Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1743
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Lutz Habermann
Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures
Springer
Author Lutz Habermann Institute of Mathematics and Computer Science Ernst-Moritz-Arndt-University Jahnstr. 15a 17487 Greifswald, Germany E-mail:
[email protected]
Cataloging-in-PublicationData applied for Die Deutsche Bibliothek- CIP-Einheitsaufnahme Habermann,Lutz: Riemannianmetricsof constantmass and modulispacesof conformal structures/ LutzHabermann.- Berlin ; Heidelberg; New York ; Barcelona; HongKong; London; Milan; Paris ; Singapore; Tokyo : Springer,2000 (Lecture notesin mathematics; 1743) ISBN 3-540-67987-1
Mathematics Subject Classification (2000): 58D27, 53A30, 53C20 ISSN 0075- 8434 ISBN 3-540-67987-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10724216 41/3142-543210 - Printed on acid-free paper
Preface
In understanding spaces of complex structures, it has proved to be useful to construct "canonical" Riemannian metrics on complex manifolds. For such a "canonical" metric g j , it is required that this metric is uniquely determined by the underlying complex structure J, depends smoothly on that structure, and satisfies f * g g = g f . j for any diffeomorphism f of the manifold M under consideration. Moreover, it is expected that g j has an analyzable behavior as the underlying structure degenerates in some explicit manner. Such a metric gg for each complex structure J then gives rise to a Riemannian metric on the regular part ff~* of the corresponding moduli space 9~ by identifying tangent vectors to !)Y~* with harmonic sections of a certain vector bundle on M and taking the L2-product of such sections with respect to the canonical metric. The best known example of a canonical metric is the hyperbolic metric on compact Riemann surfaces of genus p > 1, whose existence is a consequence of the uniformization theorem. In this case the induced L2-metric on the moduli space is the Weil-Petersson metric. The asymptotic behavior of the hyperbolic metric for degenerating Riemann surfaces and its consequences for the geometry of the moduli space was analyzed by Masur [50]. Other examples for canonical metrics on Riemann surfaces are the Bergman and the Arakelov metric (see Section 2.1), where the second one is of special interest for the following. Namely, this metric is defined in terms of the asymptotic expansion of a Green function near its singularity, and we shall adapt this construction principle to the situation which we are interested in. In higher dimensions, such canonical metrics include the K~hler-Einstein metrics constructed by Yau [68], Aubin [8], and others. The present book deals with the investigation of moduli spaces of conformal structures on the basis of a similar principle, i.e. by constructing and studying canonical metrics can(C) for conformal structures C. The metric presented in this book is not the first canonical metric associated to a conformal structure. For example, by the resolution of the Yamabe problem by Trudinger [67], Aubin [7], and Schoen [56], each conformal structure on a compact manifold can be represented by a Riemannian metric of constant scalar curvature. In the case of positive scalar curvature, however, even after normalizing, that metric in general is not unique (cp. [57], [58]). Other canonical metrics,
VI
Preface
which, however, can be constructed only for fiat structures, were introduced by Apanasov [2] and Kulkarni-Pinkall [43] and by Nayatani [51]. The first of these metrics is based on the Kobayashi construction in the context of MSbius geometry, whereas Nayatani's metric is strongly related to the P a t t e r s o n Sullivan measure on the limit set of a Kleinian group (see Section 2.4). The construction of a canonical metric for conformal structures on a compact manifold M of dimension n _> 3 presented here employs the (suitably normalized) Green function G of the conformal Laplacian L with respect to a Riemannian metric g in the given conformal class C. Since L is conformally invariant up to some conformal factor, so is G. This function exists, if L is invertible, which is the case if we assume that the scalar curvature S of g is positive. Now the idea for the construction of the canonical metric can(C) is to put c~(p) :-- lim
q--~p
(G(p,q)- dist2-n(p,q)) 1/(n-2)
for
p E M,
where dist denotes the geodesic distance for g, and then
can(C) := a2g. To transfer this idea into a rigorous definition, one has to clarify the following points: 1. Is can(C) canonical, i.e. is the expression of the Riemannian metric g E C? 2. Does the limit in the description of 3. Does
a(p) ~ 0 hold
a2g independent
of the choice
a(p) exist?
for all p e M ?
For the first point, the conformal invariance of the Green function G mentioned above is crucial. In order to assure the existence of G, we shall assume on principle that the conformal structure C admits a Riemannian metric with positive scalar curvature, which is equivalent to saying that the Yamabe invariant of C is positive. Concerning the second point, one easily observes that, for n > 3, the limit exists not for all g E C. T h a t is the reason why we shall first restrict to the case that the structure C is flat. As we shall see, the third point is strongly related to the positive mass theorem of Schoen and Yan [59], [60], [61], and these considerations will allow us to define the metric can(C) in dimensions 3, 4, and 5 also for non-flat C. As already indicated, the definition ansatz for can(C) is motivated by a construction of a Riemannian metric on a compact Riemann surface due to Arakelov. A similar procedure can be found also in a different context, namely in Hersch's [30] (cp. also [10]) definition of the conformal radius for domains f2 in Euclidean space E n. Leutwiler observed that this can be turned into
Preface
VII
the construction of a Riemannian metric g~ on [2 invariant under conformal transformations of E n. We recall these constructions of Arakelov and Leutwiler in Section 2.1. Section 2.2 deals with the conformal Laplacian of a Riemannian manifold. The most important part of Chapter 2 is Section 2.3. Here we define the canonical metric can(C) for flat conformal structures C. Furthermore, we prove some properties of this metric. We show that can(C) is invariant under conformai diffeomorphisms and that conformal maps are locally distance non-decreasing with respect to the canonical metric. By the latter, our metric differs from the metrics of Apanasov-Kulkarni-Pinkall and Nayatani. A detailed comparison of these metrics can be found in the last section of Chapter 2. By a result of Schoen and Yau [62], the proof of which in dimension n = 3 involves the positive mass theorem, any flat conformal manifold whose conformal structure C can be represented by a Riemannian metric of positive scalar curvature is conformally diffeomorphic to a Kleinian manifold, i.e. the quotient ~(F)/T' of the discontinuity domain 12(F) of a Kleinian group F. In Section 3.1 we use this to express can(C) in terms of F. For this we apply a result of Nayatani [51] which, refining a result of Schoen and Yau, says that, for a conformal structure C as above, the critical exponent 5(F) of the corresponding Kleinian group F satisfies the inequality 5(/") < (n - 2)/2. From the description of can(C) by means of F, we immediately obtain that can(C) nowhere vanishes and indeed is a Riemannian metric, if (M, C) is not conformally diffeomorphic to the standard sphere (S n, Cs). On the other hand, one has can(Cs) -= 0, which can also be derived from the conformal invariance of can(Cs). The last property is of importance for a "natural" compactification of the moduli space of (flat) conformal structures by means of our canonical metrics. Namely, since one may locally "bubble off" a sphere from any conformal manifold, the vanishing of can(Cs) is a necessary condition for the Hansdorff property of the compactified moduli space. Section 3.2 contains some basic facts on moduli spaces of conformai structures. Among other things, it is explained how canonical metrics for conformal structures induce an L2-metric on the corresponding moduli space. Furthermore, an explicit parametrization of the moduli space B0 (S 1 x S 2) of flat conformal structures on S 1 x S 2 is given. Since any flat conformal structure on S 1 x S 2 has positive Yamabe invariant, the canonical metrics can(C) give rise to a Riemannian metric on/3o (S 1 x $2). We study this metric in Section 3.3 to get some indications of what one can expect for the geometry of the moduli space of flat conformal structures on general manifolds. In particular, we show that the metric completion of B0 (S 1 x S 2) differs from B0 (S 1 X S 2) by only one point and that this point corresponds to the unique element of the moduli space of flat conformal structures on S 3 with two punctures. In Chapter 4 we discuss the geometry of the moduli space B+o(M)of flat conformal structures with positive Yamabe invariant on a connected and closed
VIII
Preface
manifold of dimension n > 3. First we study the behavior of can(C) under surgery type degenerations and show that the limit of the canonical metrics yields the canonical metric on the limit space (see Proposition 4.1.10). In other words, we can track our canonical metric through a change of topological type. In Section 4.3 we derive from this asymptotics of the L2-metric on B+(M). In particular, we see that the points which corresponds to "degenerated" flat conformal structures have finite distance to inner points of B+(M) and hence B+(M) is not complete. Thus, we obtain a similar picture as in Masur's work [50] on the geometry of the Mumford-Deligne compactification of the space of stable Riemann surfaces. In this sense the analysis in Chapter 4 can be viewed as a first step towards understanding a natural compactification of the moduli space B0+ (M). The things described above concern the geometry of the moduli space in directions of increasing Yamabe invariant. Related to the behavior of the L 2metric for decreasing Yamabe invariant, we investigate B0+ (M) near boundary points which can be represented by Riemannian metrics with vanishing scalar curvature. We make the observation that, for a sequence of compact flat conformal structures with scalar curvature going to zero, our canonical metric tends to a non-compact metric of infinite diameter and vanishing scalar curvature (see Proposition 4.2.2). It is clear that this must be so. Namely, since the canonical metric can(C) is unique in its conformal class C, it fixes a scaling factor. In the case of zero scalar curvature, however, there can be no natural scaling factor. For the geometry of B+o(M) we conclude that, in dimensions 3 and 4, any boundary point given by a Riemannian metric with vanishing scalar curvature, has infinite distance from inner points. Our calculations indicate that, in dimensions n _> 5, the distance of a boundary point of B+ (M) as above from the interior is finite, although we have no rigorous proof for this. We start Chapter 5 with a new interpretation of the canonical metric can(C). For this we define the mass re(g) of a Riemannian metric g E C as that function on M whose value at p is the (suitably normalized) mass of the asymptotically flat metric G~/(n-2)g on M \ {p}, where Gp is obtained from the Green function G of the conformal Laplacian of (M,g) by Gp(q) := G(p, q). As explained in Section 5.1, we than have can(C) = m2/(n-2)(g) g . (Similar considerations were done by Schoen-Yau and Lee-Parker in the context of the proof of the positive mass theorem and the resolution of the Yamabe problem. Compare [45].) Therefore, our canonical metric is characterized as the unique metric in the conformal class which has constant mass 1. The advantage of the definition of the canonical metric can(C) by the mass function re(g) consists in the fact that, in dimensions n = 3, 4, 5, it is also
Preface
IX
applicable to conformal structures which are non-flat. (In particular, in dimension 4, we may define the canonical metric of a half conformally flat structure which admits a compatible metric with positive scalar curvature. The induced L2-metric could serve as a tool for investigating the corresponding moduli space. See e.g. [37].) The statement that can(C) for any non-flat conformal structure with positive Yamabe invariant in fact is a Pdemannian metric rests on the positive mass theorem of Schoen and Yau. Namely, by this theorem, the function re(g) is positive, if (M, g) is not conformally diffeomorphic to the standard sphere. The restriction to n < 6 is needed to apply Bartnik's result [11] that the mass of an asymptotically fiat Riemannian manifold (N, h) is an invariant of the metric h which requires T > (n - 2)/2 for the order T of (N, h) (cf. Theorem 5.1.4). It is not clear whether one can carry through the construction of can(C) for non-flat conformal structures C in dimensions n _> 6. In Section 5.3 we verify that, also in the general case, can(C) is of class C °O. For this we use the asymptotic expansions of the heat kernel for the conformal Laplacian. In the last chapter we generalize some of the considerations of Chapter 4 to the moduli space B+(M) of all conformal structures on M with positive Yamabe invariant, where M is a connected and closed manifold of dimension n = 3, 4, 5. We show that, for sequences of conformal structures whose Yamabe invariants tend to zero, the generalized canonical metric behaves as in the flat case. From this we obtain that, for n -- 3, 4, the boundary points of 13+(M) which correspond to Riemannian metrics with vanishing scalar curvature form infinite distance components of the boundary. On the other hand, if n = 5 and g is a Riemannian metric on M with vanishing scalar curvature and non-vanishing Ricci curvature, then the Diff(M)-equivalence class of the conformal class of g is a boundary point of B+(M) with finite distance from the interior. In this context, it would be of interest to clarify whether there are examples of five-dimensional closed manifolds which carry a Ricci-flat Riemannian metric as well as a metric of positive scalar curvature. To my knowledge, this question is still open. It is my pleasure to express my deep thanks to Jiirgen Jost for the fruitful and stimulating cooperation. Also parts of this book are based on a joint work [27] with him, which was continued with [28], [29]. I am also grateful to my wife and colleague Katharina Habermann for her help and support. Greifswald, March 2000
Lutz Habermann
Contents
1.
2.
3.
4.
5.
Preliminaries
............................................
1
1.1
Geometric preliminaries
1.2
Analytic preliminaries ..................................
A canonical
metric
................................
for flat conformal
1 6
manifolds
..........
2.1
M o t i v a t i o n of t h e a p p r o a c h
2.2
The conformal Laplacian
2.3
Definition a n d b a s i c p r o p e r t i e s
2.4
T h e m e t r i c s of A p a n a s o v - K u l k a r n i - P i n k a l l a n d N a y a t a n i
Kleinian
groups
and moduli
.............................
16
..........................
spaces
22 . ..
.......................
D e s c r i p t i o n of t h e canonical m e t r i c b y K l e i n i a n g r o u p s
3.2
M o d u l i spaces of fiat c o n f o r m a l s t r u c t u r e s
3.3
T h e g e o m e t r y of ]3+(S 1 × S 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . The fiat case
12
...............................
3.1
Asymptotics:
11
25 33
.....
................
...............................
33 38 45 55
4.1
D e g e n e r a t i n g fiat c o n f o r m a l s t r u c t u r e s . . . . . . . . . . . . . . . . . . . .
55
4.2
T h e b e h a v i o r n e a r s c a l a r fiat s t r u c t u r e s . . . . . . . . . . . . . . . . . . .
70
4.3
C o n s e q u e n c e s for t h e g e o m e t r y of B + ( M )
74
Generalization
in low dimensions
.................
........................
5.1
T h e c a n o n i c a l m e t r i c as m e t r i c of c o n s t a n t m a s s
5.2
T h e c a n o n i c a l m e t r i c : T h e general case
5.3
T h e r e g u l a r i t y of t h e c a n o n i c a l m e t r i c
...........
83 83
...................
88
....................
97
XII
.
Contents
The moduli space of all conformal structures
.............
6.1
T h e generalized m e t r i c n e a r scalar flat s t r u c t u r e s
..........
6.2
Infinite distance c o m p o n e n t s of the b o u n d a r y . . . . . . . . . . . . . .
101 101 105
References ....................................................
109
Index .........................................................
115
1. P r e l i m i n a r i e s
In this chapter we collect some notations and well-known facts from differential geometry and analysis on manifolds. For details and proofs, we refer to [13], [20], [34], [41] [44], [45], [631, [64].
1.1 Geometric preliminaries Let M be a manifold of dimension n J We denote the space of vector fields on M, i.e. the space of smooth sections of the tangent bundle TM, by :~(M) and the space of symmetric (2, 0)-tensor fields on M, i.e. the space of smooth sections of the symmetric product S2T*M of the cotangent bundle T'M, by ,52(M). The Kulkarni-Nomizu product of hi, h2 E 3 2(M) is the (4, 0)-tensor field hi (~ h2 defined by
(hl(~h~) (vl,v~,v3,v4) := hi (Vl,V3) h2 (v2,v4) + hi (v2,v4) h~ (vl,v3) - h t (vl, v4) h2 (v2, va) - hx (v2, va) h2 (Vl, v4) for v l , v 2 , v 3 , v 4 E TpM. Let g be a Pdemannian metric on M and let V denote the Levi-Civita connection of g. The Riemannian curvature of g is the (4, 0)-tensor field R defined by
n(xl,x
,x3,x4)
:= 9
(V[xl,x lX3 -
Vx X3 + Vx Vx X3,X,)
for X1, X2, Xa, X4 E :E(M). (Here we use the sign convention of [13].) The Pdcci curvature Ric E $a (M) of g is obtained from the Riemannian curvature R by n
Ric (vl, v2) := ~ R (ei, Vl, el, V2) i----1
for vl,v= E TpM, where {el,... ,en} is any orthonormal basis of TpM. The scalar curvature of g is the function S on M determined by 1 Unless otherwise stated, all manifolds, all tensor fields on it, and all maps are assumed to be smooth, i.e. of class C ~.
2
1. Preliminaries n
s ( p ) :=
Ric
e,) ,
i=l
where { e l , . . . , en} is as above. The Weyl curvature of g is given by
W := R - g ~ H
,
where H is the Schouten tensor, i.e.
H := ~
R i c - 2(n--1) g
"
Finally, the volume element of g is the density d#[g] on M given by d#[g] = ~
(gij)[d xl A . . . A dxn[
in local coordinates x 1, . . . , x n. Two Riemannian metrics g and ~ on M are said to be conformally equivalent, if there exists a function ¢ on M such that ~ = e2¢g. By a conformal class or conformal structure we understand a class of conformally equivalent Riemannian metrics. If C is a eonformal structure on M, then the pair (M, C) is called conformal manifold.
Proposition 1.1.1 Let g be a Riemannian metric and ¢ a .function on M. As above, let V, R, Ric, S, W, and d#[g] denote the various invariants of g. Then the corresponding invariants of ~ = e2¢ g can be expressed in terms of those of g and the derivatives of ¢ by the following relations: (i) (Levi-Civita connection) ~ 7 x Y = V x Y + d ¢ ( X ) Y + d ¢ ( Y ) X - g(X, Y ) V ¢ for X, Y e ~(M) (ii) (Riemannian curvature) ] ~ = e 2¢
(
R-g@
(
1
Vd¢-d¢®d¢+~g(V¢,V¢)g
))
(iii) (Ricci curvature) Ric = Ric - (n - 2)(Vd¢ - d e ® de) + (A¢ - (n - 2) g(V¢, V¢)) g (iv) (scalar curvature) = e-2¢(S + 2(n - 1)A¢ - (n - 2)(n - 1) g(V¢, V¢))
1.1 Geometric preliminaries
3
(v) (Weyl curvature) I/V = e2¢W
(vi) (volume element) d#[.~] = enCdp[g]
Here, V ¢ is the gradient of ¢ and A = d*d the Laplacian with respect to g. [] Let gE denote the Euclidean metric on/l~n. A conformal structure C on M is called flat, if it can be locally represented by flat Riemannian metrics. This means that, for each p E M, there exist a neighborhood U of p with coordinates x = ( x l , . . . , x n) : U --+ V C ]~n and a Riemannian metric g in C such that n
g=x*gE,
i.e.
g=~dx
i®dx i
on
U.
i=1
We call a conformal manifold (M, C) flat, if C is a flat conformal structure. A Riemannian metric g is said to be locally conformally flat, if its conformal class is a flat conformal structure. If n = 2, then by the existence of isothermal coordinates, any conformal structure on M is flat. Therefore, a flat conformal structure on a manifold of arbitrary dimension is a natural generalization of the conformal (or complex) structure of a Riemann surface. For n > 3, the conformal flatness can be dedected by the Weyl-Schouten theorem stated below. T h e o r e m 1.1.2 Let g be a Riemannian metric on a manifold of dimension
n>_3. (i) If n = 3, then g is locally conformally flat if and only if the Schouten tensor H satisfies ( V x H ) (Y, Z) = ( V y H ) (Z, Z)
for all X, Y, Z e ~(M). (ii) If n > 4, then g is locally conformally flat if and only if the Weyl curvature W of g identically vanishes. [] Let (M1, C1) and (M2, C2) be two conformal manifolds. By a conformal map f : (M1,C1) ~ (M2,C2) we understand a map f : M1 -~ M2 such that f ' C 2 = C1. The last means that, given gl E C1 and g2 E C2, there exists a function ¢ on M1 such that
4
1. Preliminaries f'g2 = e2¢gl •
(1.1.1)
We say that (M1, C1) and (M2, C2) are conformally diffeomorphic, if there exists a conformal diffeomorphism f : (M1,C1) --+ (M2,C2). A conformal diffeomorphism f : (M, C) ~ (M, C) is referred to as a conformal transformation of (M, C). R e m a r k 1.1.3 According to (1.1.1), any conformal map is a local diffeomorphism. [] Let gs denote the standard metric on the sphere S ~ and Cs the conformal class of gs. Since the stereographic projection a : S n \ {(0,..., 0, 1)} -r ]~n is conformal with respect to Cs and the canonical flat structure on Rn, i.e. the conformal class of gE (cp. Example 2.2.8), the conformal structure Cs is flat. Let Conf(S n) denote the group of all conformal transformations of (S n, Cs). This group is also referred to as MSbius group and its elements as M6bius transformations. P r o p o s i t i o n 1.1.4 Each transformation f G Conf(S n) can be written as f = f l o f 2 , where fl is an isometry of(Sn,gs), i.e. fl E O ( n + l ) , and after identifying S ~ with R~ U {c~) via stereographic projection, f2 takes the form f2 (x) = )~x - b for some )~ > O and b G R n.
[]
In particular, the last proposition gives that the group Conf(S n) is not compact. By the following theorem of Lelong-Ferrand and Obata [46], [53], this property characterizes the standard sphere among all manifolds which are closed, i.e. compact and without boundary. T h e o r e m 1.1.5 Let (M, C) be a closed conformal manifold of dimension n. If the group of all conformal transformations of (M, C) is not compact, then (M, C) is conformally diffeomorphic to (S n, Cs). [] The following proposition is also due to Obata [53].
Proposition 1.1.6 If g is a Riemannian metric on S n which is conformaUy equivalent to the standard metric gs and has constant scalar curvature, then up to a constant scale factor, g is obtained from gs by a conformal transformation of (S n, Cs). [] The theorem below is known as Liouville theorem. It says that, on (S n, Cs) with n > 3, any local conformal transformation is already given by a global conformal transformation.
1.1 Geometric preliminaries
5
T h e o r e m 1.1.7 Letn >_3 and let ~ be an open subset ofR "~. If f : J2 -+ R n is conformal with respect to the canonical fiat structure of l~~ , then f is the restriction of a transformation ] • Conf(Sn). [] R e m a r k 1.1.8 From Theorem 1.1.7, it follows that a fiat conformal structure on a manifold M of dimension n _> 3 can be also thought of as an atlas of M each of whose transition maps is a restriction of a MSbius transformation. In other words, a fiat conformal structure is a (Conf(Sn), Sn)-structure (cf. e.g. [351, §4). [] The following result is due to Kuiper [40]. T h e o r e m 1.1.9 Let (M, C) be a simply connected, fiat conformal manifold of dimension n ~ 3. Then there exists a conformal immersion • : (M, C) --+
(s Cs).
[]
A direct consequence of Theorem 1.1.9 is C o r o l l a r y 1.1.10 Any closed and simply connected, fiat conformal manifold (M, C) of dimension n is conformally diffeomorphic to (S n, Cs). [] In particular, this implies that not every manifold possesses a flat conformal structure. For example, S 2 x S 2 cannot carry any flat conformal structure. We note that flat conformal manifolds are not completely classified. (See, however, Theorems 3.2.2 and 3.2.3.) At the end of this section we recall the Yamabe functional and the Yamabe invariant of a closed conformal manifold (M, C) of dimension n >_ 3. The Yamabe functional of (M, C) is the map
y : g • C ~4 vol(M,g) (2-n)/n/MSdl~[g] where S is the scalar curvature taken for g and vol(M, g) denotes the volume of M with respect to g, i.e. vol(M, g) : = / u dp[g]. The Yamabe invariant of C is defined as
, ( c ) := inf{y(9): 9 • c } . R e m a r k 1.1.11 According to the resolution of the Yamabe problem by Trudinger, Aubin, and Schoen, the Yamabe functional Y attains its minimum at a Riemannian metric of constant scalar curvature. In particular, every conformal structure on a closed manifold contains a constant scalar curvature metric. []
6
1. Preliminaries
1.2 Analytic preliminaries Let M and g be as in the preceding section. As usual, Coo(U) and C~(U) for an open subset U C M denote the spaces of smooth functions and smooth compactly supported functions on U, respectively. For two subsets [71 and [72 of M, we write U1 CC U2, if U1 is open and its closure is a compact subset
of U2. Let U be an open subset of M. For 1 < s < oo, we define the LS-norm of
u E Coo(U) by
llullL.(U);g:-- (fv lul"d~[g])1/8 If in addition m is a non-negative integer, the Win'S-norm of u is given by
where V is the Levi-Civita connection of g and
with the natural extension of g to tensors. The Lebesgue space Ls(U) and the Sobolev space W m's (U) are defined as the completions of the sets {U • Coo(U): IlztllLs(U);g < 00} and {u • C°°(U): IluHw,,.,(tr);g < oo} with respect to the L*-norm and Wm,~-norm, respectively. The space Cm(U) is the set of k times continuously differentiable functions u on U for which the norm m
llutlc-(u);g := Y~ sup Iv i-25v
19
is finite. For a Riemannian vector bundle E on M, the spaces Ls(U,E), Wrn's(U, E), and C'n(U,E) of sections of E on U are similarly defined. We note that, if U CC M, although the norms defined above clearly depend on the choice of the Pdemannian metric g on M, the corresponding Banach spaces and their topologies are independent of g. The following result is a version of the compactness theorem of RellichKondrachov. T h e o r e m 1.2.1 Let U CC M and suppose that the boundary OU of U is
smooth. Let 1 < s, r < o¢ and let k and m be non-negative integers. 1 1 m __~>__--E, r
8
n
then Wm's(u) is compactly embedded into Lr(U).
1.2 Analytic preliminaries
(iO If k<m--,
7
n 8
[]
then Wm,"(U) is compactly embedded into Ck(U).
