This completes the proof.

It is now easy to prove the

regularity of a strictly convex solution u.

Theorem 2. A strictly convex solution u of det D,Ju = dp

with the Cl,° norm depending only on the Lipschitz (dµ satisfying PI) is norm of u, and on its strict convexity. Proof. Subtract from u is supporting plane at Xo = 0. The conclusion of Lemma 2 is invariant under affine transformations, and therefore, it follows after renormalization of the level surface u = 2-k and iteration of the lemma that

h2-k <'ykhl. Since u is Lipschitz,

h1(X)
PAIrr 3. A PRIORI ESTIMATES OF SOLUTIONS TO MONGE AMPERE EQUATIONS

45

Then

h2_k(X) < (2-k)aIXI < C2-k. provided that

IXI < (2 k)(I-°). Since u(X) < h2-k (X) as long as h2-k (X) < 2-k, it follows that u(X) < 2-1; provided that IXI < C2-k(I This allows to estimate the slope, s, of a supporting plane at say IYI by

C2-k(I-a)

I8I < C2-ka < IXII-a.

This is because the vertical component of such a plane must remain smaller than 2-k on the ball of radius IYI around Y.

PART 4 Interior W2'p Estimates for Solutions of the Monge Ampere Equation In this chapter we adapt the techniques developed in Part 1 to prove interior estimates for solutions of perturbations of Monge Ampere equation. Our two main results are as follows:

Theorem 1. Let u be a (convex) viscosity solution of

Mu = det Dou = f on the convex set 1 C R". Normalize fZ so that

B1CStCB". (see Lemma 1 below) and assume further ulan a 0. Then:

a) Given p < oo, there exists an e = e(p) such that if

If-11<e then

u E W2'P(B1/2)

and

C(0b) If f is continuous and strictly positive, u E W2'P(B112) for any p < oo and IIUIIW3.P(B,,,)
with a the modulus of continuity of f. The previous result in this direction is due to Pogorelov [P11 but valid only for

n=2.

A simple consequence of our techniques is

Theorem 2. If f E CO, then u E C2'". In this direction, Urbas has similar results provided that f is Lipschitz and D,Ju satisfies some integrability conditions (see [U]). 47

48 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

1. Preliminary Results We first recall a result on solvability and a few a priori estimates. Let ft be a convex domain in R. We state a normalization theorem of John Cordoba and Gallegos (see [dGJ).

Lemma 1. Let ft be a convex bounded set with non-empty interior. Let E be the ellipsoid of smallest volume containing Cl.

B.

Then nE C ft. In particular after an affine transformation T, B1 C T(ft) C

We also recall some interior estimates and solvability of Monge Ampere equation (see [G-T]).

Theorem S. Let ft be a strictly convex smooth (normalized) domain in R" (that

is, B1Cf1CB,,). Let f > O,g be smooth (C4) functions on n, M, then the following problem has a unique smooth (C2) solution w: convex in Cl

Iw wlen = g det D;1 (w) = f

in r.

F16rthermor e, if f =- C, g =- 0,

IIw1Ic3.1(f1_6) < C(b)

(f1_6 = {x E f)/d(X,Oft) > b})

with C(b) independent of the smoothness of A.

Proof of the last estimate (Pogorelov's). By taking a slightly smaller section (Cl' = {Xlw < -C6'j) we may assume that Vw is bounded in ft', and by taking a multiple, we assume Iwl < 1.

Thus,

Iowl < C1 We bound the maximum of

fw <

C.

(Cb" - w)D11we1/2(DD)2

= = eh by a constant M, depending only on sup Jul and sup IDiul. Let M be such a maximum, taken at an interior point, say zero. IuID11ue1/2(D,u)'

We repeat Pogorelov's computation: By an affine transformation

_

x1 = x1

- D1iu(0) DI1u(0) x`

which leaves h invariant, we may assume that D;,u is diagonal at zero.

PART 4. INTERIOR W2.P ESTIMATES FOR SOLUTIONS OF MONGE AMPERE EQN.

49

Since h = log u11 + Iu? + log(-u) has a maximum at zero, the linearized operator L = ,a7 o Diih cannot be positive, i.e. Lh < 0. That is

u11 L(uu)

- (u 1)2 ui (U112)'/ (uli)2 + 1 Lu

+ u1Lu1 +

u

uei

< 0. - (u1)2 uii

Now

Lu = n, Lu1 = 0, Lull = (ulkt)2 ukkutt

Thus, we get

-

(ulkt)2 ] ull ukkult 1

+1611 + n - (ui)2 5 0. - (U11)2C-(ulli)2J uii u \uii 1

1

We now use the factors k = 1 to cancel the i terms, i.e. 1

ull and, to control

[:(ulkt)2 + ul l + n (ui)2 <0 u uii k#1 ukkull

we use that, at zero uu11i

ull

+ ululi + u1 = 0. u

Fori3& 1, this means u2

_ u2(ulli)2

uii

(ull)2uii.

Since Jul < 1, this leaves us with

ul2

ull 5 ull + n, u or multiplying by u2ulle(u1)'

0

M2 < (u1)2 M2+el/2(u1)2nciM. (u11)2

Finally, an approximation lemma on the image of the gradient mapping:

Lemma 2. Let 6 > e > 0, and u, v two convex functions in Bl with Ilu-vliL. < e. Then (with N26 the 26-neighborhood)

{VuIn1_,,,} C N26{VvlB,}.

50 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

Proof. Consider v` = v + e - 6(1 - r2). Then V*188, > u and v*IB,_C,a < U.

Therefore any plane of support of u in BI-E/6 is a plane of support of v` in B1. That is {VUIB,_E,5 } C N26 {Vvla, }

.

Remark. (Pogorelov, see [P2]. If u,,, -+ u uniformly on compact sets of B1,

mess (Vu(Bi)) < limmeas

Proof. UD Vu(BI) C where lim(Vun(Bi)) = UO° 1 nkn (Vuk(Bl)) and D is the set of

IV = VL}

with L a supporting function that coincides with u in a segment [XI, X2] with

IXII<1.,IX21=1. We show that

v, = D n Du(Bj_,)

has no Lebesgue density point.

We may assume L =0, and X2=-el. In particular u Don B1 andu=0on -el. In particular, for any X in BI_E,u(y) > u(X)+(Du(X),y-X) for any y E B1.

Taking y = -el and noticing that u(-el) = 0 = 0, u(X) > 0, we know that (Du(X ), X + e1) must be nonnegative.

Since IX I < 1 and (X + el, el) > e, if ul is negative, it must satisfy elu, l < 21Vul.

That excludes a cone (of aperture - e) around -en, and 0 is not a density point for DE. Since we have shown that DE has no Lebesgue density point, so DE has zero Lebesgue measure and therefore V = Uk 2D1 /k also has zero Lebesgue measure. It follows immediately that

meas (Vu(Bi)) < meas (lim(Dun(Bj)) < limmeas (Dun (B1)).

