ON PERELMAN’S PAPERS
161
According to [24, 25] and [53], if none of the prime factors in the Kneser-Milnor decomposition of the initial manifold are aspherical then the Ricci flow with surgery again goes extinct after a finite time. Along with Lemma 81.2, this proves the Poincar´e Conjecture. Passing to Ricci flow solutions that may not go extinct after a finite time, the main result of II.6 is the following : Corollary 81.3. (cf. Corollary II.6.8) For any w > 0 one can find τ = τ (w) > 0, K = K(w) > 0, r = r(w) > 0 and θ = θ(w) > 0 with the following property. Suppose we have a solution to the Ricci flow with (r, δ)-cutoff on the time interval [0, t0 ], with normalized initial data. Let hmax (t0 ) be the maximal surgery radius on [t0 /2, t0 ]. (If there are no surgeries on [t0 /2, t0 ] then√hmax (t0 ) = 0.) Let r0 satisfy 1. θ−1 (w)hmax (t0 ) ≤ r0 ≤ r t0 . 2. The ball B(x0 , t0 , r0 ) has sectional curvatures at least − r0−2 at each point. 3. vol(B(x0 , t0 , r0 )) ≥ wr03. Then the solution is unscathed in P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Corollary 81.3 is an analog of Corollary 55.1, but there are some differences. One minor difference is that Corollary 55.1 is stated as the contrapositive of Corollary 81.3. Namely, Corollary 55.1 assumes that −r0−2 is achieved as a sectional curvature in B(x0 , t0 , r0 ), and its conclusion is that vol(B(x0 , t0 , r0 )) ≤ wr03. The relation with Corollary 81.3 is the following. Suppose that assumptions 1 and 2 of Corollary 81.3 hold. If − r0−2 is achieved somewhere as a sectional curvature in B(x0 , t0 , r0 ) then Hamilton-Ivey pinching implies that the scalar curvature is very large at that point, which contradicts the conclusion of Corollary 81.3. Hence assumption 3 of Corollary 81.3 must not be satisfied. A more substantial difference is that the smoothness of the flow in Corollary 55.1 is guaranteed by the setup, whereas in Corollary 81.3 we must prove that the solution is unscathed in P (x0 , t0 , r0 /4, −τ r02 ).
The role of the parameter r in Corollary 81.3 is essentially to guarantee that we can use Hamilton-Ivey pinching effectively. 82. II.6.5. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and a later volume bound A simpler analog of Corollary 81.3 is the following. Lemma 82.1. (cf. Lemma II.6.5(a)) Given w > 0, there exist τ0 = τ0 (w) > 0 and K0 = K0 (w) < ∞ with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff such that 1. The parabolic neighborhood P (x0 , 0, r0, −τ r02 ) is unscathed, where τ ≤ τ0 . 2. The sectional curvatures are bounded below by − r0−2 on P (x0 , 0, r0 , −τ r02 ). 3. vol(B(x0 , 0, r0 )) ≥ wr03. Then R ≤ K0 τ −1 r0−2 on P (x0 , 0, r0 /4, −τ r02 /2).
162
BRUCE KLEINER AND JOHN LOTT
Proof. Let us first note a consequence of Corollary 45.13. Sublemma 82.2. Given w > 0, there exist τ0 = τ0 (w) > 0 and K = K(w) < ∞ with the following property. Suppose that we have S a Ricci flow with (r, δ)-cutoff such that 1. There are no surgeries on a family t∈[−τ r2 ,0] B(x0 , t, r0 ) of time-dependent balls, where 0 τ ≤ τ0 . 2. The sectional curvatures are bounded below by − r0−2 on the above family of balls. 3. vol(B(x0 , 0, r0 )) ≥ wr03. Then R ≤ K τ −1 r0−2 on
S
t∈[− 43 τ r02 ,0]
B(x0 , t, r0 /2).
Here weShave changed the conclusion of Corollary 45.13 to obtain an upper curvature S bound on t∈[− 3 τ r2 ,0] B(x0 , t, r0 /2) instead of t∈[− 3 τ r2 ,0] B(x0 , t, r0 /4), but this clearly fol0 0 4 4 lows from the arguments of the proof of Corollary 45.13. We now return to the situation of Lemma 82.1.
If τ0 is sufficiently small, then for t ∈ [−τ r02 , 0] and (x, t) ∈ B(x0 , t, 9r0 /10), the lower curvature bound Rm ≥ − r0−2 on P (x0 , 0, r0, −τ r02 ) implies that (x, 0) ∈ B(x0 , 0, r0) (more precisely, that (x, t) lies on a static curve S with one endpoint in B(x0 , 0, r0 ), or equivalently, 2 that (x, t) ∈ P (x0 , 0, r0, −τ r0 )). Thus t∈[−τ r2 ,0] B(x0 , t, 9r0 /10) ⊂ P (x0 , 0, r0 , −τ r02 ) and so 0 S Rm ≥ −r0−2 ≥ −(9r0 /10)−2 on t∈[−τ (9r0 /10)2 ,0] B(x0 , t, 9r0 /10).
Applying Sublemma 82.2 with S r0 replaced by 9r0 /10, and slightly redefining w, gives that −1 −2 R ≤ K τ (9r0 /10) on t∈[− 3 τ (9r0 /10)2 ,0] B(x0 , t, 9r0 /20). Then the length distortion 4 estimate of Lemma 27.8 implies that for sufficiently small τ0 , if (x, 0) ∈ B(x0 , 0, r0/4) then (x, t) ∈ B(x0 , t, 9r0 /20) for t ∈ [−τ r02 /2, 0]. That is,
(82.3)
P (x0 , 0, r0 /4, −τ r02 /2) ⊂
[
B(x0 , t, 9r0/20).
t∈[−τ r02 /2,0]
In applying the length distortion estimate we use the fact that the change in distance is p estimated by ∆d ≤ const. K τ −1 (9r0 /10)−2 · τ r02 /2 which, for small τ0 , is a small fraction of r0 .
Thus we have shown that R ≤ K τ −1 (9r0 /10)−2 on P (x0 , 0, r0 /4, −τ r02 /2). This proves the lemma.
The formulation of [52, Lemma II.6.5] specializes Lemma 82.1 to the case w = 1 − ǫ. It includes the statement [52, Lemma II.6.5(b)] saying that vol(B(x0 , r0 /4, −τ r02 )) is at least 1 of the volume of the Euclidean ball of the same radius. This follows from the proof of 10 Corollary 45.1(b), provided that τ0 is sufficiently small. There is an evident analogy between Lemma 82.1 and Corollary 81.3. However, there is the important difference that Corollary 81.3 (along with Corollary 55.1) only assumes a lower sectional curvature bound at the final time slice.
NOTES ON PERELMAN’S PAPERS
163
83. II.6.6. Locating small balls whose subballs have almost Euclidean volume The result of this section is a technical lemma about volumes of subballs. Lemma 83.1. (cf. Lemma II.6.6) For any b ǫ, w > 0 there exists θ0 = θ0 (b ǫ, w) such that if B(x, 1) is a metric ball of volume at least w, compactly contained in a manifold without boundary with sectional curvatures at least −1, then there exists a subball B(y, θ0) ⊂ B(x, 1) such that every subball B(z, r) ⊂ B(y, θ0 ) of any radius has volume at least (1 − b ǫ) times the volume of the Euclidean ball of the same radius. The proof is similar to that of Lemma 54.1. Suppose that the claim is not true. Then there is a sequence of Riemannian manifolds {Mi }∞ i=1 and balls B(xi , 1) ⊂ Mi with compact closure so that Rm ≥ − 1 and vol(B(xi , 1)) ≥ w, along with a sequence ri′ → 0 so that each B(xi ,1)
subball B(x′i , ri′ ) ⊂ B(xi , 1) has a subball B(x′′i , ri′′ ) ⊂ B(x′i , ri′ ) with vol(B(x′′i , ri′′)) < (1 − b ǫ) ω3 (r ′′ )n . After taking a subsequence, we can assume that limi→∞ (B(xi , 1), xi ) = (X, x∞ ) in the pointed Gromov-Hausdorff topology, where (X, x∞ ) is a pointed Alexandrov space with curvature bounded below by −1. From [12, Theorem 10.8], the Riemannian volume forms dvolMi converge weakly to the three-dimensional Hausdorff measure µ of X. If x′∞ is a regular point of X then there is some δ > 0 so that B(x′∞ , δ) has compact closure in X and b ǫ for all r < δ, µ(B(x′∞ , r)) ≥ (1 − 10 ) ω3 r 3 . Fixing such an r for the moment, for large i there are balls B(x′i , r) ⊂ B(xi , 1) with vol(B(x′i , r)) ≥ (1 − 5bǫ ) ω3 r 3 . Recalling the sequence {ri′ }, by hypothesis there is a subball B(x′′i , ri′′ ) ⊂ B(x′i , ri′ ) with vol(B(x′′i , ri′′)) < (1 − b ǫ) ω3 (ri′′ )3 . ′ ′′ ′ Clearly B(xi , r) ⊂ B(xi , r + ri ). From the Bishop-Gromov inequality, R r+ri′ sinh2 (s) ds vol(B(x′′i , r + ri′ )) 0 ≤ . R ri′′ vol(B(x′′i , ri′′ )) sinh2 (s) ds
(83.2)
0
Then (83.3)
vol(B(x′i , r))
≤
vol(B(x′′i , r
For large i we obtain (83.4)
vol(B(x′i , r))
+
ri′ ))
≤ (1 − b ǫ) ω3 R r′′ i 0
b ǫ ≤ (1 − ) ω3 · 3 2
Z
r
(ri′′ )3 2
sinh (s) ds
Z
r+ri′
sinh2 (s) ds.
0
sinh2 (s) ds.
0
Then if we choose r to be sufficiently small, we contradict the fact that vol(B(x′i , r)) ≥ (1 − 5bǫ ) ω3 r 3 for all i. Remark 83.5. By similar reasoning, for every L > 1 one may find θ1 = θ1 (b ǫ, L) such that under the hypotheses of Lemma 83.1, there is a subball B(y, θ1 ) ⊂ B(x, 1) which is LbiLipschitz to the Euclidean unit ball.
164
BRUCE KLEINER AND JOHN LOTT
84. II.6.8. Proof of the double sided curvature bound in the thick part, modulo two propositions In this section we explain how Corollary 81.3 follows from Lemma 83.1 and two other propositions, which will be proved in subsequent sections. We first state the other propositions, which are Propositions 84.1 and 84.2. Proposition 84.1. (cf. Proposition II.6.3) For any A < ∞ one can find positive constants κ(A), K1 (A), K2 (A), r(A), such that for any t0 < ∞ there exists δ A (t0 ) > 0, decreasing in t0 , with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on a time interval [0, T ], where δ(t) < δ A (t) on [t0 /2, t0 ], with normalized initial data. Assume that 1. The solution is unscathed on a parabolic ball P (x0 , t0 , r0 , −r02 ), with 2r02 < t0 . 2. | Rm | ≤ 3r12 on P (x0 , t0 , r0 , −r02 ). 0 3. vol(B(x0 , t0 , r0 )) ≥ A−1 r03 . Then (a) The solution is κ-noncollapsed on scales less than r0 in B(x0 , t0 , Ar0 ). (b) Every point x ∈ B(x0 , t0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighborhood in the sense of Definition 69.1. √ (c) If r0 ≤ r t0 then R ≤ K2 r0−2 in B(x0 , t0 , Ar0 ). Proposition 84.1(a) is an analog of Theorem 28.2. (The reason for the “3” in the hypothesis | Rm | ≤ 3r12 comes from Remark 28.3.) Propo0 sition 84.1(c) is an analog of Theorem 53.1, but the hypotheses are slightly different. In Proposition 84.1 one assumes a lower bound on the volume of the time-t0 ball B(x0 , t0 , r0 ), while in Theorem 53.1 one assumes assumes a lower bound on the volume of the time(t0 − r02 ) ball B(x0 , t0 − r02 , r0 ). In view of the curvature assumption on P (x0 , t0 , r0 , −r02 ), the hypotheses are essentially equivalent. Conclusions (a), (b) and (c) of Proposition 84.1 are similar to the conclusions of Theorem 28.2, Lemma 53.3 and Theorem 53.1, respectively. Conclusions (a) and (b) of Proposition 84.1 are also related to what was proved in Proposition 77.2 to construct the Ricci flow with surgery. The difference is that the noncollapsing and canonical neighborhood results of Proposition 77.2 are statements at or below the scale √ r(t), whereas Proposition 84.1 is a statement about much larger scales, comparable to t0 . We note that the parameter δ A in Proposition 84.1 is independent of the function δ used to define the Ricci flow with (r, δ)-cutoff. In the proof of the next proposition we will apply Lemma 83.1 with b ǫ equal to the global parameter ǫ, so we will write θ(w) instead of θ(ǫ, w).
Proposition 84.2. (cf. Proposition II.6.4) There exist τ, r, C1 > 0 and K < ∞ with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on √ the time interval [0, t0 ], with normalized initial data. Let r0 satisfy 2C1 hmax (t0 ) ≤ r0 ≤ r t0 , where hmax (t0 ) is the maximal cutoff radius for surgeries in [t0 /2, t0 ]. (If there are no surgeries on [t0 /2, t0 ] then hmax (t0 ) = 0.)
NOTES ON PERELMAN’S PAPERS
165
Assume 1. The ball B(x0 , t0 , r0 ) has sectional curvatures at least −r0−2 at each point. 2. The volume of any subball B(x, t0 , r) ⊂ B(x0 , t0 , r0 ) with any radius r > 0 is at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. Then the solution is unscathed on P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Proposition 84.2 is an analog of Theorem 54.2. However, there is the important difference that in Proposition 84.2 we have to prove that no surgeries occur within P (x0 , t0 , r0 /4, −τ r02 ).
Assuming the validity of Propositions 84.1 and 84.2, suppose that the hypotheses of Corollary 81.3 are satisfied. We will allow ourselves to shrink the parameter r in order to apply Hamilton-Ivey pinching when needed. Put r0′ = θ0 (w) r0 , where θ0 (w) is from Lemma 83.1. By Lemma 83.1, there is a subball B(x′0 , t0 , r0′ ) ⊂ B(x0 , t0 , r0 ) such that every subball of B(x′0 , t0 , r0′ ) has volume at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. As the sectional curvatures are bounded below by −r0−2 on B(x0 , t0 , r0 ), they are bounded below by −(r0′ )−2 on B(x′0 , t0 , r0′ ). By an appropriate 0 (w) , we choice of the parameters θ(w) and r of Corollary 81.3, in particular taking θ(w) ≤ θ2C 1 ′ ′ can ensure that Proposition 84.2 applies to B(x0 , t0 , r0 ). Then the solution is unscathed on P (x′0 , t0 , r0′ /4, −τ (r0′ )2 ) and satisfies | Rm | ≤ K (r0′ )−2 there, where the lower bound on Rm comes from Hamilton-Ivey pinching. With τ being the parameter of Proposition 84.2 and putting r0′′ = min(K −1/2 , τ 1/2 , 14 ) r0′ , for all t′′0 ∈ [t0 − (r0′′ )2 , t0 ] the solution is unscathed on P (x′0 , t′′0 , r0′′, −(r0′′ )2 ) and satisfies | Rm | ≤ (r0′′ )−2 there. From the curvature bound Rm ≥ − (r0′ )−2 on P (x′0 , t0 , r0′ /4, −τ (r0′ )2 ) (coming from pinching) and the fact that B(x′0 , t0 , r0′′ ) has almost Euclidean volume, we obtain a bound vol(B(x′0 , t′′0 , r0′′ )) ≥ const. (r0′′ )3 . Applying 0 Proposition 84.1 with A = 100r gives R ≤ K2 (r0′′ )−2 on B(x0 , t′′0 , 10r0 ) ⊂ B(x′0 , t′′0 , 100r0), r0′′ for all t′′0 ∈ [t0 − (r0′′ )2 , t0 ]. Writing this as R ≤ const. r0−2 , if we further restrict θ(w) to be sufficiently small then we can ensure that R ≤ const. θ2 (w) h−2 ≤ .01 h−2 . As surgeries only occur at spacetime points (x, t) where R(x, t) ∼ h(t)−2 , there are no surgeries S on t′′ ∈[t0 −(r′′ )2 ,t0 ] B(x0 , t′′0 , 10r0). Using length distortion estimates, we can find a parabolic 0 0 S neighborhood P (x0 , t0 , r0 /4, −τ r02 ) ⊂ t′′ ∈[t0 −(r′′ )2 ,t0 ] B(x0 , t′′0 , 10r0 ) for some fixed τ . This 0 0 proves Corollary 81.3. 85. II.6.3. Canonical neighborhoods and later curvature bounds on bigger balls from curvature and volume bounds We now prove Proposition 84.1. We first recall its statement. Proposition 85.1. (cf. Proposition II.6.3) For any A > 0 one can find positive constants κ(A), K1 (A), K2 (A), r(A), such that for any t0 < ∞ there exists δ A (t0 ) > 0, decreasing in t0 , with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on a time interval [0, T ], where δ(t) < δ A (t) on [t0 /2, t0 ], with normalized initial data. Assume that 1. The solution is unscathed on a parabolic neighborhood P (x0 , t0 , r0 , −r02 ), with 2r02 < t0 . 2. | Rm | ≤ r0−2 on P (x0 , t0 , r0 , −r02 ).
166
BRUCE KLEINER AND JOHN LOTT
3. vol(B(x0 , t0 , r0 )) ≥ A−1 r03 . Then (a) The solution is κ-noncollapsed on scales less than r0 in B(x0 , t0 , Ar0 ). (b) Every point x ∈ B(x0 , t0 , Ar0 ) with R(x, t0 ) ≥ K1 r0−2 has a canonical neighborhood in the sense of Definition 69.1. √ (c) If r0 ≤ r t0 then R ≤ K2 r0−2 in B(x0 , t0 , Ar0 ). Proof. The proof of part (a) is analogous to the proof of Theorem 28.2. The proof of part (b) is analogous to the proof of Lemma 53.3. The proof of part (c) is analogous to the proof of Theorem 53.1. We will be brief on the parts of the proof of Proposition 84.1 that are along the same lines as was done before, and will concentrate on the differences. 0) 0) For part (a) : we can reduce the case r0 < r(t to the case r0 ≥ r(t as in Sublemma 100 100 79.23; a canonical neighborhood of type (d) with small volume cannot occur, in view of condition 3 of Proposition 84.1. (We remark that the κ-noncollapsing that we want does not follow from the noncollapsing estimate used in the proof of Proposition 77.2, which p r(t0 ) would give a time-dependent κ.) So we may assume that 100 ≤ r0 ≤ t0 /2. For fixed t0 , this sets a lower bound on r0 .
Now suppose that (x, t0 ) ∈ B(x0 , t0 , Ar0 ), ρ < r0 , the parabolic neighborhood P (x, t0 , ρ, −ρ2 ) is unscathed and | Rm | ≤ ρ−2 there. We want to get a lower bound on ρ−3 vol(B(x, t0 , ρ)). We recall the idea of the proof of Theorem 28.2. With the notation of Theorem 28.2, after rescaling so that r0 = t0 = 1, we had a point x ∈ B(x0 , 1, A) around which we wanted to prove noncollapsing. Defining l using curves starting at (x, 1), we wanted to find a point (y, 1/2) ∈ B(x0 , 1/2, 1/2) so that l(y, 1/2) was bounded above by a universal constant. Given such a point, we concatenated a minimizing L-geodesic (from (x, 1) to (y, 1/2)) with curves emanating backward in time from (y, 1/2). Then the bounded geometry near (y, 1/2) allowed us to estimate from below the reduced volume at a time slightly less than 1/2. We knew that there was some point y ∈ M so that l(y, 1/2) ≤ 23 , but the issue in Theorem 28.2 was to find a point (y, 1/2) ∈ B(x0 , 1/2, 1/2) with l(y, 1/2) bounded above by a universal constant. The idea was to take the proof that some point y ∈ M has l(y, 1/2) ≤ 32 and localize it near x0 . The proof of Theorem 28.2 used the function h(y, t) = nondecreasing function that is one φ(d(y, t)−A(2t−1))(L(y, 1−t)+7). Here φ was a certain√ on (−∞, 1/20) and infinite on [1/10, ∞), and L(q, τ ) = 2 τ L(q, τ ). Clearly min h(·, 1) ≤ 7 and min h(·, 1/2) is achieved in B(x0 , 1/2, 1/10). The equation h ≥ −(6 + C(A))h implied that dtd min h ≥ −(6 + C(A)) min h, and so (min h)(t) ≤ 7 e(6+C(A))(1−t) .