Obviously, if M is closed, then U := M satisfies the assumptions of Theorem 1.2.1. Now we turn to the analysis of the linear partial differential operator P on M given by p:=A+¢ for some ¢ • C°°(M), where A is again the Laplacian of g. If u and v are locally integrable functions on an open set U C M, u is called a weak solution of P u = v on U, if
fu u Puo d [g] = fu vuo d#[g] for all Uo • C~°(U). In the following let go be a fixed reference metric on M, whereas the Riemannian metric g which is involved in the definition of P may vary. For U CC M, we set ~(g; U) := inf { g ( v , v ) : v • T p M with g0(v,v) = 1 and p e U} . T h e o r e m 1.2.2 Let u be a weak solution of P u = v on U CC M . I f v E w m ' s ( U ) , then u E Wm+2's(U ') for any U' CC U. If in addition u • Ls(U), then
Ilullw .+2,.(v,);go <_ co (llvllw. ,.(u);go + with a constant co depending on U, U',
a(g; u).
IlullL.(U);ao)
Ilgllc-+l(u);~o, II¢llc,,,(u);go,
and
[]
As a consequence of the last theorem, we have C o r o l l a r y 1.2.3 If u is a weak solution of P u = 0 on an open set U C M , then u e C °o (U). [] Next we state the Harnack inequality and the strong maximum principle. T h e o r e m 1.2.4 Let U CC M and let u E C2(U) be a non-negative solution of P u = 0 on U. Then, for any U' CC U, we have sup u _< cl inf u U'
with a constant Ca depending on U, U',
U'
Ilgllco(u);go, II¢llco(u);go,
and )~(g; U).
8
1. Preliminaries
T h e o r e m 1.2.5 Let U C M be open and connected and let the function ¢ be non-negative on U. Furthermore, suppose that u • C 2 (U) satisfies P u <_ 0 (resp. P u >_ 0). Then u cannot achieve a non-positive m i n i m u m (resp. nonnegative m a x i m u m ) in U unless it is constant. [] The proposition below treats removable singularities. T h e o r e m 1.2.6 Suppose that n >_ 3. Let U C M be open and p • U. I f u is a weak solution of P u = 0 on U \ {p} and u • Ls(U) for some s > n / ( n - 2), then u satisfies P u = 0 in the weak sense on all of U. [] We conclude this section with a modification of the estimate in Theorem 1.2.2 for closed manifolds. P r o p o s i t i o n 1.2.7 Let M be closed and s > 1. Then there exists a constant c2 depending on Ilgllc~(M);go, [lCHCO(M);go, and A(g; M ) such that
Ilullw=,.tM);ao < c2 (IIPulIL.(M);go + IlulIL'(M);go) for all u • W 2 ' S ( M ) .
For convenience, we shall prove this proposition. First we show L e m m a 1.2.8 Let U CC M have smooth boundary and let s > 1. Then, for each ~ > O, there exists a c(c) such that
HullL.(u);go <_ 611ullw~.,(u);go + C(~)]IuIIL~(U);go for all u • W2's(u). Pro@ Let us assume that the statement does not hold. Then there exist an E > 0 and a sequence (uk) in W 2 , s ( u ) such that, for all k • N,
[[UkllL.(U);go =
1
(1.2.1)
and 1 > ~ [luk[[w2.,(u);go + k [[u~[IL~(U);go .
(1.2.2)
From (1.2.2), it follows that the sequence (uk) is bounded in W2's(U). Thus, since W2's(u) is compactly embedded into L s ( u ) (see Theorem 1.2.1(i)), we may assume that u~-~u in L s ( u ) for some function u. Then, because of (1.2.1),
IlU[[L.(U);9o =
1,
(1.2.3)
1.2 Analytic preliminaries and, since the inclusion map Ls(U) C LI(U) is continuous, uk--+u
in
LI(U).
But (1.2.2) implies that lim Ilu~llLl(U);go = O .
k--+oo
Therefore,
[lullL'(U);go =
0 and hence u = 0, which contradicts (1.2.3).
[]
Pvoo] of Proposition 1.2.7. By Theorem 1.2.2, there exists a constant co depending on
W2,S(M),
Ilgllcl(M);go, IlCllco(M);go,
and A(9; M) such that, for all u E
[[U[lw=,,(M);oo < Co ([[PU]lL,(M);go + []UIIL'(M);9o) Moreover, applying Lemma 1.2.8 to U := M, there is a constant ~ > 1 such that, for all u C W2'S(M), 1
IlullL.(M);go < -2--~liullw2,o(M);go + OllulIL'CM);~o Consequently, we have 1
Ilullw=,.(M);~o < co[[PU[IL,(M);oo + ~llullw~,.(M);~o + coe[[UllLl(M);go and hence
[[UIIw=,'(M);9o ~ 2CO[[PUIIL°(M);go -4-2co~lJullLl(M);go < 2COO([IPuIIL*tM);go + IIulILI(M);oo) for all u E W2'~(M).
[]
2. A canonical metric for flat conformal manifolds
In this chapter we introduce a canonical Riemannian metric can(C) for fiat conformal structures C with positive Yamabe invariant. To motivate our idea for the definition of the metric can(C), we first recall the definition of the Arakelov metric on Riemann surfaces and the conformally invariant Riemannian metric for Euclidean domains introduced by Leutwiler. As we shall see, these metrics are constructed from a Green function. We shall apply this construction principle to the Green function of the conformal Laplacian on a closed manifold. Some basic facts on the conformai Laplacian are stated in Section 2.2. In particular, it is shown that, in some sense, the conformal Laplacian is conformaily invariant. If the underlying conformal structure C has positive Yamabe invariant, the conformal Laplacian is invertible and hence possesses a unique Green function. In Section 2.3, using this Green function, we construct a smooth symmetric (2,0)-tensor field can(C), provided the conformal structure C is fiat. We prove that this tensor field is canonical, i.e. can(C) depends only on the conformal structure C. It will turn out in Chapter 3 that can(C) is a Riemannian metric in C, if the conformal manifold under consideration is not conformally diffeomorphic to the sphere with its standard (fiat) structure. Having this in mind, we show that, locally, the geodesic distance with respect to can(C) does not decrease under conformal maps. This is one of the reasons why the canonical metric can(C) differs from the two canonical Riemannian metrics introduced by Apanasov-Kulkarni-Pinkall and Nayatani. The construction and basic properties of these canonical metrics are outlined in the last section of this chapter. We shall see that, although the metrics of Apanasov-Kulkarni-Pinkall and Nayatani can be defined also for conformal structures with zero or negative Yamabe invariant, their construction essentially use that the underlying conformal structure is fiat. On the other hand, in low dimensions it is possible to extend the definition of the canonical metric can(C) to non-fiat structures. This will be carried out in Chapter 5.
12 2.1
2. A canonical metric for flat conformal manifolds Motivation
of the
approach
First we explain how the Green function of the Laplacian of a closed Riemannian manifold is defined. Let M be a closed manifold and let g be a Riemannian metric on M. Let diag(M) denote the diagonal of M x M, i.e. diag(M) :-- {(p,q) • M × M : p = q}. Then the Green function of the Laplacian A of (M, g) is defined as that continuous function GA : (M x M) \ diag(M) --4 I~ which satisfies
M G A (p, q) Au(q) d#[g](q) -- u(p)
1 g) vol(M,
fM UdB[g]
for all u E C ~ (M) and
fM
Ga(p, q) dl~[g](q) = 0 .
Thus G a is the Schwartz kernel of the Green operator On : C°°(M) -+ C°°(M) for A. Recall that, since A : C ~ ( M ) ~ C°°(M) is not invertible, the Green operator ~Sa is determined by
~
o ~(u) = A o
eA(U)
---- U -- P h a r m ( U )
and o n o Pharm(U) = 0
for each u E Coo (M), where P h a r m denotes the L2-orthogonal projection onto the space of harmonic, i.e. constant functions on M. Now let Z b e a Riemann surface, i.e. an oriented, connected, two-dimensional manifold with a complex structure. We assume that E is closed and that p _> 1, where p is the genus of Z . Let { r l , . . . ,vp} be a basis of the space of holomorphic 1-forms on ,U such that rk A ~7 = ~k~ •
Then by 1
p k----1
a Hermitian metric gB on 57 is defined. This metric, which is canonical in the sense that it only depends on the complex structure, is sometimes called Bergman metric.
2.1 Motivation of the approach
13
Let GA,B denote the Green function of the Laplacian of (E, gB). For a holomorphic coordinate z : U ~ C on E and p E U, we set
gA(p) := q--+p lim [[z(p)
--
z(q)lexp(2~
G
a,B(P,q))] - 2 ,
that means log (gA~(P))=--4~r q~vlim[G,~,B(p,q)+ 2~ log , z ( p ) - z(q),] . One easily verifies that the limit gzh~(p) exists and is positive. Furthermore, the (2, 0)-tensor field gA :__ gA dz ® d2 is independent of the choice of the holomorphic coordinate z. Namely, one can show that 9A __ a2 gB with aB (p) ----q-~p lim [distB (p, q) exp(2~r GA,B (P, q))]--i , where distB denotes the geodesic distance with respect to gB. Hence, g i is again a canonical Hermitian metric on Z, called Arakelov metric. Using methods of algebraic geometry, Arakelov [4] proved T h e o r e m 2.1.1 Let dV h and dV B be the volume forms with respect to gA
and gB, respectively, and let K A be the Gauss curvature of gA. Then KAdV A = 4~r(l - p)dV B ,
(2.1.1)
i.e. the Ricci form with respect to gA is a constant multiple of the volume form with respect to gB. [] We note that 9A is determined by (2.1.1) up to a constant factor. In the second part of this section we consider an open and bounded domain C ]~n, n > 2, with smooth boundary a n . Let AE denote the Laplacian with respect to the Euclidean metric gE, i.e. AE=-
7x~
"
i=1 Let G~ be the Green function of the Dirichlet problem of AE on J~, i.e. that continuous function G~ : ( ~ × ~ ) \ diag(/2) -+ ~ which satisfies
/
G ~ ( x , y ) ~ E ~ ( y )
d,[gE](y) = u(x)
for all
u e C~(n)
14
2. A canonical metric for flat conformal manifolds
and
G~(x,y)=O
if y • 0 0 .
Here, O = O U 0/2 is the closure o f / 2 . Let wn-i denote the volume of the unit sphere S n - i . Then an application of Corollary 1.2.3 gives that 1 G~(x,y) : ~ (-loglx-
y[ - G~,res(X,y))
for
n : 2
and
G~(z,y)-
1 (n-2)~_1
([x-yl2-~-G~,reg(X,y))
for
n>3,
where Gx,~,reg(Y) := G~,reg(X, y) is a harmonic function on O for each x E O. The harmonic radius r~ : ~2 --+ ]~ is defined by
r~(x) :-- exp(-G~,reg(X,X))
for
n = 2
and i / ( 2 - n ) /~'~,'J ~ ~ r~(x) : : ~ "-'9,reg
for
n >-3
,
that means rg(x) := y--~x lim [ I x - y l e x p ( 2 ~ r G g ( x , y ) ) ]
for
n = 2
and ~ ( ~ ) := aim [Ix - y l 2-n - (n - 2 ) ~ n _ l a ~ ( x , y ) ]
1/(2-~)
for
n > 3.
y---~z
Note that, for n >_ 3, by the maximum principle (Theorem 1.2.5),
G~,reg(X,y) > 0
for all
x , y E [2.
In two dimensions, r~ is the conformed radius, which plays an important role in geometric function theory. For n _> 3, the notion of the harmonic radius was introduced by Hersch [30] (cp. also [10]). Leutwiler [47] observed that the Riemannian metric g~ on 0 given by g~ := r~2gE is conformally invariant in the following sense. P r o p o s i t i o n 2.1.2 Let Y2 and [2' be open and bounded domains in W ~, n > 2, with smooth boundary, and let f : ~ --+ [2' be a conformal diffeomorphism. Then f* g~, = ga •
2.1 Motivation of the approach
15
Proof. Assume first that n _> 3 and define the positive function 7~ E C ~ ( ~ )
by f * gE
= ~4/(n-2)gE
•
Then
/*go, = (to, o f)-~ I* gE = (to, o f ) - 2 ~4/(,~-2)g E .
Thus we have to show that to, o f = ~2/(n-2)~ o . By the Liouville theorem (Theorem 1.1.7), f is the restriction of a conformal transformation of S n. This implies (cf. [1] or [47]) that I f ( x ) - f ( Y ) l = ~Pl/(n-2)(x)cpl/(n-2)(Y)I x - Y l
for all
x , y e T2.
Moreover, it is easy to check (cp. the proof of Lemma 2.2.7) that 1 Go, (f(x), f ( y ) ) - 7~(x)cp(y) G o ( x , y ) . Hence ~:O'(f(x)) = lira [If(x) - f(y)l 2-n - (n - 2)Wn-lGO, (f(x), f(y))] 1/(2--n) y--¢x
= lim [ ~ : y l 2 - n
(n_2)~_lGO(x,y)j 1/¢2-")1
. . . . . = 7~2/(n-2) (x) y--+x lim [Ix - yl 2 - n - (n - z ) w n - l ~ o [ x ,
11/(2-~)
y)J
= ~2/("-2)(x) t o ( x ) . For n -- 2, the assertion can be proved in a similar way, using that, in this case, f is holomorphic or antiholomorphic and the Green functions satisfy G m ( f ( x ) , f ( y ) ) -- G o ( x , y ) .
[] We have seen that the Arakelov metric gA and the Riemannian metric go for a Euclidean domain ~ are obtained by means of the same construction principle. We shall adapt this principle to the situation which we are interested in. The Green function which we shall use is the one of the conformal Laplacian on a closed Riemannian manifold.
16
2. A canonical metric for flat conformal manifolds
2.2 The
conformal
Laplacian
Let M be a connected manifold of dimension n _> 3 and let g be a Riemannian metric on M. The conformal Laplacian L : C°°(M) -~ Coo(M) of the Riemannian manifold (M, g) is defined by n-1
Lu = 4-~-~-~_2Au + Su
for
u E Coo(M) .
Here, S is again the scalar curvature of g. Let ~ q04/(n-2)g for a positive function qa e Coo(M). Then Proposition 1.1.1(iv) can be written as =
(2.2.1)
= 79-(n+2)/(n-2)L79,
where S denotes the scalar curvature of the Riemannian metric ~. In particular, ~ has constant scalar curvature A if and only if Lqo =
A~9 ( n + 2 ) / ( n - 2 )
.
The last equation is the Euler-Lagrange equation of the Yamabe functional (cf. [13], Section 4.D). For that reason L is sometimes referred to as Yamabe operator. The conformal Laplacian is conformally invariant in the following sense. L e m m a 2.2.1 Let L denote the eonformal Laplaeian with respect to ~ =
qo4/(n-2) g. Then L(qou) = qo(n+2)/(n-2)L(u)
for all u E Coo(M).
Proof. Let z~ denote the Laplacian of ~. Since (cp. Proposition 1.1.1) d/z[~] = V2n/(n-2)d#[g]
(2.2.2)
for the volume elements d#[g] and d#[.O] of g and t~, respectively, we can deduce that
fM ~-(n+2)/(n-2) [A(qou) -- A(qo)u]
U0
d#[0]
= ./~ [A(vu ) - A(qo)u] Vuo d#[g] = . / , [g(d(~u), d(7~Uo)) - g(dqo, d(Vuuo)) ] d#[g]
= / ~ qa2 g(du, duo) d#[g] = . ~ qo-4/(n-2) g(du, duo) ~2n/(n-2) d#[g] = f/~(u)
JM
uo d#[~]
2.2 The conformal Laplacian
17
for u E C°°(M) and Uo E C ~ ( M ) . Thus we have ~(n+2)/(n-2) Au = A(~u) - Z~(~)u. From this and (2.2.1), it follows
L(~u) =
4n-1 n - 2A(~u) + S ~ u
= 4 n - lcp(n+2)/(n-2)Z~u + L(~)u n-2
= ~(~+2)/(n-2) [4nn-~_12~ u -k Su ] = ~(n+2)/(n-2)L(U). D An important fact in conformal geometry is P r o p o s i t i o n 2.2.2 Let M be a connected closed manifold of dimension n >_ 3 and let C be a conformat structure on M. Then one and only one of the following cases holds: C contains a Riemannian metric of (i) positive, (ii) negative, or (iii) vanishing scalar curvature.
Proof. Let g E C be arbitrary. The conformal Laplacian L of g is a formally self-adjoint, elliptic differential operator whose spectrum is bounded from below. Let A1 be the first eigenvalue of L and let ~ be an eigenfunction of L with eigenvalue A1. Since ~ nowhere vanishes, we may assume that ~ > 0. Then ~ = ~4/(n-2)g is a Riemannian metric in C the scalar curvature of which is S = Al~O-4/(n-2) by (2.2.1). In particular, S > 0 (resp. S < 0, resp. _= 0) if A1 > 0 (resp. A1 < 0, resp. A1 = 0). On the other hand, if ~ is a Riemannian metric in C with scalar curvature > 0 (resp. S < 0, resp. S = 0), then the first eigenvalue A1 of the conformal Laplacian L with respect to ~ obviously satisfies A1 > 0 (resp. A1 < 0, resp. = 0).
Since, by Lemma 2.2.1, the sign of the first eigenvalue of the conformal Laplacian does not change within a class of conformally equivalent Riemannian metrics, the statement follows. [] Definition 2.2.3 Referring to the cases (i), (ii), and (iii) of Lemma 2.2.2, we shall call a conformal structure on a closed manifold scalar positive, scalar negative, or scalar fiat, respectively. R e m a r k 2.2.4 (i) Proposition 2.2.2 is also valid for n = 2. In two dimensions, by the Gauss-Bonnet theorem, the topology of the underlying manifold
18
2. A canonical metric for flat conformal manifolds
already determines to which of the three cases the conformal structure belongs. For n > 3, this is no longer true. In particular, every closed manifold of dimension n > 3 admits a Riemannian metric with negative scalar curvature (cf. [5], [36]), even negative Ricci curvature [48]. (ii) Actually, by Remark 1.1.11, every conformal structure C on a closed manifold can be represented by a Riemannian metric having constant scalar curvature. Furthermore, for n _> 3, C is scalar positive, scalar negative, or scalar flat if and only if its Yamabe invariant t~(C) is positive, negative, or zero, respectively. [] D e f i n i t i o n 2 . 2 . 5 A continuous function G : (M x M) \ diag(M) -+ ~ is called a Green function of the conformal Laplacian L of (M, g) if
fM
G(p,q)Lu(q) d#[g](q) = cnu(p)
for all u e C ~ ( M )
(2.2.3)
with cn := 4(n - I)6on-1. We point out that this definition of a Green function includes a normalization. R e m a r k 2.2.6 Let G be as above, p E M, and set
Gp(q) := G(p,q) . Then, by Corollary 1.2.3, we have LGv=0
on
M\{p}.
(2.2.4)
Furthermore, one checks that lim G(p, q) = c¢.
q--~p
(2.2.5) []
From the conformal invariance of the conformal Laplacian, we get L e m m a 2.2.7 Let G : (M × M) \ diag(M) ~ R be a Green function of the eonformal Laplacian L of (M, g). Then G : (M x M) \ diag(M) -+ R given by 1 G(p,q) G(p'q) = qa(p)cp(q)
is a Green function of the conformal Laplaeian L with respect to the Riemannian metric ~ = cpa/(n-Z)g.
2.2 The conformal Laplacian
19
Proof. By L e m m a 2.2.1, (2.2.2),and (2.2.3),
/M G(p,q)Lu(q)d#[~](q) = ~
/M G(p,q)L(~u)(q)d#[g](q)
= cau(p)
n
for each u E C~¢ (M).
E x a m p l e 2.2.8 The fundamental solution of the Laplace equation yields a Green function GE of the conformal Laplacian n-1 LE ----4 ~ - ~ _2 AE of the Euclidean space (I~a,gE), n _> 3, which is given by
GE(Xl,X2) = lXl - X212-n
for
Xl,X2 E Ra
with
xlCx2.
Set S" := S a \ {(0, .... 0, 1)} and let a : ~a __+ i~a be the stereographic projection, i.e. •
1-
~a+l'''"
1 - ~a+l
for (~1,...,~a+1) E 3 a C l~n+l. Then, for the standard metric gs of the sphere S a, we have 4/(n-2) (a-i)* gS = ¢PS gE with
( ~os(x)=
2 ) (n-2)/2 l+lxl 2
for
xeR a.
By Lemma 2.2.7, it follows that Cs e C c° ((Sn x ~a) \ diag (~a)) defined by 1
¢S (~1,~2) ----(~S(O.(~I))~S(0.(~2))GE(°'(~I),°'(~2))
is a Green function of the conformal Laplacian of (~a, gs). By means of I~(~1) - ~ ( ~ ) [ ~
_
I~1 - ~2J 2
one obtains that ds (~i,~2) = I~I - ~l 2-" •
Thus Gs can be continued to a COO-function on (S a × S a) \ diag (Sn). N o w it is easy to verify that this continuation, i.e.the function Gs : (S n × S a) \ diag (S") ~ IR given by
20
2. A canonical metric for flat conformal manifolds
Vs
=
-
212-" =
_
2
\j=l is a Green function of the conformal Laplacian Ls of (S n, gs)- According to the considerations below, this Green function is unique. [] For the remaining part of this section we assume that the manifold M is closed. Then a Green function of the conformal Laplacian L of (M, g) exists if and only if L : C°°(M) --+ Coo(M) is invertible. Moreover, if a Green function G exists, it is uniquely determined as a constant multiple of the Schwartz kernel of L -1. Now let the scalar curvature S of g be positive. Then, obviously, L is invertible. Furthermore, since L satisfies the assumptions for the strong maximum principle (Theorem 1.2.5), it follows from (2.2.4) and (2.2.5) that G(p,q)>0
for all
(p,q) e ( M x M ) \ d i a g ( M ) .
Because of Lemma 2.2.7, the above statements continue to hold if one replaces the condition S > 0 by the condition that the conformal structure represented by g is scalar positive. Analogously to the reasoning concerning the Green function of the Laplacian (cf. e.g. [8], Chapter 4), one sees that G is symmetric, i.e.
G(p,q)=G(q,p) forall p, qeM Namely, for ul,u2 E Coo(M), one has
with
p~q.
(2.2.6)
/M L (/M 6(I9,q)ut (p) d#[g](p)) u2(q)d#[g](q) *'~/M (/M G(p,q)ul (p) d/~[g](p)) Lu2(q)d#[g](q) = /M (/M G(p'q)Lu2(q)d#[g](q)) uI(p) d#[g](p) = Cn [
JM
u2(P)Ul (P) d#[gl(p) ,
which gives
L(/MG(p,q)ul(p)d/~[g](p) ) With
= Cnul(q) .
Ul = Lu, (2.2.7) leads to L (/M G(p, q)Lu(p) d/~[g](p)) = cnLu(q) .
(2.2.7)
2.2 The conformal Laplacian
21
Since L is invertible, it follows
M G(P, q)Lu(p) d#[g](p) = ¢nu(q) for all u E C ~ ( M ) . By the uniqueness of G, this implies (2.2.6). Thus we have P r o p o s i t i o n 2.2.9 Let M be closed and let g be a Riemannian metric on M which is conformally equivalent to a Riemannian metric with positive scalar curvature. Then the conformal Laplacian L of (M, g) possesses a unique Green function G. Moreover, G is everywhere positive and symmetric. R e m a r k 2.2.10 (i) For a Riemannian metric g which represents a scalar flat conformal structure on M, a Green function of the conformal Laplacian L does not exist. To see this, assume without loss of generality that the scalar curvature of g vanishes. Then L:4n-IAI n-2
"
Hence L : C°¢(M) ~ C ~ ( M ) is not invertible. (ii) If g is a Pdemannian metric on M with scalar curvature S < 0, then a Green function G of L can exist. However, from (2.2.3) with u = 1, one gets
M G(P, q)S(q) d#[g](q) = ¢n, which implies that G has to be negative somewhere. (iii) Let g be a Riemannian metric on M the conformal class of which is scalar positive and let G be the Green function of the conformal Laplacian L of (M, g). If (uj)jeN is a complete orthonormal system of L2(M) with respect to g such that
Luj = )tjuj
and
0 <: ~I _~ ~2 ~ ' ' '
,
then
a(p, q) = in L2(M × M).