2. Approximation lemmas We first recall the definitions of weak and viscous solutions, and compare them.

PART 4. INTERIOR W2 ESTIMATES FOR SOLUTIONS OF MONGE AMPERE EQN.

b1

Definitions. Let f E L°O,u be a convex function in ft. Then (see Pogorelov [P2]) a) u is a weak solution of Mu = f if the Jacobian of the transformation

T:X -Vu, coincides, as a measure, with f, that isJfdx. mess Vu(O) =

b) u is a viscosity solution of Mu = f, as before, if it is a sub and super solution, that is: u is convex and for any C2-* convex function v, with u(Xo) = v(Xo),u < v (reap. u > v) near Xo, we have Mv(Xo) > f(Xo) (resp. Mv(Xo) < f ). We should remark that the above definition is valid if f is continuous. If f is not continuous, we only care about the case when If - 11 < e and Mu = f should be understood as 1- e < Mu < 1 + e and this is what we really need.

Lemma 3. a) A weak solution is a viscosity solution b) if f < fo (a constant) in a ball Ba and u is a viscous solution meas(DuIB.) <- foI BE

(In particular, both definitions are equivalent for continuous f).

Proof. a) Suppose that u - v has a strict local maximum at X0. We may assume u > v in a neighborhood 0 and u = v on 852. Then Du(O) C Vv(O) and mess (Vu(O)) <- mess (Ov(O)). On the other hand

mess (Vu(O)) =

oMuJ

v

My

= meas (Ov(O)), a contradiction. b) Heuristically, we compare in BE, u with w with aBe

and Mw = fo. Then w
fw, I

= aB.

smoothing of u SB. U- 18B,

Mw, = fo where u, is chosen so that uo I BB. <- u1 as.

52 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

We compare u with w, in BE where woI SBe = UaIaB. < UJOBe

and Mw, = fo. Then w, < u and mess (Vu(BB)) < meas(Vw,(Be)) = foIBFI. First an obvious lemma.

Lemma 4. ("Close data imply close solutions"). Let f1 continuous, f2 smooth, and u a viscosity solution of det D,Ju = Mu = f1 with

f2
u=0 on iin. Let w be a C2'° solution of

Mw=f2 lw=0 onOfZ. Then

(1+e)1""w
Lemma 5. ("An equation for the difference of solutions") u as before (Mu < 1+e in the viscosity sense). Let r(u - v) denote the convex envelope of (u - v) where v is a convex C2,° function. Then (X the indicator function)

MI/" (r) < [(1 +e)1l" - M11"(v)1 . Xr(u-v)=u-v-

Proof. Heuristically this follows from the fact that a) m(r) is always supported at the contact set r(u - v) = (u - v), and b) that at such points both D,j (u - v) and D;j(u - v) and Dijv are non-negative matrices and that det1l" is a concave function of non-negative matrices. Hence

M1/"(v) + M1 ""(u - v):5 M1/"(u). For a more formal argument, we first point out that is is well known that

det D,,r(u - v) is (as defined by the gradient map) a measure supported on the set

{I'(u - v) = u - v}.

PART 4. INTERIOR W2.v ESTIMATES FOR SOLUTIONS OF MONOE AMPERE EQN.

53

(See the proof of the remark after Lemma 2.) Therefore, it is enough to prove that for any ball BE(Xo),

vol (Imvr(u - v)) (B,) < {1 +0(1)}IBBI((1 +e)I/' - MIi"(v)J If we substitute u by a minorizing sequence u(k) of solutions with smooth right hand side 1 + e and smooth boundary date gk I OB, -+ ulas. the result is true for any Xo from the heuristic discussion above. Since uk converges uniformly to u°O, with u°° < u, u°D l as, = u, r(uk - v) does

so to r(uO° - v) and therefore, for any ball Be

Im (Vr(u - v))(BE) c Im(vr(u°O - v)) C limlm(vr(uk - v))(RE).

3. Tangent paraboloids in measure and level surface estimates The next lemma and its corollary are crucial for our estimate. They give us the tool to control second derivatives. Lemma 6. Let u be as above, namely, u convex, Mu = f, ulecl = 0, B, C fI C B,,. Assume that 1 < f < 1 + e. Let w be the solution of

(Mw=1 onQ t w=0 on M. Then,

I{r(u - 1w) = u - Zw} n BI/2l

> 1 - b(e)

IBI121

with b(e) = CEI/2. Where C = C(n) > 0 depends only on the dimension n.

Proof. We will estimate

{F(u_w)

=u-

2w}1'

through the image of the gradient mapping. Since

[(1+e)I""-1/2]w
)I/" - 1/2] w < r(u - 1/2w) < 2w.

From Lemma 2

N_6{Vw} 2l CN6{vi u-wlL1,2} ( /

But v2-w

54 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

is an a priori smooth set of volume exactly 1

2,,

IB1/2(1-,/6) I.

Therefore

Vol{vr(u - 1/2w)IB,1,} 2! 21 IB1/2(1_,/6)1 - Co.

{Vw}

Here we have used the fact that

IN-6{v1

HBI/20-0011>

I-C6

with C = C(n) depending only on dimension n. The above inequality relies on the fact that {VZwIBII,(1- c/b)} is a smooth deformation of B1 since we have all the derivative estimates of w. We now use Lemma 5 to conclude that 2n IB112(1-,16)1 - Cb < Vol

u-

(ImVl'

w

1B1,2)

from Lemma 5,

((I +)1/n-2}'i{r(u-,1-'wu-2w}nB1/2 <_

1

2n +

Cc) l Ir (u - 2w) = u -

5w

I

r1 B1/2We

now choose b = s1/2 and the lemma is/complete.

Remark. At any point X0 in {r(u-1/2w) = u-1/2w}, there exists by definition a linear function Lxo such that in all of fl

(u- aw) (X) Lxo(X) { Lxo(Xo) _ (u - 2w) (Xo) Therefore

Lxo(X)+iw(X)


1 Lxo(Xo) + lw(Xo) = u(Xo).

Corollary 1. For any point Xo E B1/2 in {r(u - 1w) = u - 1w}, u has a global tangent paraboloid by below of the form

Lxo +

I (X - Xo)2

where N is a universal constant. Our next objective is to study how the level surfaces of u deteriorate around a point. The idea is the following:

PART 4. INTERIOR W2.p ESTIMATES FOR SOLUTIONS OF MONGE AMPERE EQN.

55

Suppose that u is a non negative solution, with u(O) = 0, and let us look at the level sets say {u < 2-k} = Sk. From the general renormalization theory Sk looks like an ellipsoid of volume approximately equal to (2-k)n/2, since when we renormalize Sk to Sk, inf u' becomes of order one (thus, from the renormalization formula, T = Tk).

(detT)2/n , 2k, Since we do not expect u to be C", that means that the eccentricity of Tk will deteriorate. By approximating u, and iterating, we will prove that the eccentricity of Tk deteriorates as slowly as we want if f is closer and closer to a constant.