In the present case, if one knew that the possible contribution of a barely admissible curve to h(y, t) was greater than 7 e(6+C(A))(1−t) + ǫ then one could still apply the maximum principle to find a point (y, 1/2) with h(y, 1/2) ≤ 7 e(6+C(A))/2 . For this, it suffices to know that the possible contribution of a barely admissible curve to L(q, τ ) can be bounded below by a sufficiently large number. However, Lemma 79.3 only says that we can make the contribution of a barely admissible curve to L large √ (using the lower scalar curvature bound to pass from L+ to L). Because of the factor 2 τ in the definition of L(q, τ ), we cannot necessarily say that its contribution to L(q, τ ) is large. To salvage the argument, the idea
NOTES ON PERELMAN’S PAPERS
167
is to redefine h and redo the proof of Theorem 28.2 in order to get an extra factor of min h.
√
τ in
(The use of Lemma 79.3 is similar to what was done in the proof of Proposition 77.2. However, there is a difference in scales. In Proposition 77.2 one was working at a microscopic scale in order to construct the Ricci flow with surgery. The function δ(t) in Proposition 77.2 was relevant to this scale. In the present case we are working at the macroscopic scale √ r0 ∼ t0 in order to analyze the long-time behavior of the Ricci flow with surgery. The function δA (t) of Proposition 84.1 is relevant to this scale. Thus we will end up further reducing the surgery function δ(t) of Proposition 77.2 in order to be able to apply Proposition 84.1.) By assumption, | Rm | ≤ 1 at t = 0. From Lemma 79.11, R ≥ − t ∈ [t0 − r02 /2, t0 ], (85.2)
R
r02
1 3 . 2 t+1/4
Then for
r02 3 r02 3 ≥ − = −1. ≥ − 2 t0 − r02 /2 2 2r02 − r02 /2
After rescaling so that r0 = 1, the time interval [t0 − r02 , t0 ] is shifted to [0, 1]. Then for t ∈ [ 21 , 1], we certainly have R ≥ −3. √ Rτ √ ˆ τ) ≡ From this, if 0 < τ ≤ 21 then L(y, τ ) ≥ −6 τ 0 v dv = −4τ 2 , so L(y, √ L(y, τ ) + 2 τ > 0. Putting ˆ τ) h(y, τ ) = φ(dt (x0 , y) − A(2t − 1)) L(y, √ √ and using the fact that dtd τ = − dτd τ = − 2√1 τ , the computations of the proof of Theorem 28.2 give
(85.3)
√ 1 h ≥ − L + 2 τ C(A) φ − 6 φ − √ φ τ 1 φ. = − C(A)h − 6 + √ τ
(85.4)
Then if h0 (τ ) = min h(·, τ ), we have d h0 (τ ) 1 1 φ 1 −1 dh0 (85.5) log √ = h0 − ≤ C(A) + 6 + √ − dτ τ dτ 2τ τ h0 2τ 1 1 1 = C(A) + 6 + √ √ − 2τ τ L+2 τ √ 6 τ + 1 1 . = C(A) + √ − 2τ τ L + 2τ As L ≥ − 4τ 2 , (85.6)
d dτ
√ h0 (τ ) 6 τ + 1 1 50 √ − log √ ≤ C(A) + ≤ C(A) + √ . 2 τ 2τ − 4τ τ 2τ τ
168
BRUCE KLEINER AND JOHN LOTT
As τ → 0, the Euclidean space computation gives L(q, τ ) ∼ |q|2 , so limτ →0 Then √ √ (85.7) h0 (τ ) ≤ 2 τ exp(C(A)τ + 100 τ ). √ This estimate has the desired extra factor of τ .
h0 (τ ) √ τ
= 2.
It now suffices to show that for a barely admissible curve γ that hits a surgery region at time 1 − τ , Z τ √ √ 2 v R(γ(1 − v), v) + |γ(v)| ˙ dv ≥ exp(C(A)τ + 100 τ ) + ǫ, (85.8) 0
where 0 < τ ≤ 12 . Choosing δ A (t0 ) small enough, this follows from Lemma 79.3 along with the lower scalar curvature bound. Then we can apply the maximum principle and follow the proof of Theorem 28.2. In our case the bounded geometry near (y, 1/2) ∈ B(x0 , 1/2, 1/2) comes from assumptions 2 and 3 of the Proposition. The function δ A is now determined. After reintroducing the scale r0 , this proves part (a) of the proposition.
The proof of part (b) is similar to the proofs of Lemma 53.3 and Proposition 77.2. Suppose that for some A > 0 the claim is not true. Then there is a sequence of Ricci flows Mα which together provide a counterexample. In particular, some point (xα , tα ) ∈ B(xα0 , tα0 , Ar0α ) has R(xα , tα ) ≥ K1α (r0α )−2 but does not have a canonical neighborhood, where K1α → ∞ as α → ∞. Because of the canonical neighborhood assumption, we must have K1α (r0α )−2 ≤ r(tα0 )−2 . Then 2K1α ≤ K1α tα0 (r0α )−2 ≤ tα0 r(tα0 )−2 . Since K1α → ∞ and the function t → tr(t)−2 is bounded on any finite t-interval, it follows that tα0 → ∞. Applying point selection to each Mα and removing the superscripts, there are points x ∈ B(x0 , t, 2Ar0 ) with t ∈ [t0 −r02 /2, t0 ] such that Q ≡ R(x, t) ≥ K1 r0−2 and (x, t) does not have a canonical neighborhood, but each point (x, t) ∈ P with R(x, t) ≥ 4Q does have a canonical neighborhood, where −1 1/2 −1/2 P = {(x, t) : dt (x0 , x) ≤ dt (x0 , x) + K1 Q , t ∈ [t − 14 K1 Q , t]}. From (a), we have −1 noncollapsing in P . Rescaling by Q , we have bounded curvature at bounded distances from x; see Lemma 70.2. Then we can extract a pointed limit X∞ , which we think of as a time zero slice, that will have nonnegative sectional curvature. (The required pinching for the last statement comes from the assumption that 2r02 < t0 , along with the fact that K1α → ∞.) The fact that points (x, t) ∈ P with R(x, t) ≥ 4Q have a canonical neighborhood implies that regions of large scalar curvature in X∞ have canonical neighborhoods, from which one can deduce as in Section 46 that the sectional curvatures of X∞ are globally bounded above by some Q0 > 0. Then for each A, Lemmas 27.8 and 70.1 imply that for large α, the −1/2 −1 , −ǫη −1 Q−1 parabolic neighborhood P (x, t, A Q 0 Q ) is contained in P . (Here ǫ is a small parameter, which we absorb in the global parameter ǫ.) In applying Lemma 27.8 we use the curvature bound near x0 coming from the hypothesis of the proposition along with the curvature bound near x just derived; cf. the proof of Lemma 53.3. In addition, we claim −1/2 −1 that P (x, t, A Q , −ǫη −1 Q−1 0 Q ) is unscathed. This is proved as in Section 80. Recall −1/2 −1 that the idea is to show that a surgery in P (x, t, A Q , −ǫη −1 Q−1 0 Q ) implies that (x, t) lies in a canonical neighborhood, which contradicts our assumption. In the argument we use the fact that tα0 → ∞ implies δ(tα0 ) → 0 in order to rule out surgeries; this is the replacement α for the condition δ → 0 that was used in Section 80.
NOTES ON PERELMAN’S PAPERS
169
We extend X∞ to the maximal backward-time limit and obtain an ancient κ-solution, which contradicts the assumption that the points (x, t) did not have canonical neighborhoods. This proves part (b) of the proposition. To prove part (c), we can rescale t0 to 1 and then apply Lemma 70.2; see the end of the proof of Theorem 53.1. The Φ-pinching that we use comes from the Hamilton-Ivey estimate of (B.4). We recall that in the proof of Theorem 53.1 we need to get nonnegative curvature in the region W near the blowup point; this comes from the fact that r α → 0 in the contradiction argument, along with the Hamilton-Ivey pinching. This proves the proposition. In what follows, we will want to apply it freely for arbitrary A, provided that t0 is large enough. To do so, we reduce the function δ used to define the Ricci flow with (r, δ)-cutoff, if necessary, in order to ensure that δ(t) ≤ δ 2t (2t). Here δ 2t (2t) is the quantity δ A (2t) from Proposition 84.1 evaluated at A = 2t. 86. II.6.4. Earlier scalar curvature bounds on smaller balls from lower curvature bounds and volume bounds, in the presence of possible surgeries In this section we prove Proposition 84.2. We first recall its statement. Proposition 86.1. (cf. Proposition II.6.4) There exist τ, r, C1 > 0 and K < ∞ with the following property. Suppose that we have a Ricci flow with (r, δ)-cutoff on √ the time interval [0, t0 ], with normalized initial data. Let r0 satisfy 2C1 hmax (t0 ) ≤ r0 ≤ r t0 , where hmax (t0 ) is the maximal cutoff radius for surgeries in [t0 /2, t0 ]. (If there are no surgeries on [t0 /2, t0 ] then we put hmax (t0 ) = 0.) Assume 1. The ball B(x0 , t0 , r0 ) has sectional curvatures at least −r0−2 at each point. 2. The volume of any subball B(x, t0 , r) ⊂ B(x0 , t0 , r0 ) with any radius r > 0 is at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. Then the solution is unscathed on P (x0 , t0 , r0 /4, −τ r02 ) and satisfies R < Kr0−2 there. Proposition 84.2 is an analog of Theorem 54.2. However, the proof of Proposition 84.2 is more complicated, due to the need to deal with possible surgeries. The idea of the proof is to put oneself in a setting in which one can apply Lemma 82.1. To do this, one needs to first show that the solution is unscathed in a parabolic region P (x0 , t0 , r0 , −τ0 r02 ) and that one has Rm ≥ −r0−2 there. Proof. The constants C1 , K and τ are fixed numbers, but the requirements on them will be specified during the proof. The number r will emerge from the proof, via a contradiction argument. We first dispose of the case when r0 ≤ r(t0 ). Suppose that r0 ≤ r(t0 ). We claim that R ≤ r0−2 on B(x0 , t0 , r20 ). If not then R(x, t0 ) > r0−2 for some x ∈ B(x0 , t0 , r20 ), and so R(x, t0 ) > r(t0 )−2 . This implies that R(x, t0 ) is in a canonical neighborhood, which contradicts the almost-Euclidean-volume assumption on subballs of B(x0 , t0 , r0 ).
170
BRUCE KLEINER AND JOHN LOTT
1 −1 2 Thus R ≤ r0−2 on B(x0 , t0 , r20 ). Lemma 70.1 implies that R ≤ 16r0−2 on P (x0 , t0 , 12 r0 , − 16 η r0 ). Furthermore, if (*.1) C1 ≥ 100 1 −1 2 then R ≤ 2h12 on P (x0 , t0 , r0 /4, − 16 η r0 ). As surgeries only occur when R ≥ h−2 , there cannot be any surgeries in the region. Hence if we have (*.2) K ≥ 200 and 1 −1 (*.3) τ ≤ 16 η then there are no counterexamples to the proposition with r0 ≤ r(t0 ).
Continuing with the proof of the proposition, suppose that we have a sequence of Ricci flows with (r, δ)-cutoff Mα satisfying the assumptions of the proposition, with r α → 0, so that the conclusion of the proposition is violated for each Mα . Let tα0 be the first time when the conclusion is violated for Mα and let B(xα0 , tα0 , r0α) be a time-tα0 ball of smallest radius which provides a counterexample. (Such a ball exists since we have already shown that the proposition holds if r0α ≤ r(tα0 ).) That is, either there is a surgery in P (xα0 , tα0 , r0α/4, −τ (r0α )2 ) or R ≥ K(r0α )−2 somewhere on P (xα0 , tα0 , r0α /4, −τ (r0α )2 ). From the previous paragraph, r0α > r(tα0 ).
Let τb be the supremum of the numbers τe with the property that for large α, P (xα0 , tα0 , r0α, −e τ (r0α )2 ) is unscathed and Rm ≥ − (r0α )−2 there.
Lemma 86.2. τb is bounded below by the parameter τ0 of Lemma 82.1, where we take w = 1 − ǫ in Lemma 82.1.
Proof. Suppose that τb < τ0 . Put b tα = tα0 − (1 − ǫ′ ) τb (r0α )2 , where ǫ′ will eventually be taken to be a small positive number. Applying Lemma 82.1 to the solution on P (xα0 , tα0 , r0α , − (1 − 1 of the ǫ′ ) τb (r0α )2 ) (see the end of Section 82), the volume of B(xα0 , b tα , r0α /4) is at least 10 volume of the Euclidean ball of the same radius. From Lemma 83.1, there is a subball B(xα1 , b tα , r α ) ⊂ B(xα0 , b tα , r0α ) of radius r α = θ0 (1/10) r0α /4 with the property that all of its subballs have volume at least (1 − ǫ) times the volume of the Euclidean ball of the same radius. The sectional curvature on B(xα1 , b tα , r α ) is bounded below by − (r α )−2 . As we can apply the conclusion of the proposition to this subball (in view of its earlier time or smaller radius than B(xα0 , tα0 , r0α )), it follows that the solution on P (xα1 , b tα , r α /4, −τ (r α )2 ) is unscathed and has R < K(r α )−2 . As
(r α )2 (r α )2 (r α )2 (r α )2 (r0α )2 /tα0 ≥ α 2 α 0 α 2 = α 2 , b (r0 ) t0 − τ0 (r0 ) (r0 ) 1 − τ0 (r0α )2 /tα0 tα the fact that rα → 0 as α → ∞ implies that the Hamilton-Ivey pinching improves with α. In particular, for large α, | Rm | < K(r α )−2 on P (xα1 , b tα , r α /4, −τ (r α )2 ). Putting re0α = K −1/2 r α , if 1 (*.4) K −1 ≤ 10 τ α −2 then | Rm | ≤ (e r0 ) on the parabolic ball P (xα1 , e tα , e r0α, −(e r0α )2 ) for any e tα ∈ [b tα − 12 τ (r α )2 , b tα ]. α α Taking A = 100r0 /e r0 , Proposition 84.1(c) now implies that for large α, we have R ≤ α −2 α eα K2 (A) (e r0 ) on B(x1 , t , 100r0α). Provided that 1 C12 (*.5) K2 (A) K (θ0 (1/10))−2 ≤ 1000 1 −2 α −2 we will have K2 (A) (e r0 ) < 2 h and so there will not be any surgeries on such balls. Then the length distortion estimates of Lemma 27.8 imply that there is some c > 0 so that (86.3)
NOTES ON PERELMAN’S PAPERS
171
R ≤ K2 (A) (e r0α )−2 on P (xα0 , r0α , b tα , −c(r0α )2 ). Hamilton-Ivey pinching now implies that for large α, Rm ≥ − (r0α )−2 on P (xα0 , r0α , b tα , −c(r0α )2 ). As c can be taken independent of the ′ ′ small number ǫ , taking ǫ → 0 we contradict the maximality of τb. We can now apply Lemma 82.1 to obtain R ≤ K0 τ0−1 (r0α )−2 on P (xα0 , tα0 , r0α /4, −τ0 (r0α )2 /2). This will give a contradiction provided that (*.6) K0 τ0−1 < K/2, (*.7) τ < τ0 /2 and (*.8) K0 τ0−1 ≤ C12 , where the last condition rules out surgeries in P (xα0 , tα0 , r0α /4, −τ (r0α )2 ).
We choose τ to satisfy (∗.3) and (∗.7). We choose K to satisfy (∗.2), (∗.4) and (∗.6). Finally, we choose C1 to satisfy (∗.1), (∗.5) and (∗.8). This proves the proposition.
Remark 86.4. In subsequent sections we will want to know that for any w > 0, with the notation of Corollary 81.3, we have θ−1 (w) hmax (t0 ) ≤ r(t0 ) if t0 is sufficiently large (as a function of w). We can always achieve this by lowering the function δ(·) used to define the (t0 ) Ricci flow with (r, δ)-cutoff so that limt0 →∞ hmax = 0. We will assume hereafter that this r(t0 ) is the case. 87. II.7.1. Noncollapsed pointed limits are hyperbolic In this section we start the analysis of the long-time decomposition into hyperbolic and graph manifold pieces. In the section, M will denote a Ricci flow with (r, δ)-cutoff whose initial time slice (M0 , g(0)) is compact and has normalized metric. From Lemma 81.1, if g(0) has positive scalar curvature then the solution goes extinct in a finite time. From Lemma 81.2, these manifolds are understood topologically. If g(0) has nonnegative scalar curvature then either it acquires positive scalar curvature or it is flat, so again the topological type is understood. Hereafter we assume that the flow does not become extinct, and that Rmin < 0 for all t. − 3 Lemma 87.1. V (t) t + 14 2 is nonincreasing in t.
Proof. Suppose first that the flow is nonsingular. In the case the lemma follows from Lemma 79.11 and the equation Z dV (87.2) = − R dV ≤ − Rmin V. dt M
If there are surgeries then it only has the effect of causing further decrease in V . − 3 2 ˆ Definition 87.3. Put V = limt→∞ V (t) t + 41 2 and R(t) = Rmin (t) V (t) 3 .
Lemma 87.4. On any time interval which is free of singular times, and on which Rmin (t) ≤ 0 for all t (which we are assuming), we have Z ˆ dR 2 ˆ −1 (87.5) ≥ RV (Rmin − R) dV. dt 3 M
172
BRUCE KLEINER AND JOHN LOTT
Proof. From (B.1), (87.6)
dRmin dt
ˆ dRmin dR = V dt dt
≥ 2 3
2 3
2 Rmin . Then
1 dV 2 2 2 ≥ Rmin V + Rmin V − 3 3 dt 3
2 3
1 2 − Rmin V − 3 3
from which the lemma follows.
Z
R dV,
M
ˆ is nondecreasCorollary 87.7. If Rmin (t) ≤ 0 for all t (which we are assuming) then R(t) ing. Proof. If M is a nonsingular flow then the corollary follows from Lemma 87.4. If there are ˆ (since surgeries then it only has the effect of decreasing V (t), and so possibly increasing R(t) Rmin (t) ≤ 0). ˆ Put R = limt→∞ R(t). Lemma 87.8. If V > 0 then R V
−2/3
= − 23 .
Proof. Suppose that V > 0. Using Lemma 79.11, (87.9) − 32 !− 23 2 3 1 1 −2/3 Rmin (t) ≥ − . RV = lim Rmin (t)V (t) 3 V (t) t + = lim t + t→∞ t→∞ 4 4 2 In particular, there is a limit as t → ∞ of t + 14 Rmin (t). Suppose that 3 1 −2/3 Rmin (t) = c > − . (87.10) RV = lim t + t→∞ 4 2
Combining this with (87.2) gives that for any µ > 0, V (t) ≤ const. tµ−c whenever t is sufficiently large. Then V (t)t−3/2 ≤ const. t−(c+3/2−µ) . Taking µ = 21 (c + 23 ), we contradict the assumption that V > 0. From the proof of Lemma 87.8, if V > 0 then Rmin (t) ∼ −
3 . 2t
The next proposition shows that a long-time limit will necessarily be hyperbolic.
Proposition 87.11.√Given the flow M, suppose that we have a sequence of parabolic neighborhoods P (xα , tα , r tα , −r 2 tα ), for tα → ∞ and some fixed r ∈ (0, 1), such that the scalings of the parabolic neighborhoods with factor tα smoothly converge to some limit solution (M∞ , (x, 1), g∞ (·)) defined in a parabolic neighborhood P (x, 1, r, −r 2 ). Then g∞ (t) has constant sectional curvature − 4t1 . Proof. Suppose first that the flow is surgery-free. Because of the assumed existence of the limit (M∞ , (x, 1), g∞(·)), the original solution M has V > 0. We claim that the scalar curvature on P (x, 1, r, −r 2) is spatially constant. If not then there are numbers c < 0 and s0 , µ > 0 so that Z ′ (87.12) (Rmin (s) − R(x, s)) dV ≤ c B(x,s,r)
NOTES ON PERELMAN’S PAPERS
173
′ whenever s ∈ (s0 −µ, s0 +µ) ⊂ [1−r 2 , 1], where Rmin (s) is the minimum of R over B(x, s, r). Then for large α, Z c√α ′ α α (87.13) (R (st ) − R(x, st )) dV < t , min √ 2 B(xα ,stα ,r tα ) √ ′ where Rmin (stα ) is now the minimum of R over B(xα , stα , r tα ). Thus Z c√α t . (Rmin (stα ) − R(x, stα )) dV < (87.14) √ 2 B(xα ,stα ,r tα )
After passing to a subsequence, we can assume that
tα+1 tα
>
s0 +µ s0 −µ
for all α. From (87.5),
(87.15)
Z Z 2 ∞ ˆ −1 ˆ R(t) V (t) (Rmin (t) − R(x, t)) dV (x) dt R − R(0) ≥ 3 0 M Z Z 2 X α s0 +µ ˆ α α −1 ≥ t R(st ) V (st ) (Rmin (stα ) − R(x, stα )) dV (x) ds 3 α s0 −µ M Z Z s +µ 0 2 X α ˆ α ) V (stα )−1 (Rmin (stα ) − R(x, stα )) dV (x) ds. t R(st ≥ √ 3 α s0 −µ B(xα ,stα ,r tα ) 2
Using the definitions of V and R = − 23 V 3 , along with (87.14), it follows that the right-hand side of (87.15) is infinite. This contradicts the fact that R < ∞.