Z []
22
2. A canonical metric for flat conformal manifolds
2.3 Definition
and
basic
properties
Let M be a connected and closed manifold of dimension n > 3 with a scalar positive conformal structure C, and let g be an arbitrary Riemannian metric in C. By Proposition 2.2.9, the conformal Laplacian L of (M, g) possesses a unique Green function G. The idea for the definition of a canonical Riemannian metric can(C) associated to C, which is motivated by the considerations in Section 2.1, is to put can(C) := a2g with
a(p) := ~imp[G(p,q)-dist2-n(p,q)[ 1/(n-2) ,
(2.3.1)
where dist denotes the geodesic distance with respect to g. As we shall see in Section 5.2, for n = 3, (2.3.1) actually defines a function on M and a2g is conformally invariant. However, for n >__4, the above approach does not work in general. In particular, additional requirements on the Riemannian metric g E C are needed to assure the existence of the limit in (2.3.1). For that reason we shall now restrict to the conformally flat case. Later, in Chapter 5, we shall reformulate the definition of the canonical metric given below. This will allow us to extend the definition to non-flat conformal structures in dimensions 3, 4, and 5. So let C be a scalar positive, flat conformal structure on M and let g E C such that n
i
g= ~
dx i ® dx ~
(2.3.2)
i----1
on a neighborhood U C M with coordinates x = (xl, . .. , x n) : U ~ V C R n. Then, for the Green function G of the conformal Laplacian L of (M, g), we have L e m m a 2.3.1 Under the above assumptions, the map (p,q) E (U x U) \ diag(U) ~ G(p,q) - I x ( P ) - x(q)[ 2-'~ E R []
can be continued to a C °°-function on U x U. Proof. Equation (2.3.2) implies L=_4n-l~-~( n - 2 i= 1
O ) 2
on
U
and hence
f u iX(P) - x( q) i2-n Lu( q) d#[g](q) = ¢nu(P )
2.3 Definition and basic properties
23
for each u • C ~ ( U ) . Since, by Proposition 2.2.9, the Green function G of L is symmetric, it follows that
u
(G(p,q) -[x(p) -x(q)[ 2-n) (L(v) + L(q))w(p,q)dtt[g@g](p,q) = 0 ×U
for each w • C ~ ( U × U), where L(p) and L(q) mean that L is applied with respect to the variables p and q, respectively. By Corollary 1.2.3, this gives the assertion. [] In particular, we obtain that the limit
a(p) := lim [ G ( p , q ) - dist2-n(p,q)[ 1/(n-2) q-~p
= lim IG(p, q) - Ix(p) - x ( q l l 2 - n l l / ( n - 2 ) q-.+p exists for each p • U. P r o p o s i t i o n 2.3.2 Setting, with the above assumptions, can(C) := a29
on
U,
we obtain a well-defined symmetric (2, 0)-tensor field on M which is of class C °° and depends only on the conformal structure C. Proof. The smoothness of can(C) follows from the fact that, by Lemma 2.3.1, a 2 : U -~ n~ is a C~-function. It remains to check that can(C) does not depend on the choice of g. To see this, let ~ = ~4/(n-2)g for a positive function 7~ • C °¢ (M) and let n
= E
d~i ® d£:i
i=1
on a neighborhood 0 C U for coordinates ~ = ( ~ 1 , . . . , 5:,~) : ~- ~ 1~'. Then x o ~-1 : ~) ~ x(O) C V is a conformal diffeomorphism and hence, by the Liouville theorem (Theorem 1.1.7), the restriction of a conformal transformation of S n. Using Lemma 2.2.7, one can deduce as in the proof of Proposition 2.1.2 that &(p) = lim G(p,q) - I&(p) - &(q)l 2 - " 1/(n-2) q-.-}p = ~2/(2-~)(p)a(p) for each p E/~" and hence
24
2. A canonical metric for flat conformal manifolds []
This proves the proposition.
We shall now discuss the behavior of the tensor field can(C) under conformal maps. P r o p o s i t i o n 2.3.3 Let Mi for i = 1, 2 be a connected and closed manifold
of dimension n > 3 with a scalar positive, fiat conformal structure Ci and let f : (M1, C1) --~ (M2, C2) be a conformal map. Then (f'can(C2)) (v, v) >_ can(Cl)(v, v)
for each tangent vector v E T M1. Proof. Since f is a local diffeomorphism and M1 and M2 are connected and closed, f is a finite regular covering. We fix p E Mz and choose g2 E C2 such that g2 is flat on a neighborhood of f(p). Then gl := f'g2 is also flat on a neighborhood of p. Let disti denote the geodesic distance with respect to gi and Gi the Green function of the conformal Laplacian of (Mi,gi). Since f is a local isometry with respect to gl and g2, dist2(f(p), f(q)) = distl (p, q) for q E M1 near p. Furthermore, using that the covering transformation group of f : M1 --~ M2 consists of isometries of (M1, gl), one sees that G2 (f(p), f(q)) = Z
Gz (p, ")'q)
"yEF for each q E M1 with f(q) ~ f(p). It follows that
a~ -2 (f(p)) ----~im IG2 (f(p), f(q)) - dist2-n(f(p), f(q))l = q-+p lim Gl(p,q)-dist~-n(p,q)l +
=~r-2(P) + Z
~er\{z}
~
GI(p, Tq)
~er\{i}
vl(p,~q)
__>OL~-2(p) and hence
(f'can(C2)) (v, v) --- a2(f(p))(f'g2)(v, v) _> c~ (p)g~ (v, v) = can(cl)(v, v)
for v e TpMI. From the last proposition, one immediately obtains
[]
2.4 The metrics of Apanasov-Kulkarni-Pinkall and Nayatani
25
Corollary 2.3.4 Let (Mi, Ci), i = 1,2, be as in Proposition 2.3.3 and let f : (M1, C1) -4 (M2, C2) be a conformal diffeomorphism. Then f'can(C2) = can(C1) . In particular, for (M1, C1) = (M2,C2) = (M, C), can(C) is invariant under the group of con]ormal transformations of (M, C). Yq
The most important example of a scalar positive, fiat conformal structure is the conformal class Cs of the standard metric gs of the sphere S u. A direct consequence of the description of can(C) by means of Kleinian groups in the next chapter (see Theorem 3.1.2) will be (cp. Remark 3.1.3) T h e o r e m 2.3.5 Let M be a connected and closed manifold of dimension n >_3 with a scalar positive, flat conformal structure C. If (M, C) is conformally diffeomorphic to (Sn, Cs), then can(C) vanishes identically. In all other cases, can(C) is a Riemannian metric in C. For simplicity, we shall refer to can(C) in any cases as the canonical metric of C. R e m a r k 2.3.6 (i) Rephrasing Proposition 2.3.3, we have that conformal maps f : (M1, C1) -~ (M2, C2) are locally distance non-decreasing with respect to the canonical metrics can(C1) and can(C2). (ii) By Corollary 2.3.4, f*can(Cs) -- can(Cs) for every conformal transformation f of (S n, Cs). As one easily checks, this already implies can(Cs)
- 0.
Thus the first assertion of Theorem 2.3.5 can also be derived from Corollary 2.3.4. (iii) Let (M, C) be not conformally diffeomorphic to (S n, Cs). Then, by Theorem 2.3.3, can(C) is a Riemannian metric and, by Corollary 2.3.4, the group of conformal transformations of (M, C) coincides with the group of isometries of (M, can(C)). Thus we obtain a proof of the theorem of Lelong-Ferrand and Obata (Theorem 1.1.5) in the considered cases. []
2.4 The metrics of Apanasov-Kulkarni-Pinkall and Nayatani First we shall describe a construction by Apanasov [2] and Kulkarni and Pinkall [43] which bases on the Kobayashi construction (see [39], cp. also
26
2. A canonical metric for flat conformal manifolds
[23]) in the context of MSbius geometry. For this we consider a fiat conformal manifold (M, C). Let B " := {= •
: Ixl < 1},
and let diSthyp denote the geodesic distance on B n with respect to the hyperbolic metric 4 n g"YP " -
(1 -
dxi ® dx'.
I=1=) = i=1
For p, q • M, we define distKob(p, q) as the infimum of all numbers diSthyp (Xl, Yl ) + " " + diSthyp (xm, Ym) with X l , . . . , x m , Y l , . . . , y m • B n such that there exist conformal maps fl,...,fm : B n --+ M satisfying f l ( x l ) = p, fi(Y~) = fi+l(Xi+l) for i = 1,... , m - 1, and fm(Ym) = q. It is not difficult to see that the pseudodistance diStKob : M x M -+ ~ given that way is trivial, if (M, C) is conformally diffeomorphic to a spherical or Euclidean space form. On the other hand, one has the following result ([2], [43]).
Theorem 2.4.1 Let (M, C) be a flat conformal manifold which is not conIorreally diffeomorphic to a spherical or Euclidean space form. Then M admits a complete Riemannian C 1,1-metric giob compatible with C such that diStKob is the geodesic distance with respect to gKob. []
From the definition of diStKob one deduces, in contrast to the properties of our canonical metric (see Proposition 2.3.3 and Remark 2.3.6(i)),
Proposition 2.4.2 Let (M~, C~), i = 1, 2, be fiat con]ormal manifolds and let f : (M1,CI) --+ (M2,C2) be a conformal map. Then
distgob(f(p),f(q)) _< distgob(p,q)
for all p,q • M1 •
That means that con]ormal maps are distance non-increasing with respect to distgob. []
The second thing which we want to describe in this section is a construction due to Nayatani [51] which assigns a canonical metric to every so-called Kleinian manifold. Before defining Nayatani's metric, we have to recall some concepts of the theory of Kleinian groups. For more details we refer to [3] and [49]. Let F be a subgroup of the group Conf(S n) of all conformal transformations
of (s-, Cs).
2.4 The metrics of Apanasov-Kulkarni-Pinkall and Nayatani
27
2.4.3 A point ~ E S n is called a limit point of F provided there exist a point ~ E S n and different elements "h,'Y2,... E F such that
Definition
= .lim l--~oo
"/i'~'
•
The set A(F) of all limit points of F is called the limit set of F. The limit set A(I') is closed and invariant under F. The open set
n ( r ) := s"
\ A(r)
is the maximal domain in S n on which/~ acts properly discontinuously. For that reason JT(F) is called the discontinuity domain o f / ' . 2.4.4 A subgroup F C Conf(S n) is called a Kleinian group if g)(F) is non-empty. A Kleinian group F is called elementary if A(F) consists of at most two points and non-elementary otherwise. It is said to be geometrically finite if it possesses a finite-sided fundamental polyhedron.
Definition
Let F be a Kleinian group, and suppose that F leaves an open set ~2 C JT(/') invariant and acts freely there. Then the manifold JT/l" is called a Kleinian manifold. We restrict the following considerations to Kleinian manifolds for which J? = ~ ( F ) and denote by Cr the flat conformal structure on J7(1")/I" induced by the standard structure Cs on S n. According to the next theorem, which was proven by Schoen and Yau (see [62], Theorem 4.5), Kleinian manifolds form a large class of examples of fiat conformal manifolds. 2.4.5 Let M be a connected closed manifold of dimension n >_ 3 and let C be a fiat conformal structure on M. If C is scalar positive or scalar fiat, then there exists a Kleinian group F C Conf(S n) isomorphic to the fundamental group 7rl (M) of M such that (M, C) is confurmally diffeomorphie to (n(r)/r, o r ) . [] Theorem
For a conformal transformation T E Conf(Sn), we denote by [Tq that positive function on S ~ which is determined by
T*gs = iT'[ 2 g s .
(2.4.1)
D e f i n i t i o n 2.4.6 The critical exponent 6(1") of a Kleinian group 1" C Conf(S n) is defined by
5(F) := inf { s > O : Z where ~ E/2(/").
['~'(~)[" < ° ° }
28
2. A canonical metric for flat conformal manifolds
The number 5(F) does not depend on the particular choice of ~ E J?(F) and is bounded from above by <
Furthermore, 5(F) > 0 if F is non-elementary. Proofs of these assertions can be found e.g. in [52]. For Nayatani's construction the following result of Patterson and Sullivan (see [55], [65], [66]) is crucial. T h e o r e m 2.4.7 There exists a probability measure # r supported on A(F) which satisfies 7*~r --17'l~(r)pr (2.4.2)
]or each 7 E F. If F is non-elementary and geometrically finite, then such a #r is unique. A measure # r as in Theorem 2.4.7 is called a Patterson-Sullivan measure. For a Kleinian manifold (J?(F)/F, Cr), the construction of Nayatani's metric goes now as follows. We first assume that F is non-elementary. Then ~(F) > 0. We consider a Patterson-Sullivan measure # r and the Green function Gs of the conformal Laplacian Ls of (S n, gs) and define the function C r E C~(T2(F)) by
Since, by Lemma 2.2.7 and (2.4.1), a s (T(~),T(~')) = IT'(~)I (2-n)/2 IT'(~')I (2-n)/: Gs(~, ~') for T E Conf(Sn), it follows, by means of (2.4.2), that
= 17'(()1-1¢r(()
for 7 E F. Thus the Riemannian metric
(2.4.3)
2.4 The metrics of Apanasov-Kulkarni-Pinkall and Nayatani
29
on f2(F) is invariant under F and induces a Riemanian metric g r on 12(I')/F which lies in the conformal class C r . Now let /" be elementary. If A(F) = {51,52} or A(F) = {~o}, we define Cr e C°°(n(F)) by
¢F(5) := c~/(~-2)(6, 5)a~/(~-2)(52, 5) in the first case and
¢r(5) := G~/(~-2)(5o, 5) in the second case. Let T E Conf(S ~) with T({~1,52}) = {51,52}. Because of (2.4.3), Gs(51,52) = G s ( T ( ~ ) , T(52)) = ]T'(51)] (2-=)/2
IT'(52)1(2-=)/2 Gs(6,52)
and hence IT'(51)1 IT'(52)1 = 1.
(2.4.4)
For the second case, consider T E Conf(S n) with T(50) = 50 and 5 e S n with # 50. Then IT'(5) I(2-=)/2
IT'(5o)l (2-~)/2 as(5, 50)
-- Gs (T(~), T(5o)) --= Gs (T(5), T2(50))
= IT'(5)] (2-n)/2 ]T'(T(So))] (2-n)/2 IT'(So)l (2-n)/2 Gs(5, ~o) = IT'(5)I (2-n)/2 IT'(5o)] 2-'~ Gs(5, 50) and hence IT'(50)l = 1.
(2.4.5)
From (2.4.3), (2.4.4), (2.4.5) and the invariance of A(F) under F, it follows in both cases that again
Cr (~) = Iv'(5) l-lCr (5) for all 7 E F. Thus we can proceed as above to obtain a Riemannian metric gr E C r . It is easy to verify that, in the case A(F) = {50}, the Riemannian metric .@ on f2(F) = S n \ {5o} is a constant multiple of the pull-back of the Euclidean metric gE via the streographic projection S '~ \ {~0} --+ ]~n. If A(F) is empty, then (~(F)/F, CF) is conformally diffeomorphic to a spherical space form (cf. [41], §6). In this case let g r E C r be a Riemannian metric of constant sectional curvature. The Riemannian metric gr has some interesting properties. In particular, the following theorem ([51], Theorem 3.3) gives a refinement of a result of Schoen and Yau ([62], Theorem 4.7).
30
2. A canonical metric for flat conformal manifolds
T h e o r e m 2.4.8 Let O ( F ) / F be a Kleinian manifold, and suppose that the limit set A(F) is empty or contains a least two points. Then (i) gr has positive (resp. vanishing, resp. negative) scalar curvature if and only i/S(1") < ( n - 2 ) / 2 (resp. 5(F) = ( n - 2 ) / 2 resp. 5(F) > ( n - 2 ) / 2 ) . (ii) If 5(F) > n - 2, then gr has negative Ricei curvature.
For further properties and applications of Nayatani's metric we refer to [51], [32], and [33]. As we shall illustrate in the next section (cp. Example 3.1.5), it does not hold in general that can(C£), if defined, has positive scalar curvature everywhere. This is in contrast to the properties of gr described in Theorem 2.4.8. In the following we want to explain two other differences between our canonical metric and that of Nayatani. For this we consider a Kleinian manifold n(V)/V. Let F ' be a subgroup of F with finite index. Then A(F') = A(F) and
Then A(FT) = T ( A ( F ) ) and 6(FT) = 6(F). Furthermore, we have L e m m a 2.4.9 Let # r be a Patterson-Sullivan measure associated to the Kleinian group 1" and set
:: (T-')" (c+' IT'l'+.r) with CT ::-
fA(r) IT'I~(r) d # r
•
Then t~T is a Patterson-Sullivan measure associated to FT. Proof. Let ")'T
----
T T T -1 for 7 E F. Then i +l = iT' o T o T - 1 1 1 7 ' o T - ' [
IT' o T - t l
2.4 The metrics of Apanasov-Kulkarni-Pinkall and Nayatani
31
By means of this and (2.4.2), one deduces
v~#Tr : ( T - 1 ) * 7 . T . #T
=
(c+,
o
= (T-l)" (c+ 1 IT' o'~1~
(T-l)* # r
Since 5 (FT) = 5(F) and
-~1, the statement follows.
[]
Now let # r be a Patterson-Sullivan measure associated t o / ' , and let #T and CT as in Lemma 2.4.9. For the Pdemannian metrics ~r and grr formed with respect to # r and # r r = #T, respectively, one calculates by means of (2.4.3) that T*OrT = CT:Or • Hence one has
f~rgrr : cT2 gl"
(2.4.6)
for the conformal diffeomorphism
fT : ( ~ ( r ) / r , v r ) -+ ( ~ ( r T ) / r ~ , c r ~ ) induced by the transformation T. Since, in general, CT ~ 1, (2.4.6) is in contrast to Corollary 2.3.4. Furthermore, because of (2.4.6), one has to normalize Nayatani's metric in order to construct an L2-metric on moduli spaces of flat conformal structures by means of it (cp. Section 3.2). We summarize that the Riemannian metrics gKob and gr presented in this section differ strictly from our canonical metric can(C). The first two metrics can be defined for a larger class of flat conformal structures than can(C). For the construction of gKob and gr, however, it is essential that the conformal structure under consideration is flat. On the other hand, the definition of our canonical metric can be generalized to non-flat conformal structures at least in low dimensions.
3. K l e i n i a n
groups and moduli
spaces
As mentioned in the last section, any flat conformal manifold (M, C) whose conformal structure C is scalar positive is a Kleinian manifold. In the first section of this chapter we use this to describe the canonical metric can(C) more explicitly, namely in terms of the underlying Kleinian group. In particular, this description proves that can(C) indeed is a Riemannian metric provided that (M, C) is not conformally diffeomorphic to the standard sphere. Furthermore, we give examples of flat conformal manifolds (M, C) such that the scalar curvature of can(C) is not constant, even not everywhere positive. In the second section we start our investigation of moduli spaces of conformal structures. Besides collecting some known facts concerning such spaces and explaining how canonical Riemannian metrics induce a Riemannian L 2metric on the corresponding moduli space, we parametrize the moduli space B0 (S 1 × S 2) of flat conformal structures on S 1 x S 2. We use this parametrization in Section 3.3 to study the geometry of B0 (S 1 × S 2) induced from our canonical metrics can(C). In particular, we determine the metric completion of this space.
3.1 Description of the canonical metric by Kleinian groups We again consider a connected and closed manifold M of dimension n _> 3 and a scalar positive, flat conformal structure C on M. Our next aim is to find an explicit expression for can(C). By Corollary 2.3.4 and Theorem 2.4.5, we may assume that M = f 2 ( F ) / P and C = C r for a Kleinian group F C Conf(Sn). Let 7rr : ~2(F) ~ f2(F)//" be the canonical projection, and let g be a Riemannian metric in C. Then 7r~g is conformally equivalent to the restriction of the standard metric gs of S n to ~ ( F ) . Let the positive function on ~2(F) be determined by 7r~g = ~4/(n-2)g S .
(3.1.1)
The Green function G of the conformal Laplacian of ( ~ ( F ) / F , g ) can be expressed by @ and the Green function Gs of the conformal Laplacian of (S '~, gs) in the following way.
34
3. Kleinian groups and moduli spaces
L e m m a 3.1.1 With the above assumptions and notations,
1 G(~rr(~l),rr(~2)) - ~(~1)~5(~2) Z J"Y'(52)j(n-2)/2 Gs(51'?~2) "yEF
(3.1.2)
/or ~1,~2 • I)(F) with ~rr(~l) ~ ~r(~2). Proof. By Lemma 2.2.7, the function G given by
is a Green function of the conformal Laplacian of (~2(F), ~r~g). Since the Riemannian metric 7r~g is invariant under 1", o "y = [,.yt[(2--n)/2~
for all 7 • F .
From this and (2.4.3), one obtains that the right-hand side of (3.1.2) coincides with the series "yEF and is invariant under transformations
for ~/1,~2 • 1". Hence it remains to prove that Z 1"/'(~2)1(n--2)/2 GS(~I'~2) <~ (30 "yET
(3.1.3)
for all ~1, ~2 • ~(1") with r r ( ~ l ) ¢ ~rr(~2). The last relation can be deduced from Theorem 2.4.8. First we observe that A(F) is empty or contains at least two points, since otherwise ([2(F)/F, Cr) would be conformally diffeomorphic to a Euclidean space form (cp. [41], §6) and hence Cr would not be scalar positive. Thus, by Theorem 2.4.8, the critical exponent 5(1") of 1" satisfies n-2 < -5This implies (see Definition 2.4.6) that Z I~/(~)1(n-2)/2 < oo for all ~ • f2(1") . -yEt Moreover, since 1" acts properly discontinously on ~2(1"), for any ~1, ~2 • ~(1") with ~rr(~l) ~ ~rr(~2), i.e. ~1 ¢ ~ 2 for all ~/ ~ 1", there exists a constant c(~1, ~2) > 0 such that Gs(~l,'y~) _< c(~,~2)
for all ~/• 1".
This shows (3.1.3), and the lemma is proven. We use Lemma 3.1.1 to prove
[]
3.1 Description of the canonical metric by Kleinian groups
35
T h e o r e m 3.1.2 Let 12(F)/F be a connected and closed Kleinian manifold
of dimension n > 3 and suppose that the flat conformal structure Cr is scalar positive. Then ~7.can(Cr) = a~,gs with
~r(5) =
~
1"/(5)l("-~)/~Gs(5, 75)
~r\{~} Proof. We fix ~ • 12(F) and choose g • Cr such that r*rg = a~gE on a neighborhood ~r of ~, where a~ : S n \ {-~} ~ 11~n is the stereographic projection. Let G be the Green function of the conformai Laplacian of (~2(F)/F, g). Then, by definition, we have can(C/-) --- a2g
(3.1.4)
with
~(~F(O) = ~)2~ ]a(~-r(~),~F(()) -Io~(~) - ~'~(~')I~-"]~/~"-~) Let the positive function 0 E C°°(O(F)) be given by (3.1.1). From .~can(Cr) = ~gs
and (3.1.4), it follows that ~r(~) =
~-/("-~)(~)~(~r(~))•
(3.1.5)
Furthermore (cp. Example 2.2.8), Gs(~I, ~2) = ~(~1)~(~2)1o'~(~1) - or~(~2)l2-n
(3.1.6)
for ~1,~2 • U with ~1 ~ ~2. From (3.1.2) and (3.1.6), we obtain that
C(~r(~,), ~r(~2)) - I~(~1) - ~(~2)12-" 1
Z
I"f(~2)l("-"~/" cs(a' 7e")
for ~I,~2 • 0 with ~1 # ~/~2for all 7 • _r\ {1}. Consequently,
This and (3.1.5) yield the assertion.
O
36
3. Kleinian groups and moduli spaces
R e m a r k 3.1.3 (i) Corollary 2.3.4, Theorem 2.4.5, and Theorem 3.1.2 imply Theorem 2.3.5. Namely, let (M, C) be as in Theorem 2.3.5. Then, by Corollary 2.3.4 and Theorem 2.4.5, we may assume without loss of generality that (M, C) = (J?(F)/F, Cr). If (J2(F)/T', C r ) is conformally diffeomorphic to (S n, Cs), then the Kleinian group F is trivial and we obtain by means of Theorem 3.1.2 that a r -= 0 and hence can(Cr) = 0. If/~ is non-trivial, then Theorem 3.1.2 implies that a r is everywhere positive and hence can(Cr) is a Riemannian metric in C r . (ii) Let (J?(F)/F, C r ) be as in Theorem 3.1.2. Let G be the Green function of the conformal Laplacian and dist the geodesic distance with respect to a Riemannian metric g E C r which is supposed to be flat on a neighborhood of p E T2(F)/F. Then from the proof of Theorem 3.1.2, it follows that lim (G(p, q) - dist2-n(p, q)) > 0
q--+p
in the case of non-trivial F. This makes it possible to resolve the Yamabe problem for closed Kleinian manifolds by Schoen's method (cf. e.g. [63], in particular §V.4) without applying the positive mass theorem of Schoen and Yan (see Theorem 5.1.6). Since in addition Theorem 2.4.5 can be proven for n > 4 without the positive mass theorem, one obtains a resolution of the Yamabe problem for closed flat conformal manifolds which does not base on the last mentioned theorem. I-1 If the fiat conformal manifold (M, C) is homogeneous, then, according to Corollary 2.3.4, so is the canonical metric can(C) and hence, provided that can(C) is a Riemannian metric, it has constant curvature. We shall now illustrate that the last does not hold in general. For this we consider S 3 as the unit sphere in C 2 , i.e.
and prove L e m m a 3.1.4 Let ¢ be a positive function on S 3 which depends only on and suppose that the Riemannian metric ¢2gs on S 3 has constant scalar curvature, then ¢ is constant.
I¢11
Proof. Since the Riemannian metric ¢2gs has constant scalar curvature, it follows by Proposition 1.1.6 that there exist a conformal transformation f E Conf(S 3) and a real number c > 0 such that ¢2g S = c2f*gs . According to the description of conformal transformations of (S n, Cs) given in Proposition 1.1.4, this means that there exist a vector b E ~3 and real numbers c > 0 and A > 0 such that
3.1 Description of the canonical metric by Kleinian groups
¢(~) =cA where a : S 3 \ {(0,i)} assumption, ¢ depends {(¢1,¢2) e $ 3 : ~i -- 0}, lows that b = 0 and A-:
I~(~)l~÷1
(3.1.7)
,
A2 la(~) - b[2 + 1
37
-~ ll~3 is the stereographic projection. Since, by only on I~11, the restrictions of ¢ to the subsets i -- 1,2, are constant. From this and (3.1.7), it fol1. Thus ¢ = c. []
E x a m p l e 3.1.5 We consider the lens spaces L(k,l) = S3/F(k,l). Here k and l are coprime integers and F(k, l) ~- Zk is that subgroup of the isometry group of (S 3, gs) which is generated by the transformation 7k,l: ('1,~2) E $3 ~ (exp ( ~ ) ¢ l , e x p
(2kl---~i) ~2) E $3 .
The canonical fiat conformal structure CF(k,l) of itive. By Theorem 3.1.2 and Example 2.2.8,
OtF(k,l)(~)
k-1 E I¢ m----1
L(k, l)
is clearly scalar pos-
m -1
Since
m 2 -- 4 it - ~/k,t¢l
(sin 2 (--~-)I~112 7rm
and
+ sin2
(~-~-) 1~212)
l~1[ 2 + ]~212 = 1,
the function ar(k,~) depends only on [~1[. On the other hand, aF(k,0 for e.g. k = 5 and l : 2 is not constant, as one easily sees from
at(5,2) = (a2 + (al a2)[~llU)-l/u _
+
with al=sin2(5
)
and
a2=sin2(~)
.