Lemma 7. Let u be a non negative solution of

Mu=f, If-1I<<e with a) u(0) = 0.

Q) Ulan=1. 'y) fZ ", BI Let

So={X :u<µ},e<posmall. Then, there exists a quadratic polynomial

P with

(xi? ai

a) II!=1 b) 0
such that u - PI s,, < C(µ3/2 + (µe)1/2 +e). If furthermore we assume the more precise bound

Bvi_e(Xo) C fE C B f+s(Xo). Then, we get the more precise bounds b')

(a;-2I
Iu - PI < C[9µ312 + (µe)1/2 + e).

Proof. We use the approximation of u by w, solution of

w1an=1, Mw=1. Then Iw-ui < e, and w is C',I and thus C2'° (since M becomes uniformly elliptic) and thus C3 by Shauder estimates, in say the section {X : w < 7/8}. Thus, near the origin. w = w(0) + (Vw(0), X) +

-, x + 0(IXI3)

56 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

P will be precisely E e x?. We estimate Iw(o)I = tw(o) - u(O) <_ e and

IVw(O)I < eI/2

since w> -e in all of fl. In particular S. c {X : w < p + c} that for p small is contained in Bc(µ+o) Substituting in X, we get the desired estimate. To improve the estimate in case ry', we notice that, with that hypothesis

11X-Xo12C95w< lIX-XoI2+C9. Therefore 1 v=w-_IX-Xo12

satisfies a concave, fully non linear uniformly elliptic equation

det(D2v + 1) = 1 that carries C2'2, and thus C3, estimates: 11w - IX - X01211c9({w<7/8}) <- Co.

In particular, we can improve the estimate of the term O(IX13) by 9(µ + We now use the previous estimates to produce C,° and C""l (thus C2,Q, thus C3) estimates. c)3/2.

Lemma 8. Assume that If -Illsµ 5 e(p) where we are interested in the two cases. a) e(µ) < co, and b) e(µ) < le for some s. Then, (p < µo, small but independent of e) there exists polynomials

Pk = E xa, such that for p = (po)k, we have

Ju - PII S,. <_ CA(e(p)°) (for some o) and

sup ai(k)

supa;(k - 1)

< 1 + Ce°.

Proof. There is a distinct first step, before starting induction to carry the shape of Cl to be almost a ball.

PART 4. INTERIOR

ESTIMATES FOR SOLUTIONS OP MONGE AMPERE EQN.

57

Step 1. We are in hypothesis 'y, and thus on Sµ, Iu - PI <- C(µ3"2 + (1+EO)112 + CO)

with

0
i

Then, on 8(Sµ)' we now have

u'=1, andl(u'-IXI2)= 11

601/2

- 2IXI2I 5

A

We chose µ

/2 +

JA

and we get now our starting configuration: (we can forget about *'s) u = (old u') = 1 on OS1 (old BS,,).

B f+Cel/4 Now we do the inductive step. Assume that we have proven step k. As before, we renormalize in two steps, first by making P = IX a,2; x, and then 2 I2 i.e. x, -+ expanding by 141/2, i.e. Us = MU(pI/2X), (this does not change P = P' =

2IXI2).

From the inductive hypothesis B,-2-CC'(µ) C S' C B f+Cc,(µ)' and

I(f -1)1-< E(µ) Thus, we are under hypothesis', and on S;. = {u' < po} we have

Iu' - PI 5 C[E°(,

)µo/2

+

1/2E(µ1/2

)+

and

16t; - 215 C(c°(µ))-We renormalize and we get for µ = µoµ

Iu - Pk+,l 5 µ?7[E°(µ)µo/2 + po/2E1/2(14) + e({'1)].

If co is much smaller than µo, we can say Iu - Pk+I I < Cppoe in case 1, and

lu - Pk+1I 5 Cµµo(µµo)° in case 2.

About the coefficients of Pk+I, they are the renormalization of P by the transformation that carries zlxl2 into Pk and thus the new largest a;(k + 1) is smaller or equal than sup a, (k + 1) < sup(a; - 2) sup aj (k).

i

W L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

Corollary. a) The gfne transformation of determinant one that makes Su equivalent to a ball of radius µI/2 has norm (for µo < µ < µo-I) 1ITlhII:5 H

b) If the infinite product above is convergent, for instance if e(p) < p", u is

C" at the origin. c) Given a < 1 there exists eo(a), so that if e(p) < co, u at the origin is Cl"°. Finally, before entering into the W2,P estimates; a consequence about the geometry of sections. We recall that if Lx,, is the supporting plane to u at Xo, we define the section

St,x0={u
The previous approximation theorem, says that as e becomes small, sections look like ellipsoids whose eccentricity may deteriorate, but at a slower and slower geometric rate as we let c become smaller. We are now ready to prove the W2,p estimates. At this point it is convenient to review the geometric properties of the graph of u, solution of

a) Mu=f

b) If-1I<e(p)

c) u4,=0,BICIlcB,,. We recall that we denote by Sa,xa the section

{X .u - LXo < µ}. S', u' denote the corresponding renormalizations

S' = T(S).

U* = u(T-'(X)). where S* - Bi

and

Mu' = f(T'I(X)) - 1. We also can decompose T = µI 12Q where

detQ = 1. By now, we have proven the following properties (for e small, some power a, small, independent of e).

Property 1. B

C T(S) = S' C

B2--Ce.

PART 4. INTERIOR W2"P ESTIMATES FOR SOLUTIONS OF MONGE AMPERE EQN.

Property 2. I IQI I, I IQ-' 115

59

p-T(E)

with r(e) as small as we want for a small. (lemma 8).

Property 3. (Rescaling of property 2.)

If q
Property 4. (Adjacent sections.)

IfyES,,(X), IIQµ(y)QA'(x)II,IIQr.(X)Qµl (Y)11<- (I+Ce°).

(This follows from the fact that the quadratic approodmation to u is SK,,(Xo) as being almost a circle.) determines, for K a large, constant, the shape of

Property 5. (Corollary of Property 3.) If we normalize S,,,& to be a unit ball and q < p, B(n)l/2+,(.) C T,,(S7) C Bl.2ilI2-,(<)

Property 6. If u has a tangent paraboloid by below, (A large)

u>

Ixf

then u has a tangent paraboloid u < ,\n-11X12

by above.

Proof. The paraboloid by below puts a uniform bound IIQ,.II < A.

Since det Q = 1, this gives a bound by below. We finally recall lemma 6 and Corollary 1 combined.

Property 7. If we renormalize S,, to S* - B, and it to u', then u' has a tangent paraboloid by below

IX-X012 in a set E C S,, of measure 15,,\E1 < Ce112

for some universal A0, and the last tool is a finite overlapping theorem:

60 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION

Property S. Let A be a measurable subset of 0, so that for every X in A we chose a section S,,,x. For any 5, there is an a such that, we can extract a countable 11 < e) covering of A, S,,k,xk, so that Sµk (1-6),xk

only overlaps a finite number of times depending only on dimension. of decreasing paProof. We extract the standard subcovering by sections rameter µk, such that Xk does not belong to the previous sections in the covering. Let now X0 be a point that belongs to S,(1_6),x for a sequence of sets which we relabel S{.1,X1, ... , Spk,xk.