Thus R is spatially constant on P (x, 1, r, −r 2). As Rmin (t) ∼ − 2t3 on M, we know that the scalar curvature R at (x, t) ∈ P (x, 1, r, −r 2 ), which only depends on t, satisfies R(t) ≥ − 2t3 . It does not immediately follow that the scalar curvature on P (x, 1, r, −r 2 ) equals − 2t3 , as Rmin (t) is the minimum of the scalar curvature on all of M. However, if the scalar curvature is not identically − 2t3 on P (x, 1, r, −r 2 ) then again we can find c < 0 and s0 , µ > 0 so that for large α, (87.14) holds for s ∈ (s0 − µ, s0 + µ) ⊂ [1 − r 2 , 1]. Again we get a contradiction using (87.5). Thus R(t) = − 2t3 on P (x, 1, r, −r 2 ). Then from (B.1), each time-slice of P (x, 1, r, −r 2) has an Einstein metric. Thus the sectional curvature on P (x, 1, r, −r 2 ) is − 4t1 . The argument goes through if one allows surgeries. The main ingredient was the monotonicity formulas, which √ still hold if there are surgeries. Note that for large α there are no surgeries in P (xα , tα , r tα , −r 2 tα ) by assumption. 88. II.7.2. Noncollapsed regions with a lower curvature bound are almost hyperbolic on a large scale √ In this section it is shown that for fixed A, r, w > 0 and large time t0 , if B(x0 , t0 , r t0 ) ⊂ 3
3 2 M+ curvatures at least −r −2 t−1 t0 has volume at least wr t0 and sectional√ 0 then the Ricci flow 2 on the parabolic neighborhood P (x0 , t0 , Ar t0 , Ar t0 ) is close to the flow on a hyperbolic manifold.
We retain the assumptions of the previous section.
174
BRUCE KLEINER AND JOHN LOTT
Lemma 88.1. (cf. Lemma II.7.2) √ (a) Given w, r, ξ > 0 one can find T = T (w, r, ξ) > ∞ such that if the ball B(x0 , t0 , r t0 ) ⊂ 3
3 2 −2 −1 M+ t0 at some time t0 ≥ T has volume at least wr t0 and sectional curvatures at least −r t0 then the curvature at (x0 , t0 ) satisfies
(88.2)
|2tRij (x0 , t0 ) + gij |2 = (2tRij (x0 , t0 ) + gij )(2tRij (x0 , t0 ) + g ij ) < ξ 2.
(b) Given in addition√A < ∞ and allowing T to depend on A, we can ensure (88.2) for all points in B(x0 , t0 , Ar t0 ). √ (c) The same is true for P (x0 , t0 , Ar t0 , Ar 2 t0 ). Note that the time T will depend on the initial metric. Proof. To prove (a), suppose that there is a sequence of points (xα0 , tα0 ) with tα0 → ∞ that provide a counterexample. We wish to apply Corollary 81.3 with the parameter r0 of the √α corollary equal to r t0 . Putting ! Rx 2 sinh (s) ds 0 (88.3) w b = min w, R1 x∈[0,1] x3 sinh2 (s) ds 0 b where r is from Corollary 81.3, then the hypotheses of the lemma will still be if r > r(w), b Thus after redefining w, if necessary, we satisfied upon replacing w by w b and r by r(w). may assume that r ≤ r(w). As√the function hmax (t) is nonincreasing, if tα0 is sufficiently large then θ−1 (w)hmax (tα0 ) ≤ r tα0 . Using Corollary 81.3 with a redefinition of w, we can take a convergent pointed subsequence as α → ∞ of the tα0 -rescalings, whose limit is defined in an abstract parabolic neighborhood. From Proposition 87.11 the limit will be hyperbolic, which is a contradiction.
−2 For part (b), Corollary 81.3 gives a bound R ≤ Kr√ 0 in the unscathed parabolic neighbor′ 2 hood P (x0 , t0 , r0 /4, −τ r0 ), where r0 = min(r, r(w )) t0 . We apply Proposition 84.1 to the −2 ′ ′ 2 ′ −2 parabolic neighborhood P (x Proposition 84.1(b), √0 , t0 , r0 , − (r0 ) ) where K r0 = (r0 ) . By ′ each point y ∈ B(x0 , t0 , Ar t0 ) with scalar curvature at least Q = K (A)r0−2 has a canonical neighborhood. Suppose that there is such a point. From part (a) we have √ R(x0 , t0 ) < 0, ′ so along a geodesic from x0 to y there will be some point x0 ∈ B(x0 , t0 , Ar t0 ) with scalar curvature Q. It also has a canonical neighborhood, necessarily of type (a) or (b). We can apply part (a) to a ball around x′0 with a radius on the order of (K ′ (A))−1/2 r0 , and with a value of w coming from the canonical neighborhood condition, to get a contradiction for √ ′ −1/2 large t0 . (Note r0 is proportionate to t0 .) Thus R ≤ K ′ (A)r0−2 on √ that (K (A)) B(x0 , t0 , Ar t0 ). If T is large enough then the Φ-almost nonnegative curvature implies that | Rm | ≤ K ′ (A)r0−2 . Then the noncollapsing in Proposition 84.1(a) gives a lower local volume bound. √ Hence we can apply part (a) of the lemma to appropriate-sized balls in A B(x0 , t0 , 2 r t0 ). As A is arbitrary, this proves (b) of the lemma.
For part (c), without loss of generality we can take ξ small. Suppose that the claim is not true. Then there is a point (x0 , t0 ) that√satisfies the hypotheses of the lemma but for which ξ. Without there is a point (x1 , t1 ) ∈ P (x0 , t0 , Ar t0 , Ar 2 t0 ) with |2t1 Rij (x1 , t1 ) + gij | ≥ √ loss of generality, we can take (x1 , t1 ) to be a first such point√in P (x0 , t0 , Ar t0 , Ar 2 t0 ). By part (b), t1 > t0 . Then |2tRij + gij | ≤ ξ on P (x0 , t0 , Ar t0 , t1 − t0 ). If ξ is small then this region has negative sectional curvature and there are no surgeries in the region.
NOTES ON PERELMAN’S PAPERS
175
′ Using the length distortion estimates of Section 27, we can find r ′ = r√ (r, A) > 0 so that ′ −2 −1 ′ the sectional curvature is bounded below by − (r ) t1 on B(x0 , t1 , r t1 ). Also, by the evolution of volume under Ricci flow, there will be a w ′ = w ′ (r, w, ξ, A) so that the volume √ 3 of B(x0 , t1 , r ′√ t1 ) is bounded below by w ′ (r ′)3 (t1 ) 2 . Thus for large t0 we can apply (b) to B(x0 , t1 , A′ r ′ t1 ) with an appropriate choice of A′ to obtain a contradiction.
89. II.7.3. Thick-thin decomposition This section is concerned with the large-time decomposition of the manifold in “thick” and “thin” parts. Definition 89.1. For x ∈ M+ t , let ρ(x, t) be the unique number ρ ∈ (0, ∞) such that inf B+ (x,t,ρ) Rm = − ρ−2 , if such a ρ exists, and put ρ(x, t) = ∞ otherwise. The function ρ(x, t) is well-defined because M+ t is a compact smooth Riemannian manifold, so for fixed (x, t) ∈ M+ the quantity inf B + (x,t,ρ) Rm is a continuous nonincreasing t function of ρ which is negative for sufficiently large ρ if and only if (x, t) lies in a connected component with negative sectional curvature somewhere; on the other hand the function −ρ−2 is continuous and strictly increasing. We note that when it is finite, the quantity ρ(x, t) may be larger than the diameter of the component of Mt containing (x, t).
As an example, if M is the flow √ on a manifold M with spatially constant negative curvature then for large t, ρ(x, t) ∼ 2 t uniformly on M. The “thin” part of Mt , in the sense of hyperbolic geometry, can then be characterized as the points x so that vol(B(x, t, ρ(x, t)) < w ρ3 (x, t), for an appropriate constant w. Lemma 89.2.√For any w > 0 we can find ρ = ρ(w) > 0 and T = T (w) such that if t ≥ T and ρ(x, t) < ρ t then (89.3)
vol(B(x, t, ρ(x, t))) < wρ3 (x, t).
Proof. If the lemma is not true then there is a sequence (xα , tα ) with tα → ∞, ρ(xα , tα )(tα )−1/2 → 0 and vol(B(xα , tα , ρ(xα , tα ))) ≥ wρ3 (xα , tα ). The first step is to apply Corollary 81.3, but we need to know that for large α we have ρ(xα , tα ) ≥ θ−1 (w) hmax (tα ), where θ(w) and hmax are from Corollary 81.3. Suppose that this is not the case. Then after passing to a subsequence we have ρ(xα , tα ) < θ−1 (w) hmax (tα ) ≤ r(tα ) for all α, where we used Remark 86.4 in the last inequality. There are points xα,′ ∈ B(xα , tα , ρ(xα , tα )) with a sectional curvature equal to − ρ−2 (xα , tα ). Applying the Hamilton-Ivey pinching estimate of (B.4) with X α = ρ−2 (xα , tα ), and using the fact that limα→∞ tα X α = ∞, gives (89.4)
lim R(xα,′ , tα ) ρ2 (xα , tα ) = ∞.
α→∞
We claim that the curvatures at the centers of the balls satisfy (89.5)
lim R(xα , tα ) ρ2 (xα , tα ) = ∞.
α→∞
Suppose not. Then there is some number C ∈ (0, ∞) so that after passing to a subsequence, R(xα , tα ) ρ2 (xα , tα ) ≤ C for all α. By continuity and (89.4), after passing to another subsequence we can assume that there is a point xα′′ on a time-tα geodesic segment between
176
BRUCE KLEINER AND JOHN LOTT
xα and xα′ so that R(xα′′ , tα ) ρ2 (xα , tα ) = 2C, for all α. We now apply Lemma 70.2 around (xα′′ , tα ) to get a contradiction to (89.4). (More precisely, we apply a version of Lemma 70.2 that applies along geodesics, as in Claim 2 of II.4.2.) In applying Lemma 70.2, we use the −2 (xα ,tα )) fact that limα→∞ tα ρ−2 (xα , tα ) = ∞ in order to say that limα→∞ Φ(2Cρ = 0, with 2Cρ−2 (xα ,tα ) the notation of Lemma 70.2. Making a similar argument centered at other points in B(xα , tα , ρ(xα , tα )), we deduce that (89.6) lim inf R B(xα ,tα ,ρ(xα ,tα )) ρ2 (xα , tα ) = ∞. α→∞ α α
In particular, since ρ(x , t ) ≤ r(tα ), if α is large then each point y α ∈ B(xα , tα , ρ(xα , tα )) is the center of a canonical neighborhood. As α → ∞, the intrinsic scales R(y α , tα )−1/2 become arbitrarily small compared to ρ(xα , tα ). However, by Lemma 83.1 there is a subball B α,′ of B(xα , tα , ρ(xα , tα )) with radius θ0 (w)ρ(xα , tα ) so that every subball of B α,′ has almost Euclidean volume. This contradicts the existence of a small canonical neighborhood around each point of B(xα , tα , ρ(xα , tα )).
We now know that for large α, ρ(xα , tα ) ≥ θ−1 (w) hmax (tα ). Thus we can apply Corollary 81.3 to get an unscathed solution on the parabolic neighborhood P (xα , tα , ρ(xα , tα )/4, −τ ρ2 (xα , tα )), with R < K0 ρ(xα , tα )−2 there. Applying Proposition 84.1(c), along with the fact that limα→∞ tα ρ−2 (xα , tα ) = ∞, gives an estimate R ≤ K2 ρ(xα , tα )−2 on B(xα , tα , ρ(xα , tα )). But then for large α, the Hamilton-Ivey pinching gives Rm > − 21 ρ(xα , tα )−2 on B(xα , tα , ρ(xα , tα )), which is a contradiction. Remark 89.7. Another approach to the above proof would be to use the canonical neighborhood at the center (xα , tα ) of the ball, along with the Bishop-Gromov inequality, to contradict the fact that vol(B(xα , tα , ρ(xα , tα ))) ≥ wρ3 (xα , tα ). For this to work we would 3 1 have to know that the relative volume (ǫ2 R(xα , tα )) 2 vol(B(xα , tα , ǫ−1 R(xα , tα )− 2 ) of the canonical neighborhood (of type (a) or (b)) around (xα , tα ) is small compared to w. This will be the case if we take the constant ǫ to be small enough, but in a w-dependent way. Although ǫ is supposed to be a universal constant, this approach will work because when characterizing the graph manifold part in Section 92, w can be taken to be a small but fixed constant. Definition 89.8. The w-thin part M − (w, t) ⊂ M+ t is the set of points x ∈ M so that either ρ(x, t) = ∞ or vol(B(x, t, ρ(x, t))) < w (ρ(x, t))3 .
(89.9)
− The w-thick part is M + (w, t) = M+ t − M (w, t).
Lemma 89.10. Given w > 0, there are w ′ = w ′ (w) > 0 and T ′ = T ′ (w) < ∞ so that taking √ r = ρ(w) (with reference to Lemma 89.2), if x0 ∈ M + (w, t) and t0 ≥ T ′ then B(x0 , t0 , r t0 ) 3
has volume at least w ′ r 3 t02 and sectional curvature at least − r −2 t−1 0 .
+ Proof. Suppose that x0 ∈ M √ (w, t0 ). From Lemma 89.2,−2if t0 is big enough (as a function of w) then ρ(x0 , t0 ) ≥ r t0 . As √ Rm ≥ − ρ(x0 , t0 ) on B(x0 , t0 , ρ(x0 , t0 )), we3 have Rm ≥ − r −2 t−1 on B(x , t , r t0 ). As vol(B(x0 , t0 , ρ(x0 , t0 ))) ≥ w (ρ(x0 , t0 )) , the 0 0 0 √ −3 √ Bishop-Gromov inequality gives a lower bound on r t0 vol(B(x0 , t0 , r t0 )) in terms of w.
NOTES ON PERELMAN’S PAPERS
177
90. Hyperbolic rigidity and stabilization of the thick part Lemma 89.10 implies that Lemma 88.1 applies to M + (w, t) if t is sufficiently large (as a function of w). That is, if one takes a sequence of points in the w-thick parts at a sequence of times tending to infinity then the pointed time slices subconverge, modulo rescaling the metrics by t−1 , to complete finite volume hyperbolic manifolds with sectional curvatures equal to − 14 . (The − 41 comes from the Ricci flow equation along with the equation g(t) = t g(1) for the rescaled limit, which implies that g(1) has Einstein constant − 12 .) In what follows we will take the word “hyperbolic” for a 3-manifold to mean “constant sectional curvature − 14 ”. The next step, following Hamilton [34], is to show that for large time the picture stabilizes, i.e. the limits are unique in a strong sense. Proposition 90.1. There exist a number T0 < ∞, a nonincreasing function α : [T0 , ∞) → (0, ∞) with limt→∞ α(t) = 0, a (possibly empty) collection {(H1 , x1 ), . . . , (Hk , xk )} of complete connected pointed finite-volume hyperbolic 3-manifolds and a family of smooth maps k [ 1 (90.2) f (t) : Bt = B xi , −→ Mt , α(t) i=1 defined for t ∈ [T0 , ∞), such that
1. f (t) is close to an isometry:
(90.3)
kt−1 f (t)∗ gMt − gBt k
1
< α(t),
C α(t)
2. f (t) defines a smooth family of maps which changes slowly with time: (90.4)
1 |f˙(p, t)| < α(t)t− 2
for all p ∈ Bt , where f˙ refers to the time derivative (as defined with admissible curves), and
3. f (t) parametrizes more and more of the thick part: M + (α(t), t) ⊂ im(f (t)) for all t ≥ T0 . Remark 90.5. The analogous statement in [52, Section 7.3] is in terms of a fixed w. That is, −1 ∞ for a given w one considers pointed limits of {(M+ tj , (xj , tj ), tj g(tj ))}j=1 with limj→∞ tj = ∞ and the basepoint satisfying xj ∈ M + (w, tj ) ⊂ M+ tj for all j. Considering the possible limit spaces in a certain order, as described below, one extracts complete pointed finite-volume hyperbolic manifolds {(Hi, xi )}ki=1 with xi in the w-thick part of Hi . There is a number w0 > 0 so that as long as w ≤ w0 , the hyperbolic manifolds Hi are independent of w. For Sk ′ ′ any w > 0, as time goes on the w -thick part of i=1 Hi better approximates M + (w ′ , t). Hence the formulation of Proposition 90.1 is equivalent to that of [52, Section 7.3]. Rather than proving Proposition 90.1 using harmonic maps as in [34], we give a simple proof using smooth compactness and a smoothing argument. Roughly speaking, the idea is to exploit a variant of Mostow rigidity to show that for large t, the components of the wthick part change slowly with time, and are close to hyperbolic manifolds which are isolated (due to a refinement of Mostow-Prasad rigidity). This forces them to eventually stabilize.
178
BRUCE KLEINER AND JOHN LOTT
Definition 90.6. If (X, x) and (Y, y) are pointed smooth Riemannian manifolds and ǫ > 0 then (X, x) is ǫ-close to (Y, y) if there is a pointed map f : (X, x) → (Y, y) such that (90.7)
f |B(x,ǫ−1 ) : B(x, ǫ−1 ) → Y
is a diffeomorphism onto its image and (90.8)
kf ∗ gY − gX kC ǫ−1 < ǫ,
where the norm is taken on B(x, ǫ−1 ). Note that nothing is required of f on the complement of B(x, ǫ−1 ). Such a map f is called an ǫ-approximation. We will sometimes refer to a partially defined map f : (X, x) ⊃ (W, x) → (Y, y) as an ǫ-approximation provided that W contains B(x, ǫ−1 ) and the conditions above are satisfied. By convention we will permit X and Y to be disconnnected, in which case ǫ-closeness only says something about the components containing the basepoints. We say that two maps f1 , f2 : (X, x) → Y (not necessarily basepoint-preserving) are ǫ-close if (90.9)
sup
dY (f1 (p), f2 (p)) < ǫ.
p∈B(x,ǫ−1 )
We recall some facts about hyperbolic manifolds. There is a constant µ0 > 0, the Margulis constant, such that if X is a complete connected finite-volume hyperbolic 3-manifold (orientable, as usual), µ ≤ µ0 , and (90.10)
Xµ = {x ∈ X | InjRad(X, x) ≥ µ}
is the µ-thick part of X, then Xµ is a nonempty compact manifold-with-boundary whose complement U is a finite union of components U1 , . . . , Uk , where each Ui is isometric either to a geodesic tube around a closed geodesic or to a cusp. In particular, Xµ is connected and there is a one-to-one correspondence between the boundary components of Xµ and the “thin” components Ui . For each i, let ρi : Ui → R denote either the distance function from the core geodesic or a Busemann function, in the tube and cusp cases respectively. In the latter case we normalize ρi so that ρ−1 i (0) = ∂Ui . (The Busemann function goes to − ∞ as one goes down the cusp.) The radial direction is the direction field on Ui − core(Ui ) defined by ∇ρi , where core(Ui ) is the core geodesic when Ui is a geodesic tube and the empty set otherwise. Lemma 90.11. Let (X, x) be a pointed complete connected finite-volume hyperbolic 3manifold. Then for each ζ > 0 there exists ξ > 0 such that if X ′ is a complete finitevolume hyperbolic manifold with at least as many cusps as X, and f : (X, x) → X ′ is a ξ-approximation, then there is an isometry fˆ : (X, x) → X ′ which is ζ-close to f . This was stated as Theorem 8.1 in [34] as going back to the work of Mostow. We give a proof here. The hypothesis about cusps is essential because every pointed noncompact finite-volume hyperbolic 3-manifold (X, x) is a pointed limit of a sequence {(Xi , xi )}∞ i=1 of compact hyperbolic manifolds. Hence for every ξ > 0, if i is sufficiently large then there is a ξ-approximation f : (X, x) → (Xi , xi ), but there is no isometry from X to Xi .