Thus, by Lemma 3.1.4, the scalar curvature of the Riemannian metric ~2(5,2)g S is not constant and hence so is the scalar curvature of the canonical metric can (Cr(5,2)) of L(5, 2). Recall that A s ¢ = ( 1 1 s2
-2s2)
~ss
1 - s2
2s2
-~s
for any function ¢ E C °~ (S 3) depending only on s : I 11, where As denotes the Laplacian with respect to gs- Applying this and (2.2.1), the scalar curvature of a~(kj)gs can be explicitly computed for any given coprime integers k and l. Doing this, one gets that the scalar curvature of the canonical metric can (Cr(k,O) for e.g. k = 7 and l = 2 is not everywhere positive. []
38
3. Kleinian groups and moduli spaces
3.2 Moduli
spaces
of flat conformal
structures
Let M be a connected and closed manifold of dimension n. Let Co(M) denote the space of flat conformal structures on M, and let C+(M) be the subspace of scalar positive structures in C0(M). On both spaces the diffeomorphism group Diff(M) of M acts via pulling back. The orbit spaces of these actions are Bo(M) := Co(M)/Diff(M) and B+(M) := C+(M)/Diff(M). We are now going to state some properties of Bo(M) and I3+(M). For n = 2, every conformal structure on M is flat. Therefore, in this case Bo(M) is the Riemann moduli space, and B+(M) # 0 if and only if M is the sphere S ~ or the real projective space ]ICJ~2 , where in both cases B + ( M ) contains only one element. For the further considerations let again n _> 3. As already mentioned in Section 1.1, then it can be happen that 13o(M) is empty. The next theorem due to Gromov and Lawson (see [25]) gives an obstruction to the existence of scalar positive conformal structures. T h e o r e m 3.2.1 Any closed manifold of dimension n >_ 3 which carries a Riemannian metric of non-positive (resp. negative) sectional curvature cannot carry a metric with positive (resp. non-negative) scalar curvature. In particular, this implies that I3+(M) is empty if M is a hyperbolic or Euclidean space form. According to the following result of Izeki (see [31]), in three dimensions the closed manifolds admitting a scalar positive, flat conformal structure are completely classified. T h e o r e m 3.2.2 Let M be an orientable, connected, closed manifold of dimension n = 3. Then M admits a scalar positive, fiat conformal structure if and only if M is diffeomorphic to a connected sum Mx~... ~Mm, where Mi for i = 1,... , m is either a spherical space form S 3 / F or S 1 × S 2. From Theorem 6.2 in [32] and Theorem 2.4.8, one deduces a similar result in four dimensions. T h e o r e m 3.2.3 Let M be an orientable, connected, closed manifold of dimension n = 4 and suppose that the fundamental group ZCl(M) of M is torsion free. Then M carries a scalar positive, fiat conformal structure if and only if M is diffeomorphic to the connected sum k (S 1 x S 3) of k copies of S 1 ×S 3forsomek>O.
3.2 Moduli spaces of flat conformal structures
39
If (M1, C1) and (M2, C2) are scalar positive, flat conformal manifolds of the same dimension, then, as it is well-known (see [38], cp. also Section 4.1), also the connected sum MI~M2 admits a scalar positive, flat conformal structure. Thus the essential statement of Theorems 3.2.2 and 3.2.3 is that there are no other manifolds with the specified property. Now we shall examine the space B + (M) for particular M. First we consider the case that M is a spherical space form, i.e. M = S n / F for a subgroup F C O(n + 1) acting freely on S '~. Let C be a flat conformal structure on M and let g E C. Then, by Theorem 1.1.9, there exists a diffeomorphism q5 of S n such that e Vs.
By the resolution of the Yamabe problem, we may assume that the scalar curvature of g is constant. Hence, by Proposition 1.1.6, there exist a transformation f e Conf(S n) and a real number c > 0 such that = c2/*gs,
that means = c2 (f o
as.
Thus, choosing q5 E Diff(S n) appropriately, we have ~r~g = ~*gs • Then (~oTo4~-l)*gs=gs
forall
vEF,
that means • F4 ~-t C O(n + 1). In particular, S n / ( ~ F ~ -1) is again a spherical space form, and ~ induces an isometry from (M, g) onto Sn/(q~Fq~ -1) with the Riemannian metric induced by gs. Since, by a result of de Rham (see [15]), spherical space forms which are diffeomorphic are also isometric with respect to the Riemannian metrics induced by gs, we have proven
Proposition 3.2.4 Let M be a spherical space form. Then 13+o(M) consists of exactly one element. Moreover, B+o(M) = Bo(M).
[]
Our next goal is to describe the space B+ (S 1 x $2). For this we shall use the following observation. Let TC+(M) denote the set of conjugacy classes IT'] of Kleinian groups F C Conf(S n) isomorphic to the fundamental group 7rl (M) of M with the property that Y2(F)/F is diffeomorphic to M and C r is scalar positive. Further, for C E Co(M), let [C] denote the Diff(M)-equivalence class of C. For [F]E TC+(M), we set ~P([F]):= [f~Cr] e B+(M) ,
40
3. Kleinian groups and moduli spaces
where f r is any diffeomorphism from M onto ~ ( F ) / F . This gives a welldefined map ~ : n + ( i ) -+ B+(M). Indeed, if 9(F1)/F1 and ~(/"2)/F2 are n-dimensional Kleinian manifolds and the groups F1 and/"2 are conjugate, i.e. 1"2 = TF1T -1 for some T E Conf(S n) , then the restriction of T to ~(/"1) induces a conformal diffeomorphism
f r : (n(rl)/rl, c a ) -~ (~(r2)/r2, cr:) P r o p o s i t i o n 3.2.5 The map ~P : TC+(M) -+ B+(M) is bijective. Proof. Let C1, C2 6 C0+(M). Then, by Theorem 2.4.5, there exist Kleinian groups /"i C Conf(Sn), i = 1,2, with Pi ~ 7rl(M) such that (M, Ci) is conformally diffeomorphic to (J?(Pi)//"i, Cr~ ). This shows that @ is surjective. To see that • is also injective, assume that
[cl] = [ c 2 ] , i.e. there exists a conformal diffeomorphism
f : (a(F1)/rl,Cn) -~ (a(r2)/r2,cr~). Let .f : ~2(F1) ~ 12(F2) be a lift of f. Then / is conformaI with respect to the conformal structure Cs. Consequently, by the Liouville theorem (Theorem 1.1.7), f is the restriction of a transformation T I E Conf(Sn). Moreover, since ] is a lift of f, we have Tf'/TflEF2
forall
7EF1.
Hence the groups I"1 and/"2 are conjugate. We now want to describe the set T~+ (S 1 × $2). Let F C Conf (S 3) be a Kleinian group isomorphic to rl (S 1 × $2), i.e. F ~ Z, and let 3'0 be a generator of F. Assume that f 2 ( F ) / F is diffeomorphic to S 1 × S 2. Then, as one easily sees (cp. e.g. [3], §1.5 or [42], §2), "r0 has to be loxodromic. Identifying S 3 with R 3 U {co) via stereographic projection, this means that 3'0 is conjugate to a conformal transformation of (S 3, Cs) given by x E ~ 3 ~ AAx E ~ 3
for a real number A > 0 with A ¢ 1 and a matrix A E 0(3). Since S 1 x S 2 is orientable, ")'o has to be orientation preserving. Hence A E SO(3). Since the matrices \/'c°s(0)O - sin(0) 0 ) AO := [sin(0) cos(0) 0 with O E (-Tr, Tr1 0 1
3.2 Moduli spaces of flat conformal structures
41
form a maximal torus of SO(3), it follows that 7o is conjugate to a transformation 3%0 E Conf (S a) defined by ")'x,o(x) := i~Aox
for
x e IRa ,
where A > 0 with A ~ 1 and/9 E (-Tr, 7r]. Further one verifies that Vxl,ol and Vx2,02 are conjugate if and only if (A2,02) agrees with (A1,81), (l/A1,81), (A1,-81), or (l/A1,-81). Thus the conjugacy classes of orientation preserving, loxodromic transformations of (S 3, Cs) are uniquely represented by 7x,e with A > 1 and 0 E [0, r]. Let F(A, 0) for A > 1 and O e (-Tr, 7r] denote the subgroup of Conf (S a) generated by the transformation ~/x,e. Then it is clear that A(F(A, 8)) = {0, c~} and hence YI(F(A, 0)) = IRa \ {0}, and one easily sees that I2(F(A, O))/F(A, O) is diffeomorphic to S 1 x S 2. Consider the cylinder metric gz on IR3 \ {0}. This Riemannian metric is given by 1
gz := ~-~gE • It is invariant under the action of F(A, 8) and hence induces a Riemannian metric g(,~, 0) on O(F()~, O))/F(A, 8) which represents the flat conformal structure Cr(x,0). Since the scalar curvature of gz is constantly 2 on IR3 \ {0}, the structure Cr(x,0) is scalar positive. Altogether, it follows that the elements of T~+ (S 1 x S 2) are uniquely represented by the Kleinian groups F(A, 0) with A > 1 and 0 e [0, 7r]. Combining this with Proposition 3.2.5, we get P r o p o s i t i o n 3.2.6 The moduli space B + (S 1 × S 2) can be parametrized by pairs ()~, O) of real numbers with )~ > 1 and O E [0, It], where the corresponding bijection ~0: (1, oc) x [0, ~r] ~ B + (S 1 X S 2)
is given by =
for any diffeomorphism fx,o : S 1 × S 2 ~ 12(F()~, O))/F(£, 8). R e m a r k 3.2.7 (i) Analogously, a parametrization of /30+ (S 1 x S n - l ) for n > 3 can be obtained. (ii) The above considerations and Theorem 2.4.5 imply that there do not exist scalar fiat, fiat conformal structures on S 1 x S n-1 for n >_ 3. Moreover, since ~rl (S 1 x S n - l ) = Z is abelian and hence amenable, it follows by a result of Schoen and Yau ([62], Proposition 1.2) that S 1 × S n-1 cannot carry a scalar negative, flat conformal structure. Thus, for n _> 3, the moduli space B0+ ( s 1 × s agrees with (S 1 × S n - 1 )
42
3. Kleinian groups and moduli spaces
(iii) Since S 1 x S 2 admits scalar positive, flat conformal structures, also the connected sum k (S 1 × S 2) of k copies of S 1 × S 2 with k > 2 possesses such structures. To compute the dimension of/30+ (k (S 1 x $2)), observe first that the fundamental group 7h (k (S 1 × $2)) of k (S 1 × S 2) is freely generated by k elements. Second, the loxodromic conformal transformations of (S 3, Cs) form an open subset of Conf(S 3) (cp. [42], (2.5)). Now, using that dim {V • Conf ($3): 77x,0V -1 = Vx,0} = 2 for A > 1 and 0 • (0, 7r) and dim Conf (S ~) = 10, one deduces that dim/3 + (k (S 1 × $ 2 ) ) = 1 0 ( k - 1)
for
k>2. 71
We shall not discuss the topological structure of the spaces Bo(M) and /3+ (M) for general M. This subject is studied e.g. in [22] and [35] (cp. also [19] and [37]). We want to point out, however, that/30(M) and/3+(M) are always finite dimensional spaces, but with singularities in general. In the rest of this section we shall explain how the canonical metrics can(C) yield a Riemannian metric on/3+(M). To illustrate the scheme of this construction, and also for later purposes, we shall consider a more general situation. In the context of Teichm/iller theory, i.e. for n = 2, such construction was carried out by Fischer and Tromba (see [17], [18]). Let M (M) denote the space of all Riemannian metrics on M and let 7r~c : M ( M ) ~ C(M) be the canonical projection onto the space C(M) of all conformal structures on M. Let C~(M) be a subspace of C(M) which is supposed to be invariant under the action of Diff(M). We set MS(M) := 7r~c (Cb(M)) . In what follows we assume that the spaces M ( M ) and C(M) as well as the groups Diff(M) and
C~(M):={~•C~(M):~(p)>O
for all p • M }
are completed with respect to appropriate Sobolev norms and, with respect to these topologies, the spaces 2k4~(M) and C~(M) are submanifolds of J~4(M) and C(M), respectively. Furthermore, we suppose that the canonical projection from 2k4~(M) onto
3.2 Moduli spaces of flat conformal structures
43
13b(M) := C~(M)/Diff(M) is a submersion. For the analytical details we refer to [16] and [17]. Let (hl,h2)g for hi,h2 • S2(M) and g • 2~4(M) denote the L2-product of hi and h2 associated to g. We recall that (hi, h2)g :=
/M g (hi, h2)
d#[g] ,
where the function g (hi, h2) on M is given by n
g (hi, h2) (p) := ~
hl(ei,
ej)h2(ei, ej)
i,j=l
for a basis { e l , . . . ,en} of TpM orthonormal with respect to g. Using that the tangent space Tg./~4(M) of A//(M) at a point g agrees with 82(M), a Riemannian metric ~1~ on the manifold M (M) is defined by g~(hl,h2) := (hi,h2)9
for
hi,h2 • TgJ~4(M) =
82(M).
One easily checks that gM is invariant under the action of Diff(M), i.e.
g~ (f'h1, f'h2)
= g ~ ( h l , h2)
hi, h2 e TgM(M). : C~(M) --+ .A4~(M) be a Diff(M)-equivariant
for all f E Diff(M) and all Now let s jection 7r~c, i.e.
7r~cos(C)=C
and
section of the pro-
~(f*C)=f*s(C)
C E O~(M) and all f e Diff(M). We identify the tangent space T[c]I3~(M) of t3~(M) at the Diff(M)-equivalence class [C] of C E C~(M) with the orthogonal complement of T~(c) (C~(M) . s(C)) + Tz(c)(Diff(M). a(C))
for all
with respect to the restriction of the L2-product associated to ~(C) to set
Tz(c).A4~(M) and
I}~(hl,h2) := g2vl(hl,h2)
for
hi,h2 E T[c]B~(M) C T~(c)M~(M)
•
Since s is Diff(M)-equivariant and gM is Diff(M)-invariant, the last formula gives a well-defined Riemannian metric I~ on B~(M). To describe T[c]B~(M) under the above identification in more details, we introduce the following notations. We denote the space of all 1-forms on M by E21(M). Let £:x denote the Lie derivative along X e 3C(M). For g E Jvl(M), let Trg : 82(M) ~ C°°(M) be the trace and let 5g : 82(M) ~ ~ I ( M ) be the divergence operator with respect to g. This means that
44
3. Kleinian groups and moduli spaces n
(Trgh) (p) := ~
h(ei,ei)
and
(~gh) (v) := - ~ ( V e i h ) ( e l , v )
i=1
i=1
for h • S2(M), p • M and v • TpM, where { e l , . . . ,en} is as above. We set 7-/(g) := {h • $ 2 ( M ) : Trgh = 0 and ~gh = 0} .
(3.2.1)
As it is well-known (cp. e.g. [13], Lemma 1.60),
F~xg = 2 ~ ; X b for X • • ( M ) ,
(3.2.2)
where fig : ~21 (M) -~ S 2 (M) is the formal adjoint of 8g with respect to g and X ~ • ~21 (M) is determined by
X~(Y) = g ( Z , Y )
for all
Y • X(M).
Since
Tg(Diff(M). g) = {£.xg : X • 3~(M)}, (3.2.2) implies that h • $2(M) is L2(g)-orthogonal (i.e. orthogonal with respect to the L2-product associated to g) to Ta(Diff(M ) • g) if and only if 5gh = 0. Moreover, since Tg (C~(M). g) = C°°(M) • g, it is clear that h is L2(g)-orthogonal to Tg (C~(M). g) if and only if Trgh = O. Thus 7/(g) is the L2(g)-orthogonal complement of Tg (C~(M). g) + Tg(Diff(M). g). Altogether, we obtain that
T[c]B~(M) =_H(s(C)) nT~(c).hd~(M)
for
C • C~(M) .
(3.2.3)
R e m a r k 3.2.8 If t ~ Ct is a smooth curve on £~(M), then under the identification (3.2.3),
d[C,]
{
dt = P~(c,) \ ~
] •
Here and in the sequel, Pg : $2(M) ~ 7-/(g) for g e A4(M) denotes the L 2 (g)-orthogonal projection onto 7-/(g). [] Applying the above considerations to the case that the section ~ is given by s(C) = can(C)
for
C e g0+(M),
we obtain a Riemannian metric on the moduli space B+o(M), which we simply denote by [. Of course, strictly speaking, [} is defined only on the regular part B +'* (M) of B + (M). However, the geodesic distance on B +'* (M) with respect to b extends to a distance on B+o(M).
3.3 The geometry of 13+o(S 1 x S~) 3.3 The
geometry
45
o f B o + ( S 1 × S 2)
In this section we shall study the Riemannian metric b on I3+(M) for the case that M -- S 1 × S 2, using Theorem 3.1.2 and the parametrization k~o: (1, c~) x [0, 7r] -+/30+ (S' x S 2) of B + (S 1 x S 2) = /~o (S 1 x S 2) presented in the preceding section (see Proposition 3.2.6). We shall prove T h e o r e m 3.3.1 For the Riemannian metric b on 13+ (S 1 × $2), we have U 1 (~, O) )~2 d A ® d A + u 2 ( A , O ) d O ® d O
~t~b with
1 ui(A,O) >_ const. 1A ' o4g-
on
(1,2) x [0,1r]
and
1 on ( 2 , ~ ) x [0,Tr] A3/2 logA for i = 1, 2, where const, means a positive constant independent of A and O. ui(A,0) <_ const,
The proof of Theorem 3.3.1 uses the following two lemmas. The first lemma will also be relevant to later considerations. L e m m a 3.3.2 Let M be a manifold of dimension n > 2, and let g and = c~2g with a E C ~ ( M ) be conformally equivalent Riemannian metrics on M . Then we have
( ~ (c~2h)) (X) = (6gh)(X) - n a - l h ( V a , X ) + a - a d a ( X ) T r g h
(3.3.1)
for h e S 2 (M) and X E •(M), where V a denotes the gradient of the function a with respect to g, 7-/(~) ----a ~-n. 7-/(g), (3.3.2) and (a2hl, a2h2)~ = (anhl, h2)g for ha, h2 E S 2 (M). Proof. The relation (3.3.1) can be checked by means of (X) = - Z
X)
i----1
= - ~ i=1
[ei(h(e~, X)) - h (Veiei, X ) - h(ei, VeiX)]
(3.3.3)
46
3. Kleinian groups and moduli spaces
for a local frame e l , . . . , en orthonormal with resepct to g and (see Proposition 1.1.1) (Tx Y = V x Y + o~-1 [da(X)Y + d a ( Y ) X - g(X, Y)Va] for X , Y E :~(M), where V and V are the Levi-Civita connections with respect to g and ~ = c~2g, respectively. To verify (3.3.2), let h E 7-/(g). One only has to show that then a2-nh E 74(.~). Obviously, a~ (~-"h)
= o .
Secondly, by (3.3.1),
(~ (~2-"h)) (x) = (6. (a-"h))(Z)- n~-"-lh(W, X) n
=nZ
e~(a)a-"-lh(ei' X ) + (Sgh)(X) - n a - " - l h ( V a , X)
i=l ~0.
Hence a2-nh e 7-/(.~), as desired. Finally, (3.3.3) is a consequence of {t (a2hl, a2h2) = g (hi, h2)
and
#[0] = an#[g] • []
As in the preceding section, let F(A, 0) for A > 1 and 0 E (-~r, lr] be the Kleinian group induced by 7x,0 --- AAo and let g(A,0) be the Riemannian metric on M(A, O) := (R3 \ {0}) IF(A, O) induced by the cylinder metric gz- The last means that r~,eg(A, 0) = gz, where ~rx,0 : Ra \ {0} --+ M(A, 8) is the canonical projection. By C(A, 8) we denote the canonical flat conformal structure on the Kleinian manifold M(A, 8), i.e. the conformal class of g(A, 8). L e m m a 3.3.3 For the canonical metric can(C(A, 0)) of C(A, 0), we have can(C(A, 0)) = a~,og(A , O) with
,~/21z I k=l
3.3 The geometry of B+o(S1 x S 2)
47
Proof. We again identify S a and R 3 U {oo} via the stereographic projection. By Theorem 3.1.2, we then have that can(C()~,/9)) = a 2~,og(,~, O) with ~,o
o ~,o(x)
(~L)' (x) ~/~ Gs (x,~,0x) ~(x)Ixl
=
(3.3.4)
k~Z\{0} for x E ]~3 \ {0}, where qos • C ~ (I~z) is determined by gs=qo~gE
on
~3.
From
('~,e)* gE = ~2kge
for
k
e
Z,
it follows that
(~L)' (x) =
~(x)
(3.3.5)
Moreover (cp. Example 2.2.8), 1
Gs(x,, x2) = ~s(x,)~s(x2) Ix1 - x21 for Xl, x2 E using that
1~3
(3.3.6)
with Xl ~ x2. Substituting (3.3.5) and (3.3.6) into (3.3.4) and
~-k/~lx I A~/21x I I x _ )~-kA;kxl -- I x _ XkA~xI ' we obtain that
oo
a),,a o 7rx,o(x) = 2 ~ Ak/2Iz[ k=l Ix - ~kao~xl " [] We are now ready to prove Theorem 3.3.1.
Proof of Theorem 3.3.1. We define the maps f~,i : ~3 \ {0} --~ ~3 \ {0} for s E ~ and i = 1,2 by
L,I(x)
:= IxlSx
and
•,2(x) :-- Asloglxix.
is,1 (AAox) =/~ , + i Aofs,l(X) ^
and
]8,2 ()~Aox) = )~Ao+slogX]s,2(x) ,
Since
48
3. Kleinian groups and moduli spaces
L,1 and •,2 induce diffeomorphisms fs,l: M(A, 0) --+ M (As+l, O) and fs,2: M(A, 8) -e M(A, 0 + s log A), respectively. We set h(1) d . g ()~s+l O) s=o A,O :---- -~8fs,1
and
h(2) d . I " A,O := --f~(~dsJ"2g'A'0+sl°gA'l $-~O For shortness, we write 0x, O0 for the canonical frame on B + (S 1 x S 2) with respect to the parametrization ~Po. Since Pca,(C(x,0)) ( d
.
(C (As+l
= Pca.(C(x,o))
-~s
,o o
,0)) .=o) g(A, O) + "x,o"x,o) s:O
{_2 ~,(1)'~
= Pcan(C(X,O)) \~,,O'°X,O) ,
we then have (cp. also Remark 3.2.8) 1
{ 2 ~(1)
1
{ 2 ~(2) Pcan(C(X,o)) (ax,o"A,o]
CO),()~,O) = A log----'~Pcan(C(X,e)) (°t),,o"),,o) •
(3.3.7)
Analogously, c00(A, 0) = ~
.
(3.3.8)
Let l, w 1, w 2 be the coordinates on ~3 \ {0} given by the parametrization
(t,W 1,w 2) e (0,(20) X (0,2") X (--Tr/2,Tr/Z) (e ~cos (w 1) cos (w2), e' sin (w 1) cos (w2), e' sin (w2)) E ]i~ \ {0}. In these coordinates, fs,, ( l , w ' , w 2)
=
( ( s + 1 ) l , w ' , w 2) ,
L,2 (t, w 1,
= (z,
+ st,
,
and
gz = dl ® dl + coQ (w 2) dw ~ ® dw I + dw 2 ® dw 2 . One easily computes that -~s d f ^.S,1 gz s=O = 2 dl ® dl
and
(3.3.9)
3.3 The geometry of B+o(S1 x S 2)
49
d ^.
/s,2gz s=o = c°s~ ( ~ ) (dl ® d~ 1 + dw 1 ® d0 = 2 cos2(w 2) dl ® dw 1 ,
where ® denotes the symmetric product. Consequently, zr* ;`(1) = 2 d I Q d l
and
~r* ;`(2) = 2 cos 2 (w 2) dl ® dw 1 )~,0,~,0
(3.3.10)
Now we want to prove that a2 ;.(2) ~,0,o~,0 £ 7-/(can(C(A,8))) .
(3.3.11)
By (3.3.10), Tr~(~,0)h(2,~ = 0 and hence [ 2 ;`(2)
Wrcan(C()~,8)) ~C~,8,~X,8 ] = 0 .
Thus for (3.3.11) it remains to show that
(a2 ;`(2)h
(3.3.12)
6can(C(h,0)) k X,O'"X,O] = O .
Let V z be the Levi-Civita connection and
~z: s 2 ( ~ \ {0}) -~ ~1 (R~ \ {0}) the divergence operator with respect to gz. From (3.3.9), one deduces VZdl = 0 and v z l dw 1 = 0, vz dwl_
e2
sin (w 2) dw 2
cos~ (w2)
'
VZ3dw I = tan (w 2) dw 1 with the gz-orthonormal frame
(el, e2, e3) =
(oN cos(w~) 1 oo) Owl' 0~
"
From this and 3
2gz ( d / ® d w ' ) = - E i=1 3
- E i:l
3
(vzi d/) (ei) dwl -- E
d/(¢i) VZidw 1
izl 3
(vzi dwl) ( e i ) d / - E i:1
dwl(¢i) vzi d / '
50
3. Kleinian groups and moduli spaces
one gets 6 z ( d / ® d w 1) = 0 . Because of tiz (cos 2 (w 2) dl ® dw 1) = cos 2 (w 2) 6z (d/® dw 1) 3
1 -2 Eei
(cos 2 (w2)) (d/(¢i) dw I +
dwl(ei) dl)
i=1
= cos ~ (w 2) 6z ( d / ® d w ' )
,
it follows that 5z (cos ~ (w 2) dl Q d w 1) = 0 and hence
%(x,0)'~,0 = O.