We renormalize the first one to make it a ball, by the affine transformation T, ,x, , and transform all the other by the same T,,,,x let us call these new sections Sµ.X, where now p1.X1 N B1

and all the pj < 1 (the new pj are the old p,/µ1.) If all the sections S,,,x, would satisfy, say Br, /4 C

C

4r,

for some r., they would overlap a finite number of times

M - 4" according to standard differentiation theory. From property 4 and property 5, this will happen for those p, such that

µ-r(e) < 4. Now, since X0 E S; (1-6),X1 Sµt,X1

B1

and XS V S;1,X,

any other section that contains Xo must have diam S' - 5. From properties 4 and 5, this means 1/2-r(e) > 5.

So, given 5, if we chose r(e) small, the inequality µ1/2-r(e) >

guarantees that

p-r(e) <

and the property is proven.

We are now ready to prove theorem 1.

4

PART 4. INTERIOR W2.v ESTIMATES FOR SOLUTIONS OF MONGE AMPERE EQN.

61

Proof. Our starting configuration is a renormalized section SI,0, such that for any point Y such that S1,y touches S1,o, all properties P1 to P7 are satisfied. (This is achieved by renormalizing to size one a small section around any given point X well inside fl). At this point, we retrieve the N from Corollary 1, and define A = N". Our good sets Ba., will be those Xo for which the transformations needed to renormalize S,,x0 are all of norm less than Ak, i.e.

Ba. = {Xo : (IQµfl < Ak, for every p < po}. In particular for such an Xo, u has tangent paraboloids by above and below of the form

-X012
Lxo

IX-&F

aklX The bad set Aak will be the complement of Ba., and we will show fast polynomial decay of the measure of AA.. First we point out that if a section S,,,,, has the property that IIQaII > t then p-r(e) > t (property 2), i.e. 14:5 t= t-M(0 with M large. In particular

diam(Sp) < p1/2p-r(e) < t-. We now find an inductive relation between A,,.+i n Bk+1 and Aak n Bk.

where Bk is the ball with radius vfl- = 1/2, rk+1 = rk - CA-

4M(t)

For X E Aa.+I n Bk+1, let S,,,x be the first (largest) section for which IIQµII > '\k-1-

We note the following properties: If we renormalize S4,,, after renormalization we have that every point Y' in S;,, except for a set of measure c' /2, u has a tangent paraboloid by below of the form

Ly.+IY-Y"I2=Ly.+A"IY-Y'I2 and by above

Ly. -}- AI Y - Y' I2.

This means after renormalizing back that, except for a set of measure a"/2ISpI every

point Y, in Sµ belongs to Ak-2\Ak+I

Indeed, the renormalization Q,,,x transforms already S,yinto a set trapped between two circles

A-I < IY - Y'I2 < AI/". Therefore the renormalization Q,,,y differs from Q,,,x by at most a factor of A that is

Ak-2 < IIQ,,.YII < \k+1

62 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE

EQUATION

As for those sections S,,,y with r1 not in 2p, that implies that rl > 2u and then, property 4 says that

(1+Ce°), thus IIQ,,(Y)II < IIQn(X)II(1 + Ce") < Ak-I(1 +Cea). Let us now extract a covering of B,.k+,nA*+, by these sections S,,,x, so that S,(1_6) has finite overlapping. Notice that Sµ,x is contained in Br,, since

diam SS,,x < a- "4L' < f -

k -+1

and that rk \ r,o > a if we chose e small (remember that J1 is universal). Then, the standard computation yields I Ak+I n Brk+l I <- > IAk+I n Brk+, n S,,,x, I

j :5 ECI12IS/+,.Xf1 !5 EE1/2(1+6)n I'S"(1-6).X, I : Ee112(1 + 6)n(l + e'12) I "Sµ,(1-6).X, n (AA,-2 n Brk_2)I < CC'/2(1 + 6)"(1 + 61/2)4nIAk-2 n Brk-2

(from finite overlapping, property 7). This completes the proof of the theorem.

Theorem 4. Let r be a convex set in R"

B1crcB". Assume that, in the viscosity sense, or satisfies

Q(X) = f(X,v) for 9 the Gauss curvature of or at X, where

a) 0
b) f is continuous in X and v. Then, or is locally a W2. p graph for every p.

If b') f is C" in X and v, then Or is locally a C2"0 graph. Proof. Or is strictly convex and C' from [C2]. Theorem 5. In Theorems 1 and 2, replace Ulan = 0 by

Ulan = 9

with &I E C'+" and g E C'+", with a > 1 - n; then, the same conclusion holds. Proof. We may localize, by Corollary 4 of Part 2.

PART 4. INTERIOR W2.p ESTIMATES FOR SOLUTIONS OF MONGE AMPERE EQN.

63

References [A]

[B]

[Cl] [C2] (C3]

[C-L] [dG]

A. D. Alexandrov, Majorization of solutions of second-order linear equations, Vestnik Leningrad 21, 1, (1966) Russian. English 'Translation in A.M.S. Transl. (2) 68, (1968), 120-143. Y. Brenier, Polar factorization and monotone rearrangement of vector valued functions, CPAM Vol XLIV (1991), 375-417. L.A. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Annals of Mathematics 130(1989), 189-213. L.A. Caffarel i, Elliptic second order equations, Rendiconti del Seminario Mathematico e Fisico di Milano, Vol LVIII (1988), 253-284. L.A. Caffarelli, Interior W2,p estimates for solutions of the Monge Ampere equation, Annals of Mathematics 131 (1990), 135-150. M. C. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1-42. Miguel de Guzman, Differentiation of Integrals in R", Lecture Notes 481, Springer-Verlag.

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333-363. [G-T] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of [E]

Second Order, 2nd Ed., Springer-Verlag, New York, 1983. [I]

[J)

[Je] (K-S]

[Pl]

H. Ishii, On uniqueness and viscosity solutions of fully nonlinear second order elliptic P.D.E.'s, Comm. Pure Appl. Math. XLII, no 1 (1989), 15-46. R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), no 1, 1-27. D. Jerison, Harmonic Measure in Convex Domains, Bull. Am. Math. Soc. N. V. Krylov and M. V. Safonov, An estimate on the probability that a diffusion hits a set of positive measure, Soviet math. 20 (1979), 253-256. A.V. Pogorelov, Monge Ampere Equations of Elliptic Type, Noordhoff,

[P2]

Groningen, 1964. A.V. Pogorelov, Extrinsic geometry of convex surfaces, Translations A.M.S., Vol. XXXV, 1993.

(Pu]

C. Pucci, Limitzioni per solution di equation elliptiche, Ann. Mat. Pura

[U]

Appl. 74 (1966), 15-30. J. Urbas, Regularity of generalized solutions of Monge-Ampere equation, preprint.