NOTES ON PERELMAN’S PAPERS
179
Proof. The main step is to show that for the fixed (X, x), if ξ is sufficiently small then for any ξ-approximation f : (X, x) → X ′ satisfying the hypotheses of the lemma, the manifolds X and X ′ are diffeomorphic. The proof of this will use the Margulis thick-thin decomposition. The rest of the assertion then follows readily from Mostow-Prasad rigidity [47, 54]. Pick µ1 ∈ (0, µ0 ) so that X − Xµ1 consists only of cusps U1 , . . . , Uk . The thick part Xµ1 is compact and connected. Given ξ > 0, let f : (B(x, ξ −1 ), x) → (X ′ , x′ ) be a ξ-approximation as in (90.8). The intuitive idea is that because of the compactness of Xµ1 , if ξ is sufficiently small then f Xµ is close to being an isometry from Xµ1 to its image. Then f (Xµ1 ) is close 1 to a connected component of the thick part Xµ′ 1 of X ′ . As f (Xµ1 ) and Xµ′ 1 are connected, this means that f (Xµ1 ) is close to Xµ′ 1 . We will show that in fact Xµ1 is diffeomorphic to Xµ′ 1 . The boundary components of Xµ1 correspond to the cusps of X and the boundary components of Xµ′ 1 correspond to the connected components of X ′ − Xµ′ 1 . As X ′ has at least as many cusps as X by assumption, it follows that the connected components of X ′ − Xµ′ 1 are all cusps. Hence X and X ′ are diffeomorphic. In order to show that Xµ1 is diffeomorphic to Xµ′ 1 , we will take a larger region W ⊃ Xµ1 that is diffeomorphic to Xµ1 and show that f (W ) can be isotoped to Xµ′ 1 by sliding it inward along the radial direction. More precisely, in each cusp Ui put Vi = ρ−1 i ([−3L, −L]), where −1 L ≫ 1 is large enough that every cuspidal torus ρi (s) with s ∈ [−3L, −L] has diameter much less than one. Let W ⊂ X be the complement of the open horoballs at height −2L, i.e. (90.12)
W =X−
k [
i=1
ρ−1 i (−∞, −2L).
When ξ is sufficiently small, f will preserve injectivity radius to within a factor close to 1 for points p ∈ B(x, ξ −1 ) with d(p, ∂B(x, ξ −1 )) > 2 InjRad(X, p). Therefore when ξ is small, f will map each Vi into X ′ − Xµ′ 1 , and hence into one of the connected components Uk′ i of ′ ′ X ′ − Xµ′ 1 . Let Zi′ be the image of Zi = ρ−1 i (−2L) under f . Note that d(core(Uki ), Zi ) & L (if core(Uk′ i ) 6= ∅), for otherwise f −1 (core(Uk′ i )) would be a closed curve with small diameter and curvature lying in Ui , which contradicts the fact that the horospheres have principal curvatures − 21 (because of our normalization that the sectional curvatures are − 41 ). Thus for each point z ′ ∈ Zi′ there is a minimizing radial geodesic segment γ ′ passing through z ′ with d(z ′ , ∂γ ′ ) & L. The preimage of γ ′ under f is a curve γ ⊂ X with small curvature and length & L passing through Zi . This forces the direction of γ to be nearly radial and so transverse to Zi . Hence Zi′ is transverse to the radial direction in Uk′ i . Combining this with the fact that Zi′ is embedded implies that Zi′ is isotopic in Uk′ i to ∂Uk′ i . It follows that f (W ) is isotopic to Xµ′ 1 . Then by the preceding argument involving counting the number of cusps, X and X ′ are diffeomorphic. We apply Mostow-Prasad rigidity [47, 54] to deduce that X is isometric to X ′ . We now claim that for any ζ > 0, if ξ is sufficiently small then the map f is ζ-close to an isometry from X to X ′ . Suppose not. Then there are a number ζ > 0 and a sequence of 1i -approximations fi : (X, x) → (Xi′ , x′i ) so that none of the fi ’s are ζ-close to any isometry from (X, x) to (Xi′ , x′i ). Taking a convergent subsequence of the maps fi gives a
180
BRUCE KLEINER AND JOHN LOTT
′ limit isometry f∞ : (X, x) → (X∞ , x′∞ ). From what has already been proven, for large i ′ ′ we know that Xi is isometric to X, and so isometric to X∞ . This is a contradiction.
Recall the statement of Lemma 88.1. Definition 90.13. Given w > 0, let Λw be the space of complete pointed finite-volume −1 ∞ hyperbolic 3-manifolds that arise as pointed limits of sequences {(M+ ti , (xi , ti ), ti g(ti ))}i=1 + with limi→∞ ti = ∞ and the basepoint (xi , ti ) satisfying (xi , ti ) ∈ M + (w, ti ) ⊂ Mti for all i. The space Λw is compact in the smooth pointed topology. Any element of Λw has volume at most V , the latter being defined in Definition 87.3. The next lemma summarizes the content of Lemma 88.1. Lemma 90.14. Given w > 0, there is a decreasing function β : [0, ∞) → (0, ∞] with lims→∞ β(s) = 0 such that if (x, t) ∈ M + (w, t) ⊂ M+ t , and Zt denotes the forward time −1 slice M+ rescaled by t , then t 1. Some (X, x) ∈ Λw is β(t)-close to (Zt , (x, t)). √ 2. B(x, t, β(t)−1 t) ⊂ M+ t is unscathed on the interval [t, 2t] and if γ : [t, 2t] → M is a static curve starting at (x, t), t¯ ∈ [t, 2t], then the map √ √ (90.15) B(x, t, β(t)−1 t) → P (x, t, β(t)−1 t, t) ∩ Mt¯ defined by following static curves induces a map (90.16) satisfying (90.17)
it,t¯ : (Zt , (x, t)) ⊃ (B(x, t, β(t)−1 ), (x, t)) → (Zt¯, γ(t¯)) k (it,t¯)∗ gZt¯ − gZt kC β(t)−1 < β(t).
Proof. This follows immediately from Lemma 88.1.
Proof of Proposition 90.1. If for some w > 0 we have Λw = ∅ then M + (w, t) = ∅ for large t. Thus if Λw = ∅ for all w > 0, we can take the empty collection of pointed hyperbolic manifolds and then 1 and 2 will be satisfied vacuously, and α(t) may be chosen so that 3 holds. So we assume that Λw 6= ∅ for some w > 0.
Since every complete finite-volume hyperbolic 3-manifold has a point with injectivity radius ≥ µ0 , there is a w0 > 0 such that the collections {Λw }w≤w0 contain the same sets of underlying hyperbolic manifolds (although the basepoints have more freedom when w is small). We let H1 be a hyperbolic manifold from this collection with the fewest cusps and we choose a basepoint x1 ∈ H1 so that (H1 , x1 ) ∈ Λw0 . Put w1 = w20 . Note that x1 lies in the w1 -thick part of H1 . In what follows we will use the fact that if f is a ǫ-approximation from H1 , for sufficiently small ǫ, then f (x1 ) will lie in the .9w0 -thick part of the image. The idea of the first step of the proof is to define a family {f0 (t)} of δ-approximations (H1 , x1 ) → Zt , for all t sufficiently large, by taking a δ-approximation (H1 , x1 ) → Zt , pushing it along static curves, and arguing using Lemma 90.11 that one can make small
NOTES ON PERELMAN’S PAPERS
181
adjustments from time to time to keep it a δ-approximation. The family {f0 (t)} will not vary continuously with time, but it will have controlled “jumps”. More precisely, pick T0 < ∞ and let ξ1 , . . . , ξ4 > 0 be parameters to be specified later. We assume that T0 is large enough so that 2β(T0 ) < ξ1 , where β is from Lemma 90.14. By the definition of Λw1 , we may pick T0 so that there is a point (x0 , T0 ) ∈ M + (w1 , T0 ) ⊂ M+ T0 and a ξ1 -approximation f0 (T0 ) : (H1 , x1 ) → (ZT0 , x0 ). To do the induction step, for a given j ≥ 0 suppose that at time 2j T0 there is a j point (xj , 2j T0 ) ∈ M + (w1 , 2j T0 ) ⊂ M+ 2j T0 and a ξ1 -approximation f0 (2 T0 ) : (H1 , x1 ) → (Z2j T0 , xj ). As mentioned above, if ξ1 is small then in fact xj ∈ M + (.9w0 , 2j T0 ).
By part 2 of Lemma 90.14, provided that T0 is sufficiently large we may define, for all t ∈ [2j T0 , 2j+1T0 ], a 2ξ1-approximation f0 (t) : (H1 , x1 ) → Zt by moving f0 (2j T0 ) along static curves. Provided that ξ1 is sufficiently small we will have f0 (2j+1T0 )(x1 ) ∈ M + (w1 , 2j+1T0 ) and then part 1 of Lemma 90.14 says there is some (H ′ , x′ ) ∈ Λw1 with a β(2j+1T0 )approximation φ : (H ′, x′ ) → (Z2j+1 T0 , f0 (2j+1 T0 )(x1 )). Provided that β(2j+1T0 ) and ξ1 are sufficiently small, the partially defined map φ−1 ◦ f0 (2j+1 T0 ) will define a ξ2 -approximation from (H1 , x1 ) to (H ′ , x′ ). Hence provided that ξ2 is sufficiently small, by Lemma 90.11 the map will be ξ3 -close to an isometry ψ : (H1 , x1 ) → H ′. (In applying Lemma 90.11 we use the fact that H1 is also a manifold with the fewest number of cusps in Λw1 .) Put φ1 = φ◦ψ. Provided that ξ3 is sufficiently small, f0 (2j+1T0 ) and φ1 will be ξ4 -close as maps from (H1 , x1 ) to Z2j+1 T0 . Since φ1 is a β(2j+1T0 )-approximation precomposed with an isometry which shifts basepoints a distance at most ξ3 , it will be a 2β(2j+1T0 )-approximation provided that ξ3 < 1 and β(2j+1T0 ) < 21 . We now redefine f0 (2j+1T0 ) to be φ1 and let xj+1 be the image of x1 under φ1 . This completes the induction step. In this way we define a family of partially defined maps {f0 (t) : (H1 , x1 ) → Zt }t∈[T0 ,∞) . From the construction, f0 (2j T0 ) is a 2β(2j T0 )-approximation for all j ≥ 0. Lemma 90.14 then implies that there is a function α1 : [T0 , ∞) → (0, ∞) decreasing to zero at infinity such that for all t ∈ [T0 , ∞), f0 (t) is an α1 (t)-approximation, and for every t¯ ∈ [t, 2t] we may slide f0 (t) along static curves to define an α1 (t)-approximation h(t¯) : (H1 , x1 ) → Zt which is α1 (t)-close to f0 (t¯). One may now employ a standard smoothing argument to convert the family {f0 (t)}t∈[T0 ,∞) into a family {f1 (t)}t∈[T0 ,∞) which satisfies the first two conditions of the proposition. If condition 3 fails to hold then we redefine the Λw ’s by considering limits of only those −1 + ∞ + {(M+ ti , (xi , ti ), ti g(ti ))}i=1 with ti → ∞ and xi ∈ M (w, ti ) ⊂ Mti not in the image of f1 (ti ). Repeating the construction we obtain a pointed hyperbolic manifold (H2 , x2 ) and a family {f2 (t)} defined for large t satisfying conditions 1 and 2, where im(f2 (t)) is disjoint from im(f1 (t)) for large t. Iteration of this procedure must stop after k steps for some finite number k, in view of the fact that V < ∞ and the fact that there is a positive lower bound on the volumes of complete hyperbolic 3-manifolds. We get the desired family {f (t)} by taking the union of the maps f1 (t), . . . , fk (t).
182
BRUCE KLEINER AND JOHN LOTT
91. Incompressibility of cuspidal tori By Proposition 90.1, we know that for large times the thick part of the manifold can be parametrized by a collection of (truncated) finite volume hyperbolic manifolds. In this section we show that each cuspidal torus maps to an embedded incompressible torus in Mt . (An alternative argument is given in Section 93.) The strategy, due to Hamilton, is to argue by contradiction. If such a torus were compressible then there would be an embedded compressing disk of least area at each time. By estimating the rate of change of the area of such disks one concludes that the area must go to zero in finite time, which is absurd. Let T0 , α, {(H1 , x1 ), . . . , (Hk , xk )}, Bt , and f (t) be as in Proposition 90.1. We will consider a fixed Hi , with 1 ≤ i ≤ k, which is noncompact. Choose a number a > 0 much smaller than the Margulis constant and let {V1 , . . . , Vl } ⊂ Hi be the cusp regions bounded by tori of diameter a. Each Vj is an embedded 3-dimensional submanifold (with boundary) of Hi and is isometric to the quotient of a horoball in hyperbolic 3-space H3 by the action of a copy of Z2 sitting in the stabilizer of the horoball. The boundary ∂Vj is a totally umbilic torus whose principal curvatures are equal to 21 everywhere. We let Y ⊂ Hi be the closure S of the complement of lj=1 Vj in Hi . Let Ta < ∞ be large enough that BTa (defined as in Proposition 90.1) contains Y . In order to focus on a given cusp, we now fix an integer 1 ≤ j ≤ l and put
(91.1)
Z = ∂Vj ,
Zˆt = f (t)(Z) Yˆt = f (t)(Y ),
for every t ≥ Ta . The objective of this section is:
ˆ t = M+ ˆ W t − int(Yt )
Proposition 91.2. The homomorphism (91.3)
π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t , f (t)(⋆))
is a monomorphism for all t ≥ Ta . Proof. The proof will occupy the remainder of this section. The first step is: Lemma 91.4. The kernels of the homomorphisms (91.5)
π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t , f (t)(⋆)),
ˆ t , f (t)(⋆)) π1 (f (t)) : π1 (Z, ⋆) → π1 (W
are independent of t, for all t ≥ Ta . Proof. We prove the assertion for the first homomorphism. The argument for the second one is similar. The kernel obviously remains constant on any time interval which is free of singular times. − Suppose that t0 ≥ Ta is a singular time. Then the intersection M+ t0 ∩ Mt0 includes into + Mt0 and, by using static curves, into Mt for t 6= t0 close to t0 . By Van Kampen’s theorem, these inclusions induce monomorphisms of the fundamental groups. Therefore for t close to t0 , the kernel of (91.5) is the same as the kernel of (91.6)
− π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t0 ∩ Mt0 ),
which is independent of t for times t close to t0 .
NOTES ON PERELMAN’S PAPERS
183
We now assume that the kernel of (91.7)
π1 (f (t)) : π1 (Z, ⋆) → π1 (M+ t , f (t)(⋆))
is nontrivial for some, and hence every, t ≥ Ta . By Van Kampen’s theorem and the fact that the cuspidal torus Z ⊂ Y is incompressible in Y , it follows that the kernel K of ˆ t , f (t)(⋆)) (91.8) π1 (f (t)) : π1 (Z, ⋆) → π1 (W ˆ t ; R) → H1 (∂ W ˆ t ; R) is a Lain nontrivial for all t ≥ Ta . By Poincar´e duality, Im H1 (W ˆ t ; R). In particular, Im H1 (W ˆ t ; R) → H1 (Z; R) has rank one. grangian subspace of H1 (∂ W c Dually, Ker H1 (Z; R) → H1 (Wt ; R) has rank one and so K, a subgroup of a rank-two free abelian group, has rank one. We note that for all large t, Zˆt is a convex boundary compoˆ t . The main theorem of [43] implies that for every such t, there is is a least-area nent of W compressing disk ˆ t , Zˆt ). (91.9) (N 2 , ∂N 2 ) ⊂ (W t
t
We recall that a compressing disk is an embedded disk whose boundary curve is essential ˆ t is a compact manifold even when t is a singular time. in Zˆt . We note that by definition, W −1 The embedded curve f (t) (∂Nt ) ⊂ Z represents a primitive element of π1 (Z) which, since K has rank one, must therefore generate K. It follows that modulo taking inverses, the homotopy class of f (t)−1 (∂Nt ) ⊂ Z is independent of t. We define a function A : [Ta , ∞) → (0, ∞) by letting A(t) be the infimum of the areas of such embedded compressing disks. We now show that the least-area compressing disks avoid the surgery regions.
Lemma 91.10. Let δ(t) be the surgery parameter from Section 73. There is a T = T (a) < ˆ t is ∞ so that whenever t ≥ T , no point in any area-minimizing compressing disk Nt ⊂ W in the center of a 10δ(t)-neck. Proof. If the lemma were not true then there would be a sequence of times tk → ∞ and ˆ t , Zˆt ), along with a for each k an area-minimizing compressing disk (Ntk , ∂Ntk ) ⊂ (W k k point xk ∈ Ntk that is in the center of a 10δ(tk )-neck. Note that the scalar curvature near ∂Ntk is comparable to − 2t3k . We now rescale by R(xk , tk ), and consider the map of ˆ t , Zˆt , xk ) where the domain is equipped with pointed manifolds fk : (Ntk , ∂Ntk , xk ) ֒→ (W k k the pullback Riemannian metric. By [57] and standard elliptic regularity, for all ρ < ∞ and every integer j, the j th covariant derivative of the second fundamental form of fk is uniformly bounded on the ball B(xk , ρ) ⊂ Ntk , for sufficiently large k. Therefore the pointed Riemannian manifolds (Ntk , ∂Ntk , xk ) subconverge in the smooth topology to a pointed, complete, connected, smooth manifold (N∞ , x∞ ). Using the same bounds on the derivatives of the second fundamental form, we may extract a limit mapping φ∞ : N∞ → R × S 2 which is a 2-sided isometric stable minimal immersion. By [58, Theorem 2], φ∞ is a totally geodesic immersion whose normal vector field in M has vanishing Ricci curvature. It follows that φ∞ is a cover of a fiber {pt}×S 2 . This contradicts the fact that N∞ is noncompact. We redefine Ta if necessary so that Ta is greater than the T of Lemma 91.10.
184
BRUCE KLEINER AND JOHN LOTT
We can isotope the surface Z by moving it down the cusp Vj . In doing so we do not change the group K but we can make the diameter of Z as small as desired. The next lemma refers to this isotopy freedom. Lemma 91.11. Given D > 0, there is a number a0 > 0 so that for any a ∈ (0, a0 ),√ if R diam(Z) = a and t is sufficiently large then ∂Nt κ∂Nt ds ≤ D2 and length(∂Nt ) ≤ D2 t, where κ∂Nt is the geodesic curvature of ∂Nt ⊂ Nt . Proof. This is proved in [34, Sections 11 and 12]. We just state the main idea. For the ˆ t the metric t−1 g(t). First, ∂Nt is the intersection of Nt purposes of this proof, we give W with Zˆt . Because Nt is minimal with respect to free boundary conditions (i.e. the only constraint is that ∂Nt is in the right homotopy class in Zˆt ), it follows that Nt meets Zˆt ˆt orthogonally. Then κ∂Nt = Π(v, v), where Π is the second fundamental form of Zˆt in W and v is the unit tangent vector of ∂Nt . Given a > 0, let Z be the horospherical torus in Vj of diameter a. By Proposition 90.1, for large t the map f (t) is close to being an isometry of pairs (Y, Z) → (Yˆt , Zˆt ). As Z has principal curvatures 21 , we may assume that Π(v, v) is close to 21 . This reduces the problem to showing that with an appropriate choice of a0 , if a ∈ (0, a0 ) then for large values of t the length of ∂Nt is guaranteed to be small. The ˆ t is close to being the standard cusp Vj , a large piece of the minimal intuition is that since W disk Nt should be like a minimal surface N∞ in Vj that intersects Z in the given homotopy class. Such a minimal surface in Vj essentially consists of a geodesic curve in Z going all the way down the cusp. The length of the intersection of N∞ with the horospherical torus of diameter a is proportionate to a. Hence if a0 is small enough, one would expect that if a < a0 and if t is large then the length of ∂Nt is small. In particular, the length of ∂Nt is uniformly bounded with respect to a. A detailed proof appears in [34, Section 12]. R Rescaling from the metric t−1 g(t) to the original metric g(t), ∂Nt κ∂Nt ds is unchanged √ and length(∂Nt ) is multiplied by t. Lemma 91.12. For every D > 0 there is a number a0 > 0 with the following property. Given a ∈ (0, a0 ), suppose that we take Z to be the torus cross-section in Vj of diameter a. Then there is a number Ta′ < ∞ so that as long as t0 ≥ Ta′ , there is a smooth function A¯ ¯ 0 ) = A(t0 ), A¯ ≥ A everywhere, and defined on a neighborhood of t0 such that A(t 3 1 ′ (91.13) A¯ (t0 ) < A(t0 ) − 2π + D. 4 t0 + 41 Proof. Take a0 as in Lemma 91.11. For t0 > Ta , we begin with the minimizing compressing disk Nt0 ⊂ M+ t0 . If t0 is a surgery + − time and Nt0 intersected the surgery region M+ − (M ∩ M ) then N t0 would have to pass t0 t0 t0 through a 10δ(t0 )-neck, which is impossible by Lemma 91.10. Thus Nt0 avoids any parts added by surgery. For t close to t0 we define an embedded compressing disk St ⊂ M+ t as follows. We take Nt0 + ′ and extend it slightly to a smooth surface Nt0 ⊂ Mt0 which contains Nt0 in its interior. The surface Nt′0 will be unscathed on some open time interval containing t0 . If we let St′ ⊂ M+ t be the surface obtained by moving Nt′0 along static curves then for some b > 0, the surface
NOTES ON PERELMAN’S PAPERS
185
ˆ t transversely for all t ∈ (t0 − b, t0 + b). Putting St′ will intersect ∂ W ˆt St = St′ ∩ W
(91.14)
ˆ t. defines a compressing disk for Zˆt ⊂ W Define A¯ : (t0 − b, t0 + b) → R by
¯ = area(St ). A(t)
(91.15)
¯ 0 ) = A(t0 ) and A¯ ≥ A. Clearly A(t
For the rest of the calculation, we will view St as a surface sitting in a fixed manifold M ˆ t ) with a varying metric g(·), and put S = St0 = Nt0 . (a fattening of W
By the first variation formula for area, Z Z d ′ ¯ (91.16) A (t0 ) = hX, ν∂S i ds + dvolS , ∂S S dt t=t0 where X denotes the variation vector field for Zˆt , viewed as a surface moving in M, and ν∂S is the outward normal vector along ∂S. By Proposition 90.1, there is an estimate −1 |X| ≤ α(t0 )t0 2 , where α(t0 ) → 0 as t0 → ∞. Therefore Z −1 hX, ν∂St0 i ds ≤ α(t0 ) t0 2 length(∂Nt0 ). (91.17) ∂St0 By Lemma 91.11, the right-hand side of (91.17) is bounded above by
D 2
if t0 is large.