Now, using Lemma 3.3.2(i) and the fact that, by Lemma 3.3.3, aa,o o ~ra,0 only depends on w 2, one derives (3.3.12). The relations (3.3.8) and (3.3.11) imply
0o(~,0) = c~'° ~(2)
(3.3.13)
1-~g ~ '~)~,O "
Applying Lemma 3.3.2(iii), we deduce from (3.3.7), (3.3.10), (3.3.11), and (3.3.13) that
(oh(x, o), o0(x, 0)) _
2
(1)
^2
~(2)~
A log 2 A _
-
-
1
{~2
~(1)_2
~(2)~
log 2 A I,~'x'°'°x'°' ~'~,a'~,o) ca,(c(~,o)) 1 {~3 1.(1) 1.(2) ), log 2 A ~'-'x,o'o~,,o, '°~,o)go,,e )
4
)~ log 2 A
f~3
(~,,o)
&3~,o cos 2 ( ~ 2 )
gz
(dl®dl, dl®dw 1) d#[gz]
,
where
M(A,O):=
{ x e R 3 : 1 < Ixl <
~}
and (~A,O :---- 00,,0 o 71-A,0 .
Together with gz (dl ® dl, dl ® dw 1) = O, this yields (0h, 00) = o .
(3.3.14)
3.3 The geometry of
B+(S1 x S 2)
51
Analogously, the expression D(0e, 00) can be computed. To do this, observe first that 1 gz (d/® dw 1, dl ® dw I ) =
2 cos 2 (w 2) "
Second, recall that d#[gz] can be written as cos (w 2) dl A dw 1 A dw 2 . Then deduce that 0(0e(A,O),0e(A,8))-
1 / a .(2) .(2)~ log2 A
(,ax'onx'e' nx'°)g(x,e)
=
^3
log A j _ . / 2 ax'° c°s3 (w2) dw2 '
(3.3.15)
using once more that &x,o depends only on w 2. To handle the expression b (0x,0x) is somewhat more complicated. We see that 1 [~2 ~(1)~2 D(0),(A, 8), 0A(A, 8)) <_ A2 l°g 2 A ~*A,e'%x,e,
~(1)~
"A,e'%X,e)can(C(,X,O) )
__
_
1 [a3 ~(1) h 0)~ A2 log2A ~, ~,o"~,0, x,e)9(~,o ) s.
["/=~l,ocos(~2) d~2.
A2 log A s - . / 2
To derive an estimate from below, we consider
h(3) ~,e :=a-1 ~,,e \(h(1) ~,e- ~ g(~,o) )
.
As one easily sees, 2
,o~,oh(1)_ ~g(~, 0) ~ u(g(~, e)) By Lemma 3.3.2(ii), it follows that h(3) x,e E ~/(can(Ca,e)) . Then
(o~(~,e),o~(~,e))
h(2, ) = b t,.,,o,,.,,o]
and
(3.3.16)
52
3. Kleinian groups and moduli spaces
1
{'~2 #..(1) ~--1 (h(1)
i
g(X,o)
AlogA ~,'°x'°''°'x'e -
32rr 3A" Together with
~-a (h(1)
(h(3) h(3)~
b V-x,e,.o~,U =
~,0 l. x,e -
16.10gA
_
3
g(A,0)
f.12
#1(1)
,'ox,0 -
~g(,0) g(x,o)
a ; J c°s (~=) d ~ = ,
J-.12
this implies
t/ rrrl2
64
[/ b (0x(A,O),0x(A,0)) _> 3),2 logA \s-,12
cos (w=) dw
)--1 (3.3.17)
According to (3.3.14), (3.3.15) (3.3.16), and (3.3.17), to conclude the proof, it remains to estimate &~,0. By means of Lemma 3.3.3, we deduce that ~,0(x)
< ~.o(x)
: 2~
k=l
~
A~/2 :1 "
Hence, for A >_ 2, we have
)~-1/2
CO
&x,0(x) _< 4~--~ ~-k/2 = 4 I - A-I/2 -< const. A-1/2 .
(3.3.18)
k----I On the other hand,
Ak/2
~,o(x) >_a~,,~(x) >_ 2 ~_, I + ~k
k=l
and hence
~'9(X) ~---Z )~-k/2 : /~1/2 __ 1 k=l Since
log A lim ~1-72-- - 2 x~l 1 '
it follows that 1 &~,0(x) > const. 1-~gA for This proves the theorem.
1 < A < 2.
(3.3.19) []
3.3 The geometry of B+(S 1 x S 2)
53
C o r o l l a r y 3.3.4 The metric completion B + (S 1 x S 2) of 13+ (S 1 x S 2) with respect to the geodesic distance of b differs from ]3+ (S 1 x S 2) by exactly one point.
Proof. We consider the following curves on B0+ (S 1 x $2). We set KO,1 := k9o((1,2) X {8})
and
Ko,2 := kVo((2, oo) x {O})
for 8 6 [0, ~r] and g ~ := k0o({A} x [0, zr]) for A > 1. Let £~(K0,1), ~(K0,2), and £~(K~) denote the lengths of these curves with respect to b. Prom Theorem 3.3.1, we obtain that ~0(K0,1) = c~
and
~ ( K 0 , 2 ) < oo
for all 0 6 [0, zr] as well as lim £~(Kx) = c~
)~-~1
and
lim £,~(Kx) = O.
)~--+oo
Since in addition the canonical frame associated to ~Po is D-orthogonal, the unique point in the difference of B + (S 1 x S 2) and Bo+ (S 1 x S 2) is represented by every sequence (~O(~k,Ok))k6Nwith Ak --~ oo. [] R e m a r k 3.3.5 (i) By construction of the parametrization kVo, the unique point in the difference of B + (S 1 × S 2) and ~+ (S 1 x S 2) corresponds to the fiat conformal manifold (S3,Cs) endowed with the two punctures 0, co E $3 _-- 1~3 U {oo}. On the other hand, (Cs,0, oo) represents the unique point in the moduli space Bo,2 (Sn), n _> 2, of two-punctured fiat conformal structures on S n. (To check that Bo,2 (S n) contains only one element, use 13o (S n) = {[Cs]} and the fact that, given two pairs (~1,~2),(~[,~) E (S n x S n) \ diag(Sn), there exists a transformation f 6 Conf(S '~) such that f(~l) ~- ~ and f(~2) --- ~.) (ii) The estimates (3.3.18) and (3.3.19) imply lim vol(M(A, 0), can(C(A, 0)) = 0
X--+oo
and lim vol(U()~, 0), can(C(A, 0))) = e c , A--~I
where vol(M(A, 0), can(C(A, 8))) again denotes the volume of M(A, O) with respect to can(C()~,0)). []
4. Asymptotics:
The
flat
case
In this chapter we discuss asymptotics of the canonical metric can(C) near boundary components of the space of scalar positive, flat conformai structures and deduce conclusions for the geometry of the corresponding moduli space. In the first section we study the behavior of the canonical metric under surgery type degenerations. We shall see that the canonical metrics converge to the canonical metric on the limit space. To show this we derive estimates of the Green function of the conformal Laplacian. In the second section we prove that, for any sequence of scalar positive, flat conformal structures Ck which converges to a scalar flat conformal structure C, after an appropriate normalization, the canonical metrics can(Ck) converge to the unique Riemannian metric in C with vanishing scalar curvature and unit volume. As a consequence for the L2-geometry of the moduli space of scalar positive, flat conformal structures, we obtain in Section 4.3 that the degenerated fiat conformal structures of the type considered in Section 4.1 are contained in the metric completion of the moduli space. Thus this moduli space is not complete. On the other hand, we show that, in dimensions 3 and 4, any boundary point of the moduli space which is represented by a scalar flat structure has infinite distance from inner points.
4.1 Degenerating
fiat conformal
structures
In this section we shall examine the behavior of the canonical metric can(C) for degenerating C. We start with describing the type of degenerations which we want to consider. Let M0 be a connected and closed manifold of dimension n with two punctures P l , p 2 or the disjoint union of two connected closed n-dimensional manifolds
M(1),M(2) with one puncture P i e M(i) each, where again n >_ 3, and let Co be a scalar positive, flat conformal structure on M0. Here a conformal structure on a disconnected closed manifold is said to be scalar positive if each of its restrictions to connected components is scalar positive. Let disjoint neighborhoods U1, U2 of the punctures Pl, P2 and local coordinates x1
. xn
Bn(2) ,
i = 1,2,
56
4. Asymptotics: The flat case
be chosen such that xi(pi) = 0 and
gE,i := XigE
n = j=l E
is contained in the restriction of Co to Ui. Here and in the sequel, we use the notation B"(6) := {x e S" : Ixl < 6} for 6 > 0. For 0 < s < t < 2 and i = 1,2, we set Ui(s,t) := {p e Ui : s < Ix~(P)I < t)
and
V~(t) := {p e Us : Ix~(p)l < t } .
We form an n-dimensional manifold Mr,A for t E (0, 1) and A E SO(n) by removing the closed balls Ui(t), i = 1,2, from M0 and identifying Ul(t, 1) and U2(t, 1) along the map ~t,A : Ul(t, 1) --+ Us(t, 1) given by t x2 o ~ , A ( P )
-- ixl(P)l 2
AXl(p).
R e m a r k 4.1.1 An alternative way to construct Mt,A is to remove the open bails Ui (vfi), i = I, 2, from Mo and to indentify {p E UI: lxl(p)l = V~} with {p E Us: Ix2(p)l = v~} along ~,A- From this one realizes that, as t tends to O, M~,A degenerates to that manifold with a node which arises from Mo by identifying the punctures pl and P2. Clearly, this manifold with a node can be understood as Mo together with the punctures Pl, P2. [7 Obviously, the manifolds Ut,A are connected, closed, and diffeomorphic to each other. Let
7rt,A :Mo \ (Ui(~)(.JU2(t)) ~ Mt,A be the canonical projection. Since Ot,A is conformal with respect to Co, there is a unique conformal structure on Mt,A whose pull-back along ~rt,A agrees with the restriction of Co to Mo \ (Ul(t)U U---~)." We denote this conformal structure, which is clearly fiat, by Ct,A. According to the remark above, we can think of the pairs (Mt,A,Ct,A) for t E (0, 1) und A E SO(n) as a family of fiat conformal manifolds which degenerates to the fiat conformal manifold (Mo, Co) with the punctures Pl and P2 as t --} 0. %
By a result of O. Kobayashi [38] (cf. also [21]), the proof of which rests on estimates of the Yamabe invariant, Ct,A is scalar positive at least for small t. For our purposes, it will be useful to derive this statement in a more explicit manner. To do so, we first prove the following lemma. It says that we may
4.1 Degenerating flat conformal structures
57
locally interpolate between any locally conformally fiat Riemannian metric with positive scalar curvature and a cylindrical one by a metric of positive scalar curvature within the given conformal class. We note that, in contrast to this, in the interpolation lemma of Gromov and Lawson [24] the conformal class may change. L e m m a 4.1.2 Let ~ be a positive function on the open ball B n = B'~(1) with n > 3 and suppose that the Riemannian metric W4/(n-2)gE on B n has positive scalar curvature. Then, for each Co E (0, 1), there exist real numbers a > 0 and Cl E (0,eo) and a positive function ¢ defined on B n \ {0) such that
(i) (iO
¢ ( ~ ) -- a ~ ( ~ ) ¢(~) = I~1(2-~)/:
for
co < I~1 < 1. for
0 < Ixl < c~.
(iii) The Riemannian metric ¢4/(n-2)gE on B n \ {0) has positive scalar curvature. Proof. Let 771 and ~2 be non-negative smooth functions on the open interval
(0,1) such that ~l(t) = 1 for t < c2, ~?l(t) = 0 for t > Co, ~2(t) = 0 for t < cl, and y2(t) = 1 for t >_ c2, where cl and c2 are real numbers with 0 < Cl < c2 < Co < 1. We define a positive smooth function ¢ on B n \ {0} by setting ¢(x) := ~l(IXl)Ixl( 2-~)/2 + a~2(lxl)~(x) (4.1.1) with a positive real number a. Then ¢ satisfies (i) and (ii) by definition. We fix e0, c2, and ~?1 and show that we can choose a, cl, and 712 such that (iii) is also valid, which is equivalent to AE¢>0
on
B n \ {O)
by (2.2.1). Equation (4.1.1) and the assumptions on 72 imply that
a ~ ¢ = aE (nl(lxl)lxl (~-n)/~) + a a E w
on B n \ B=(c=).
Moreover, since ~4(n-2)gE has positive scalar curvature, we have AE~>0
on
B n.
(4.1.2)
Thus we can choose a > 0 such that AE¢ > 0 on B '~ \ B'~(c2). Now recall that
~E-
02
or2
n-1
r
0
Or +
r~
~s
on ~ ' \ { 0 )
58
4. Asymptotics: The flat case
with r = Ixl, where here As is the Laplacian of (S n-l, gs). Putting I = log r, we obtain that
o
A E = - - e -2t
~+(n--2)~-~--As
)
on
IRn \ { 0 } .
We use this to derive from (4.1.2) and ¢ ( x ) = e ~2-")~/2 + a~2(Z)~(x)
for
0 < Ixl < ¢2
with fl2(/) := 72 (et) that AE¢ _> a e_21 ((2-4an)2 e(2-n)l/202~2012
~o- 2 Ofl201O~Ol (n - 2) --~0~/2~ )
on Bn(¢2) \ {0}. Thus, since the functions ~ and
Ol
Or
are bounded on Bn(¢2), we can choose ¢1 and 72 such that AE¢ > 0 on Bn(¢2) \ {0}. This completes the proof. D Let go be a Riemannian metric in the conformal class Co with scalar curvature So > 0 and let M~ :-- Mo \ {pl,P2} • By Lemma 4.1.2, we may assume without loss of generality that there exists a positive function ~Oo E C °o (M~) such that the Pdemannian metric g~ := ~04/(n - 2 ) ^ o uo on M~ has positive scalar curvature and 1
g~o=~-~gE,i on U i ( 0 , 3 / 2 ) for i = 1,2. The last means that the Pdemannian manifold (M~,g~) has cylindrical ends. It is easy to see that the gluing map ~Pt,A is isometric with respect to g~. Hence g~ induces a Riemannian metric gt,A E Ct,A for t E (0, 1) and A E SO(n) by the rule that on
(4.1.3)
Uo\
Let S~ (resp. St,A) denote the scalar curvature ofg~ (resp. gt,A). Since S~ > 0 and [xi]-2gE,i f o r i ~ 1, 2 is a Riemannian metric of constant scalar curvature, there exists a constant Co > 0 such that S~(p)>Co By (4.1.3), it follows that
forall
peM~.
4.1 Degenerating flat conformal structures inf S t , A > c o
M't, A
for all
rE(0,1)
and
AESO(n).
59 (4.1.4)
For the further considerations we introduce the following notations. For t E (0, 1), we set Mt := Mt,~ ,
gt := gt,1,
St := St,1,
Ct :=Ct,1,
and
rt:=~rt,1,
where 1 is the unit element of SO(n). We denote the conformal Laplacian of (M~, gt) for 0 < t < 1 by L~ and its Green function by G~. Furthermore, let G~ be that Green function of the conformal Laplacian L~ of (M~,g~) which is given by
1
G~o(p,q) - ~o(p)~o(q) Go(p,q)
for p, q E M~
with
p ¢ q
(cp. Lemma 2.2.7). R e m a r k 4.1.3 In the case that M0 is the disjoint union of two connected manifolds M(1) and M(2), let G(i) for i = 1,2 denote the Green function of the conformal Laplacian on M(i) with respect to go. Then G0(p, q) = G(i)(p, q), if p, q E M(i), and Go(p, q) = 0, ifp and q are contained in different components of Mo. [] We now want to study the asymptotic behavior of the Green functions Gt as t tends to 0. To do this, for 0 < s < t < 1, we embed
into Ms along the restriction of ~r, to M~. Then a set U CC M~ is also a subset of M~ for small t. We start with
Proposition 4.1.4 For any domain U CC M~ and m = 0, 1, 2,..., Gt - Go -* O in
c m ( U x U)
as t -~ O. Proof. Let U CC M~ be arbitrary. We choose domains Vi, i = 1,2,3, with smooth boundary 0Vi such that
u c c Yl c c v2 c c ½ c c M~. Let GD be the Green function of the Dirichlet problem of L~ on ½, i.e. that continuous function GD : (~33x ~33) \ diag (V3) --4 ]R which satisfies
60
4. Asymptotics: The fiat case
fvsGD (p, q)Ltou(q) d#[g~] (q)
=
fn u
(p)
for all
u E C ~ (V3)
and GD = 0
on
0 (V3 x V3) = (Va x
Let to C (0, 1) be so small that V3 C embedded into Mr. We set
F[(p,q)
:=
Gt(p,q) - G~o(p,q)
Mto,
OVa) U (OV3 x
(4.1.5)
and let t E (0, to). Then V3 is also
Ft(p,q)
and
V3) .
:=
Gt(p,q) - GD(p,q)
for p, q E V3. Since gt = g ~
and hence
Lt =L~o
on
V3,
similarly to the proof of Lemma 2.3.1, elliptic regularity (Corollary 1.2.3) implies that Ft~ and Ft are smooth functions on V3 x V3. Moreover, (L;,(,) + L;,(,))
Ft(p,q) = O,
(4.1.6)
i ) and L~o,(q)again mean the actions of L~ with respect to p and q, w h ere Lo,(p respectively. Since the scalar curvature So of go is positive, for the operator L~o,(p) q- L~o,(q) the strong maximum principle (Therorem 1.2.5) holds. Thus from (4.1.5), (4.1.6), and Gt > 0 (see Proposition 2.2.9), it follows that Ft > 0.
(4.1.7)
By the Harnack inequality (Theorem 1.2.4), we get from (4.1.6) and (4.1.7) that there exists a cl > 0 such that sup F ~ < c l inf Ft -- V2xV2
V2xV2
for all
tE(0,to).
(4.1.8)
Using (2.2.3) with u - 1 and (4.1.4), we deduce that Cn
= f _ Gt(p,q)St(q)d#[g~](q) JM
t
>_co/i_ Gt(p,q) JM
d#[g~](q)
(4.1.9)
t
for all q E Mt and t E (0, 1). Again using that G~ > 0, it follows that there exists a constant c2 > 0 such that
v2Ft(p, q) d#[g~](q) for all q E V2 and t E (0, to). Consequently,
< c2
4.1 Degenerating flat conformal structures inf Ft <
C2
for all
vol(V2,g )
61
t • (O, to).
(4.1.10)
t • (0,t0) •
(4.1.11)
Combining (4.1.8) and (4.1.10), we obtain sup Ft < ClC2 v2×y2 vol (Y2, g~)
for all
The inequalities (4.1.7) and (4.1.11) imply that {Ft : t • (0, to)} is bounded in L 2 (V2 x V2). Then, by (4.1.6) and Theorem 1.2.2, this set is also bounded in W m,2 (V1 x Vl) for each integer m > 0. Since the boundary cOV1 of V1 is smooth, we can apply Theorem 1.2.1 to derive that, for each sequence (tk) of real numbers tk • (0, to) with tk --+ 0 and for each integer m _> 0, there exists a subsequence (tk,) of (tk) such that (Ft~,) converges in C m (V1 x V1). This means that (Ft) and hence (F[) converge in C m (V1 x VI) for m = 0 , 1 , 2 , . . . as t -+ 0. Thus it remains to show that
F~ :-- t---~0 lim F [ is identically zero. Since U CC M~ is chosen arbitrarily, F~ can be extended to a smooth function on M~ x M~ satisfying L~o,(p)F~(p, q) = 0. (4.1.12) Let 0 < a < 1. We fix a point q • Mo \ ( U - - ~ U U2(a)) and set
Gq,t(p) := Gt(p,q)
for p • Mt
and
t • (0,a 2)
and F~,o (p) :-- F~ (p, q)
for
p•M~.
We have to show that F~,o = 0. First we show that Fq,o is bounded. Since g~ agrees with the cylinder metric on Ui( 0, 3/2 ) and
LtGq,t = 0
on
Mt\{q},
the Harnack inequality gives that there exists some constant c3 > 0 such that sup
,,(U~(s,sla))
Gg,t < Cs -
for all t E (0,a3), s E [tin, a2], and i :
vol
inf
~,(Vi(s,8/a))
Gq,t
1,2. Moreover,
82)), g,) = vol = w,-1 (log sa - log 81)
for 0 < sl < s2 < 1. This and (4.1.9) yield that
62
4. Asymptotics: The fiat case
1 ~ aq,t d#[gt] inf G. t < .,(Vi(s,s/a)) ~' wn-i I logal dU~(s,s/a)) -
Cn
cown-i I logal "
Consequently,
Gq,t <_
sup lrt ( U , ( t / a , a ) )
C3 Cn
co w~-t I logal
for all t • (0, aa). Since in addition a~(p, q) --> 0
for p -+ Pi,
it follows that Fq,o • C ~ (M~) is bounded. To see that this function is even identically zero, we proceed as follows. As a consequence of Lemma 2.2.1 and (4.1.12), Lo (qooF~,o) = 0
on
U~.
Moreover, since Fq,o is bounded and ¢Po = 0 (Ixil (2-'~)12) k
as
p - + Pi ,
/
we have that
~OoF~,o • L s (Mo) if i _< s < 2 n / ( n - 2 ) . Then, by Theorem 1.2.6, it follows that Lo (~ooF~,o) = 0 on Mo in the weak sense. In particular, by elliptic regularity, ~ooFq,o extends to a smooth function on Mo. Since Lo : C°°(Mo) -~ C~(Mo) is invertible, we arrive at ~a0Fq,o -- 0 and hence F~,0 = 0. This concludes the proof. [] The following lemma says that, using the embeddings of Mg into Ms for 0 < s _< t to compare the Green functions Gt with each other, the family (Gt) is monotonically increasing in t.
L e m m a 4.1.5 For all p, q • M~ with p # q,
dGt (p, q) >_O . Proof. We fix t E (0,1), p e M~, and ut • C c¢ (Mr). For 0 < s < t, let Us • C ¢¢ (Ms) be such that Us=Ut Then, because of gs = gt on Mg,
on
Mr.
4.1 Degenerating flat conformal structures
63
¢n ut(P) = fM Gs(p, q)Lsus(q) d#[gs](q) 6
-~ fM Gs(p, q)Ltut(q) dp[gt](q) 2
(4.1.13)
+~ fu,(~,vq) Gs(p, q)Lsus(q) d#[gs](q)
for 0 < s < t. Let
S~(r) :=
(q ~ Ui: Ix~(q)l - r} ,
and let ~i,~ : Si(r) -~ S ~-'
be defined by
~,~ (p) := ~i(p) r Then
fu~(v~,vq) Gs(p, q)Lsus(q) d#[g~](q) = f(l°gt)/2 ~S
~*.e' ] ( q ) d l . J(logs)/2 i(ez)Gs(p,q)Lsus(q) d# [.[gsj__
Identifying (4.1.14) with S n - l , it follows that
d fv~ ( ,z, ,z) Gs(p,q)Lsus(q) d#[gs](q) 2t
,(vq)
= __1 2t fg , Gt(p,q)Ltut(q) d#[gs](q) Thus, differentiating (4.1.13) in s at s = t, we obtain
fM --~-(p, dGt q)ntut(q) d#[gt](q) = "~ l fK Gt(p, q)ntut(q) d#[gs](q) . Hence, since
Lt : C °O(Mr) ~ Coo (Mr) is surjective,
fM, ~-~(p,q)u(q) d#[gt](q) = ~ fKt Gt(p,q)u(q)d#[gs](q) for
all u E Coo (Mr). Because
of
Gt >
0, this gives the assertion.
As a consequence of Proposition 4.1.4 and Lemma 4.1.5, we get
O
64
4. Asymptotics: The flat case
C o r o l l a r y 4.1.6 For all t E (0, 1) and all p, q E M~ with p ~ q,
C~(p, q) > c'o(p, q) . [] Next, we want to derive uniform estimates from above for the Green functions Gt near the curves Kt defined in (4.1.14). For this we need some preparations. Let
Z"(b, ~):= {x e ~": b-l~ < Ixl < b~} and S "-1(~) := {x e ~" : Ixl = ~} for b > 1 and A > 0. Let Gz : (l~n \ {0}) × (~n \ {0}) ~ ]~U {co} be given by .--2
az(~,y) := \ I~- yl ] We define Gz,b: (I~" \ {0}) x (~" \ {0}) --~ I~ U {co} for b > 1 by
Vz,b(~,~) := ~ Vz (x, b2ky) kEZ
and Gx,z,b : h a \ {0} --+ R U (co} for fixed x e ]~n \ {0} by
Gx,z,b(y) := Gz,b(x,y) • By Lemma 2.2.7 and Example 2.2.8, Gz is a Green function of the conformal Laplacian Lz of li(" \ {0} endowed with the cylinder metric gz := ]x]-29E • By means of Lemma 3.1.1, one verifies that Gz,b is the pull-back of the Green function of the conformal Laplacian of the Riemannian manifold (M (b2, 0), g (b2, 0)) introduced in Section 3.3. As one easily sees,
az
~, i - ~ y ) = az(~,y)
for all x,y e e" \ {0}.
With that one checks that Gx,z,b is invariant under the reflections
b2k Ixl2 y • ~" \ {o} ~ -F~]Ty e ~" \ {o} on S "-~ (bklxl), k e Z. I n p a r t i c u l a r , w i t h r = IYl,
OG
Orr x,Z,b = 0
on
oZn(b, lx]) = S n-1 (blxl) u S "-x
(b-~lxl)
Thus the restriction of Gx,z,b to Zn(b, Ix]) is the Green function with singularity at x of the Neumann problem of Lz on Zn(b, Ix[).
4.1 Degenerating flat conformal structures
65
Now, for ~ > 0, we set
mb,+(A) := sup {G~,z,b(y) : Y • S"-I(Alxl)} and mb,-(A) := inf {Gx,z,b(y) : y e Sn-I(Alxl)} • Since G z ( f ( x ) , f ( y ) ) = Gz(x, y) for all isometries f of (]I(n \ {0}, gz), which are given by f ( x ) = ~Ax with )~ > 0 and A E O(n), the numbers mb,+(,k) and mb,-(A) do not depend on x E ]Rn \ {0}. Moreover, since Gx,z,b is invariant under the reflection on Sn-I(Ixl), we have mb,+ (A -1) ----mb,:l: (~) . (4.1.15) L e m m a 4.1.7 For each c > 1, there exists a bo > 1 such that
mb,+(b) <_ mb,_(cb)
for all
b >_ bo .