The Moser-Trudinger Inequality and Applications to Some Problems in Conformal Geometry Sun-Yung Alice Chang

IAS/Park City Mathematics Series Volume 2. 1996

The Moser-Trudinger Inequality and Applications to Some Problem in Conformal Geometry Sun-Yung Alice Chang Introduction This is the set of lecture notes for a graduate course which the author gave at the Summer Geometry Institute at Park City, Utah in the summer of 1992. The main theme of the notes is the study of extremals of Sobolev inequalities with applications to some non-linear partial differential equations arising from problems in conformal geometry. Some effort has been made to make the notes self-contained. However, the intention is to keep the material at basic and introductory level, thus many recent research articles in the subject are not covered or mentioned. The notes are organized as follows: In Lecture 1, we cover some background material concerning the notion of eigenvalue, the heat kernel and Weyl's asymptotic formula. In Lecture 2, we discuss the trace of the heat kernel, the concept of logdeterminant of the Laplacian operator and the Ray-Singer/Polyakov formula for the zeta-function determinant on compact surfaces. In Lecture 3, we discuss a limiting form of the Sobolev inequality called the Trudinger-Moser-Onofri inequality, we also have as an application of this inequality the isospectral problem on compact surfaces. In Lecture 4, we establish the existence of extremals for Moser's inequality. In Lecture 5, we discuss the higher dimensional analogue of Moser's inequality and application to the extremal and compactness problem for zeta-function determinant of 4-manifolds. In Lecture 6, we return to the "isospectral" problem on three dimensional manifolds, we discuss the relation of the problem to the "Yamabe" problem, and the role played by the first eigenvalue in the spectral compactness problem. Finally in Lecture 7 we discuss the role of Moser-TYudinger type inequalities to the problem of prescribing curvature equations in conformal geometry. The author benefited from discussions and suggestions during the preparation of the lecture notes from many colleagues and friends. She would like in particular to thank Matt Gursky, Jie Qing, Wensheng Wang and Paul Yang for their input in the notes. IMathematics Department, U.C.L.A., Los Angeles, CA 90024

E-mail address: changiath.ucla.edu © 1996 American

67

Soci, Iv

LECTURE I Some Background Material 1. Preliminaries Let M be a compact R.iemannian manifold with boundary OM (OM may be empty). In terms of local coordinates (x', ... , xn) the metric can be expressed as ds2 = E gi,dx'dx3 and the Laplace operator is defined by

a

a)

A= 1 E 8x' (f9ii 8x3 where (gi3) = (g.3)-', g = det(9i3)

Given f E COO (M), define a Hilbert space norm by

If

Ifl2dx+ fMIvfl2dx}j,

and denote by W"2(M) the Sobolev space obtained by taking the closure of COO (M)

in the above norm, and by W0I'2(M) the closure of Co (M) (C°° functions with compact support in M) in the above norm. (a). If OM is empty, then A is a self-adjoint operator on W',2(M). According to the spectral theory of self-adjoint operators, A has discrete eigenvalues,

0=Ao
-AO. = k0i where 4i E C°O(M)fW1"2(M) can be chosen to form an orthonormal basis of Wl'2(M) (Parseval Theorem). (b). When OM is not empty, we should specify some boundary conditions so that A is self-adjoint. There are two standard conditions: 69

70

S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

(1) Dirichlet boundary conditions, in which case Dom(i) = W0'2(M), and the eigenvalues

0
-Aoi = Ai0i 4)iIaM = 0

Oi E C°°(M).

(2) Neumann boundary conditions, in which case Dom(i) = W1"2(M), and the eigenvalues

and the corresponding eigenfunctions satisfy -o-Oi = AiWi ao` I8M = 0,

Oi E C°°(M).

Some Basic Examples. i = 1, 2, ... , and the 1. Let M = [0, a]. Then the eigenvalues are Ai = corresponding eigenfunctions are sin i' for the Dirichlet boundary conditions; and the eigenvalues are Ai = ce, i = 0, 1, ... , and the corresponding eigenfunctions are cos? for the Neumann boundary conditions. 2. If M is a domain in R", then in Euclidean and polar coordinates,

0=

n

82

=T1_" 19 (r"'1

)+

agn-t.

In the special case when M = D the unit disk in R2, using separation of variables, we consider u(r, B) = f (r)h(B). Then Du + Au = 0 is equivalent to the following equation: r2(f"(r) + ,-1,f'(r) + Af(r))

=

h"(B) = c = constant. h(B)

f(r)

Since h(B) is periodic in B with period 2-7r, c = n2 and h = acosnO + bsinnO. Set f (r) = y; we have (*)

r2y" + ry' + (r2A - n2)y = 0

To find A we solve (*) with y continuous at the origin and y(1) = 0. Let

rV=p(A t0). We setA=k2,thenrk=pand

z +--+(1- P )y=0 d2 y

1 dy

up-

P pup

n2

LECTURE 1. SOME BACKGROUND MATERIAL

71

Solutions of (**) are called Bessel functions J(p). Actually all eigenfunc-

tions are of the form J(fr)(acosnB+bsinn8). Remark. It is a classical result of Faber-Krahn that the ball realizes the smallest Al among domains (in R") of the same volume. A recent result of AshbaughBerguria [A-B], using special properties of the Bessel functions, proved that Al (for the Dirichlet problem) is maximal if and only if the domain Sl is a ball.

3. Let S"(r) = {x E R"+1, IxI = r}, and S" = S"(1). Given X E R"+1 write x = rt, for r E [0, oo) and C E S". It is simple to show that AR-+i F = r-"C7r(r"8rF) + AS"(r)(FI S..(,.)).

See [C, p 34], or [S-W, p 137].

If F(x) = rkg(t) (F is homogeneous of degree k), then ARr+1 F = rk-2(Asng + k(k + n - 1)g)

Thus F is harmonic on R1+1 if and only if g is an eigenfunction on S" with eigenvalues k(k + n - 1). It turns out that the space of homogeneous harmonic polynomials of degree k when restricted to S" is the eigenspace corresponding to the eigenvalue k(k + n - 1) of S" with dimension Ckn+k - +2 -2 [S_yq

Corollary. A1(S") = n with eigenfunctions {x1 Isn, x2ls., ... , xn+1 I Sn }.

2. Rayleigh Quotients We define the space H in the following way:

(1) If aM = 0, then H = If E W1'2(M)I fey fdx = 0}. (2) If OM 36 0, then for the Dirichlet problem take H = Wo'2(M), and for the Neumann problem take H as in (1). For

0=1\0C1\1 we have

Al

fEH of

Ai = inf {

fM I 0 f j2dx fM fa

IM I y f I2dx If E H, rM fOjdx = 0, for j = 1,2,...,i - 1}. fm f2dx

In particular, Al is the smallest value c such that ;

CMfJi2dx, for any f EH, M

72

S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

which is called the Poincare inequality [C, p.16]. In general, one likes to estimate Al from below by simple geometric quantities of the manifold, such as the isoperimetric constant, diameter of the manifold, volume

of the manifold, and so on. The reader is referred to [C, Lecture 3) for some examples in this direction.