We turn to the second term in (91.16). Pick p ∈ S and let e1 , e2 , e3 be an orthonormal basis for Tp M with e1 and e2 tangent to S. Then 2 d 1 X dg 1 dvolS = (91.18) (ei , ei ) = − Ric(e1 , e1 ) − Ric(e2 , e2 ). dvolS dt t=t0 2 i=1 dt t=t0
Now
− Ric(e1 , e1 ) − Ric(e2 , e2 )) = −R + Ric(e3 , e3 ) = −R + K(e3 , e1 ) + K(e3 , e2 ) R R = − − K(e1 , e2 ) = − − KS + GKS , 2 2 where KS denotes the Gauss curvature of S and GKS denotes the product of the principal curvatures. Applying the Gauss-Bonnet formula Z Z (91.20) κ∂S ds = 2π − KS volS , (91.19)
∂S
S
the fact that GKS ≤ 0 (since S is time-t0 minimal) and the inequality 1 3 (91.21) Rmin (t) ≥ − 2 t + 41
from Lemma 79.11, we obtain Z Z Z d 3 1 (91.22) dvolS ≤ dvolS + κ∂S ds − 2π. t0 + 41 S dt t=t0 S 4 ∂S
186
BRUCE KLEINER AND JOHN LOTT
By Lemma 91.11, if a ∈ (0, a0 ), diam(Z) = a and t0 is sufficiently large then Using (91.16), (91.17) and (91.22), if t0 is large then 3 1 ′ ¯ (91.23) A (t0 ) < A(t0 ) − 2π + D. 4 t0 + 41
R
∂S
κ∂S ds ≤
This proves the lemma.
D . 2
Proof of Proposition 91.2. Pick D < 2π. Let a < a0 and Ta′ be as in Lemma 91.12. By Lemma 91.12, A is bounded on compact subsets of [Ta′ , ∞). By Lemma 91.10, for ′ ′ any t ∈ [Ta′ , ∞) we can find a compact set Kt ⊂ M+ t so that for all t ∈ [Ta , ∞) sufficiently ′ close to t, the compressing disk Nt′ lies in Kt . If (Kt , g(t)) and (Kt , g(t )) are eσ -biLipschitz equivalent then A(t) ≤ e2σ area (Nt′ ) = e2σ A(t′ ) and A(t′ ) ≤ e2σ area (Nt ) = e2σ A(t). It follows that A is continuous on [Ta′ , ∞). For t ≥ Ta′ , put
(91.24)
F (t) =
1 t+ 4
− 43
1 1 4 A(t) + 4(2π − D) t + . 4
We claim that F (t) ≤ F (Ta′ ) for all t ≥ Ta′ . Suppose not. Put t0 = inf{t ≥ Ta′ : F (t) > F (Ta′ )}. By continuity, F (t0 ) = F (Ta′ ). Consider the function A¯ of Lemma 91.12. Put − 3 1 1 4 1 4 ¯ ¯ A(t) + 4(2π − D) t + . (91.25) F (t) = t + 4 4 Then F¯ (t0 ) = F (t0 ) and in a small interval around t0 , we have F¯ ≥ F . However, (91.13) implies that F¯ ′ (t0 ) < 0. There is some σ > 0 so that for t ∈ (t0 , t0 + σ), we have 1 (91.26) F (t) ≤ F¯ (t) ≤ F¯ (t0 ) + F¯ ′ (t0 ) (t − t0 ) < F¯ (t0 ) = F (t0 ) = F (Ta′ ), 2 which contradicts the definition of t0 . Thus if t ≥ Ta′ then F (t) ≤ F (Ta′ ). This implies that A(t) is negative for large t, which contradicts the fact that an area is nonnegative. We have shown that the homomorphism (91.3) is injective if t is sufficiently large. In view of Lemma 91.4, the same statement holds for all t ≥ Ta . 92. II.7.4. The thin part is a graph manifold This section is concerned with showing that the thin part M − (w, t) is a graph manifold. We refer to Appendix I for the definition of a graph manifold. We remind the reader that this completes the proof the geometrization conjecture. The next two theorems are purely Riemannian. They say that if a 3-manifold is locally volume-collapsed, with sectional curvature bounded below, then it is a graph manifold. They differ slightly in their hypotheses. Theorem 92.1. (cf. Theorem II.7.4) Suppose that (M α , g α ) is a sequence of compact oriented Riemannian 3-manifolds, closed or with convex boundary, and w α → 0. Assume that
NOTES ON PERELMAN’S PAPERS
187
(1) for each point x ∈ M α there exists a radius ρ = ρα (x) not exceeding the diameter of M α such that the ball B(x, ρ) in the metric g α has volume at most w αρ3 and sectional curvatures at least − ρ−2 . (2) each component of the boundary of M α has diameter at most w α , and has a (topologically trivial) collar of length one, where the sectional curvatures are between − 14 − ǫ and − 14 − ǫ. Then for large α, M α is diffeomorphic to a graph manifold.
Remark 92.2. A proof of Theorem 92.1 appears in [64, Section 8]. The proof in [64] is for closed manifolds, but in view of condition (2) the method of proof clearly goes through to manifolds-with-boundary as considered in Theorem 92.1. The statement of the theorem in [52, Theorem II.7.4] also has a condition ρ < 1, which seems to be unnecessary. Theorem 92.3. (cf. Theorem II.7.4) Suppose that (M α , g α ) is a sequence of compact oriented Riemannian 3-manifolds, closed or with convex boundary, and w α → 0. Assume that (1) for each point x ∈ M α there exists a radius ρ = ρα (x) such that the ball B(x, ρ) in the metric g α has volume at most w αρ3 and sectional curvatures at least − ρ−2 . (2) each component of the boundary of M α has diameter at most w α , and has a (topologically trivial) collar of length one, where the sectional curvatures are between − 41 − ǫ and − 14 − ǫ. (3) for every w ′ > 0 and m ∈ {0, 1, . . . , [ǫ−1 ]}, there exist r = r(w ′ ) > 0 and Km = Km (w ′ ) < ∞ such that for sufficiently large α, if r ∈ (0, r] and a ball B(x, r) in the metric g α has volume at least w ′ r 3 and sectional curvatures at least − r −2 then |∇m Rm |(x) ≤ Km r −m−2 . Then for large α, M α is diffeomorphic to a graph manifold.
Remark 92.4. The statement of this theorem in [52, Theorem II.7.4] has the stronger assumption that (3) holds for all m ≥ 0. In the application to the locally collapsing part of the Ricci flow, it is not clear that this stronger condition holds. However, one does get a bound on a large number of derivatives, which is good enough. Remark 92.5. As pointed out in [52, Section 7.4], adding condition (3) simplifies the proof and allows one to avoid both Alexandrov spaces and Perelman’s stability theorem. (A proof of Perelman’s stability theorem appears in [38]). Proofs of Theorem 92.3 are in [9], [40] and [46]. Remark 92.6. Comparing Theorems 92.1 and 92.3, Theorem 92.1 has the extra assumption that ρα (x) does not exceed the diameter of M α . Without this extra assumption, the Alexandrov space arguments could give that for large α, M α is homeomorphic to a nonnegatively curved Alexandrov space [64, Theorem 1.1(2)]. This does not immediately imply that M α is a graph manifold. Remark 92.7. We give some simple examples where the collapsing theorems apply. Let (Σ, ghyp ) be a closed surface with the hyperbolic metric. Let S 1 (µ) be a circle of length µ and consider the Ricci flow on S 1 × Σ with the initial metric S 1 (µ) × (Σ, c0 ghyp ). The Ricci flow solution at time t is S 1 (µ) × (Σ, (c0 + 2t)ghyp ). In the rest of this example we consider the rescaled metric t−1 g(t). Its diameter goes like O(t0 ) and its sectional curvatures go like O(t0 ). If we take ρ to be a small constant then for large t, B(x, t, ρ) is approximately a circle bundle over a ball in a hyperbolic surface of constant sectional curvature − 21 , with
188
BRUCE KLEINER AND JOHN LOTT
circle lengths that go like t−1/2 . The sectional curvature on B(x, t, ρ) is bounded below by − ρ−2 , and ρ−3 vol(B(x, t, ρ)) ∼ t−1/2 . Theorems 92.1 and 92.3 both apply.
Next, consider a compact 3-dimensional nilmanifold that evolves under the Ricci flow. Let θ1 , θ2 , θ3 be affine-parallel 1-forms on M which lift to Maurer-Cartan forms on the Heisenberg group, with dθ1 = dθ2 = 0 and dθ3 = θ1 ∧ θ2 . Consider the metric (92.8)
g(t) = α2 (t) θ12 + β 2 (t) θ22 + γ 2 (t) θ32 . 2
Its sectional curvatures are R1212 = − 34 αγ2 β 2 and R1313 = R2323 = 41 is 1 γ2 (92.9) Ric = − α2 θ12 − β 2 θ22 + γ 2 θ32 . 2 2 2 α β The general solution to the Ricci flow equation is of the form (92.10)
γ2 . α2 β 2
The Ricci tensor
α2 (t) = A0 (t + t0 )1/3 ,
β 2 (t) = B0 (t + t0 )1/3 , A0 B0 γ 2 (t) = (t + t0 )−1/3 . 3 In the rest of this example we consider the rescaled metric t−1 g(t). Its diameter goes like t−1/3 , its volume goes like t−4/3 and its sectional curvatures go like t0 . If we take ρ(x) = diam then ρ−3 vol(B(x, ρ)) ∼ t−1/3 , so both Theorem 92.1 and Theorem 92.3 apply. We could also take ρ(x) to be a small constant c > 0, in which case Theorem 92.3 applies. In general, among the eight maximal homogeneous geometries, the rescaled solution for a ^ compact 3-manifold with geometry H 2 × R or SL 2 (R) will collapse to a hyperbolic surface 1 of constant sectional curvature − 2 . The rescaled solution for a Sol geometry will collapse to a circle. The rescaled solution for an R3 or Nil geometry will collapse to a point. We remark that although these homogeneous solutions are collapsing in the sense of Theorem 92.1, there is no contradiction with the no local collapsing result of Theorem 26.2, which only rules out local collapsing on a finite time interval. Returning to our Ricci flow with surgery, recall the statement of Proposition 90.1. If the b i (t) be the collection {H1 , . . . , Hk } of Proposition 90.1 is nonempty then for large t, let H result of removing from Hi the horoballs whose boundaries are at distance approximately 1 b i (t) = Hi .) Put α(t) from the basepoint xi . (If there are no such horoballs then H 2 (92.11)
b b Mthin (t) = M+ t − f (t)(H1 (t) ∪ . . . Hk (t)).
Proposition 92.12. For large t, Mthin (t) is a graph manifold.
Proof. We give two closely related proofs, one using Theorem 92.3 and one using Theorem 92.1. If the proposition is not true then there is a sequence tα → ∞ so that for each α, Mthin (tα ) is not a graph manifold. Let M α be the manifold obtained from Mthin (tα ) by throwing away connected components which are closed and admit metrics of nonnegative sectional curvature, and put g α = (tα )−1 g(tα ). Since any closed manifold of nonnegative
NOTES ON PERELMAN’S PAPERS
189
sectional curvature is a graph manifold by [35], for each α the manifold M α is not a graph manifold. We first show that the assumptions of Theorem 92.3 are verified. Lemma 92.13. Condition (3) in Theorem 92.3 holds for the M α ’s. ′ Proof. With w ′ being a parameter as in Condition (3) in Theorem 92.3, let r = r(w√ ) be the parameter of Corollary 81.3. It is enough to show that for large α, if r ∈ (0, r tα ], α α ′ 3 xα ∈ M+ tα , and B(x , t , r) has volume at least w r and sectional curvatures bounded below by − r −2 , then |∇m Rm |(xα , tα ) ≤ Km r −m−2 for an appropriate choice of constants Km .
To prove this by contradiction, we assume that after passing to a subsequence if necessary, √ α α α ′ α 3 α α α there are r ∈ (0, r t ] and x ∈ M+ tα such that B(x , t , r ) has volume at least w (r ) α −2 α m+2 m α α and sectional curvature at least −(r ) , but limα→∞ (r ) |∇ Rm |(x , t ) = ∞ for −1 some m ≤ [ǫ ].
In the notation of Corollary 81.3, if r α ≥ θ−1 (w ′ )hmax (tα ) for infinitely many α then for these α, Corollary 81.3 gives a curvature bound on an unscathed parabolic neighborhood P (xα , tα , r α /4, −τ (r α )2 ) and hence, by Appendix D, derivative bounds at (xα , tα ). This is a contradiction. Therefore we may assume that r α < θ−1 (w ′ )hmax (tα ) ≤ r(tα ) for all α, where we used Remark 86.4 for the last inequality.
Suppose first that R(xα , tα ) ≤ (r(tα ))−2 . By Lemma 70.1, there is an estimate R ≤ 1 −1 α −2 α α 1 −1 α α 2 16 (r(t )) on the parabolic neighborhood P x , t , 4 η r(t ), − 16 η (r(t )) . A surgery in this neighborhood could only occur where R ≥ hmax (tα )−2 . For large α, hmax (tα )−2 >> r(tα )−2 by Remark 86.4. Hence this neighborhood is unscathed. Appendix D now gives bounds of the form |∇m Rm |(xα , tα ) ≤ const. (r(tα ))−m−2 ≤ const. (r α )−m−2 , which is a contradiction.. Suppose now that R(xα , tα ) > (r(tα ))−2 . Then (xα , tα ) is in the center of a canonical m+2 neighborhood and there are universal estimates |∇m Rm |(xα , tα ) ≤ const.(m) R(xα , tα ) 2 for all m ≤ [ǫ−1 ]. Hence in this case, it suffices to show that R(xα , tα ) is bounded above by a constant times (r α )−2 , i.e. it suffices to get a contradiction just to the assumption that limα→∞ (r α )2 R(xα , tα ) = ∞. So suppose that limα→∞ (r α )2 R(xα , tα ) = ∞. We claim that limα→∞ (r α )2 inf R = B(xα ,tα ,r α )
∞. Suppose not. Then there is some C ∈ (0, ∞) so that after passing to a subsequence, there are points xα′ ∈ B (xα , tα , r α ) with (r α )2 R(xα′ , tα ) < C. Considering points along the time-tα geodesic segment from xα′ to xα , for large α we can find points xα′′ ∈ B (xα , tα , r α ) with (r α )2 R(xα′′ , tα ) = 2C. Applying Lemma 70.2 at (xα′′ , tα ), or more precisely a version that applies along geodesics as in Claim 2 of II.4.2, we obtain a contradiction to the assumption that limα→∞ (r α )2 R(xα , tα ) = ∞. In applying Lemma 70.2 we use that r α < r(tα ) and Φ(2C(r α )−2 ) limα→∞ r(tα ) = 0, giving limα→∞ (r α )−2 tα = ∞, in order to say that limα→∞ 2C(rα )−2 = 0. Hence for large α, every point x ∈ B (xα , tα , r α ) is in the center of a canonical neighbor1 hood of size comparable to R(x, tα )− 2 , which is small compared to r α . On the other hand,
190
BRUCE KLEINER AND JOHN LOTT
from Lemma 83.1, there is a ball B ′ of radius θ0 (w ′) r α in B (xα , tα , r α ) so that every subball of B ′ has almost-Euclidean volume. This is a contradiction. This proves the lemma.
We continue with the proof of Proposition 92.12. By construction there is a sequence 1 w α → 0 so that conditions (1) and (2) of Theorem 92.3 hold with ρα (xα ) = (tα )− 2 ρ(xα , tα ). Hence for large α, M α is diffeomorphic to a graph manifold. This is a contradiction to the choice of the M α ’s and proves the theorem. We now give a proof that instead uses Theorem 92.1. Let dα denote the diameter of 1 M α . If we take ρα (xα ) = (tα )− 2 ρ(xα , tα ) then we can apply Theorem 92.1 as long as that the diameter statement in condition (1) of Theorem 92.1 is satisfied. If it is not satisfied then there is some point xα ∈ M α with ρα (xα ) > dα . The sectional curvatures of M α are bounded below by − ρα (x1α )2 , and so are bounded below by − (dα1 )2 . If there is a subsequence with
ρα (xα ) dα
(92.14)
≤ C < ∞ then
vol(M α ) = vol(B(xα , ρα (xα ))) ≤ w α ρα (xα )3 ≤ w α C 3 (dα )3
and we can apply Theorem 92.1 with ρα = dα , after redefining w α . Thus we may assume α) α α → 0 then we can apply that limα→∞ ρ d(xα ) = ∞. If there is a subsequence with vol(M α 3 (d ) α
) Theorem 92.1 with ρα = dα . Thus we may assume that vol(M is bounded away from zero. (dα )3 After rescaling the metric to make the diameter one, we are in a noncollapsing situation with the lower sectional curvature bound going to zero. By the argument in the proof of Lemma 92.13 (which used Corollary 81.3) there are uniform L∞ -bounds on Rm(M α ) and its covariant derivatives. After passing to a subsequence, there is a limit (M∞ , g∞ ) in the smooth topology which is diffeomorphic to M α for large α, and carries a metric of nonnegative sectional curvature. As any boundary component of M∞ would have to have a neighborhood of negative sectional curvature (see the definition of Mthin and condition (2) of Theorem 92.1), M∞ is closed. However, by construction M α has no connected components which are closed and admit metrics of nonnegative sectional curvature. This contradiction shows that the diameter statement in condition (1) of Theorem 92.1 is satisfied. The other conditions of Theorem 92.1 are satisfied as before.
b b Thus for large t, M+ t has a decomposition into a piece f (t)(H1 (t) ∪ . . . Hk (t)), whose interior admits a complete finite-volume hyperbolic metric, and the complement, which is a graph manifold. In addition, by Section 91 the cuspidal tori are incompressible in M+ t . By Lemma 73.4, the initial (connected) manifold M0 is diffeomorphic to a connected sum of the connected components of Mt , along with some possible additional connected sums with a finite number of S 1 × S 2 ’s and quotients of the round S 3 . This proves the geometrization conjecture of Appendix I.