Proof. From
n-2
it follows that
rob,+ (~) = Gx,z,b (~x)
n--2
=Z 1-b2k ) kEZ and
mb,-()~) ----Gx,z,b(--}kx) n-2
Hence
mb,+(b) ----2 k----0E~kf ----b-'~TI ] and
rob,- ( cb ) = k=oE
1 + cb2k+1
( clbk l
"}" i 7 c -1 b2k----~1]
J
66
4. Asymptotics: The flat case
One concludes the proof by verifying that, for each c > 1, there exists a b0 > 1 such that
1--~-~]
f
n--2
n--2
+ ~,l+c-'s) []
for all s > b0.
The following lemma compares the Green function of the conformal Laplacian on a closed manifold with the Green function of the Neumann problem on an embedded cylinder. L e r n m a 4.1.8 Let G be the Green function of the conformal Laplacian of a
closed Riemannian manifold (M, g) of dimension n > 3 with positive scalar curvature. Let the cylinder (Zn(cb,)~), gz) with c, b > 1 and )~ > 0 be isometrically embedded into (M,G) and suppose that mb,+(b) < mb,_(cb). Then
G(z, y) < Gx,Z,b(y) for each x 6 Sn-I(A) C Zn(cb,)~) and for all y 6 Zn(cb, A). Proof. We fix x 6 Sn-I(A), set Gx(y) := G(x,y), and assume that min ( G x , z b - Gx) < 0 . z~(cb,X) ' -Since g is supposed to have positive scalar curvature and
L(Gz,z,b- Gz) = 0 on
Zn(cb,$),
by the maximum principle this minimum is attained at a boundary point Yo of Zn(cb, $). Because of (4.1.15) and mb,+(b) <_mb,_(cb), we can deduce that
ax(yo) > Gx,z,b(yo) - G~,z,b(y) + V~(y) >_mb,_(cb) - mb,+(b) + Gz(y) >_ vx(y) for each y in the boundary of Zn(b, )~). Therefore, Gx E C°°(M \ {x}) has a maximum in the open set M \ Zn(b, )~). Since LGx = 0, this contradicts the maximum principle. [] We are now able to estimate the functions Gt in the following way.
Proposition 4.1.9 There exists a &o E (0, 1) such that, for i = 1, 2, all t e (0,&0), and all p,q e U~ satisfying v/t < Ixi(p)l < v~o and Ixi(p)l 2 _< Ixi(q)l <_ 1, we have
v~(~(p),~(q)) < Gz,b(xi(p),x~(q)) with b = Ixi(p)1-1
4.1 Degenerating flat conformal structures
Proof. We define f~ : Z" ( 3 / ( 2 ~ ) , v q ) ~ i ~ by f~(x) t < Ixl <_ 3/2, where f~(x) • Ul(t,2) is given by X1 0 f : ( X )
67
:= ~ o f:(x), if
= X,
and ]t(x) := r t o f2(x), if 2t/3 _< Ix I < 1, where f2(x) • U2(t, 2) is given by X2
of2(x)=~X.
By construction of (M~, gt), the map ft is an embedding and
flgt = gz • Let 5o • (0,1) and p • [/1 such that V~ _< ]xl(p)] <_ v~o. Then, with b = ]Xl(p)] -1 and A = Ixl(p)], the cylinder Zn(3b/2, A) is embedded into Mt along ft and
f,(Z"(b,A)) = ~, (U1 (Ixl(p)l~, 1)) Thus, setting 50 = bo 2, where bo is chosen according to Lemma 4.1.7 with c = 3/2, and applying Lemma 4.1.8, we obtain the assertion for i = 1. For i = 2, one proceeds completely analogously.
[]
Now we turn to the description of the asymptotic behavior of the canonical metrics can(Ct). To do this, let at • C ~ (Mr) for 0 g t < 1 be determined
by can(C,) = a~gt and ai,o • C ~ (U~) for i --- 1, 2 by can(C0) = a is, O g E , i P r o p o s i t i o n 4.1.10
•
(i) For any domain U CC M~ and m = O, 1, 2,...,
can(Ct) -+ can(C0)
in
C m (U, S2T*M;)
as
t-+ O .
(ii) There exists a constant d such that
ai,o _< ~at < c'
on
ui (v/t, 1)
]or all t • (0,1) and i = 1, 2. Proof. (i) Let U CC M~. We have to show that, for each positive integer m~_0, a t - ~ a o in C re(U) as t--+O.
68
4. Asymptotics: The flat case
We choose ~ E C~(Mo) such that the Riemannian metric ~p4/(n-2)go is flat on a neigborhood U ~ C U and denote by disto,~ the geodesic distance with respect t o ~4/(n-2)g O. Let p E U ~. Then, by the definition of the canonical metric (cp. also Remark 3.1.3(ii)) and Lemma 2.2.7,
(
ao(p) = ~2/(n-2)(p) lim q-~v ~(p
~(q) GO(p,q) -- disto2~n(p, q) ) l/(n--2)
= ~2/(n-2)(p) q-~p lim (~0(P)~0(q) \ ~
, G°(p'q) -
1/(n--2)
dist~'~(p,q) )
and, for sufficiently small t > 0, ' a~(p) = ~2/(n-2)(p) ]imp (~ ~0~ (P)~/:~0(q) G t(p,q) -dist2~(p,q))\
1/(n--2)
It follows that
q--+p(Gt(p, q) - G~o(p,q)) ,
a~-2(p) _ a~-2(p) -_ ~2(p) lim and Proposition 4.1.4 yields the assertion.
(ii) Let i = 1, 2 and let p E Ui (v~, 1). Since ~,~ = I~l~g0
on
U~ \ {p~},
we have ai,o (P) = lim (Ixi (p) l(2- n)/2 ix i (q) 1(2-n)/2 G~ (p, q)
q--~p
-]xi(p) - zi(q)l 2-n) and, embedding Ui (v/t, 1) into at(p) = Ix~(p), ~im -Ixi(p) -
Mt
1/(n-2)
along ~rt,
(Ix,(p)l(=-'~)/~lx~(q)l(2-~)/2et(p,q) xi(q)[ ~-") 1/(,~-u) .
Hence, by Corollary 4.1.6, ai,o (P)
< at (p)
-Ix~(p)l "
To derive the second part of the assertion, set b = ]xi(p)1-1. Then, by Proposition 4.1.9,
4.1 Degenerating flat conformal structures
~(p)
< lim
- -
I~(p) l
if
q--.~p
69
(Ixi(p)l(2-n)/21xi(q)l(2-n)/2Gz,b(xi(p),xi(q))
-Iz~(p) - x~(q)l 2-") 1/(.-2),
Ixi(p)l<_V~o.Moreover, ( ~/'xi(p)'2k+l'xi(q)') Gz,b(xi(p),xi(q))= ~ xi(p----)) : 'xi(P)-------~2~xi(----q) "
n--2
From this one gets that oo
"~(P___A) Ix~(p) l <- 2 Ix,(p)l ~-~ ~
(Ix~(p)l -k - I~,(p)l~) 2-~
k=l
for p • Ui (V~, v~o). Now, deducing that, for s • (0, v~0), oo
oo
s2-O ~2 (s-k_ sk)2-~ _<s2-~ ~2 (s-k _ ~0s-~)2-~ k=l
k=l oo
= (l--~O)2-n~sk(n-2) k=0
(1 - ~o) 2-~
1
-
8 n-2
< (1 - ~o)2-n -
I - 6~ n - 2 ) / 2
and using that, by (i), ---+ai,o
in
Ui
,1
as
t--+0,
Xi
one obtains the desired estimate.
[]
R e m a r k 4.1.11 The classification of the flat conformal structures on S 1 x S 2 in Section 3.2 shows that each of these structures can be obtained from the flat conformal manifold (S3,Cs) with the punctures 0 and oo in the way considered in this section. Therefore, taking into consideration t h a t the canonical metric can(Cs) is identically zero, one can conclude lim v o l ( M ( A , 0), c a n ( C ( A , 0))) = O, stated in Remark 3.3.5(ii), also from Proposition 4.1.10.
[]
70
4. Asymptotics: The flat case
4.2 The
behavior
near
scalar flat structures
If one looks for a canonical metric in a scalar negative or scalar flat conformal class C on a connected closed manifold, one can choose a conveniently normalized Yamabe metric, i.e. a Riemannian metric of constant scalar curvature. Namely, by the proposition below, in these cases there is, up to multiplication by a positive constant, only one such metric in C. If C is scalar positive, this is no longer true in general (cf. [57], §2 and [58]). P r o p o s i t i o n 4.2.1 Let M be a connected and closed manifold of dimension n > 3 and let g be a Riemannian metric on M of constant scalar curvature S < O. Suppose that the Riemannian metric [? = qva/(n-2)g for some positive function ~o E C ~ ( M ) has also constant scalar curvature. Then qo has to be constant. Proof. Assume first that S = 0. Then, by Proposition 2.2.2, the scalar curvature of ~ also vanishes. Hence, by (2.2.1), A~=0, where A is the Laplacian with respect to g. Since M is connected and closed, this implies that ~ is constant, as desired. Now let S < 0. Then we may assume without loss of generality that the scalar curvature of g as well as the scalar curvature of ~ are - 1 . By (2.2.1), it follows that 4 nn - 1 2 A~ = ~ ( 1 - ~4/(n-2)) " (4.2.1) If ~ attains its maximum at Pmax E M, then (A~) (Pmax) _> 0 and hence, by (4.2.1), ~ (Pmax) _< 1. Analogously, if ~ attains its minimum at Pmin E M, then (A~)(Pmin) < 0 and hence qa (Pmin) > 1. Consequently, ~v has to be constantly 1. This completes the proof. [] The next proposition relates our canonical metrics to normalized Yamabe metrics in scalar flat conformal structures. P r o p o s i t i o n 4.2.2 Let M be a connected closed manifold of dimension n > 3. Let gk, k E N, be locally conformally flat Riemannian metrics on M with constant scalar curvature Sk > 0 and vol (M, gk) = 1, let g be a Riemannian metric on M with vanishing scalar curvature, and suppose that gk --r g
in
C m (M,S2T*M)
for m > (3n + 2)/2. Then S2/(n-2)can(Ck) -+
c2/(n-2)g in C' (M, S2T*M)
4.2 The behavior near scalar flat structures
71
for 0 ~ l < m - ( 3 n + 2 ) / 2 , where Ck denotes the conformal class of gk and Cn is the constant from Definition 2.2.5, which depends only on the dimension
Proof. Let Lk for k E N be the conformal Laplacian of (M, gk), i.e. L k : 4 n~ - l A
k+Sk,
(4.2.2)
where Ak is the Laplacian with respect to gk, and let Gk denote the Green function of Lk. Let (Uk,j)j=O,X,2,... be a complete orthonormal system of L 2 (M) with respect to gk such that Z~kUk, j = )~A,k,jUk,j ,
where 0 = Az~,k,o < )~A,k,1 _< •A,k,2 _< . . . . Then, by (4.2.2), LkUk,j = ,~k,jUk,j
with )~k,j := Since
4n-1 ),a,kj + Sk n--2
oo 1 Gk(P, q) = Cn Z "V---Uk'j(P)Ukj(q) j = 0 Ak'J
in L2(M x M), the L2-norm of Gk - cn/Sk with respect to the Riemannian metric gk (~ gk on M x M can be estimated by Gk
~Cn k 2 L2(M×M);g~$gk
2 oo
= Cn Z
-2
Ak,J
j----1 O0
< ~ Z -- (n - 2) 2 wn--1
a,kj )~--2
j=l =
( n - 2)2
_1 IIG Z~,kllL2(MxM);gkOg~ 2
where GA,k denotes the Green function of the Laplacian Ak (cp. Section 2.1). Moreover, since by assumption gk ~ g in C m (M, S2T*M), there exist constants cl > 0 and c2 > 1 with ][GA,k[[L2(MxM);g~eg~ < Cl
for all
kEN
(cp. [611 and 1 [[ullL~(MxM);gkeg~ < [[Ul[L2(M×M);gSg < C2 [[U[IL2(M×M);gkSg~ C2
72
4. Asymptotics: The flat case
for all u E L2(M x M) and all k E 1% Thus there exists a c3 > 0 such that Gk
--~kk 6n
L2(M×M);g~g _
kEN.
(4.2.3)
Now let 7ruc : 57/--+ M be the universal covering of M. By Theorem 1.1.9, for each k E N, there is an immersion ~k : M -4 S n such that =
iVs
If ~ would be surjective for some k, then M would be diffeomorphic to the quotient of S n by a finite subgroup of O(n + 1) (cf. [62], Theorem 4.5 and [41], §6). Such a quotient does not admit a scalar flat, flat conformal structure (see Section 3.2). In view of Theorem 1.1.2, our assumptions, however, imply that g is locally conformally flat and hence the conformal class of g is scalar flat and flat. Therefore, we may assume that
~k (]~/[) C ~n C S n
forall
kEN,
where S n is again identified with ]~n U {oo} via stereographic projection. Let Po E M be fixed. We consider a simply connected neighborhood Uo of Po and choose a connected component 8to of 7rue1(Uo). Then the restriction fo : Cro ~ U0 of ~ru¢ to/)0 is a diffeomorphism. Moreover, there exist positive functions ~k E C°°(M), k E N, such that ( f o 1 ) * ~ g E = ~4/(n-2)gk
on
Uo.
(4.2.4)
Composing ~k with an appropriate scaling of Rn if necessary, we may assume that qOk(Po) ---- 1 for all k E N . (4.2.5) Let the Riemannian metrics gk on M be given by ~k := ~4kl('~-2)gk • Because of (4.2.4), each of the metrics gk is flat on U0. Hence Lemma 2.2.1 implies that Lkqa~ = 0 on U0. (4.2.6) Let U~ und U~' be neighborhoods of P0 with smooth boundary such that
vg c c
c c Vo.
Then, by (4.2.6) and the Harnack inequality (Theorem 1.2.4), there exists a constant c4 > 0 such that sup qak < e4 inf ~k u5 us
for all
k E N.
(4.2.7)
4.2 The behavior near scalar flat structures
73
By means of (4.2.5), it follows that sup ~h < c4 for all v~
k E N.
In particular, the sequence (~h)heN is bounded in L 2 (U~). By Theorem 1.2.2, this, (4.2.6), and the assumption that gk -'+ g in C m (M, S 2 T * M ) imply that (7~h)heN is also bounded in W m'2 (U~'). Then, applying Theorem 1.2.1, we get that, for each integer ml with 0 < m l < m - n/2, there exists a sequence (k') of positive integers such that
C m' (U~')
~h' --~ 7~ in
(4.2.8)
for some function 7~ E C ml (U~'). Because of (4.2.5) and (4.2.7), 1 inf~oh>-U~
Thus we also have
--
1
for all
kEN.
C4
1
--+-
cml(u~').
in
(4.2.9)
7~h' Let exph : TpoM --+ M denote the exponential map at po with respect to 0h. For the further considerations we identify the Euclidean vector space (TpoM, (0h)po) with (Rn,gE). Furthermore, we suppose that ml _> 1. Let Ek be the injectivity radius at Po with respect to gh, i.e. the maximum of all ¢ > 0 for which the restriction of exp h to Bn(¢) is an embedding. Then, since Oh'-'+ +4/(n-~)g
C ml (U~',S2T'M)
in
(4.2.10)
by (4.2.8), we may choose a neighborhood U1 of P0 such that
u1 c v~'n Nexph, (Bn(eh,))
•
h'
Set
Vk, := exp,-,1 (U1) N Bn(~k ,) and let fh' : U1 --+ Vk, be the inverse of the restriction of exp h, to Vh,. Since 0h, is flat on U~', we have
1;,~
=
Oh,
•
Now let Gk denote the Green function of the conformal Laplacian /~h of (M,~h). We define Fh, E C °° (U1 × U1) by ~
Fk,(p,q) := Gk,(p,q) -
Cn
~k' (P) Tk' (q) Sk,
(cp. Lemma 2.3.1). According to (2.2.1),
-- ]fk'(P) -- fk'(q)l 2-n
74
4. Asymptotics: The fiat case
1 ) = ~pk(n+2)/(n_2)Sk. Thus
Lv,(p) + Lk,,(q)) Fk, (p, q) _
_~,(n+2)/(n-2)(p)
1
-(n+2)/(n-2) (q~
1
(4.2.11)
J
-
Since
,, _ 1 (Gk(p,q)_ C, ) G~ (P' q) - (Pk(P) (Pk(q) Sk ~k (P) cpk (q) _ by Lemma 2.2.7, it follows from (4.2.3), (4.2.9), and (4.2.10) that the sequence (Fk,) is bounded in L 2 (U1 × U1). Analogously to the reasoning for (~k), using (4.2.9), (4.2.10), and (4.2.11), it follows that (Fk,) is bounded in C m2 (U2 × U2) for each integer m2 with 0 _< m2 < ml - n and for any domain U2 CC U1 with smooth boundary. Recall that can(Ck, ) = &~,gk, with 5v(p) = lim ( G v ( p , q ) - [ A , ( P ) - . f v ( q ) 1 2 - n )
q--~p \
/
1~(n-a)
for p e U1.
Since, by assumption, Sk --~ 0 and Cn
Sk, &'~;-'(p) = Sk, Fk, (p,p) + V2k,(p--~ , (4.2.9) and the boundedness of (Fk,) in C m2 (U2 x Us) imply that s tk'/ ( n - 2 ) O~k ~ , "-~ ¢~/(n-2)~p2/(2-n)
in
Cra2(U2)
and hence =
,/(--')_,
~k' k ~k' Yk --~ C~/(n-2)g in C 'n2 (U2,S2T*M)
.
Since in the above considerations (k) can be replaced by any subsequence of (k), this concludes the proof. []
4.3 Consequences
for the geometry
of B+(M)
First we consider the situation described in Section 4.1. Since all manifolds
Mt,A for t E (0, 1) and A E SO(n) are diffeomorphic to each other, we can
4.3 Consequences for the geometry of B+(M)
75
think of Ct,A as conformal structures on a fixed manifold M. Let so(n) denote the Lie algebra of SO(n). For t • (0, 1) and X • ~o(n), we define tangent vectors Xt and Xt,x to B+(M) at [Ct] by d [Ct,exp(sX)] Xt := d [Ct] and Xt,x := ~ss
s=o
The length of the vectors Xt and Xt,x with respect to the Riemannian metric 0 on B+(M) induced by our canonical metrics can be estimated as follows. T h e o r e m 4.3.1 As t --+ O,
~(x~,x~) =o(t
-~)
and, for each X • so(n), i~(Xt,x, Xt,x) = 0 (log-2(t)) . Proof. For t • (0,1) and - t < s < l - t ,
we define a homeomorphism
~t,8 : Mt ~ Mt+s as follows. We set
~t,s(p):=p
for
peMo\(Ul(1)nU2(1)).
I f p • U~ (x/7, 1) for i = 1,2, then let ~t,s(p) be that point in Ui ( tvq'-+--s, 1) which is determined by xi o ~,~(p) = (1 - t 4 ~ - ~ ) Ix,(p)l - vq + 4 t + 8 ~i(v) •
(1 - vq)Ix,(P)l Here we use the embeddings M~ ~ Mt and M~+s ~ Mt+s. We set d~,
I
(18
Is=0
ht := "7- t,sgt+s As one easily computes, 1
h,= v q ( v q - 1) ~,3 dri ® dri on Ui (v~, 1) with ri = Ixi[. Hence
can(c,) ( ~ h , , ~ h , )
= g, (h,, h,) 1
t ( 1 - , , a ) ~ , -2, on Ui (vq, 1), where again can(Ct) = Oqgt. 2 Clearly,
76
4. Asymptotics: The flat case
ht=O
on
M o \ ( U I ( 1 ) nU2(1)) .
Thus, by Proposition 4.1.10(ii), it follows that
(°~2ht'°L2ht)can(Ct)
l ~-'~/u ~ a~ r~n-2 d#[gE,i] - t(]. -~ V~) 2 i=l (vz~,l) < 2__t_.~V~) n Wn-1 .-f'zr'~-3dr
=o(t-') as t-~ O. Now choose diffeomorphisms ~t,8 : Mt
-4 Mr+8 such that
~;,sgt+8 - ~;,sgt+8 L2(M,);g,=
0 (S 2)
s -4 0.
as
(Such diffeomorphisms can be constructed by smoothing ~t,8 on small neigborhoods of the boundaries of Ui (Vq, 1).) Then d ~,
d--s t,sgt+8 8=0 = ht and hence
xt = Pc~n(c,) ( d ~, 8can(Ct+8)
) 8=0
We conclude that
=o(t -1) as t --+O. The second part of the theorem can be proved in a similar way. We fix X E so(n) and consider the homeomorphisms $t,x,8 : Mt --+ Mt,exp(sX) defined by ~t,x,8(P)=P for pE Mo \ (UI(1) fqU2(1)) and
xio~t,x,8(P)=exp
( sl°glxi(p)[ ) (_l)/log(t)X
xi(p),
if
(
peUi v~,l
)
4.3 Consequences for the geometry of B0+(M~)
77
for i = 1, 2. We set d -.
ht,x := "~8~t,x,sgt,exp(sX) s=0 and assume without loss of generality that the only non-vanishing coefficients of the matrix X are X12 = - 1 and X21 = 1. Then, putting x i1 : li COS(Wi) und x i2 = li sin(wi), we have ~,x,sgt,exp(sX)
= ~;,x,s
gE,i
1[
= -L~ dli ® dli + E dxk ® dzk ri
k:3 s
( - 1 ) i log(t) ri dri
)]
and hence
2 l~ dri ® dwi ht,x = (_1) i log(t) r i3 on Ui (v~, 1). Since the vector fields riO/Ori and (ri/li)O/Owi are gtorthonormal on Ui (v~, 1), it follows that, on Ui (V~, 1), can(Ct) (a2 ht,x, a~ ht,x ) = gt (ht,x, ht,x ) _
212
log 2 (t) r i2 < - -2 -
log2(t) •
Proceeding as above, we arrive at
I~(Xt,x,Xt,x) = O (log-2(t))
as
t --+ 0 . []
An immediate consequence of the last theorem, which generalizes the A -~ oc part of Theorem 3.3.1, is the following statement analogous to Corollary 3.3.4.
Corollary 4.3.2 The curve t • (0, 1) ~-+ [Ct] • B+(M)
on the moduli space B+ ( M) of scalar positive, fiat conformal structures on M has finite length with respect to b. In particular, (B+( M), ~) is not complete. Moreover, all sequences ([Ct~,a~])~eN with t~ ~ 0 and Ak E SO(n) represent the same point in the metric completion of (B+(M), t}). []
78
4. Asymptotics: The flat case
Theorem 4.3.1 deals with the geometry of the moduli space of scalar positive, flat conformal structures in directions of increasing Yamabe invariant. Now we wish to study the L2-metric D in directions of decreasing Yamabe invariant. For this we consider a curve k : t 6 (0,T] ~ gt 6 ,~4+(M) C S2(M) for some T > 0, where M is again a connected and closed manifold of dimension n > 3 and .h40+(M) denotes the space of those Riemannian metrics on M the conformal class of which is scalar positive and flat. We suppose that, in C m (M, S2T*M) for sufficiently large m, k is differentiable and gO := lim gt t--+0
is a Riemannian metric on M with vanishing scalar curvature. In the following, we are interested only in the length of the projection a : (0,T] --~ B+o(M) of k. Therefore, we may assume that vol (M,g °) = 1. Let G ( M ) denote the space of all Riemannian metrics g on M with constant scalar curvature and vol(M, g) = 1. Since the restriction of the map
(~o,g) 6 C ~ ( M ) x ®(M) ~ ~g 6 J~4(M) to a sufficiently small neigborhood of (1, gO) is a diffeomorphism onto a neighborhood of gO (cf. [13], Theorem 4.44), we may assume in addition that each gt, t 6 (0,T], is a Riemannian metric with constant scalar curvature S t and vol (M, gt) = 1. T h e o r e m 4.3.3 With the above assumptions and notations, we have:
(i) For n = 3, 4, the curve a has infinite length with respect to I~. (ii) Let n >_ 5 and suppose that < c~
su, te(0,T) ~ , ~ - '
--
and
g,
dS t inf > 0 te(O,T) --~
Then the length of ~ is finite. Proof. For t E (0,T], let C t denote the conformal class of gt and write can (C t) = (~t)2 gt. With that,
-aT(t) =
--
4.3 Consequences for the geometry of B+o(M)
79
Let Ric t E 8 2 (M) denote the Ricci curvature of gt, and set
h t := 1 S t g t _ Ric t "
n
Then (cp. [13], 4.17 und 4.19), for all t E (0, T],
dS~ = ( i s~¢ _ ni~,, ~ ) g~
dt
and (cp. (3.2.1)) h ~ e ~ (g~) By Lemma 3.3.2(ii), the last relation implies that
(at)2-nh t En(can(Ct)) .
(4.3.1)
Furthermore, because of vol (M, gt) = 1,
dg t
t'~
(-~,g )9,
= 0
Using Lemma 3.3.2(iii), it follows that
ds, dt
g~ -- ( ( a t ) 2-n h t, (at) ? dgt~ dt ] can(C')
(4.3.2)
By Proposition 4.2.2, there exists a constant O > 1 such that
± (s~) 1/(2-n) < ~ ) < cl (s~) '/(2-n) Cl
(4.3.3)
for all t E (0, T] and all p E M. Using again Lemma 3.3.2(iii), we deduce that
((o~t) 2-n h t, (t~t) 2-n ht)can(G,) = ((O~t)-n/2 h t, (o~t)-n/2 hi)g, < c~ (s~)~/(~-2) (h',h~)9, for all t E (0,T]. Since, by assumption, gt ._4 gO in C m (M, S 2 T * M ) as t --~ O, it follows that
((at)2-nht,(at)2-nht)can(C~)
<_c~ (St) n/(n-2)
(4.3.4)
80
4. Asymptotics: The flat case
for a constant c2 > 0 independent of t E (0, T]. From (4.3.1), (4.3.2), and (4.3.4), we obtain that (d~ t
dn
-Ti( 1,--~(t)) ~.(c,) d~ .t) h 2
1 (s,)°/~_°)(dS"~ c-~ t-aT)
~
for all t E (0, T] and hence 0 t~-~(), ~-(t) d~
/
/? snl(4_2n)
d t _ > - -1
C2
ds.