Examples. 1. For the dumb-bell region, Al is very small when the neck is either thin or long.

2. (Hersch Theorem) For (S2, g) with any metric g with Area(M) = 4n we have Al (g) < 2 and Al (g) = 2 if and only if g is isometric to the standard metric on S2.

Exercise. If 0 is a conformal map between compact surfaces (M, gl) and (M, g2), then IM

< Dlf, V19 > dv1 =

JM

< V2(f - 0-1), 02(9.0-1) > dv2.

Exercise. Verify the following: N

N

fM uk d 2

L 1Ak=sup k=1 fM I V ukI2dv k=1

(*)

where the sup is taken over functions u1, us,

, uk

fM ukdv = 0 and JM < puk, pui > dv = 0 if 1

satisfying the conditions k. Equality holds if and

only if Uk is the k-th eigenfunction.

Proof of the Hersch Theorem. We take the standard coordinate functions x1, x2, x3 as the test functions in (*). However, to meet the restrictions it will be necessary to find a conformal transformation of S2 so that fs, (xk o O)dv = 0 for k = 1, 2, 3. This can be done by a topological argument. The remaining orthogonality condition follows from conformal invariance of the Dirichlet product. Thus we find, using fs, I V X1. I2dv = 3 for each k, 3 Al

3

>

Al

+

A2

A3

>

811

J5,(x)dv

Hence Al < 2. On the other hand, if Al = 2, equality must hold everywhere in the formula above. Hence x1i x2, x3 must be the eigenfunctions of the metric. This means that the metric is the standard sphere.

Remark. N. Korevaar [1(o[ has recently proved Ak < Ck on compact surfaces, and also an extension of this result to conformal metrics on higher dimensional manifolds.

LECTURE 1. SOME BACKGROUND MATERIAL

73

3. Weyl's Asymptotic Formula Let M be a compact Riemannian manifold (or bounded domain in RI). Let's denote N(A) = # of eigenvalues < A, where we count multiplicity. Then N(i1) - w,,Vol(M) (2

r)n

as A --+ oo, where wn is the volume of the unit ball in Rn. If A = Ak, then we get (14)

^ w(V 1(M)

Remarks. (1) In the case M is a bounded smooth domain, Weyl's formula can be derived by using the max-min principle (Rayleigh quotient). [C, p 32-33]. nk and (p")$ < W(2oi" for plane do(2) Polya conjecture: (Ak) > mains, where µk denotes the Neumann eigenvalues. Polya proved his conjecture for "plane-tiling" domains in 1961 [P]. (3) The formula means that N(A) = w- Vol(M) (27r)-

lim n-oo \t

i.e., Vol(M) is determined by the spectral counting function N(A). A sharper study of the asymptotic distribution of the eigenvalues is through the study of the trace of the heat kernel. Let's write Z(t) = > e-a,t = Trace a-tA.

Theorem ([MP, MS]). When t

Z(t) -

0+, Z(t)

(41rt)-i Vol(M). Actually,

(47rt) - 21 (ao + a j ti + a, t +

)

as t -+ 0+, where ao = Vol(M), and a; are given by the integrals of the local invariants of g. We will return to this theorem in Lecture 2. If we write

Z(t) = /

e-tadN(A)

0

then we see that Weyl's formula implies ao = Vol(M). It turns out that ao = Vol(M) also implies Weyl's asymptotic formula through an application of Karamata's Tauberian Theorem.

74

S.-Y. CHANG, THE MOSER.TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Remark. One cannot mention this subject without mentioning the problem of M. Kac, "Can one hear the shape of a drum?" N. Let (I be a domain in R2 with smooth boundary. I s ) determined b y its spectrum A1, X2 ,--- ,? i.e., can

we hear St (a membrane of a drum) if we had perfect pitch? Weyl's formula shows

that the area of 1 can be heard. Kac [K] proved that if 11 is convex, then L = the length of boundary of fl can be heard. In fact, L appears in the coefficient a4 in the asymptotic formula of Z(t) :

0 Zt

e

`

i=o

I

C _L

1

I

2rrt

4

for t - 0. Since a disk is the only region with L2 = 41rlntl, we have

Corollary. Circular drums can be heard.

Remark. Kac's originally problem was answered last year in the negative by C. Gordon, D. Webb and S. Wolpert [G-W-W] based on a method of Sunada [S]. The main statement is that there is a pair of non-congruent simply connected domains in the Euclidean space which have the same spectrum for both the Dirichlet and Neumann problems.

References The basic reference for this chapter is [C)

I. Chavel, Eigenvalues in Riemannian Geometry, Chapter 1 and 2, Academic Press, 1984.

Additional references: [G-W-W] C. Gordon, D.Webb, and S.Wolpert, Isospectral Plane Domains and Surfaces via Riemannian Orbifolds, (preprint). [K] M.Kac, Can one hear the shape of a drum?, A.M.S. Monthly 73 (1966), 1-23. [Ko) N.Korevaar, Upper bounds for eigenvalues of conformal metrics , JDG 1993. [M-P]

[M-S] [P]

S.Minakschisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Can. J. Math. 1 (1949), 242-256. H.Mckean and I. Singer, Curvature and eigenvalues of the Laplacien, J. Diff. Geom. (1967), 43-70.

G.Polya, On the eigenvalues of vibrating membranes, Proc. London Math. Soc. 11 (1961), 419-433.

[S]

T.Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 248-277.

[S-W]

E.Stein and G.Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton University Press, 1971.

LECTURE 2 Ray-Singer-Polyakov Formula on Compact Surfaces 1. Heat kernel In chapter 1 we have mentioned Weyl's formula: (Vol(Mk

C(n)

)

as k -+ oo. If we let Z(t) = E' e-)kt, Z(t) tends to (4it)- (ao) as t -+ 0+, is where ao = Vol(M). Z(t) is called "the trace of the heat kernel". Assume a sequence of eigenfunctions which forms an orthonormal basis for L2(M), then

Z(t) = j Ee-'I'Oj(x)O.(x)dV(x) M i=0

= J K(x, x, t)dV(x) M

where K(x, y, t) = E e-",'Oi(x).i(y) is called the heat kernel. Proposition. K(x, y, t) is the unique fundamental solution of the heat equation on M. It is continuous in M x M x (0, oo), C2 in x, y, and C' in t. That is, for any given bounded continuous function f on M, the function u(x) = f K(x, y, t) f (y) dV(y) satisfies the following equation:

ju(x) = Du(x) t

lim+ u(x, t) = f (x) on M.

75

78

S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Examples.

1. On S' = {z E C : IzI = 1}, the Laplacian has spectrum {n2,eine}0n0_--00 and dV = r'r . So 00

K(x, y, t) = E e-n2teinxe-tny. n=-o0

In particular, K(x, x, t) = Z(t) _

00

e-n2 which tends to sat

as t -'

0+.