NOTES ON PERELMAN’S PAPERS
191
93. II.8. Alternative proof of cusp incompressibility The goal of this section is to prove Perelman’s Proposition II.8.2, which gives a numerical characterization of the geometric type of a compact 3-manifold. It also contains an independent proof of the incompressibility of the cuspidal ends of the hyperbolic piece in the geometric decomposition. We recall from Section 7 that λ(g) is the first eigenvalue of −4△ + R, and can also be expressed as R (4|∇Φ|2 + R Φ2 ) dV M R (93.1) λ(g) = inf . Φ∈C ∞ (M ) : Φ6=0 Φ2 dV M From Lemma 7.11, if g(·) is a Ricci flow and λ(t) = λ(g(t)) then d 2 λ(t) ≥ λ2 (t). dt 3
(93.2) 2
From Lemma 8.1, λ(t) V (t) 3 is nondecreasing when it is nonpositive. For any metric g, there are inequalities (93.3)
min R ≤ λ(g) ≤
R
R dV , vol(M, g) M
where the first inequality follows directly from (93.1) and the second inequality comes from using 1 as a test function in (93.1). 2
Perelman’s proof of his Proposition II.8.2 uses the functional λ(g)V (g) 3 . The functional 2 2 b = Rmin (t)V (t) 3 Rmin V (g) 3 plays a similar role. For example, from Corollary 87.7, R(t) is nondecreasing when it is nonpositive. We first give a proof of an analog of Proposition 2 2 II.8.2 that uses Rmin V (g) 3 instead of λ(g)V (g) 3 . The technical simplification is that when 2 Rmin V (g) 3 is nonpositive, it is nondecreasing under a surgery, as surgeries are only done in regions of large positive scalar curvature, so Rmin doesn’t change, and a surgery reduces 2 volume. (A possible extinction of a component clearly doesn’t change Rmin V (g) 3 .) We show that a minimal-volume hyperbolic submanifold of M has incompressible tori, which gives a different approach to Section 91. 2
Perelman’s alternative approach to Section 91 uses the functional λ(g)V (g) 3 instead of 2 2 2 Rmin V (g) 3 . Our use of Rmin V (g) 3 and the sigma-invariant σ(M), instead of λ(g)V (g) 3 and λ, is inspired by [4]. 2
We then give the arguments using λ(g)V (g) 3 , thereby proving Perelman’s Proposition 2 II.8.2. The main technical difficulty is to control how λ(g)V (g) 3 changes under a surgery. 93.1. The approach using the σ-invariant. We first give some well-known results about the sigma-invariant. We recall that the sigma-invariant of a closed connected manifold M of dimension n ≥ 3 is given by R R(g) dvol(g) (93.4) σ(M) = sup inf M , n−2 C g∈C vol(M, g) n
192
BRUCE KLEINER AND JOHN LOTT
where C runs over the conformal classes of Riemannian metrics on M. From the solution to the Yamabe problem, the infimum in (93.4) is realized by a metric of constant scalar curvature in the given conformal class. It follows that if σ(M) > 0 then M admits a metric with positive scalar curvature. Conversely, suppose that M admits a metric g0 with positive scalar curvature. Let C be the conformal class containing g0 . Then R 4(n−1) R 2 2 |∇u| + R(g0 ) u dvolM (g0 ) R(g) dvol(g) M n−2 = inf (93.5) inf M n−2 n−2 R u>0 g∈C 2n n vol(M, g) n n−2 dvolM (g0 ) u M is positive, in view of the Sobolev embedding theorem, and so σ(M) > 0. We claim that if σ(M) ≤ 0 then (93.6)
2
σ(M) = sup Rmin (g) V (g) n . g
To see this, as the infimum in (93.4) is realized by a metric of constant scalar curvature 2 in the given conformal class, it follows that σ(M) ≤ supg Rmin (g) V (g) n . Now given a Riemannian metric g, the infimum in (93.4) within the corresponding conformal class C 4 2 e V (e e Then equals R g ) n for a metric e g = u n−2 g with constant scalar curvature R. R 4(n−1) |∇u|2 + R u2 dvolM n−2 M 2 e n (93.7) R V (e g ) = inf . R n−2 u>0 2n n u n−2 dvolM M As
(93.8)
R 2 2 |∇u| + R u dvolM u2 dvolM n−2 M ≥ R (g) min R R n−2 n−2 2n 2n n n n−2 dvol n−2 dvol u u M M M M
R 4(n−1) M
2 2 e V (e and Rmin (g) ≤ 0, Holder’s inequality implies that R g ) n ≥ Rmin (g) V (g) n . It follows 2 that σ(M) ≥ supg Rmin (g) V (g) n .
The next proposition answers conjectures of Anderson [2].
Proposition 93.9. Let M be a closed connected oriented 3-manifold. (a) If σ(M) > 0 then M is diffeomorphic to a connected sum of a finite number of S 1 × S 2 ’s and metric quotients of the round S 3 . Conversely, each such manifold has σ(M) > 0. (b) M is a graph manifold if and only if σ(M) ≥ 0. 3 (c) If σ(M) < 0 then − 32 σ(M) 2 is the minimum of the numbers V with the following property : M can be decomposed as a connected sum of a finite collection of S 1 ×S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) complete finite-volume manifold N with constant sectional curvature − 41 and volume V which can be embedded in M ′ so that the complement M ′ − N (if nonempty) is a graph manifold. 3 Moreover, if vol(N) = − 23 σ(M) 2 then the cusps of N (if any) are incompressible in M ′.
NOTES ON PERELMAN’S PAPERS
193
Proof. If σ(M) > 0 then M has a metric g of positive scalar curvature. From Lemmas 81.1 and 81.2, M is a connected sum of S 1 × S 2 ’s and metric quotients of the round S 3 . Conversely, if M is a connected sum of S 1 × S 2 ’s and metric quotients of the round S 3 then M admits a metric g of positive scalar curvature and so σ(M) > 0. Now suppose that σ(M) ≤ 0. If M is a graph manifold then M volume-collapses with bounded curvature, so (93.6) implies that σ(M) = 0. Suppose that M is not a graph manifold. Suppose that we have a given decomposition of M as a connected sum of a finite collection of S 1 ×S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) finite-volume complete manifold N with constant sectional curvature − 14 which can be embedded in M ′ so that the complement (if nonempty) is a graph manifold. Let Vhyp denote the hyperbolic volume of N. We do not assume that the cusps of N are incompressible in M ′ . For any ǫ > 0, we claim that there is a metric gǫ on M with R ≥ − 6 · 41 − ǫ and volume V (gǫ ) ≤ Vhyp + ǫ. This comes from collapsing the graph manifold pieces, along with the fact that the connected sum operation can be performed while decreasing the scalar curvature arbitrarily little and increasing the volume arbitrarily 2 2/3 2/3 little. Then Rmin (gǫ ) V (gǫ ) 3 ≥ − 23 Vhyp − const. ǫ. Thus σ(M) ≥ − 32 Vhyp . Let Vb denote the minimum of Vhyp over all such decompositions of M. (As the set of volumes of complete finite-volume 3-manifolds with constant curvature − 41 is well-ordered, there is a minimum.) Then σ(M) ≥ − 23 Vb 2/3 .
Next, take an arbitrary metric g0 on M and consider the Ricci flow g(t) with initial metric g0 . From Sections 90 and 92, there is a nonempty manifold N with a complete finite+ volume metric of constant curvature − 14 so that for large t, there is a decomposition M t = 1 M1 (t)∪M2 (t) of the time-t manifold, where M1 (t) is a graph manifold and (M2 (t), t g(t) M2 (t) ) is close to a large piece of N. In terms of condition (c) of Proposition 93.9, we will think of 3 M ′ as being M+ t . Because of the presence of N, we know that t Rmin (t) ≤ − 2 + ǫ(t) and V (t) ≥ t2/3 Vhyp (N) − ǫ(t) for a function ǫ(t) with limt→∞ ǫ(t) = 0. The monotonicity of Rmin (t) V 2/3 (t), even through surgeries, implies that 3 3 (93.10) Rmin (g0 ) V 2/3 (g0 ) ≤ − Vhyp (N)2/3 ≤ − Vb 2/3 . 2 2 Thus σ(M) ≤ − 3 Vb 2/3 . 2
This shows that σ(M) = − 23 Vb 2/3 . Now take a decomposition of M as in condition (c) of Proposition 93.9, with Vhyp (N) = Vb . We claim that the cuspidal 2-tori of N are incompressible in M ′ . If not then there would be a metric g on M with R(g) ≥ − 32 and vol(g) < Vhyp (N) [3, Pf. of Theorem 2.9]. This would contradict the fact that σ(M) = − 23 Vhyp (N)2/3 . 93.2. The approach using the λ-invariant.
Proposition 93.11. (cf. II.8.2) Let M be a closed connected oriented 3-manifold. (a) If M admits a metric g with λ(g) > 0 then it is diffeomorphic to a connected sum of a finite number of S 1 × S 2 ’s and metric quotients of the round S 3 . Conversely, each such
194
BRUCE KLEINER AND JOHN LOTT
manifold admits a metric g with λ(g) > 0. (b) Suppose that M does not admit any metric g with λ(g) > 0. Let λ denote the supremum 2 of λ(g)V (g) 3 over all metrics g on M. Then M is a graph manifold if and only if λ = 0. 3 (c) Suppose that M does not admit any metric g with λ(g) > 0, and λ < 0. Then − 32 λ 2 is the minimum of the numbers V with the following property : M can be decomposed as a connected sum of a finite collection of S 1 × S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) complete manifold N with constant sectional curvature − 14 and volume V which can be embedded in M ′ so that the complement M ′ − N (if nonempty) is a graph manifold. 3 Moreover, if vol(N) = − 32 λ 2 then the cusps of N (if any) are incompressible in M ′ . Proof. We first give the argument for Proposition 93.11 under the pretense that all Ricci flows are smooth, except for possible extinction of components. (Of course this is not the case, but it will allow us to present the main idea of the proof.) If λ(g) > 0 for some metric g then from (93.2), the Ricci flow starting from g will become 3 extinct within time 2λ(g) . Hence Lemma 81.2 applies. Conversely, if M is a connected sum 1 2 of S × S ’s and metric quotients of the round S 3 then M admits a metric g of positive scalar curvature. From (93.3), λ(g) > 0. Now suppose that M does not admit any metric g with λ(g) > 0. If M is a graph manifold then M volume-collapses with bounded curvature, so (93.3) implies that λ = 0. Suppose that M is not a graph manifold. Suppose that we have a given decomposition of M as a connected sum of a finite collection of S 1 × S 2 ’s, metric quotients of the round S 3 and some other components, the union of which is denoted by M ′ , and there exists a (possibly disconnected) complete manifold N with constant sectional curvature − 41 which can be embedded in M ′ so that the complement (if nonempty) is a graph manifold. Let Vhyp denote the hyperbolic volume of N. We do not assume that the cusps of N are incompressible in M ′ . For any ǫ > 0, we claim that there is a metric gǫ on M with R ≥ − 6 · 14 − ǫ and volume V (gǫ ) ≤ Vhyp + ǫ. This comes from collapsing the graph manifold pieces, along with the fact that the connected sum operation can be performed while decreasing the scalar curvature arbitrarily little and increasing the volume arbitrarily little. Then (93.3) implies 2 2/3 2/3 that λ(gǫ ) V (gǫ ) 3 ≥ − 23 Vhyp − const. ǫ. Thus λ ≥ − 23 Vhyp .
Let Vb denote the minimum of Vhyp over all such decompositions of M. (As the set of volumes of complete finite-volume 3-manifolds with constant curvature − 14 is well-ordered, there is a minimum.) Then λ ≥ − 32 Vb 2/3 .
Next, take an arbitrary metric g0 on M and consider the Ricci flow g(t) with initial metric g0 . From Sections 90 and 92, there is a nonempty manifold N with a finite-volume complete metric of constant curvature − 14 so that for large t, there is a decomposition of the time-t + 1 manifold Mt = M1 (t) ∪ M2 (t) where M1 (t) is a graph manifold and (M2 (t), t g(t) M2 (t) ) is
NOTES ON PERELMAN’S PAPERS
195
close to a large piece of N. As N has finite volume, a constant function on N is squareintegrable and so inf spec(−△N ) = 0. Equivalently, R |∇f |2 dvolN N R (93.12) inf = 0. f ∈Cc∞ (N ),f 6=0 f 2 dvolN N
Taking an appropriate test function Φ on M with compact support in M2 (t) gives t λ(t) ≤ − 32 +ǫ1 (t), with limt→∞ ǫ1 (t) = 0. In terms of condition (c) of Proposition 93.11, we will think 2/3 of M ′ as being M+ Vhyp (N) − ǫ2 (t), with t . From the presence of N, we know that V (t) ≥ t limt→∞ ǫ2 (t) = 0. As we are assuming that the Ricci flow is nonsingular, the monotonicity of λ(t) V 2/3 (t) implies that (93.13)
λ(g0 ) V 2/3 (g0 ) ≤ −
Thus λ ≤ − 32 Vb 2/3 .
3 3 Vhyp (N)2/3 ≤ − Vb 2/3 . 2 2
This shows that λ = − 23 Vb 2/3 . Now take a decomposition of M as in condition (c) of Proposition 93.11, with Vhyp (N) = Vb . We claim that the cuspidal 2-tori of N are incompressible in M ′ . If not then there would be a metric g on M with R(g) ≥ − 23 and vol(g) < Vhyp (N) [3, Pf. of Theorem 2.9]. Using (93.3), one would obtain a contradiction to the fact that λ = − 32 Vhyp (N)2/3 . 2
To handle the behaviour of λ(t) V 3 (t) under Ricci flows with surgery, we first state a couple of general facts about Schr¨odinger operators. Lemma 93.14. Given a closed Riemannian manifold M, let X be a codimension-0 submanifoldwith-boundary of M. Given R ∈ C ∞ (M), let λM be the lowest eigenvalue of −4△ + R on M, with corresponding eigenfunction ψ. Let λX be the lowest eigenvalue of the corresponding operator on X, with Dirichlet boundary conditions, and similarly for λM −int(X) . Then for all η ∈ Cc∞ (int(X)), we have (93.15)
λM ≤ min(λX , λM −int(X) )
and (93.16)
λX ≤ λM + 4
R
|∇η|2 ψ 2 dV . η 2 ψ 2 dV M
M R
Proof. Equation (93.15) follows from Dirichlet-Neumann bracketing [55, Chapter XIII.15]. To prove (93.16), ηψ is supported in int(X) and so R (4|∇(ηψ)|2 + R η 2 ψ 2 ) dV R λX ≤ M (93.17) η 2 ψ 2 dV M R (4|∇η|2 ψ 2 + 8h∇η, ∇ψiηψ + 4η 2 |∇ψ|2 + R η 2 ψ 2 ) dV R . = M η 2 ψ 2 dV M As −4△ψ + Rψ = λM ψ, we have
196
(93.18)
BRUCE KLEINER AND JOHN LOTT
λM
Z
2
Z
2
Z
2
η ψ dV = − 4 η ψ△ψ dV + Rη 2 ψ 2 dV M M Z M Z 2 = 4 h∇(η ψ), ∇ψi dV + Rη 2 ψ 2 dV ZM Z M Z 2 2 = 4 η |∇ψ| dV + 8 h∇η, ∇ψiηψ dV + Rη 2 ψ 2 dV. M
M
M
Equation (93.16) follows.
The next result is an Agmon-type estimate. Lemma 93.19. With the notation of Lemma 93.14, given a nonnegative function φ ∈ C ∞ (M), suppose that f ∈ C ∞ (M) satisfies 4|∇f |2 ≤ R − λM − c
(93.20)
on supp(φ), for some c > 0. Then
kef φψk2 ≤ 4 c−1 kef △φk∞ + kef ∇φk∞ (λM − min R)1/2 kψk2 .
(93.21)
Proof. Put H = − 4△ + R. By assumption,
φ(R − 4|∇f |2 − λM ) φ ≥ c φ2
(93.22)
and so there is an inequality of operators on L2 (M) : (93.23)
φ(H − 4|∇f |2 − λM ) φ = 4φd∗ dφ + φ(R − 4|∇f |2 − λM ) φ ≥ c φ2 .
In particular,
Z
(93.24)
f
2
f
e ψφ (H − 4|∇f | − λM ) φe ψdV ≥ c
M
Z
φ2 e2f ψ 2 dV.
M
For ρ ∈ C ∞ (M), (93.25) and so
ef H(e−f ρ) = Hρ + 4 ∇ · ((∇f )ρ) + 4 h∇f, ∇ρi − 4 |∇f |2ρ Z
(93.26)
f
−f
ρe H(e ρ) dV =
M
f
Z
M
ρ(H − 4 |∇f |2 )ρ dV.
Taking ρ = e φψ gives Z Z 2f (93.27) e φψH(φψ) dV = ef φψ(H − 4 |∇f |2)ef φψ dV. M
M
From (93.24) and (93.27), (93.28) Z Z f 2 2f cke φψk2 ≤ e φψ(H − λM )(φψ) dV = e2f φψ[H, φ]ψ dV = hef φψ, ef [H, φ]ψi2 . M
M
Thus
(93.29)
ckef φψk2 ≤ kef [H, φ]ψk2 .
NOTES ON PERELMAN’S PAPERS
197
Now (93.30) Then (93.31) Finally,
ef [H, φ]ψ = − 4 ef (△φ)ψ − 8 ef h∇φ, ∇ψi. kef [H, φ]ψk2 ≤ 4 kef △φk∞ kψk2 + 8 kef ∇φk∞ k∇ψk2 .
(93.32) and so (93.33) This proves the lemma.
4
k∇ψk22
=
Z
M
(λM − R) ψ 2 dV
2k∇ψk2 ≤ (λM − min R)1/2 kψk2 .
Clearly Lemma 93.19 is also true if f is just assumed to be Lipschitz-regular. We now apply Lemmas 93.14 and 93.19 to a Ricci flow with surgery. A singularity caused by extinction of a component will not be a problem, so let T0 be a surgery time + and let M+ = M+ T0 be the postsurgery manifold. We will write λ instead of λM+ . Let + − + − Mcap = M+ T0 − (MT0 ∩ MT0 ) be the added caps and put X = M+ − Mcap = MT0 ∩ MT0 . For simplicity, let us assume that Mcap has a single component; the argument in the general case is similar. From the nature of the surgery procedure, the surgery is done in an ǫ-horn extending from Ωρ , where ρ = δ(T0 )r(T0 ). In fact, because of the canonical neighborhood assumption, we can extend the ǫ-horn inward until R ∼ r(T0 )−2 . Appplying (93.1) with a test function supported in an ǫ-tube near this inner boundary, it follows that λ+ ≤ c′ r(T0 )−2 for some universal constant c′ >> 1. In what follows we take δ(T0 ) to be small. As R is much greater than r(T0 )−2 on Mcap , it follows that λM −int(X) is much greater than r(T0 )−2 . Then from (93.15), λ+ ≤ λX . We can apply (93.16) to get an inequality the other way. We take the function η to interpolate from being 1 outside of the h(T0 )-neighborhood Nh Mcap of Mcap , to being 0 on Mcap . In terms of the normalized eigenfunction ψ on M+ , this gives a bound of the form R ψ 2 dV Nh Mcap R . (93.34) λX ≤ λ+ + const. h(T0 )−2 1 − Nh Mcap ψ 2 dV R We now wish to show that Nh Mcap ψ 2 dV is small. For this we apply Lemma 93.19 with c = c′ r(T0 )−2 . Take an ǫ-tube U, in the ǫ-horn, whose center has scalar curvature roughly 200 c′ r(T0 )−2 and which is the closest tube to the cap with this property. Let x : U → (− ǫ−1 , ǫ−1 ) be the longitudinal parametrization of the tube, which we take to be increasing in the direction of the surgery cap. Let Φ : (−1, 1) → [0, 1] be a fixed nondecreasing smooth function which is zero on (−1, 1/4) and one on (1/2, 1). Put φ = Φ◦x on U. Extend φ to M+ by making it zero to the left of U and one to the right of U, where “right of U” means the connected component of M+ − U containing the surgery cap. Dimensionally, |∇φ|∞ ≤ const. r(T0 )−1 and |△φ|∞ ≤ const. r(T0 )−2 . Define a function f to the right of x−1 (0) by setting it to be the distance from x−1 (0) with respect to the metric 41 (R − λ+ − c) gM+ . (Note that to the right of x−1 (0), we have R ≥ 200c′r(T0 )−2 ≥
198
BRUCE KLEINER AND JOHN LOTT
λ+ + c.) Then equation (93.21) gives |ef φψ|2 ≤ const.(T0 ). The point is that const.(T0 ) is independent of the (small) surgery parameter δ(T0 ). Hence (93.35)
Z
Nh Mcap
ψ 2 dV ≤
sup e−2f Nh Mcap
!Z
Nh Mcap
e2f ψ 2 dV ≤ const. sup e−2f . Nh Mcap
To estimate supNh Mcap e−2f , we use the fact that the ǫ-horn consists of a sequence of ǫ-tubes stacked together. In the region of M+ from x−1 (0) to the surgery cap, the scalar curvature ranges from roughly 200 c′ r(T0 )−2 to h(T0 )−2 . On a given ǫ-tube, if ǫ is sufficiently small then the ratio of the scalar curvatures between the two ends is bounded by e. Hence in going from x−1 (0) to the surgery cap, one must cross at least N disjoint ǫ-tubes, with 1 2 −2 eN = 200c . Traversing a given ǫ-tube (say of radius r ′ ) in going towards the ′ r(T0 ) h(T0 ) R ǫ−1 r′ surgery cap, f increases by roughly const. −ǫ−1 r′ (r ′ )−1 ds, which is const. ǫ−1 . Hence near the surgery cap, we have (93.36)
sup e−2f ≤ const. e− const. N ǫ
Nh Mcap
−1
−1
= const. (r(T0 )2 h(T0 )−2 )− const. ǫ .
Combining this with (93.34) and (93.35), we obtain (93.37)
−1
λX ≤ λ+ + const. h(T0 )−2 (r(T0 )2 h(T0 )−2 )− const. ǫ .