Consequently, for n = 3, 4,
/o
T7
[ d ~ t dn
/
I I
This proves (i). To prove (ii), we first note that
( dn t dn -~-~( ),-d-~(t))can(c,) < dgt _ ((at) 2 --~-,
Since, by assumption, .~r(dg'ldt,
dgt (at) l d-T/canCC, )
dg'ldt),,: t ~ (O,T)} is bounded, it follows
by means of (4.3.3) that there exists a constant c3 > 0 such that
~-(),-~(t)
ca.(c<)
-
Furthermore, by assumption, there exists a e4 > 0 such that
dS t
-->c4 dt -
for all
tE(0,T).
We derive that d~ an) L T 7 I~(-~(t),-~(t)
dt
T
(St)nl(4-2~)at
dS t < e_~ IT (s~)°i~,_2.~--d~ --
_
--
dt
C4 dO
c__33[ C4
s T
Jo
s nl(4-2n) ds
4.3 Consequences for the geometry of B+o(M)
81
For n > 5, this yields
T~
/ d~ t
da
!
and hence the desired assertion.
[]
As a consequence of Theorem 4.3.3(i), we abtain C o r o l l a r y 4.3.4 Let M be a connected and closed manifold of dimension n = 3, 4 and let C be a scalar fiat, fiat conformal structure on M. If [C] is a boundary point of I3+o(M), then [C] has infinite distance from 13+o(M) with respect to the Riemannian metric [~. [] R e m a r k 4.3.5 We cannot deduce from Theorem 4.3.3 that, for n _> 5, a boundary point [C] of B+(M) as in Corollary 4.3.4 has finite distance from B+(M) with respect to [), because we do not know whether there exists a curve ~ : (0, T] -+/3+ (M) satisfying lim ~(t) = [C]
t--+O
and the assumptions of part (ii) of the theorem at all. (See also Remark 6.2.3.) [] We conclude this section with an example which shows that there are indeed manifolds M of dimension n = 3, 4 such that B+(M) possesses boundary points [C] which can be represented by Riemannian metrics with vanishing scalar curvature. E x a m p l e 4.3.6 L e t / 3 1 , / 3 ~ , . . . , Bk, B~ be mutually disjoint closed geodesic balls in S n and let "Y1,..., Vk be orientation preserving elements of Conf(S n) such that ~/i(Bi) = S n \ [~ for i = l , . . . , k , where Bi is the interior of Bi- Identifying S n with R '~ U {co} and assuming without loss of generality that Bi and Bi for fixed i are centered at 0 and co, respectively, the transformation Vi takes the form x E S n ~ )~Ax E S n for appropriate A > 1 and some A E SO(n). Hence the transformations 7 1 , . . . , 7k are loxodromic. Furthermore, as it is well-known, if1,..., Vk freely generate a Kleinian group F, which is called a Schottky g r o u p . Obviously, the Kleinian manifold ~ ( F ) / F is diffeomorphic to the connected sum k (S 1 x S '~-1) of k copies of S 1 x S n-1. In [51]_Nayatani shows that, for n = 3,4 and k >_ 2, one can choose the balls B1,/3~,. • ., Bk, - B- 'k and the conformal transformations V1,- .. , ~/k such that the canonical fiat conformal structure Cr on ~2(1")/F is scalar negative.
82
4. Asymptotics: The flat case
On the other hand, Lemma 4.1.2 (cp. also [38]) implies that Cr is scalar positive provided the balls B1, B ~ , . . . , / ~ k , / ~ are sufficiently small. Thus, for n = 3, 4 and k > 2, starting with B1, B ~ , . . . , / ~ k , / ~ and ~/1,..., ~'k for which Cr is scalar negative, and then decreasing the radii of B1, B~,..., Bk, B~, one obtains a smooth path of flat conformal structures on k (S 1 x S n-l) from a scalar negative to a scalar positive structure. Since the Yamabe invariant is continuous with respect to the C2-topology (cf. [13], Proposition 4.31), such path passes through the space of scalar flat, flat conformal structures. As one easily sees, this construction can be done such that the projection of the resulting path onto/~0 (k (S 1 x Sn-1)) avoids the singular strata of this moduli space. I~
5. G e n e r a l i z a t i o n in low d i m e n s i o n s
The goal of this chapter is to extend the definition of the canonical metric can(C) to non-fiat conformal structures C. The first step is a reformulation of the construction of can(C) for fiat C. This will be done in Section 5.1. Here, using the notion of mass of an asymptotically flat manifold, we describe the canonical metric can(C) as the unique Riemannian metric in C of constant mass 1. As carried out in Section 5.2, this description allows us to generalize can(C) to non-fiat C in dimensions 3, 4, and 5. Then we explain how, using so-called conformal normal coordinates, also in the general case can(C) can be constructed by means of the singularities of Green functions of conformal Laplacians. In dimensions 3 and 4, we do this more explicitly. We conclude this section by proving that can(C) is smooth also for any non-fiat C. The proof relies on the asymptotic expansions of the heat kernel of the conformal Laplacian.
5.1 T h e c a n o n i c a l m e t r i c a s m e t r i c
of constant
mass
We start with recalling the definition of an asymptotically fiat Riemannian manifold and its mass (see e.g. [45]). Definition 5.1.1 An n-dimensional Riemannian manifold (N, h) is called asymptotically flat of order T > O, if there exist a compact subsetNo C N and a diffeomorphism N \ No -+ {z E ~n :lzl > do} for some do > 0 such that, in the coordinates z l , . . . ,z n induced on N \ No, hit = 5ij + 0 (p-T) ,
Oh, Oz k = 0 ( p - T - l ) ,
and
02 hij OzkOz' = 0 (p-T-2)
as p := Izl -+ oc for i , j , k , l = 1 , . . . ,n. The coordinates z l , . . . ,z n are then called asymptotic coordinates. Definition 5.1.2 Suppose that (N, h) is an asymptotically fiat Riemannian manifold of dimension n > 3 with asymptotic coordinates z l , . . . , z n. The mass of (N, h) is that real number mass(h) which is given by
84
5. Generalization in low dimensions
mass(h)::
1
lim f
~-~(Oh,,
0.in_ 1 d-+oo J{lzl=d} i , j = l ~X~ Z /
Ohii'~ , c~zJ ] dsj
where dS j := ( - 1 ) J + l d z 1 ^ ... A d z j-1 Adz j+l A ... Adz n . R e m a r k 5.1.3 Let dz := dz x A ... Adz ~. Then
dz = pn- ldp A d( n = pn-2 E zidzi A d~ i=1
in polar coordinates (p,~) = of (S n-l, gs). Since
(Izh z/lzl),
where d~ denotes the volume form
dSJ ( X 1 , . . . , X , - 1 ) = dz ( ~ - ~ , X 1 , . . . , X , - I )
,
it follows that dS j = d n - 2 z j d ~
on
{Iz[=d}.
Thus, with the assumptions of Definition 5.1.2, mass(h)-
1 limdn_2 f ~ (Oh,j ~n--1 d--+~ J{Izl=a)/ , j = l \ Ozi
Ohii~ OzS ] zjd~" []
The definition of mass involves a choice of asymptotic coordinates. But the following theorem, due to Bartnik (cf. [11], [45]), says that the mass is in fact an invariant of the Riemannian metric. T h e o r e m 5.1.4 If the Riemannian manifold (N, h) of dimension n > 3 is asymptotically flat of order T > (n - 2)/2, then mass(h) depends only on the
Riemannian metric h.
[]
L e m m a 5.1.5 Let (N, h) be as in Theorem 5.1.4 and let A be a positive real
number. Then mass (A2h) = An-2 mass(h).
Proof. Clearly, if z l , . . . , z n are asymptotic coordinates of (N, h), then 5i :__ Az i, i -- 1,..., n, are asymptotic coordinates of (N, A2h). With it, the assertion directly follows from Definition 5.1.2.
[]
For the construction of the canonical metric for non-flat conformal structures, we shall apply the positive mass theorem of Schoen and Yau ([59], [60], [61], cp. also [45]) which we state now.
5.1 The canonical metric as metric of constant mass
85
T h e o r e m 5.1.6 Let (N, h) be a Riemannian manifold of dimension n >>_3 with non-negative scalar curvature and suppose that (N, h) is asymptotically fiat of order T > (n - 2)/2. Then mass(h) _> 0, with mass(h) = 0 if and only if (N, h) is isometric to (~n, gE). R e m a r k 5.1.7 We point out that we shall use Theorem 5.1.6 only for n = 3,4,5. [] Now let us again consider a connected and closed manifold M of dimension n _> 3 and a scalar positive, flat conformal structure C on it. For a Riemannian metric g E C and a point p E M, we define a symmetric (2, 0)-tensor field h(g, p) on M \ {p} by h(g,p) := G4/(n-2) g where the function Gp on M \ {p} is again formed from the Green function G of the conformal Laplacian L of (M, g) by the rule
Gp(q) := a(p, q) . From Lemma 2.2.7, it follows that h(~, p) = ~o4/(2-n) (p)h(g, p)
(5.1.1)
for ~ = 7~4/(n-2)g. Since C is scalar positive, G is positive (see Proposition 2.2.9) and hence h(g,p) is a Riemannian metric on M \ {p}. Since C is also flat, the Riemannian manifold (M \ {p}, h(g,p)) is asymptotically fiat of order n - 2 (cp. [45] or the proof of the next theorem). Thus, by Theorem 5.1.4, mass(h(g,p)) is well-defined. D e f i n i t i o n 5.1.8 We define the mass of the Riemannian metric g as that ]unction re(g) on M which is given by 1
m(g)(p) := 4 ( n - 1) mass(h(g,p))
for p E M .
The key to the generalization of our canonical metric is T h e o r e m 5.1.9 Let (M, C) be as above. Then can(C) = m2/(n-u)(g) g
for any g E C. Proof. According to Lemma 5.1.5 and (5.1.1), for conformally equivalent Riemannian metrics g and ~ = 7~4/(n-2)g in C, =
re(g)
86
5. Generalization in low dimensions
Thus, the (2, 0)-tensor field m 2/(n-2) (g)g does not depend on the choice of
9EC. To show that the canonical metric can(C) agrees with m 2/(n-~) (g)g, we fix a point p E M and choose a Riemannian metric g E C such that n
g= ~
dx i ® dx i
i----1
on a neigborhood U of p with local coordinates x = ( x l , . . . , xn). We assume that x(p) = 0 and set r := Ix[. Then, by elliptic regularity or Lemma 3.1.1, the function Gp formed as above satisfies
Gp=r 2-n+a+u
on
U\{p}
(5.1.2)
for some constant a and a function u E C ~ ( U ) with u(p) = 0. We set Xi
z i:=r- ~
for
i=l,...,n
and
p : = ] z I.
Then 0
cgz = p-2
(Sij - 2p-2ziz j) OxJ j=l
which implies that G
Oq)=p_
4 n
g ~ z~' O~J
Z
(&ik - 2P-2Z izk) (~jk - 2P-2zjzk)
k=l =
P
-4
~ij
on U. Furthermore, by (5.1.2),
a~/("-~)=p'
l + n _ 4a 2 p2-n + ~ )
on V\{p}
with fi = O" (pl-n), which means that
=O(p'-~)
,
0fi 0z k = O ( p - " )
,
and
0%2 - 0zk0z,
-
o
(p-"-')
as p --+ c~ for k, l = 1 , . . . , n. It follows that, on U,
h(g,P)~j:= h(g,P) ( O~ , Oj ) 4a 2-" + o'' (p~-n) ) ~j. = ( 1+h--~_20
(5.1.3)
5.1 The canonical metric as metric of constant mass
87
Therefore, z l , . . . , z n are asymptotic coordinates of the asymptotically flat aiemannian manifold (M \ {p}, h(g,p)). We use these coordinates to compute the mass of (M \ {p}, h(g,p)). From (5.1.3), we get
i,j=l
OzJ
=(1-n)~"z j 0 z..., OzJ
1+
] 4a p2-~ 0 " n - 2 + (pl-n)
j=l
°(
= ( 1 - n)p-~p
1 + -~-~_2p
+
(Ol - n )
)
= 4(n - 1)ap 2-n + O (pl-n) as p ~ co. In view of Remark 5.1.3, this gives
mass(h(g,p)) = 4(n - 1)a and hence m(g)(v)
= a
On the other hand, if we write can(C) = c~2g, then by the definition of the canonical metric, Remark 3.1.3(ii), and (5.1.2),
a(p) = lim (Gv(q) - r2-n(q)) 1/(n-2) q-+p = all(n-2)
.
[]
This concludes the proof. The last theorem can be rephrased as
T h e o r e m 5.1.10 Let M be a connected closed manifold of dimension n >_ 3 and let C be a scalar positive, fiat eonformal structure on M . If (M, C) is not conformally diffeomorphic to (S n, Cs), then can(C) is the unique Riemannian metric in C which has constant mass 1. [] R e m a r k 5.1.11 Equations (2.2.1) and (2.2.5) imply that the Riemannian metric h(g,p) has vanishing scalar curvature. Thus Theorem 5.1.6 can be applied to (M \ {p}, h(g,p)). By means of Theorem ~.1.9, this gives! another [3 proof of Theorem 2.3.5.
!
88
5. Generalization in low dimensions
5.2 The
canonical
metric:
The
general
case
We shall now extend the definition of the canonical metric to non-fiat structures in dimensions n = 3, 4, 5. Let C be any scalar positive conformal structure on a connected and closed manifold M of dimension n > 3 and let the Riemannian metric h(g,p) on M \ {p} for a Riemannian metric g E C and a point p E M defined as in the preceding section. Then the following holds (cp. [45]).
Proposition 5.2.1 The Riemannian manifold (M \ {p}, h(g,p)) is asymptotically flat of order 1, if n = 3, and order 2, if n >_4. [] We now assume that n = 3,4,5. Then, by Theorem 5.1.4 and Proposition 5.2.1, mass(h(g,p)) is well-defined and hence the mass re(g) of g can be defined as in the flat case (see Definition 5.1.8). Hence we can use Theorem 5.1.9 to generalize our canonical metric. We then have
T h e o r e m 5.2.2 L e t C be a scalar positive conformal structure o n a c o n n e c t e d closed manifold of dimension n = 3, 4, 5. Then the (2, O)-tensor field can(C) on M defined by can(C) := m :/("-2) (g) g
for a Riemannian metric g E C depends only on the conformal structure C. If (M, C) is not conformaUy diffeomorphic to (S n, Cs), then can(C) is a Riemannian metric. Proof. Exactly as in the flat case, the first part of the statement follows from Lemma 2.2.7 and Lemma 5.1.5. The proof of the second part is an application of the positive mass theorem. Indeed, this theorem gives (cp. Remark 5.1.11) that the function re(g) for any g E C is positive everywhere, provided (M, C) is not conformally diffeomorphic to (S n, Cs). (We note that, also for non-flat structures C, the canonical metric can(C) is of class C °°. The proof of this will be carried out in the next section.) [] R e m a r k 5.2.3 It remains the question whether one can generalize the canonical metric can(C) in dimensions n _> 6. For this, among other things, it would be to clarify whether one can break loose from Theorem 5.1.4. In what follows it will turn out that, at least in dimensions 3 and 4, it can be verified without Theorem 5.1.4 that can(C) is well-defined also in the general case. On the other hand, this rests on considerations which essentially use that n < 6 (see (5.2.8) in the proof of Proposition 5.2.5, cp. also the asymptotic expansions of the Green function in [45], Lemma 6.4). []
5.2 The canonical metric: The general case
89
We shall now explain that, also in the present situation, can(C) can be constructed by means of the singularities of the Green function G. We start with recalling the notion of conformal normal coordinates (see [45], Theorem 5.1, cp. also 126]). T h e o r e m 5.2.4 Let (M, C) be a conformal manifold and fix p E M . For each v E N, there is a metric g E C such that det (gij) = 1 + 0 (r ~)
in g-nor'real coordinates x = ( x l , . . . ,x n) at p as r := Ixl -+ o.
[]
Such coordinates x l , . . . , x n as in Theorem 5.2.4 are called conformal normal coordinates of order v at p. Note that any normal coordinates are conformal normal coordinates of order 2. The next proposition says that, using conformal normal coordinates for the construction of the canonical metric, we can return to the approach (2.3.1). P r o p o s i t i o n 5.2.5 Let M be a closed and connected manifold of dimension n = 3, 4, 5 with a scalar positive conformal structure C. Let the Riemannian metric g E C be such that g-normal coordinates x = ( x l , . . . ,x n) at p E M are conformal normal coordinates of sufficiently large order v. Then can(C) = ot2g with, a(p) = q--+p lim (G(p, q) - Ix(q)12-~) 1/("-2)
,
where G is the Green function of the conformal Laplacian of (M, g). Proof. Because of n = 3, 4, 5, in the conformal normal coordinates x = ( x l , . . . , x n) at p the function Gp(q) := G(p, q) has the asymptotic expansion (cp. [45]~ Lemma 6.4) Gp = r 2-" + a + O"(r) (5.2.1) with a constant a as r := txl ~ 0. Here, u = O" (r k) means that
u=O(rk)
,
Ou
Ox,=O(rk-'),
and
02u
Ox'OxJ
- O (r ~-~)
as r --+ 0 for i , j = 1 , . . . , n . Therefore, we have to show that
mass(h(g,p)) = 4(n - 1)a.
(5.2.2)
Since the inverted conformal normal coordinates z i := r - 2 x i, i = 1 , . . . , n, are asymptotic coordinates of (M \ {p}, h(g,p)) (cp. again [45]), by Remark 5.1.3 we have
90
5. Generalization in low dimensions
mass( h(g, p) ) Oh(g,p)ii ~ zjd~ wn-1 d--*~ _
1
Oz~
IzI=d} i,j=l
Oh(g, P)ii ~ xj de.
~Oh(g,p)ij
limd_n~ {
w n - 1 ,t--,o
I~l=d} ~ ~
]
Oz~
\
J
From
0 0 = r2 ~n (5~ - 2r-2xix j) OxJ Ozi
(5.2.3)
j=l
and the fact that g-normal coordinates xl,... ,x n satisfy (see e.g. [13], Theorem 1.45)
E g i J x J = xi for i = 1 , . . . , n , j=l
(5.2.4)
we get 0,
0)
g 0--;~ ~ J
4
=rg~
and hence
h(g,p)ij = ct/(n-2)gij
with ¢ := rn-2Gp.
By means of (5.2.3) and (5.2.4), one verifies that
~-~ cQh~ziP)iJx j
2 -4n r 3¢ -10¢ ~r + r2¢4/(n-2)(n- ~)
-
i,j=l
with
n
:= ~ gii • i=1
Moreover,
~ i,j= l
Oh(g,p)ii xj = - r 3 ~ Oh(g,p)ii OzJ Or i=1 4 r3¢_lO¢¢_r3¢4/(n_2)O ~
_
2 --~
~
or "
Consequently,
i,j=l
( Oh(g,p)ij Ozi 4
Oh(g,p)ii) x j OzJ
3 -10¢ C
= n _ 2r ¢
-~r (~ -
1 ) + r 2 ¢ 4/(~-~)
(
n
-
°')
~+r~rr
.
(5.2.5)
5.2 The canonical metric: The general case
91
The expansion (5.2.1) implies that ¢ = 1 + o (r
,
¢-1 = 1 + O (r ~-=) ,
and
(5.2.6)
O~b = (n - 2)at n-3 + 0 (r n-2) Or as r -+ 0. Since x 1, . . . , x n are g-normal coordinates,
g~j = ~j + O" (r 2) and hence
det(gij) = l - n + { + O" (r 4) . Using that, by assumption, det (gij) = 1 + 0 (r") for sufficiently large ~,, it follows that, as r -+ 0, { = n + O" (r 4) •
(5.2.7)
Substituting (5.2.6) and (5.2.7) into (5.2.5), we obtain that, as r -+ 0,
( Oh(~ziP)iJ i,j=l
Oh(g'P)ii ) xJ OZJ
= 4(n - 1)ar n + 0 (r n+l) q- O (r 6) .
(5.2.8)
Because of n = 3, 4, 5, this yields (5.2.2), and the proposition is proven.
[]
R e m a r k 5.2.6 Theorem 5.1.9 and Proposition 5.2.5 are reformulations of Lemma 9.7 in [45]. [] We apply Proposition 5.2.5 to prove the following generalization of Proposition 2.3.3. P r o p o s i t i o n 5.2.7 For i = 1, 2, let Mi be a connected and closed manifold of dimension n = 3, 4, 5 and let Ci be a scalar positive con.formal structure on Mi. Suppose that the map f : (M1,CI) -+ (M2,C2) is con formal. Then (f'can(C2)) (v, v) >_ can(C1)(v, v)
]or each tangent vector v E T M1. Proof. We fix p E M1, choose 92 E C2 such that g2-normal coordinates at p are conformal normal coordinates of sufficiently large order u, and set gl := f'g2. Then, obviously, gl-normal coordinates at p are also conformal normal coordinates of order u. Now, using Proposition 5.2.5, one can proceed exactly as in the proof of Proposition 2.3.3 to derive the assertion. []
92
5. Generalization in low dimensions
Corollary 5.2.8 Let (Mi, Ci), i = 1, 2, be as in the preceding proposition and let f : (M1,C1) ~ (M2,C2) be a conformal diffeomorphism. Then f'can(C2) = can(Or) . [] In Proposition 5.2.5 we have considered conformal normal coordinates of sufficiently large order v. The following proposition specifies this v. This may be helpful for calculating canonical metrics explicitly. For simplicity, we shall restrict to the dimensions 3 and 4. P r o p o s i t i o n 5.2.9 Let M be a connected and closed manifold of dimension n = 3, 4 and let C be a scalar positive conformal structure on M. Let g E C and p E M . If n = 4, assume in addition that the Ricci curvature Ric of g vanishes at p. Then can(C) = a2 g with a(p) = lim (G(p, q) - dist2-n(p, q))l/(n-2) q--+p
,
where G is the Green function of the conformal Laplacian and dist the geodesic distance of ( M, g).
Of course, it is possible to obtain Proposition 5.2.9 by refining the considerations in the proof of Proposition 5.2.5. But we shall present a reasoning which also shows that we do not need Theorem 5.1.4 to construct the canonical metric can(C) (cp. Remark 5.2.3). The decisive point is L e m m a 5.2.10 Let M be a manifold of dimension n E N with conformally equivalent Riemannian metrics g and ~ = qo4g for some positive function qo. Let dist and dist denote the Riemannian distance with respect to g and [1, respectively. Then lim (~(p)qo(q) q-~p \ ~
1 dist(p, q) ] = 0.
If in addition n >_ 3 and the Ricci curvatures Ric, Ric E S2(M) of g,~ agree at p, then (~2(p)~2 (q) 1 ) lim ~ = 0. q-~P ~ dist (p,q) dist2"(P,q) Proof. We assume without loss of generality that ~0(p) = 1 and set O(t) := ~dist (p,g(t)) L
and
u(t):= ~o2(ff(t)) ,
5.2 The canonical metric: The general case
93
where ¢ -- ¢(t) is a g-geodesic on M through p which is parametrized by arc length. Let yl,... ,yn be 9-normal coordinates at p and set ¢i := yi o ¢ for i = 1 , . . . , n . Then we have
o = ~
(¢)~ , i----1
n
dO
--
i de
d--/= ~
i
N,
and
i=1 _-
°
Thus
dO
(0) = 0 .
Moreover, because of
de
it follows that d20 dt 2 (0) = 1. Now let F.~J k. and/~.k, be the Christoffel symbols for g and g, respectively, in z3 the coordinates y Z , . . . , yn, i.e. we have n
V 00yJ Oyi
n
~yk
and
k=l
Oyk
V 00yJ
Oyi
k=Z
for i,j -- 1 , . . . ,n, where V and V denote the Levi-Civita connections of g and g, respectively. By means of (cp. Proposition 1.1.1) 2
fFX Y = V x Y + ~ [d~(X)Y + dqo(Y)X - g(X, Y)V~] for X, Y E 3~(M) and n
V~ = Z
0~o 0
aij Oyi OyJ '
i,j= l
where V ~ denotes the gradient of ~ with respect to g and (aij) is the inverse of the matrix (aij) with the coefficients
94
5. Generalization in low dimensions
aij :~ g
i
'
one verifies that
?.~. = £~. 2 ~kj + 5k~ - Z., ua ~ ~J + ~ k, Y tJy t=l
7y~) /
(5.2.9)
for i,j, k = 1 , . . . ,n. Since q is a geodesic with respect to g, we have d2qk
(t) = -
~
d~ i da~j F~.(q(t))-~(t)-~(t)
n
for
k = 1,...,n.
(5.2.10)
i,j=l
Moreover,
P~(p) o =
and
aij(p) = Sij .