2. On Rn, K(x,y,t) = (41rt) e- f' . Estimates of the heat kernel for general manifolds can be found in Davies [D], and in Li-Yau's paper [L-Y] for the case when Ric > 0.

2. Asymptotic behavior of the trace of the heat kernel The main tool for studying the asymptotic behavior of the trace of the heat kernel is the Pseudo-differential calculus (see, e.g., P. Gilkey [G, Lecture 3]). In the following we will briefly mention this approach. Recall the Cauchy integral formula:

f(z) = 2iri

f

c

where f is analytic in a neighborhood of the closed curve C and z is inside C. We apply this formula to the elliptic operator D : L2(M) - s L2(M). Define

f(D)=

l

fcJD

where C is a curve avoiding the spectrum of D. Then f is analytic outside a neighborhood of the spectrum of D.

Example. f$(z) = z', s E C and D = -

on SI. Let

_ Ea,,ei"x be a

function in L2(SI). We have

(D'¢)(x) =

1(J

v'(y)e-inydy)n2'esnx

= r(> n2aein(x-y))O(y)dy So kernel of D' is K. (x, y) = E n28ein(x-y) and in particular, K,(x, x) = E n2s which converges when Re(s) < -1. K. as a function of s, which is closely related

LECTURE 2. RAY-8INGER.POLYAKOV FORMULA ON COMPACT SURFACES

77

to the Riemann zeta function, is meromorphic in all C with only a simple pole at

s= - Z Take f (s;') = e-4. It turns out that a-t° is a infinitely "smoothing" operator. Taking a "good" approximation D f of (t I - D) -', and using the pseudo differential calculus on manifolds, we can get Theorem (Minakshisundaram-Pleijel [M-P), McKean-Singer [M-S]). For any positive self-adjoint elliptic operator D of order 2, we have

K(x, x, t) ti E 00 Bk(x)t

k-dru

as t

0+

0

where Bk is computable in terms of the symbol UD of D and derivatives of or,). If

8M = 0, then Bk - 0 when k is odd Let us denote ak = f B2k(x)dV(x). Applying the above theorem to D = A on a manifold, we get

Theorem [M-K]. Suppose (M, g) is a compact Riemannian manifold without boundary. Then

Z(t) - (4irt)-i (ao + al t + a2t2 + ) as t - 0+, where ao = Vol(M)

al=3 a2

fm

180

KdV

f(10A - B + 2C)dV r

where K is the Gaussian curvature of the metric (M, g), and where A, B, C are polynomials of degree 2 in the curvature tensor R;.1kl. The formula which we will use in this lecture is the following:

Corollary. When M is a compact closed surface (dim(M) = 2), then

l,. K = 21rX(M) = 27r(2 - 2g) hence

K(x, x, t) and

Z(t)

- area(M) +

-

T7rt + X(6 M)

K(x)

T2- + 0(t) as t

+

0+

K2 +O(t2) as t -+ 0+. nI

Corollary. X(M) is audible. Remark. A computation of al can also be found in [G, Ch 61.

78

s: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

3. Ray-Singer-Polyakov log determinant formula on a compact

surface without boundary Recall Weyl's formula Ak N

0 A` converges if Re(s) >

Then Ak ° - Cnk-18. In particular

vow

d`

2M=n Now let n = 2 and define the zeta function with Re(s) > 1 as ((a) = Ek>0 Ak°.

By using dd-°A-6 = -A -° log A, we can formally write

(,(a) = -EAk°log,\k and

-('(0)" =" r` log Ak

where we must notice that the last sum diverges in general. Now we would like to formally define det A = e-0°)" = "IIk IAk if ('(0) exists. To see S'(0) is well-defined, with Re(s) > 1, x > 0 we argue as follows:

x-d =

1 r(9)

e-stt°-Idt o

where r(s) = f0 e-tt°-Idt is the gamma function. So Ak ° _ and hence °°

1

(s) =

r(s)

787 fu

e-.\ktt°-Idt

00

Z ( k=I e

Ake)t°-Idt

°O

r(s)

(Tre-t° - 1)t°-Idt e

To see the behavior as t -' 0+ of the above integral, we apply Theorem 2 and get Tre-to

4 tt+X(s

Thus for Re(s) > -1 we have

((s) = r(s) II(4jt + X(6 ) +O(t) - 1)t-Idt+ analytic function in s For a close to zero, we may write

_

(s)

1

X(M)

A

I'(s) 4R(s - 1) +

(

6

1

1

- 1) s + s + 1 + analytic function in s

Here we have used the property that r(s) = fo e-tt°-ldt = a - a I +

+analytic function ins around s = 0. Thus S(s) is well-defined for Res > -1 and

LECTURE 2. RAY-SINGER.POLYAKOV FORMULA ON COMPACT SURFACES

79

meromorphic with pole at s=+1 (it is analytic at s=0). We know ((0) = 4 1-1, so we define C(o)

C(8)

('(8)18=0 =slim

s

which is our definition of - log det A.

Ray-Singer-Polyakov Formula [Pl] [P2] [R.-S]. Suppose (M,go) is a compact closed surface, and gl = e2ugo is a metric conformal to go with vol(M, gl) _ vol(M,go). Then we have

log

det det Ao

121r

,

M

(2Kou + [Dou12)dVo

where A;, (i = 1, 2) denotes the Laplacian operators with respect to g; respectively.

Proof. Since gl = e2ugo, we have Al = e-2rAo with volume A = fM dVl = fm e2idVo and

100

(Tr(e-`°°) - Tr(e-`I ))t8_I dt.

Co(8) - Ci(s) = F1

(a)

0

Thus

I -1

det Al = lim (o(s) - (i(8) s-o s

og det A0

= lim sr(s) Jorao (Tr(e-too) - Tr(e-tn' ))t°-1dt 1

Consider the one parameter family of metrics 9a = e2°°go, where pa = e2"- = (1 - a) + ae2e, 0:5 a < 1. Denote Aa = e-2s° Ao, then the curvature K. of the metric ga = pogo satisfies

Aoua + Kae2, = Ko. Observe that A. and S. = p Q i AopQ i have the same set of eigenvalues (hence the same trace of the heat kernel) with the advantage that the domain of S. is

80

S: Y. CHANG, THE MOSER.TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

L2(dVo). Hence

I = r'(Tr(e-tno) - Tr(e-t&i))! 0

_-

o

rI

t

I .l

roc

=J

Tr(e-to')dadt

fTr(e-,&. dda )dadt 1

r Tr(P'a a-t' Pa

ZOO JJrro

r =[da =

d

daP. )

Tr(a pa a t' )dt

0

0

0

o

dtPa

flda°°

- Jo

a)dadt ( where pa =

Wt

da

pa Tr(e-t)1o

Pa

_ -I 1 da(lim Tr(PQe Ena)).