By making a single redefinition of ǫ, we can ensure that λX ≤ λ+ + const. h(T0 )4 . The last constant will depend on r(T0 ) but is independent of δ(T0 ). Thus if δ(T0 ) is small enough, we can ensure that |λX − λ+ | is small in comparison to the volume change V − (T0 ) − V + (T0 ), which is comparable to h(T0 )3 . If λ− is the smallest eigenvalue of − 4 △ + R on the presurgery manifold Mt , for t slightly less than T0 , then we can estimate |λX − λ− | in a similar way. Hence for an arbitrary positive continuous function ξ(t), we can make the parameters δ j of Proposition 77.2 small enough to ensure that (93.38)
|λ+ (T0 ) − λ− (T0 )| ≤ ξ(T0 ) (V − (T0 ) − V + (T0 ))
for a surgery at time T0 . We now redo the argument for the proposition, as given above in the surgery-free case, in the presence of surgeries. Suppose first that λ(g0 ) > 0 for some metric g0 on M. After possible rescaling, we can assume that g0 is the initial condition for a Ricci flow with surgery (M, g(·)), with normalized initial condition. Using the lower scalar curvature bound of 3 Lemma 79.11 and the Ricci flow equation, the volume on the time interval [0, λ(0) ] has an a priori upper bound of the form V (0). As a surgery at time T0 removes a volume P const. 3 3 comparable to h(T0 ) , we have h(T0 ) ≤ const. V (0), where the sum is over the surgeries and T0 denotes the surgery time. FromPthe above discussion, the change in λ due to the surgeries is bounded below by − const. T0 h(T0 )4 . Then the decrease in λ due to surgeries
NOTES ON PERELMAN’S PAPERS 3 on the time interval [0, λ(0) ] is bounded above by
(93.39)
const.
X
3 ] T0 ∈[0, λ(0)
199
h(T0 )4 ≤ const. sup h(t) V (0). 3 t∈[0, λ(0) ]
By choosing the function δ(t) to be sufficiently small, the decrease in λ due to surgeries is 3 ] coming from the increase not enough to prevent the blowup of λ on the time interval [0, λ(0) of λ between the surgeries. Hence the solution goes extinct. Now suppose that M does not admit a metric g with λ(g) > 0. Again, if M is a graph manifold then λ = 0. Suppose that M is not a graph manifold. As before, λ ≥ − 32 Vb 2/3 . Given an initial metric g0 , we wish to show that by choosing the function δ(t) small enough we can make the function λ(t)V 2/3 (t) arbitrarily close to being nondecreasing. To see this, we consider the effect of a surgery on λ(t)V 2/3 (t). Upon performing a surgery the volume decreases, which in itself cannot decrease λ(t)V 2/3 (t). (We are using the fact that λ(t) is nonpositive.) Then from the above discussion, the change in λ(t)V 2/3 (t) due from a surgery at time T0 , is bounded below by −ξ(T0 ) (V − (T0 ) −V + (T0 )) V − (T0 )2/3 . With normalized initial conditions, we have an a priori upper bound on V (t) in terms of V (0) and t. Over any time interval [T1 , T2 ], we must have X V − (T0 ) − V + (T0 ) ≤ sup V (t), (93.40) T0 ∈[T1 ,T2 ]
t∈[T1 ,T2 ]
where the sum is over the surgeries in the interval [T1 , T2 ]. Then along with the monotonicity of λ(t)V 2/3 (t) in between the surgery times, by choosing the function δ(t) appropriately we can ensure that for any σ > 0 there is a Ricci flow with (r, δ)-cutoff starting from g0 so that λ(g0 ) V 2/3 (g0 ) ≤ λ(t) V 2/3 (t) + σ for all t. It follows that λ ≤ − 32 Vb 2/3 .
This shows that λ = − 23 Vb 2/3 . The same argument as before shows that if we have a decomposition with the hyperbolic volume of N equal to Vb then the cusps of N (if any) are incompressible in M ′ .
Remark 93.41. It follows that if the three-manifold M does not admit a metric of positive scalar curvature then σ(M) = λ. In fact, this is true in any dimension n ≥ 3 [1]. Appendix A. Maximum principles
In this appendix we list some maximum principles and their consequences. Our main source is [23], where references to the original literature can be found. The first type of maximum principle is a weak maximum principle which says that under certain conditions, a spatial inequality on the initial condition implies a time-dependent inequality at later times. Theorem A.1. Let M be a closed manifold. Let {g(t)}t∈[0,T ] be a smooth one-parameter family of Riemannian metrics on M and let {X(t)}t∈[0,T ] be a smooth one-parameter family
200
BRUCE KLEINER AND JOHN LOTT
of vector fields on M. Let F : R × [0, T ] → R be a Lipschitz function. Suppose that u = u(x, t) is C 2 -regular in x, C 1 -regular in t and ∂u ≤ △g(t) u + X(t)u + F (u, t). ∂t
(A.2)
Let φ : [0, T ] → R be the solution of u(·, t) ≤ φ(t) for all t ∈ [0, T ].
dφ dt
= F (φ(t), t) with φ(0) = α. If u(·, 0) ≤ α then
There are various noncompact versions of the weak maximum principle. We state one here. Theorem A.3. Let (M, g(·)) be a complete Ricci flow solution on the interval [0, T ] with 1,2 uniformly bounded curvature. If u = u(x, t) is a Wloc function that weakly satisfies ∂u ≤ ∂t △g(t) u, with u(·, 0) ≤ 0 and Z TZ 2 (A.4) e− c dt (x,x0 ) u2 (x, s) dV (x) ds < ∞ 0
M
for some c > 0, then u(·, t) ≤ 0 for all t ∈ [0, T ].
A strong maximum principle says that under certain conditions, a strict inequality at a given time implies strict inequality at later times and also slightly earlier times. It does not require complete metrics. Theorem A.5. Let M be a connected manifold. Let {g(t)}t∈[0,T ] be a smooth one-parameter family of Riemannian metrics on M and let {X(t)}t∈[0,T ] be a smooth one-parameter family of vector fields on M. Let F : R × [0, T ] → R be a Lipschitz function. Suppose that u = u(x, t) is C 2 -regular in x, C 1 -regular in t and (A.6)
∂u ≤ △g(t) u + X(t)u + F (u, t). ∂t
= F (φ(t), t). If u(·, t) ≤ φ(t) for all t ∈ [0, T ] Let φ : [0, T ] → R be a solution of dφ dt and u(x0 , t0 ) < φ(t) for some x0 ∈ M and t0 ∈ (0, T ] then there is some ǫ > 0 so that u(·, t) < φ(t) for t ∈ (t0 − ǫ, T ]. A consequence of the strong maximum principle is a statement about restricted holonomy for Ricci flow solutions with nonnegative curvature operator Rm. Theorem A.7. Let M be a connected manifold. Let {g(t)}t∈[0,T ] be a smooth one-parameter family of Riemannian metrics on M with nonnegative curvature operator that satisfy the Ricci flow equation. Then for each t ∈ (0, T ], the image Im(Rmg(t) ) of the curvature operator is a smooth subbundle of Λ2 (T ∗ M) which is invariant under spatial parallel translation. There is a sequence of times 0 = t0 < t1 < . . . < tk = T such that for each 1 ≤ i ≤ k, Im(Rmg(t) ) is a Lie subalgebra of Λ2 (Tm∗ M) ∼ = o(n) that is independent of t for t ∈ (ti−1 , ti ]. Furthermore, Im(Rmg(ti ) ) ⊂ Im(Rmg(ti+1 ) ). In particular, under the hypotheses of Theorem A.7, a local isometric splitting at a given time implies a local isometric splitting at earlier times.
NOTES ON PERELMAN’S PAPERS
201
Appendix B. φ-almost nonnegative curvature In three dimensions, the Ricci flow equation implies that dR 2 R = △R + R2 + 2 |Rij − gij |2 . dt 3 3
(B.1)
The maximum principle of Appendix A implies that if (M, g(·)) is a Ricci flow solution defined for t ∈ [0, T ), with complete time slices and bounded curvature on compact time intervals, then (inf R)(t) ≥
(B.2) In particular, tR(·, t) > −
3 2
(inf R)(0) . 1 − 32 t(inf R)(0)
for all t ≥ 0 (compare [34, Section 2]).
Recall that the curvature operator is an operator on 2-forms. We follow the usual Ricci flow convention that if a manifold has constant sectional curvature k then its curvature operator is multiplication by 2k. In general, the trace of the curvature operator equals the scalar curvature. In three dimensions, having nonnegative curvature operator is equivalent to having nonnegative sectional curvature. Each eigenvalue of the curvature operator is twice a sectional curvature. Hamilton-Ivey pinching, as given in [34, Theorem 4.1], says the following. Assume that at t = 0 the eigenvalues λ1 ≤ λ2 ≤ λ3 of the curvature operator at each point satisfy λ1 ≥ −1. (One can always achieve this by rescaling. Note that it implies R(·, t) ≥ − 23 t+1 1 .) Given a point (x, t), put X = −λ1 . If X > 0 then 4
(B.3)
R(x, t) ≥ X (ln X + ln(1 + t) − 3) ,
or equivalently, (B.4)
1+t )−3 . tR(x, t) ≥ tX ln(tX) + ln( t
Definition B.5. Given t ≥ 0, a Riemannian 3-manifold (M, g) satisfies the time-t HamiltonIvey pinching condition if for every x ∈ M, if λ1 ≤ λ2 ≤ λ3 are the eigenvalues of the curvature operator at x, then either • λ1 ≥ 0, i.e. the curvature is nonnegative at x, or
• If λ1 < 0 and X = −λ1 then tR(x) ≥ tX ln(tX) + ln This condition has the following monotonicity property:
1+t t
−3 .
Lemma B.6. Suppose that Rm and Rm′ are 3-dimensional curvature operators whose scalar curvatures and first eigenvalues satisfy R′ ≥ R ≥ 0 and λ′1 ≥ λ1 . If Rm satisfies the time-t Hamilton-Ivey pinching condition then so does Rm′ .
202
BRUCE KLEINER AND JOHN LOTT
− 3 > 0, since otherwise the Proof. We may assume that λ′1 < 0 and log(tX ′ ) + ln 1+t t ′ condition will be satisfied (because R > 0 by hypothesis). The function 1+t (B.7) Y 7→ tY ln(tY ) + ln − 3 t 1+t − 3 is nonnegative, is monotone increasing on the interval on which ln(tY ) + ln t 1+t 1+t ′ ′ so tX ln(tX) + ln t − 3 ≥ tX ln(tX ) + ln t − 3 . Hence Rm′ satisfies the pinching condition too. The content of the pinching equation is that for any s ∈ R, if tR(·, t) ≤ s then there is a lower bound t Rm(·, t) ≥ const.(s, t). Of course, this is a vacuous statement if s ≤ − 23 .
Using equation (B.4), we can find a positive function Φ ∈ C ∞ (R) such that 1. Φ is nondecreasing. is decreasing. 2. For s > 0, Φ(s) s 3. For large s, Φ(s) ∼ lns s . 4. For all t, (B.8)
Rm(·, t) ≥ − Φ(R(·, t)).
This bound has the most consequence when s is large. We note that for the original unscaled Ricci flow solution, the precise bound that we obtain depends on t0 and the time-zero metric, through its lower curvature bound. Appendix C. Ricci solitons Let {V (t)} be a time-dependent family of vector fields on a manifold M. The solution to the equation dg = LV (t) g dt
(C.1) is (C.2)
g(t) = φ−1 (t)∗ g(t0 )
where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V , normalized by φ(t0 ) = Id. (If M is noncompact then we assume that V can be integrated. The reason for the funny signs is that if a 1-parameter family η(t) is generated by of diffeomorphisms ∗ dη−1 (t+ǫ)∗ −1 ∗ dη(t+ǫ) = − vector fields W (t) then LW (t) = η (t) η(t)∗ .) dǫ dǫ ǫ=0
ǫ=0
The equation for a steady soliton is
(C.3)
2 Ric + LV g = 0,
where V is a time-independent vector field. The corresponding Ricci flow is given by (C.4)
g(t) = φ−1 (t)∗ g(t0 ),
where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V . (Of course, in this case {φ−1 (t)} is the 1-parameter group of diffeomorphisms generated by V .)
NOTES ON PERELMAN’S PAPERS
203
A gradient steady soliton satisfies the equations ∂gij = − 2Rij = 2∇i ∇j f, ∂t ∂f = |∇f |2 . ∂t
(C.5)
It follows from (C.5) that (C.6)
∂ ij g ∂j f = 2 Rij ∇j f + g ij ∇j |∇f |2 = −2 (∇i ∇j f ) ∇j f + ∇i |∇f |2 = 0, ∂t
showing that V = ∇f is indeed constant in t. The solution to (C.5) is (C.7)
g(t) = φ−1 (t)∗ g(t0 ), f (t) = φ−1 (t)∗ f (t0 ).
Conversely, given a metric b g and a function fb satisfying
(C.8)
bij + ∇ b i∇ b j fb = 0, R
b fb. If we define g(t) and f (t) by put V = ∇
(C.9)
then they satisfy (C.5).
g(t) = φ−1 (t)∗ b g, f (t) = φ−1 (t)∗ fb
A solution to (C.5) satisfies (C.10)
∂f = |∇f |2 − △f − R, ∂t
or (C.11)
∂ −f e = − △e−f + R e−f . ∂t
This perhaps motivates Perelman’s use of the backward heat equation (5.23). A shrinking soliton lives on a time interval (−∞, T ). For convenience, we take T = 0. Then the equation is g (C.12) 2 Ric + LV g + = 0. t The vector field V = V (t) satisfies V (t) = − given by (C.13)
1 t
V (−1). The corresponding Ricci flow is
g(t) = − t φ−1 (t)∗ g(−1),
where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V , normalized by φ(−1) = Id.
204
BRUCE KLEINER AND JOHN LOTT
A gradient shrinking soliton satisfies the equations (C.14)
gij ∂gij = − 2Rij = 2∇i ∇j f + , ∂t t ∂f = |∇f |2 . ∂t
It follows from (C.14) that V = ∇f satisfies V (t) = − 1t V (−1). The solution to (C.14) is (C.15)
g(t) = − t φ−1 (t)∗ g(−1),
f (t) = φ−1 (t)∗ f (−1).
Conversely, given a metric b g and a function fb satisfying
(C.16)
bij + ∇ b i∇ b j fb − 1 gb = 0, R 2
b fb. If we define g(t) and f (t) by put V (t) = − 1t ∇ (C.17)
then they satisfy (C.14).
g(t) = − t φ−1 (t)∗ b g, f (t) = φ−1 (t)∗ fb
An expanding soliton lives on a time interval (T, ∞). For convenience, we take T = 0. Then the equation is g (C.18) 2 Ric + LV g + = 0. t The vector field V = V (t) satisfies V (t) = (C.19)
1 t
V (1). The corresponding Ricci flow is given by
g(t) = t φ−1 (t)∗ g(1),
where {φ(t)} is the 1-parameter group of diffeomorphisms generated by −V , normalized by φ(1) = Id. A gradient expanding soliton satisfies the equations (C.20)
gij ∂gij = − 2Rij = 2∇i ∇j f + , ∂t t ∂f = |∇f |2 . ∂t
It follows from (C.20) that V = ∇f satisfies V (t) = (C.21)
1 t
V (1). The solution to (C.20) is
g(t) = t φ−1 (t)∗ g(1), f (t) = φ−1 (t)∗ f (1).
Conversely, given a metric b g and a function fb satisfying
(C.22)
bij + ∇ b i∇ b j fb + 1 gb = 0, R 2
NOTES ON PERELMAN’S PAPERS
put V (t) = (C.23)
1 t
205
b fb. If we define g(t) and f (t) by ∇
g(t) = t φ−1 (t)∗ b g, f (t) = φ−1 (t)∗ fb
then they satisfy (C.20).
Obvious examples of solitons are given by Einstein metrics, with V = 0. Any steady or expanding soliton on a closed manifold comes from an Einstein metric. Other examples of solitons (see [22, Chapter 2]) are : 1. (Gradient steady soliton) The cigar soliton on R2 and the Bryant soliton on R3 . 2 . 2. (Gradient shrinking soliton) Flat Rn with f = − |x| 4t 2 3. (Gradient shrinking soliton) The shrinking cylinder R × S n−1 with f = − x4t , where x is the coordinate on R. 4. (Gradient shrinking soliton) The Koiso soliton on CP 2 #CP 2 . Appendix D. Local derivative estimates Theorem D.1. For any α, K, K ′, l ≥ 0 and m, n ∈ Z+ , there is some C = C(α, K, K ′, l, m, n) with the following property. Given r > 0, suppose that g(t) is a Ricci flow solution for 2 t ∈ [0, t], where 0 < t ≤ αr , defined on an open neighborhood U of a point p ∈ M n . K Suppose that B(p, r, 0) is a compact subset of U, that K (D.2) | Rm(x, t)| ≤ 2 r for all x ∈ U and t ∈ [0, t], and that |∇β Rm(x, 0)| ≤
(D.3) for all x ∈ U and |β| ≤ l. Then (D.4)
|∇β Rm(x, t)| ≤
r |β|+2 for all x ∈ B p, 2r , 0 , t ∈ (0, t] and |β| ≤ m. In particular, |∇β Rm(x, t)| ≤
C r |β|+2
K′
r |β|+2 C max(m−l,0) t 2
r2
whenever |β| ≤ l.
The main case l = 0 of Theorem D.1 is due to Shi [62]. The extension to l ≥ 0 appears in [41, Appendix B]. Appendix E. Convergent subsequences of Ricci flow solutions Theorem E.1. [32] Let {gi (t)}∞ i=1 be a sequence of Ricci flow solutions on connected pointed manifolds (Mi , mi ), defined for t ∈ (A, B) and complete for each i and t, with −∞ ≤ A < 0 < B ≤ ∞. Suppose that the following two conditions are satisfied : 1. For each r > 0 and each compact interval I ⊂ (A, B), there is an Nr,I < ∞ so that for
206
BRUCE KLEINER AND JOHN LOTT
all t ∈ I and all i, supBt (mi ,r) | Rm(gi (t))| ≤ Nr,I , and 2. The time-0 injectivity radii {inj(gi (0))(mi )}∞ i=1 are uniformly bounded below by a positive number. Then after passing to a subsequence, the solutions converge smoothly to a complete Ricci flow solution g∞ (t) on a connected pointed manifold (M∞ , m∞ ), defined for t ∈ (A, B). That is, for any compact interval I ⊂ (A, B) and any compact set K ⊂ M∞ containing m∞ , there are pointed time-independent diffeomorphisms φK,i : K → Ki (with Ki ⊂ Mi ) so that {φ∗K,igi}∞ i=1 converges smoothly to g∞ on I × K. Given the sectional curvature bounds, the lower bound on the injectivity radii is equivalent to a lower bound on the volumes of balls around m0 [20, Theorem 4.7]. There are many variants of the theorem with alternative hypotheses. One can replace the interval (A, B) with an interval (A, B], −∞ ≤ A < 0 ≤ B < ∞. One can also replace the interval (A, B) with an interval [A, B), −∞ < A < 0 < B ≤ ∞, if in addition one has uniform time-A bounds supBA (mi ,r) |∇j Rm(gi(A))| ≤ Cr,j . Then using Appendix D, one gets smooth convergence to a limit solution g∞ on the time interval [A, B). (Without the time-A bounds one would only get C 0 -convergence on [A, B) and C ∞ -convergence on (A, B).) There is a similar statement if one replaces the interval (A, B) with an interval [A, B], −∞ < A < 0 ≤ B < ∞.
In the incomplete setting, if one has curvature bounds on spacetime product regions I × B0 (mi , r), with I ⊂ (A, B) and r < r0 < ∞, and an injectivity radius bound at (mi , 0), then one can still extract a subsequence which converges on compact subsets of (A, B) times compact subsets of B0 (m∞ , r0 ). There are analogous statements when (A, B) is replaced by (A, B], [A, B) or [A, B], and the ball is replaced by a metric annulus.
Appendix F. Harnack inequalities for Ricci flow We first recall the statement of the matrix Harnack inequality. Put (F.1)
Pabc = ∇a Rbc − ∇b Rac , Rab 1 . Mab = △Rab − ∇a ∇b R + 2 Racbd Rcd − Rac Rbc + 2 2t
Given a 2-form U and a 1-form W , put (F.2)
Z(U, W ) = Mab Wa Wb + 2 Pabc Uab Wc + Rabcd Uab Ucd .