We deduce that, for k = 1 , . . . , n, d2q k
C5.2.11)
d t 2 (0)
L
=_
F~ (p) -~-, d~i (0), 7dqJ ~ "0"
i,j=l
~=~ i,j= l
-~- (o)-&- (o)
cly
= 2 ~ (0)-~- (0) -
(p)
For
n
d3t9 - 3 Z dt 3 --
d¢i d2¢/ + Z dt dt 2
we get
~CO)=6~
(5.2.12)
.
d e C0) ~-
"d3;i ~-~~ '
(dud
~ (o)-&- (o) -
(p)
/=1
=3 ~ du( )0 . By means of the Taylor formula, it follows that, as t -+ 0, t2
du
oct) = -f + ~ ( o ) y
t3
+ o (t 4)
and hence
~Ct) - 2 °(-~t~) = o (t ~) . b-
5.2 The canonical metric: The general case
95
This gives the first assertion. To prove the second assertion, let (~ij) be the coefficient matrix of ~ in the coordinates yX,..., yn and let (~J) be its inverse. Differentiating (5.2.10) and using (5.2.9) as well as
0 (~,a k')
0 - .-~ =0,
one checks that k
d3~k
~d~
n
-&-(Ol-&-r(o ) = -
~
0/~k
d~l
. dqi
d C j . dqk . .
-bT, (p)~(o)-di-~o-)-&-~o)-ff/-(o)
i,j,k,l=l
k=l
+ ~d2u ( ) + (0~ ( 0 ) ) du
2 -4g(V~,V~)~o).
Since y l , . . . , yn are ~-normal coordinates, 1
n
g'J = ~'J + 5 Z
~,k,~ (p)yky~ + 0 (lyl 3)
(5.2.13)
k,l=l
as lYl -* 0 for i, j = 1 , . . . , n (see e.g. the proof of Lemma 5.5 in [45]), where /~iktj are the coefficients of the Riemannian curvature/~ of .~. From (5.2.13) and
0?6
1 ( 0%~
0~
O~j
Oy t -- 2 ~,Oy-"~OyJ + Oyl - - - 7Oy
Oyl Oy k ] '
one deduces that
0~
2
+ ~,,~k(p)) •
Oy z (P) = --~ (TltOk (P) This implies that
i,j,l=l
Oy I
dt
dt
oR oR
dt
0 - ~ k 1,
=0 for k = 1 , . . . , n. Therefore, ~d~
k
_d3~ k. .
d2u..
[ d u 0)1 2
~(o)-~/r(o) = ~7~(o)+ ( , ~ ( )
k=l
Furthermore, one gets from (5.2.12) that
- 4 g(v~, vv)(p).
96
5. Generalization in low dimensions
[ d2qk
~2
k:l
It follows that d3gk d4~ (0) = 3 ~'~, /,[ ~d2~_ - ( 0k ) ) \ 2 +4~2~ d;k dt4 k=l k=l --~- (0) --~- (0)
[d2u
{du 0 \2
=4~-~-~(0)+ ~-~())
-g(V~,V~)(p)
)
•
(5.2.14)
We now assume that n >_ 3 and Ri% = Ricp. Then, since g and t~ are conformally equivalent, by Proposition 1.1.1(v) also the Riemannian curvatures R and/~ of g and .~, respectively, agree at p. Using Proposition 1.1.1(ii) and the fact that, for n _ 3, the Kulkarni-Nomizu product with a Riemannian metric is injective (cp. [13], Lemma 1.113), this implies that, at p, Vd~ + d~ ® d~ - ~ (V~, V~) ~ = O, i.e.
02~
OyicgyJ (p) = g(V~, V~)(p)5ij - ~ ( p ) ~ y ~ j (p)
(5.2.15)
By means of (5.2.12) and (5.2.15), we derive that d2u
dt 2 (0)
x2-, 02~o , ,d~ i = 2 2_, i,j=l
d~J "0"
OyiOyJ
+2 ~-~ O~ d2__~2~ (0) l ( d u )2 i=x ~--~y/(p)dt + ~ ~-(0) = -2g(v
,
2
du 0
2 .
Substituting this into (5.2.14), we arrive at d4~
d2u (0) = 6-~- (0).
Again applying the Taylor formula, we obtain that, as t --->0,
u(t) - 2~2) = O (t 3) . This yields the desired result.
[]
Proof of Proposition 5.2.9. We fix a point p E M and choose a Riemannian metric g E C. For n = 4, we assume that the Ricci curvature of g vanishes at
5.3 The regularity of the canonical metric
97
p. Let ~ = ~4/(n-2)g E C be such that ~-normal coordinates y l , . . . , yn at p are conformal normal coordinates of sufficiently large order v _> 3. Then, by Proposition 5.2.5, can(C) = 52~ with
&(p)---- lim (G(p,q)- ly(q)l2-n) 1/(.-2) q--+p \
,
/
where G is the Green function of the conformal Laplacian of (M, ~). Moreover, since v _> 3, as a consequence of (5.2.13) the Ricci curvature Ric of ~ vanishes at p. Thus, using Lemma 2.2.7 and Lemma 5.2.10, we can deduce that lim (G(p, q) - dist2-n(p, q))
q.-c.p
~2--r$
/
lira |~(p)~(q)G(p, q) - ~(p)~(q)dist
q-+p \
\
(p, q)}/
=
= ~02(p)&n-2 (p) , which yields the statement.
[]
R e m a r k 5.2.11 Let M be a connected closed manifold of dimension 4 with a scalar positive conformal structure C and let g be any Riemannian metric in C. Let ~ E C ~ ( M ) be such that, on some neighborhood o f p E M, = exp
cijxix j
\~,j=l
/
with
cij = -~Rij(p) -
1 S(p)~ij ,
/
where x 1,... , x 4 are g-normal coordinates at p and Rij are the coefficients of the Ricci curvature Ric of g in these coordinates. Then (cp. Proposition 1.1.1(iii)) the Ricci curvature of the Riemannian metric ~ = 7~2gvanishes at p. Using this metric to compute the canonical metric can(C) at p as described in Proposition 5.2.9 and proceeding as in the proof of Lemma 5.2.10, one gets that can(C) = a2g with a 2 (p) = lim (G(p, q(t)) - dist -2 (p, q(t))) t--+0
where ; is a g-geodesic through p which is parametrized by arc length.
[]
5.3 T h e regularity of the canonical metric In this section we shall verify that the canonical metric can(C) is smooth also for any non-flat structure C. For this we shall apply the asymptotic expansion of the heat kernel of the conformal Laplacian.
98
5. Generalization in low dimensions
Let M be a connected closed manifold of dimension n _> 3 and let g be any Riemannian metric on M. We set n-2
L:=--L,
1)
where L is again the conformal Laplacian of (M, g). D e f i n i t i o n 5.3.1 A heat kernel of the differential operator L is a smooth function K : M x M × ~+ -+ I~_ satisfying the following properties:
(1) K(p, q, t) as a function of p and t is a solution of the heat equation for L, i.e. (0+
f_,(v)) K(p,q,t) = 0 .
(2) For all p E M and all u E Coo(M), lim f
t--+0 J M
g(p, q, t) u(q) d#[gl(q) = u ( p ) .
It is a well-known fact that the operator L possesses a unique heat kernel. Furthermore, it is known that the heat kernel of L and the Green function of L are related as follows (cp. [12], Theorem 2.38).
Proposition 5.3.2 Suppose that the conformal class of the Riemannian metric g on M is scalar positive. Then the Green function G of the conformal Laplacian L of (M, g) can be expressed in terms of the heat kernel K of L by G(p, q) = (n - 2)w,_1
/;
K(p, q, t) d r . []
In the rest of this section, we prove
Proposition 5.3.3 Let M be a connected and closed manifold of dimension n = 3, 4, 5 and let C be a scalar positive conformal structure on M. Then the canonical metric can(C) of C is of class C °°. Proof. Let g be a Riemannian metric in C. Proceeding as in the proof of the existence of conformal normal coordinates in [45] (see also the remark below), one sees that, for each p0 E M and for each v E N, there exist a neighborhood U of P0 and a function ¢ E C ~ ( U x M) such that, for each p E U, ¢(p,p) = 1 and normal coordinates at p with respect to g[p] := Cp4/(n-2)g, where Cp(q) := ¢(p, q), are conformal normal coordinates of order
5.3 The regularity of the canonical metric v. Let t : U × M × M × ~ ~ R+ be that function for which for each fixed p E U is the heat kernel K(ql, q2, t) of
L~pt
99
t(p, ql,q2,t)
n-2 4(n - 1) L[pl '
where L[p] denotes the conformal Laplacian of (M, g[pl)" The smoothness of ¢ implies that t is smooth, too. Let dist[v ] denote the geodesic distance with respect to g[p]. Furthermore, let the function j • C~(U x V) for a sufficiently small neighborhood V of diag(M) be defined by
i(P, ql,q2):=Idet((g[p])ij(ql) ) for p E U and (ql,q2) E V, where (g[p])ij are the coefficients of g[p] in g[p]normal coordinates at q2. With this, we have the asymptotic expansions (cp. e.g. [12], Theorem 2.30) t(p, ql, q2, t)
(4rrt)-n/2j-1/2(p, ql,q2)exp
dist~pl(ql'q2) 4t
~'~a~(p,o,q2)t k
(5.3.1)
k=O
as t -+ 0. From the construction of the coefficients ak (ep. [12], Theorem 2.26), it follows that ak • C ~ (U x V). Let G[pI denote the Green function of the conformal Laplacian L[p]. Then, by Proposition 5.3.2,
alpl(qi,q~)
= ( n - 2)~on_1
/j ~(p,q~,q2,t) dt.
By means of the asymptotic expansions (5.3.1), we obtain that (cp. [54], [9]) 1 (n - 2)Wn-1 aI~l (ql, q2) = (47r)-n/2i-1/2(P, ql, q2) [q51(P, ql, q2) + ~52(P, ql, q2)] + ~50(P, qt, q2)
with ~0 E
C~(U
x V), where 4~1 and ~2 are given by [(n-3)/2]
/'dist[p](ql, q2) 2~+2-n
4~1(p, ql, q2) = and 4~2(p, ql, q2) = ~ - 2
L
a~/2-1 (p, ql, q2) log (Gist[p] (ql, q2)) 0
for even n for odd n
100
5. Generalization in low dimensions
Because of n = 3, 4, 5, we have to consider only a0 and al. For these coefficients, one has (cp. [12], Theorem 2.26) (5.3.2)
a0 - 1 and
al(p,q,v)=
-
~(p,~(.),p)d~,
where ~P(P, ql, q2) := jl/2(p, ql, q2) (Lip])(ql)j-1/2 (p, ql, q2) a n d q : [0, 1] -+ M is the girl-geodesic with ~(0) = p and ~(1) = q. The function ¢ e C~° (U x M) is such that
j(p,q,p) = 1 + 0 (disqpl(v, q))
(5.3.3)
as q --~ p. We now assume that v is sufficiently large. In particular, if v > 4, then for the scalar curvature Sip] of g[v], one has (cp. [45], Theorem 5.1)
S[p] ( q) = O (dist~](p, q)) as q --+ p. We obtain that, as q -~ p,
al(p,q,p) = 0 (dist~](p,q)) .
(5.3.4)
Using n
(n-2)wn-lF(~-1)
=4~r n/2 ,
one deduces from (5.3.2), (5.3.3), and (5.3.4) that, for n = 3,4,5 and p E U, lim
q--+p
(4~r)-n/2i-l/2(p,q,p)[el(p,q,p)+~2(p,q,p)]
d st[p] (p, q)
N - ~
=0
and hence lim (a[pl(p, q) - d i s t a i n ( p , q)~ = (n - 2)wn-1 4~0(p,p,p) . trl ]
q-4p k
Since 4~0 is of class C °o and, according to Proposition 5.2.5, the canonical metric of C is given on U by can(C) = a2g with
a(p) = lim ( V p (p,q) - d i s t ~ n ( p , q ) ) q-~p \ [ ] this proves the proposition.
1/(n-2)
for p e U []
R e m a r k 5.3.4 For n = 3, the proof of Proposition 5.3.3 can be simplified by using Proposition 5.2.9 and setting ¢ identically 1. Similarly, for n = 4, the function ¢ can be chosen such that Cp is given as the function ~o in Remark 5.2.11. []
6. T h e m o d u l i space of all conformal structures
In this chapter we consider the space of all scalar positive conformal structures on a manifold of dimension 3, 4, or 5. In Section 6.1 we show that, for sequences of scalar positive structures whose Yamabe invariants tend to zero, the generalized canonical metric behaves similarly as in the conformally flat case. In the last section we study the distance of boundary components of the moduli space formed by scalar fiat structures to inner points.
6.1 The
generalized
metric
near
scalar
fiat structures
The next proposition generalizes Proposition 4.2.2. For simplicity, we treat only C°-convergence.
Proposition 6.1.1 Let M be a connected and closed manifold of dimension n = 3,4,5 and let gk, k E N, be Riemannian metrics on M with constant scalar curvature Sk > 0 and vol(M, gk) = 1. Suppose that Sa -4 0 and
gk ~ g°
in
Cm (M,S2T*M)
for some Riemannian metric gO and sufficiently large m. Then S2/(n-2)can(Ck) ~ cV(n-2)g °
in
C o (M, S2T*M) ,
where Ca denotes the conformal class of gk and cn is the constant from Definition 2.2.5. Proof. In the following let 3(g,p) for a Riemannian metric g on M and a point p E M be the function on a neighborhood of p defined by 3(g, p)(q) := det (gij (q)) in s-normal coordinates at p. We fix p E M and v E N with n - 1 < v < m + 1. Then there exist functions ~k E C ~ ( M ) , k E N, with qok(p) = 1 such that
102
6. The moduli space of all conformal structures ~k-~o °
for some
in
Cm(M)
(6.1.1)
~o E C~ (M) and (6.1.2)
= 1 + o (eD
for all k E N as q -~ p, where gk := ~ / ( n - 2 ) g k and rk denotes the geodesic distance from p with respect to gk. This can be verified as follows. The condition (6.1.2) means that ~k-normal coordinates at p are conformal normal coordinates of order v. To achieve this, we construct ~Pk near p in the way described in the proof of the existence of conformal normal coordinates in [45] (see also Remark 6.1.2 below) and define ~ outside a neighborhood of p suitably. Then the convergence (6.1.1) can be derived from the following two facts. First, the construction of the function ~k near p depends only on gk-normal coordinates x ~ , . . . , x ~ at p and the /-th derivatives at p of the Riemannian curvature of gk with 0 < l < v - 2. Second, by means of a result of Hartman (cf. the theorem in [14]), the assumed convergence gk --~ gO in C m (M, S2T*M) implies that, choosing the gk-normal coordinates appropriately, x ~ - ~ x °'i in C re(U0) for i = 1 , . . . , n and some neighborhood Uo of p, where x ° ' l , . . . , x °,n are g°-normal coordinates at p. Let Lk denote the conformal Laplacian of (M, gk) and Gk the Green function of Lk. Fix v e C°°(M) and let Uk E Coo(M) for k E N be given by Lkuk
-= V .
Then
uk(q) = -~nl/MGk(q,q,)v(q,)dp[gk](q,)
for all
qEM
Since k(q,q)
#[gk](q) =
V~(q,
)Lk "~k (q')dp[gk](q')
Cn
Sk ' it follows that
luk(q)l < m xlvL M Gk(q, q') d#[gk](q') ~n
Sk for all q E M.
(6.1.3)
6.1 The generalized metric near scalar flat structures
103
Now, for q • M \ {p}, we set F~(q)
:= Cp,~(q)
-
%
(n - 2)Wn--lGp, A,k(q)
-
with Gp,k(q) := Gk(P,q) and Gp, A,k(q) := GA,k(p,q), where Ga,k(p,q) is the Green function of the Laplacian A k of gk (cp. Section 2.1). Then,
/M Fk vdp[gk] = ( 2 - - n ) w n - l S k / M GA,k(p,q')uk(q')dp[gk](q') .
(6.1.4)
Since the family {Gp,,a,k : k • N} is bounded in L I ( M ) (cp. e.g. [6]), (6.1.3) and (6.1.4) imply that there exists a constant cl > 0 such that
/M Fk vd.[gk]l
< cl max[v[
for all k • N and all v • Coo (M). Therefore, also the family {Fk : k • N} is bounded in L 1 (M). Consequently, there exists a constant c2 > 0 with cn
<
G p , k -- ~
LI(M);g 0 --
C2
for all
k • N.
(6.1.5)
Since Ok ~ (~0) 4/(n-2) g°
in
C m (M, S2T*M) ,
we can find a neigborhood U of p on which 0k-normal coordinates at p are defined for all k • N. We choose a function y • C °o (M) such that y _= 0 on M \ U and y _= 1 on a neighborhood U' C U of p, and set
~'k(q) := Gp,k(q)
Cn
~k(q)Sk
~(q)
~-n
(q)
for
q • M \ {p}
with Gp,k (q) := Gk (P, q), where G~ is the Green function of the conformal Laplacian
Lk = 4 ~ A k
+ ~k
of (M, gk)- From r~2-n k
,[lkU d/~[0k] =
u Ak (r k
for u • C~ ° (U), _
and (2.2.1), we get that
2)r
) dp[0k] + (n - 2)wn_tu(p)
104
6. The moduli space of all conformal structures
L~S~k = - c . ~ ; ("+2)/("-2) _ 2(n - 1) ~}-" aCi(~k,p)
O~k
3(.qk,P)
Sk-rk-2-" on U'
in the weak sense. Because of (6.1.2) with v _> n - 1 and, as a consequence,
sup ~ - " ~
< c3
U
for all k e N and some c3 > 0 (cp. [45], Theorem 5.1), it follows that there exists a constant c4 > 0 such that L k F k L'~-I(M);90 <
for all
c4
k E N.
Furthermore, from Gpk '
c,~ks~
1 (G,,k_C,) ~k
(see Lemma 2.2.7) and (6.1.5), one deduces that { F k : k E N} is bounded in
LI(M).
Thus, by Theorem 1.2.2 and Proposition 1.2.7, .{i~k : k E N }
is
bounded also in the Sobolev space W 2,n-1 (M). Using Theorem 1.2.1(ii), we obtain that Fk E C°(M) and
sup h
< c5
(6.1.6)
for all k E N and some cs > 0. We now assume that v is such that can(Ck) = &~gk with 2 - - n (q))il(,-2) &}(p) = lim (G,k(q)-r~k
q_.+p
(see Proposition5.2.5). Then, becauseof ~ok(p)= i, we have
By means of (6.1.6) and
Sk -+ O, it
lim
k-~oo
follows
Sk/("-:)&k(p)
=
,~/(,-2) .
Since M is compact and the canonical metrics can(Ck) are smooth ( P r o p + sition 5.3.3), this yields the assertion. [] R e m a r k 6.1.2 In Proposition 6.1.1, the integer m has to be such that m _> max{n - 2, 2} and there exists a positive integer v _< m + 1 which ensures the statement of Proposition 5.2.5. Thus, by Proposition 5.2.9, for n = 3, 4 it is sufficient to assume that m = 2, and then to work with v = n - 1. Hence, for n = 3, the functions ~k in the proof of Proposition 6.1.1 can be set identically 1. T h a t means, in dimension 3, we do not need the Riemannian metrics 0h. Similarly, for n = 4, the functions ~k can be chosen as described in Remark 5.2.11. []
6.2 Infinite distance components of the boundary
105
6.2 Infinite distance components of the boundary Let M be again a connected closed manifold of dimension n = 3, 4, 5. Let C+(M) denote the space of scalar positive conformal structures on M and let B+(M) := C+(M)/Diff(M) be the orbit space of the action of Diff(M) on C+(M). As explained in Section 3.2, the canonical metrics can(C), C 6 C+(M), induce a Riemannian metric on B+(M). We denote this metric, the restriction of which to B+(M) is the L2-metric b studied before, also by I•. We now transfer our considerations in Section 4.3 concerning the geometry of B+o(M) near scalar fiat structures to the moduli space B+(M). Let us consider a curve k : t 6 (0,T] ~ gt 6 ]vl+(M) C S2(M) for some T > 0, where AJ+(M) denotes the space of all Riemannian metrics the conformal class of which is scalar positive. Again we suppose that k is differentiable in C m (M, S2T*M) for sufficiently large m and gO := lim gt t-.0 is a Riemannian metric on M whose scalar curvature vanishes. As in the conformally flat case, we may assume that gt 6 6 ( M ) for t 6 (0,T]. T h e o r e m 6.2.1 With the assumptions and notations as above:
(i) If n = 3,4, then the projection a : (0,T] ~ B+(M) of k has infinite length with respect to ~. (ii) If n = 5 and we have sup ( dgt dgt < oc t6(0,T) k "~-' dt g,
and
dS t
inf > 0 t6(0,T) ~" '
where S t is the scalar curvature of gt, then the length of t¢ with respect to b is finite. Proof. The proof is completely analogous to the proof of Theorem 4.3.3, where Proposition 4.2.2 has to be replaced by Proposition 6.1.1. [3 C o r o l l a r y 6.2.2 Let M be a closed and connected manifold of dimension 3 or 4, and let C be a scalar flat conformal structure on M whose Diff(M)equivalence class [C] is a boundary point of B+(M). Then the distance of [C] to B + (M) with respect to 1~ is infinite. [] Concerning the situation in dimension 5, we make the following remark.
106
6. The moduli space of all conformal structures
R e m a r k 6.2.3 (i) If C is a conformal structure on a connected and closed manifold M of dimension n > 3 which contains a Riemannian metric g • 6 ( M ) with vanishing scalar curvature S and non-vanishing Ricci curvature Ric. Then the equivalence class [C] is a boundary point of B+(M). Moreover, there exists a smooth curve
t e ( - T , T ) ~ gt e G ( M ) with gO = g and
dS t
!
[ > 0,
dt .t=o
where S t denotes again the scalar curvature of g t To see this, we first note that, according to [13], Theorem 4.44, a sufficiently small neighborhood U of g in 6 ( M ) carries the structure of an ILH-manifold. Moreover, 32 (M) decomposes into $2(M) = TgU ~ Coo(M) . g .
(6.2.1)
On the other hand, we have the L2(g)-orthogonal decomposition S2(M) = S~;a (M) @ C ° ° ( M ) . g . Here, we have set
S~;o(M ) := {h • S2(M) : Trah = 0} . By assumption, the Ricci curvature Ric of g is a non-trivial element of S2;a(M). Consequently, the projection hRic • TaU of Ric with respect to the decomposition (6.2.1) is also non-trivial. We now choose the curve t • ( - T , T ) ~ gt • 6 ( M ) such that
dgt [ dt t=o
= -hnic
•
Then (cp. the proof of Theorem 4.3.3),
dStdt t=o = (Ric, hnic)g = (Ric, Ric)g >0. Thus, if C is as specified above and n = 5, then it follows by means of Theorem 6.2.1(ii) that the distance of [C] to B + ( M ) with respect to I} is finite. (ii) Let M as in (i) and let C be a scalar flat, flat conformal structure on M. Hence C can be represented by a Riemannian metric g with vanishing
6.2 Infinite distance components of the boundary
107
scalar curvature. If the Ricci curvature of g would vanish, then also the Riemannian curvature of g would vanish. Then, according to Theorem 3.2.1, the space B + (M) would be empty. Consequently, the Ricci curvature of each Riemannian metric with vanishing scalar curvature which represents a boundary point [C] of B+o(M)does not vanish. Therefore, (i) and Theorem 4.3.3(ii) indicate that, for n >_ 5, the distance of such [C] from B+(M) with respect to Ij is finite. To prove this rigorously, more informations concerning the embedding of B+o(M)into B+(M) are needed. []
References
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Index
-
Apanasov, 25 Arakelov, 13 metric, 13 asymptotic coordinates, 83 asymptotically flat Riemannian manifold, 83 Aubin, 5 Bartnik, 84 Bergman metric, 12
---
of the Dirichlet problem, 13, 59 of the Laplacian, 12 of the Neumann problem, 64 - operator, 12 Gromov, 38, 57 -
-
harmonic radius, 14 Harnack inequality, 7 heat equation, 98 kernel, 98 Hersch, 14 -
-
canonical metric, 25 Christoffel symbols, 93 con formal class, 2 Laplacian, 16 manifold, 2 flat, 3 - map, 3 - normal coordinates, 89 radius, 14 structure, 2 flat, 3 - - scalar flat, 17 - - scalar negative, 17 - - scalar positive, 17 transformation, 4 - - loxodromic, 40 conformally - diffeomorphic, 4 equivalent, 2 critical exponent, 27 -
injectivity radius, 73 Izeki, 38
-
-
-
-
-
-
-
-
-
-
de Rham, 39 discontinuity domain, 27 divergence operator, 43
Kleinian - group, 27 - - elementary, 27 - - geometrically finite, 27 non-elementary, 27 - manifold, 27 Kobayashi, O., 56 Kobayashi, S., 25 Kuiper, 5 Kulkarni, 25 Kulkarni-Nomizu product, 1 -
-
Laplacian, 3 Lawson, 38, 57 Lebesgue space, 6 lens space, 37 Leutwiler, 14 limit point, 27 set, 27 Liouville theorem, 4 -
-
Fischer, 42 Green function - - of the conformal Laplacian, 18 -
MSbius - group, 4 transformation, 4 -
mass
116
Index
- of a Riemannian metric, 85, 88 of an asymptotically flat Riemaanian manifold, 83
-
Sobolev space, 6 stereographic projection, 19 strong maximum principle, 7 Sullivan, 28
Nayatani, 26, 81 Ob~a, 4 Patterson, 28 Patterson-Sullivan measure, 28 Pinkall, 25 positive mass theorem, 36, 84
theorem - of Lelong-Ferrand and Obata, 4 - of Rellich-Kondrachov, 6 trace, 43 Tromba, 42 Trudinger, 5 volume element, 2
Ricci curvature, 1 Riemann moduli space, 38 Riemannian - curvature, 1 metric - - flat, 3 - - locally conformally flat, 3
weak solution, 7 Weyl curvature, 2 Weyl-Schouten theorem, 3
-
Yamabe functional, 5 invariant, 5 metric, 70 - operator, 16 - problem, 5, 36 Yau, 27, 29, 36, 41, 84 -
-
scalar curvature, 1 Schoen, 5, 27, 29, 36, 41, 84 Schottky group, 81 Schouten tensor, 2
Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Sch~iffer,Griinstadt
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