Suppose that we have a Ricci flow for t > 0 on a complete manifold with bounded curvature on each compact time interval and nonnegative curvature operator. Hamilton’s matrix Harnack inequality says that for all t > 0 and all U and W , Z(U, W ) ≥ 0 [33, Theorem 14.1]. Taking Wa = Ya and Uab = (Xa Yb − Ya Xb )/2 and using the fact that (F.3)
Rict (Y, Y ) = (△Rab )Y a Y b + 2 Racbd Rcd Y a Y b − 2 Rac Rbc Y a Y b ,
NOTES ON PERELMAN’S PAPERS
207
we can write 2Z(U, W ) = H(X, Y ), where (F.4) H(X, Y ) = − HessR (Y, Y ) − 2hR(Y, X)Y, Xi + 4 (∇X Ric(Y, Y ) − ∇Y Ric(Y, X)) 2 1 + 2 Rict (Y, Y ) + 2 Ric(Y, ·) + Ric(Y, Y ). t
Substituting the elements of an orthonormal basis {ei }ni=1 for Y and summing over i gives (F.5) X i
H(X, ei ) = − △R + 2 Ric(X, X) + 4(h∇R, Xi −
X i
∇ei Ric(ei , X)) + 2
X i
Rict (ei , ei ) + 2| Ric |2 +
1 R. t
Tracing the second Bianchi identity gives X 1 (F.6) ∇ei Ric(ei , X) = h∇R, Xi. 2 i From (F.3), (F.7)
X
Rict (ei , ei ) = △R.
X
H(X, ei ) = H(X),
i
Putting this together gives (F.8)
i
where (F.9)
H(X) = Rt +
1 R + 2h∇R, Xi + 2 Ric(X, X). t
We obtain Hamilton’s trace Harnack inequality, saying that H(X) ≥ 0 for all X.
In the rest of this section we assume that the solution is defined for all t ∈ (−∞, 0). Changing the origin point of time, we have (F.10)
Rt +
1 R + 2h∇R, Xi + 2 Ric(X, X) ≥ 0 t − t0
whenever t0 ≤ t. Taking t0 → −∞ gives (F.11)
Rt + 2h∇R, Xi + 2 Ric(X, X) ≥ 0
In particular, taking X = 0 shows that the scalar curvature is nondecreasing in t for any ancient solution with nonnegative curvature operator, assuming again that the metric is complete on each time slice with bounded curvature on each compact time interval. More generally, (F.12)
0 ≤ Rt + 2h∇R, Xi + 2 Ric(X, X) ≤ Rt + 2h∇R, Xi + 2RhX, Xi.
208
BRUCE KLEINER AND JOHN LOTT
gives If γ : [t1 , t2 ] → M is a curve parametrized by s then taking X = 21 dγ ds dR(γ(s), s) dγ 1 dγ dγ (F.13) = Rt (γ(s), s) + , ∇R ≥ − R , . ds ds 2 ds ds Integrating gives (F.14)
d ln R(γ(s),s) ds
with respect to s and using the fact that g(t) is nonincreasing in t
d2t1 (x1 , x2 ) R(x2 , t2 ) ≥ exp − R(x1 , t1 ). 2(t2 − t1 )
d2 (x1 ,x2 )
t1 whenever t1 < t2 and x1 , x2 ∈ M. (If n = 2 then one can replace 2(t by 2 −t1 ) particular, if R(x2 , t2 ) = 0 for some (x2 , t2 ) then g(t) must be flat for all t.
d2t (x1 ,x2 ) 1 .) 4(t2 −t1 )
In
Appendix G. Alexandrov spaces We recall some facts about Alexandrov spaces (see [11, Chapter 10], [12]). Given points e p (x, y) denote the comparison p, x, y in a nonnegatively curved Alexandrov space, we let ∠ angle at p, i.e. the angle of the Euclidean comparison triangle at the vertex corresponding to p. The Toponogov splitting theorem says that if X is a proper nonnegatively curved Alexandrov space which contains a line, then X splits isometrically as a product X = R × Y , where Y is a proper, nonnegatively curved Alexandrov space [11, Theorem 10.5.1]. Let M be an n-dimensional Alexandrov space with nonnegative curvature, p ∈ M, and λk → 0. Then the sequence (λk M, p) of pointed spaces Gromov-Hausdorff converges to the Tits cone CT M (the Euclidean cone over the Tits boundary ∂T M) which is a nonnegatively curved Alexandrov space of dimension ≤ n [7, p. 58-59]. If the Tits cone splits isometrically as a product Rk × Y , then M itself splits off a factor of Rk ; using triangle comparison, one finds k orthogonal lines passing through a basepoint, and applies the Toponogov splitting theorem. Now suppose that xk ∈ M is a sequence with d(xk , p) → ∞ and rk ∈ R+ is a sequence with d(xrkk,p) → 0. Then the sequence ( r1k M, xk ) subconverges to a pointed Alexandrov space (N∞ , x∞ ) which splits off a line. To see this, observe that since ( d(x1k ,p) M, p) converges to a e x (p, yk ) → π. Observe cone, we can find a sequence yk ∈ M such that d(yk ,xk ) → 1, and ∠ d(xk ,p)
k
that for any ρ < ∞, we can find sequences pk ∈ pxk , zk ∈ xk yk such that d(xk ,zk ) rk
d(xk ,pk ) rk
→ ρ,
→ ρ, and by monotonicity of comparison angles [11, Chapter 4.3] we will have e ∠xk (pk , zk ) → π. Passing to the Gromov-Hausdorff limit, we find p∞ , z∞ ∈ N∞ such that e x∞ (p∞ , z∞ ) = π. Since this construction applies for all ρ, d(p∞ , x∞ ) = d(z∞ , x∞ ) = ρ and ∠ it follows that N∞ contains a line passing through x∞ . Hence, by the Toponogov splitting theorem, it is isometric to a metric product R × N ′ for some Alexandrov space N ′ .
If M is a complete Riemannian manifold of nonnegative sectional curvature and C ⊂ M is a compact connected domain with weakly convex boundary then the subsets Ct = {x ∈ C | d(x, ∂C) ≥ t} are convex in C [18, Chapter 8]. If the second fundamental form of ∂C is
NOTES ON PERELMAN’S PAPERS
209
≥ 1r at each point of ∂C, then for all x ∈ C we have d(x, ∂C) ≤ r, since the first focal point of ∂C along any inward pointing normal geodesic occurs at distance ≤ r.
Finally, we recall the statement of the Bishop-Gromov volume comparison inequality. Suppose that M is an n-dimensional Riemannian manifold. Given p ∈ M and r2 ≥ r1 > 0, suppose that B(p, r2 ) has compact closure in M and that the sectional curvatures of B(p, r2 ) are bounded below by K ∈ R. Then R r2 n−1 sin (kr) dr R0r1 if K = k 2 , n−1 (kr) dr sin 0 n vol(B(p, r2 )) r if K = 0, (G.1) ≤ r2n 1 vol(B(p, r1 )) Rr n−1 (kr) dr R0r12 sinhn−1 if K = −k 2 . sinh (kr) dr 0
(If K = k with k > 0 then we restrict to r2 ≤ πk .) The same inequality holds if we just assume that B(p, r2 ) has Ricci curvature bounded below by (n − 1) K. Equation (G.1) also holds if M is an Alexandrov space and B(p, r2 ) has Alexandrov curvature bounded below by K. 2
Appendix H. Finding balls with controlled curvature Lemma H.1. Let X be a Riemannian manifold with R ≥ 0 and suppose B(x, 5r) is a compact subset of X. Then there is a ball B(y, r¯) ⊂ B(x, 5r), r ≤ r, such that R(z) ≤ 2R(y) for all z ∈ B(y, r¯) and R(y)¯ r 2 ≥ R(x)r 2 . Proof. Define sequences xi ∈ B(x, 5r), ri > 0 inductively as follows. Let x1 = x, r1 = r. For i > 1, let xi+1 = xi , ri+1 = ri if R(z) ≤ 2R(xi ) for all z ∈ B(xi , ri ); otherwise let ri+1 = √ri2 , and let xi+1 ∈ B(xi , ri ) be a point such that R(xi+1 ) > 2R(xi ). The sequence of balls B(xi , ri ) is contained in B(x, 5r), so the sequences xi , ri are eventually constant, and we can take y = xi , r¯ = ri for large i. There is an evident spacetime version of the lemma. Appendix I. Statement of the geometrization conjecture Let M be a connected orientable closed (= compact boundaryless) 3-manifold. One formulation of the geometrization conjecture says that M is the connected sum of closed 3-manifolds {Mi }ni=1 , each of which admits a codimension-0 compact submanifold-withboundary Gi so that • Gi is a graph manifold • Mi − Gi is hyperbolic, i.e. admits a complete Riemannian metric with constant negative sectional curvature and finite volume • Each component T of ∂Gi is an incompressible torus in Mi , i.e. with respect to a basepoint t ∈ T , the induced map π1 (T, t) → π1 (Mi , t) is injective.
210
BRUCE KLEINER AND JOHN LOTT
We allow Gi = ∅ or Gi = Mi .
A reference for graph manifolds is [42, Chapter 2.4]. The definition is as follows. One takes a collection {Pi}N i=1 of pairs of pants (i.e. closed 2-disks with two balls removed) and ′ 1 N 1 2 N′ a collection of closed 2-disks {Dj2 }N j=1 . The 3-manifolds {S × Pi }i=1 ∪ {S × Dj }j=1 have toral boundary components. One takes an even number of these tori, matches them in pairs 1 2 N′ by homeomorphisms, and glues {S 1 × Pi }N i=1 ∪ {S × Dj }j=1 by these homeomorphisms. The resulting 3-manifold G is a graph manifold, and all graph manifolds arise in this way. We will assume that the gluing homeomorphisms are such that G is orientable. Clearly the boundary of G, if nonempty, is a disjoint union of tori. It is also clear that the result of gluing two graph manifolds along some collection of boundary tori is a graph manifold. The connected sum of two 3-manifolds is a graph manifold if and only if each factor is a graph manifold [42, Proposition 2.4.3]. The reason to require incompressibility of the tori T in the statement of the geometrization conjecture is to exclude phony decompositions, such as writing S 3 as the union of a solid torus and a hyperbolic knot complement. A more standard version of the geometrization conjecture uses some facts from 3-manifold theory [59]. First M has a Kneser-Milnor decomposition as a connected sum of uniquely defined prime factors. Each prime factor is S 1 × S 2 or is irreducible, i.e. any embedded S 2 bounds a 3-ball. If M is irreducible then it has a JSJ decomposition, i.e. there is a minimal collection of disjoint incompressible embedded tori {Tk }K to isotopy, k=1 in M, unique up SK ′ with the property that if M is the metric completion of a component of M − k=1 Tk (with respect to an induced Riemannian metric from M) then
• M ′ is a Seifert 3-manifold or • M ′ is non-Seifert and any embedded incompressible torus in M ′ can be isotoped into ∂M ′ .
The second version of the geometrization conjecture reduces to the conjecture that in the latter case, the interior of M ′ is hyperbolic. Thurston proved that this is true when ∂M ′ 6= ∅. The reason for the word “geometrization” is explained in [59, 65]. An orientable Seifert 3-manifold is a graph manifold [42, Proposition 2.4.2]. It follows that the second version of the geometrization conjecture implies the first version. One can show directly that any graph manifold is a connected sum of prime graph manifolds, each of which can be split along incompressible tori to obtain a union of Seifert manifolds [42, Proposition 2.4.7], thereby showing the equivalence of the two versions. References [1] K. Akutagawa, M. Ishida and C. Lebrun, “Perelman’s Invariant, Ricci Flow, and the Yamabe Invariants of Smooth Manifolds”, Arch. Math. 88, p. 71-76 (2007) [2] M. Anderson, “Scalar Curvature and Geometrization Conjectures for 3-Manifolds”, in Comparison Geometry, MSRI Publ. 30, Cambridge University Press, Cambridge, p. 49-82 (1997) [3] M. Anderson, “Scalar Curvature and the Existence of Geometric Structures on 3-Manifolds. I”, J. Reine Angew. Math. 553, p. 125-182 (2002) [4] M. Anderson, “Canonical Metrics on 3-Manifolds and 4-Manifolds”, Asian Journal of Math. 10, p. 127-164 (2006)
NOTES ON PERELMAN’S PAPERS
211
[5] G. Anderson and B. Chow, “A Pointwise Bound for Solutions of the Linearized Ricci Flow Equation on 3-Manifolds”, Calculus of Variations 23, p. 1-12 (2005) [6] S. Angenent and D. Knopf, “Precise Asymptotics of the Ricci Flow Neckpinch”, Comm. Anal. Geom. 15, p. 773-844 (2007) [7] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of Nonpositive Curvature, Progress in Mathematics 61, Birkh¨ auser, Boston (1985) [8] W. Beckner, “Geometric Asymptotics and the Logarithmic Sobolev Inequality”, Forum Math. 11, p. 105-137 (1999) [9] L. Bessi`eres, G. Besson, M. Boileau, S. Maillot and J. Porti, “Weak Collapsing and Geometrisation of Aspherical 3-Manifolds”, preprint, http://arxiv.org/abs/0706.2065 (2007) [10] J.-P. Bourguignon, “Une Stratification de l’Espace des Structures Riemanniennes”, Compositio Math. 30, p. 1-41 (1975) [11] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics 33, American Mathematical Society, Providence (2001) [12] Y. Burago, M. Gromov and G. Perelman, “A. D. Aleksandrov Spaces with Curvatures Bounded Below”, Russian Math. Surveys 47, p. 1-58 (1992) [13] E. Calabi, “An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry”, Duke Math. J. 25, p. 45-56 (1957) [14] H-D. Cao and B. Chow, “Recent Developments on the Ricci Flow”, Bull. of the AMS 36, p. 59-74 (1999) [15] H.-D. Cao and X.-P. Zhu, “A Complete Proof of the Poincar´e and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci Flow”, Asian Journal of Math. 10, p. 165-492 (2006) Erratum to “A Complete Proof of the Poincar´e and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci Flow”, Asian Journal of Math. 10, p. 663-664 (2006) [16] A. Chau, L.-F. Tam and C. Yu, “Pseudolocality for the Ricci Flow and Applications”, preprint, http://arxiv.org/abs/math/0701153 (2007) [17] I. Chavel, Riemannian Geometry - a Modern Introduction, Cambridge Tracts in Mathematics 108, Cambridge University Press, Cambridge (1993) [18] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry. North-Holland Publishing Co., Amsterdam-Oxford (1975) [19] J. Cheeger and D. Gromoll, “On the Structure of Complete Manifolds of Nonnegative Curvature”, Ann. of Math. 96, p. 413-443 (1972) [20] J. Cheeger, M. Gromov and M. Taylor, “Finite Propagation Speed, Kernel Estimates for Functions of the Laplace Operator, and the Geometry of Complete Riemannian Manifolds”, J. Diff. Geom. 17, p. 15-53 (1982) [21] B. Chen and X. Zhu, “Uniqueness of the Ricci Flow on Complete Noncompact Manifolds”, J. of Diff. Geom. 74, p. 119-154 (2006) [22] B. Chow and D. Knopf, The Ricci Flow : An Introduction, AMS Mathematical Surveys and Monographs, Amer. Math. Soc., Providence (2004) [23] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci Flow, Graduate Studies in Mathematics 77, AMS, Providence (2006) [24] T. Colding and W. Minicozzi, “Estimates for the Extinction Time for the Ricci Flow on Certain 3Manifolds and a Question of Perelman”, J. Amer. Math. Soc. 18, p. 561-569 (2005) [25] T. Colding and W. Minicozzi, “Width and Finite Extinction Time of Ricci Flow”, preprint, http://arxiv.org/abs/0707.0108 (2007) [26] J. Dodziuk, “Maximum Principle for Parabolic Inequalities and the Heat Flow on Open Manifolds”, Indiana Univ. Math. J. 32, p. 703-716 (1983) [27] J. Eells and J. Sampson, “Harmonic Mappings of Riemannian Manifolds”, Amer. J. Math. 86, p. 109-160 (1964) [28] J.-H. Eschenburg, “Local Convexity and Nonnegative Curvature—Gromov’s Proof of the Sphere Theorem”, Inv. Math. 84, p. 507-522 (1986) [29] R. Hamilton, “Four-Manifolds with Positive Curvature Operator”, J. Diff. Geom. 24, p. 153-179 (1986)
212
BRUCE KLEINER AND JOHN LOTT
[30] R. Hamilton, “The Ricci Flow on Surfaces”, in Mathematics and General Relativity, Contemp. Math. 71, AMS, Providence, RI, p. 237-262 (1988). [31] R. Hamilton, “The Harnack Estimate for the Ricci Flow”, J. Diff. Geom. 37, p. 225-243 (1993) [32] R. Hamilton, “A Compactness Property for Solutions of the Ricci Flow” Amer. J. Math. 117, p. 545-572 (1995) [33] R. Hamilton, “The Formation of Singularities in the Ricci Flow”, in Surveys in Differential Geometry, Vol. II, Internat. Press, Cambridge, p. 7-136 (1995) [34] R. Hamilton, “Nonsingular Solutions of the Ricci Flow on Three-Manifolds”, Comm. Anal. Geom. 7, p. 695-729 (1999) [35] R. Hamilton, “Three-Manifolds with Positive Ricci Curvature”, J. Diff. Geom. 17, p. 255-306 (1982) [36] R. Hamilton, “Four-Manifolds with Positive Isotropic Curvature”, Comm. Anal. Geom. 5, p. 1-92 (1997) [37] H. Ishii, “On the Equivalence of Two Notions of Weak Solutions, Viscosity Solutions and Distribution Solutions”, Funkcial. Ekvac. 38, p. 101-120 (1995) [38] V. Kapovitch, “Perelman’s Stability Theorem,” in Surveys in Differential Geometry, vol. XI, Metric and Comparison Geometry, eds. J. Cheeger and K. Grove, International Press, Somerville, MA, p. 103-136 [39] B. Kleiner and J. Lott, webpage at http://www.math.lsa.umich.edu/˜lott/ricciflow/perelman.html [40] B. Kleiner and J. Lott, “Locally Collapsed 3-Manifolds”, to appear [41] P. Lu and G. Tian, “Uniqueness of Standard Solutions in the Work of Perelman”, preprint, http://www.math.lsa.umich.edu/˜lott/ricciflow/perelman.html (2005) [42] S. Matveev, Algorithmic Topology and Classification of 3-Manifolds, Springer-Verlag, Berlin (2003) [43] W. Meeks III and S.-T. Yau, “Topology of Three-Dimensional Manifolds and the Embedding Problems in Minimal Surface Theory”, Ann. of Math. 112, p. 441-484 (1980) [44] J. Morgan, “Recent Progress on the Poincar´e Conjecture and the Classification of 3-Manifolds”, Bull. Amer. Math. Soc. 42, p. 57-78 (2005) [45] J. Morgan and G. Tian, Ricci Flow and the Poincar´e Conjecture, Clay Mathematics Monographs 3, Amer. Math. Soc., Providence (2007) [46] J. Morgan and G. Tian, “Completion of Perelman’s Proof of the Geometrization Conjecture”, to appear [47] G. Mostow, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies No. 78, Princeton University Press, Princeton (1973) [48] L. Ni, “The Entropy Formula for Linear Heat Equation”, J. Geom. Anal. 14, p. 87-100 (2004) [49] L. Ni, “Ricci Flow and Nonnegativity of Sectional Curvature”, Math. Res. Let. 11, p. 883-904 (2004) [50] L. Ni, “A Note on Perelman’s LYH Inequality”, Comm. Anal. Geom. 14, p. 883-905 (2006) [51] G. Perelman, “The Entropy Formula for the Ricci Flow and its Geometric Applications”, http://arXiv.org/abs/math.DG/0211159 [52] G. Perelman, “Ricci Flow with Surgery on Three-Manifolds”, http://arxiv.org/abs/math.DG/0303109 [53] G. Perelman, “Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds”, http://arxiv.org/abs/math.DG/0307245 [54] G. Prasad, “Strong Rigidity of Q-Rank 1 Lattices”, Invent. Math. 21, p. 255-286 (1973) [55] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Analysis of Operators, Academic Press, New York (1978) [56] O. Rothaus, “Logarithmic Sobolev Inequalities and the Spectrum of Schr¨odinger Operators”, J. Funct. Anal. 42, p. 110-120 (1981) [57] R. Schoen, “Estimates for Stable Minimal Surfaces in Three-Dimensional Manifolds”, in Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press, Princeton, N.J., p. 111-126 (1983). [58] R. Schoen and S.-T. Yau, “Complete Three-Dimensional Manifolds with Positive Ricci Curvature and Scalar Curvature”, in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, N.J., p. 209-228 (1982) [59] P. Scott, “The Geometries of 3-Manifolds”, Bull. London Math. Soc. 15, p. 401-487 (1983) [60] V. Sharafutdinov, “The Pogorelov-Klingenberg Theorem for Manifolds that are Homeomorphic to Rn ”, Siberian Math. J. 18, p. 649-657 (1977)
NOTES ON PERELMAN’S PAPERS
213