To compute lim,.o Tr(P a-E°°), we apply the asymptotic formula for the trace of heat-kernel for Oa, and get

1,(a) = lim Tr(Pa a-i°° ) C-0 Pra

=v-0'k" lim M J Pa

(4ika)(x))2dV

where Aka), and 46ka)(x) are eigenvalues and eigenfunctions of a. Notice that I/24a) (x) are eigenvalues and eigenfunctions for Aka) = Aka), and 0ka) (x) = Pa Da. We get

,,(a)= lim

JId PR (x)-L(1 + 3Ka(x) +O(E2))dVo(x)

LECTURE 2. RAYSINGER-POLYAKOV FORMULA ON COMPACT SURFACES

81

Notice that by our assumption f p0(x)dVo(x) = A is fixed, thus fm p'(x)dVo(x) _ 0. Hence 1

1

1

1= - f dal4(a) _ -

flim fM Pp(x) .

12-KQ(x)dVo(x)

1

fi r Pa(x)

12ir Jo 1

12n 1

127r

dafM

(Ko - Aou0)(x)dVo(x) 2u(K0- Ao

)dVo(x)

1

I da fM

(2Kou0 + I VU.I2)dVo(x)

_ - 12ir 1 fM (2Kou + lVu12)dY Which finishes the proof of the formula. We will now state a easy corollary of the formula

Corollary. On (M,go), udth Ko = non-positive constant, log det A o is extremal among all metrics go on M with Vol(g1) = Vol(go). Proof. Denote g, = e2ugo, then Vol(g1) = Vol(go) implies that

exp(f 2udVo/ f dVo) <

f

e2udVo/ f dVo = 1,

hence f 2udVo < 0. Thus if Ko =- non-positive constant, we have f 2KoudVo > 0. Apply the Ray-Singer-Polyakov formula above, we have log

det01

< 0 for Al = A91; detA0 -

i.e. log det Ao is maximum among all metrics g1 with V of (g,) = V of (go ).

In the next chapter, we will discuss the case when Ko - positive constant, i.e. when M = the unit sphere S2.

References [C] [D]

I. Chavel, as in Lecture 1. A.M.Davies, Heat kernel and spectral theory, Cambridge University Press, 1989.

P.Gilkey, The index Theorem and the heat equation, Publish or Perish, 1974. [L-Y] P.Li and S.T.Yau, Acta math. 156 (1986), 153-201. [M -P) H.Mckean and I.Singer, as in Lecture 1. [M-S] S.Minakschisundaram and A. Pleijel, as in Lecture 1. [P1] A.Polyakov, Quantum Geometry of Bosonic Strings, Physics Letters 103B (1981),207-210. [G]

82

S.-Y. CHANG, THE MO8ER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

[P2]

A.Polyakov, Quantum Geometry of Ferionic Strings, Physics Letters 103B (1981), 211-213.

[R S]

D. B. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145-210.

LECTURE 3 Moser-Onofri Inequality and Applications Let's recall the Ray-Singer-Polyakov's formula on a compact surface. Suppose that gl = e2ugo and go are two conformal metrics on a compact surface M without boundary with Ol = e_2' 0. Then we have log

det Al _ det Ao

_ -12, 1 jy (2Kou + IVOU12)dVo

In the special case with M = S2 and go the standard metric of constant Gaussian curvature KO = 1, for any u with fsz e2udVo = 47r and Au = e-2aAo, log

det Au det Do

__

2

1

12-v

s,

(2u + IVU1)dVo

Onofri's inequality. (1)

4r S. e2u

<(4 J

(2u+

IVu12))

with equality iff e2ugo is isometric to go.

Corollary. Among all metrics on S2, log det Do is the maximum.

Remark. This should be compared to result of Hersch that a, < 2. There are now many different proofs of Onofri's inequality. The author knows at least four: Onofri's original proof [0] and [O-V], an independent proof by Hong [H], subsequent proof by O-P-S 10-P-S-11 and a recent proof by Beckner. Here we will present the original proof by Onofri as it connects the inequality to conformal geometry on S2. All these proofs depend on the following inequality of Moser:

Moser ([M]). If u is smooth on S2 then there is some constant Cl such that (2)

I

Js exp

a(u-u)2

l

Cfs iVui2d'o) S C1 83

84

S.Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

for any a < 47r, with 47r being the best constant, i.e., if a > 41r, the integral can be made arbitrarily large by appropriate choice of u. In the above inequality u = -L fS, udV0.

Remark 1. Trudinger first proved that there exists some constant a so that the above inequality (2) holds. Inequality (2) should be considered as the limiting case of Sobolev imbedding Theorem: Wo'9 LP with .1 = q n for 1 < q < n. If q > n, then W014 can be identified with Holder continuous function with Holder

exponent 1 - a. When q = n, W," ¢ L°°, but W01'" is contained in functions of the exponential class: Wo'"(D)'- exp "" I class, i.e., for any bounded domain D in R", there is a constant a = a(D) and constant C = C(D) so that if u is in Wo'"(D) with f D I V ul" < 1 then

I

exp(alul --n-7):5- Cmeas(D).

Later Moser improved Trudinger's inequality and obtained the value of the best constant a, in the case of plane domains a = 47r. . Remark 2. Moser's proof is quite sharp and delicate. We will not include his original proof here. (An easier proof is included in D. Adams ([A)) given by A. Garsia). But the key elements in the proof are:

Step 1. Apply "symmetrization" based on "isoperimetric inequality" and reduce the function to one variable, i.e.,

Exercise. Let u' be symmetric decreasing function with same distribution function as u. Then f IDu'I2 < f JVu12. Step 2. The one-variable inequality after the change of variable t = logr becomes the following: Given w smooth on (-oo, oo) with w > 0, f ti2dt < 1, f wpdt = 0 for some positive function p with f p = 1, P(t) < Coe-I tI ; then f _ ew'<
Remark S. We call u defined on S2 an even function if u(£) = u(-l;) for all t E S2. The best constant a in Moser's inequality for functions in the even class is a = 81r. To see this we have the following lemma. Lemma [C-Y-5]. Assume that D is a piecewise C2 domain in R2 with f D IVU12 = 1

and fD u = 0. Let LM = length of {x E DIu(x) = M} and Am = area of {x E Dju(x) > M}. If au = supM-oo,lAMl-u AM , then there exists a constant C2 (independent of u) so that

e°°"2 < C21DJ.

ID

LECTURE 3. MOSER-ONOFRI INEQUALITY AND APPLICATIONS

85

In particular, (a) when u is smooth function with compact support in D, then a > 41r. (b) When D = S2, we have the isoperimetric inequality L2 > A(41r - A), hence au > 4ir. (c) If u is even, for two small regions on S2 symmetric under antipodal map,

L = LI + L2 with LI = L2, A = AI + A2 with Al = A2i and L, > Ai(41r - Ai), thus L2 = 4L; > 4Ai(4ir - A,) = A(87r - A). So au > 8a.

Corollary. (4)

f

4a s ,

e2u < C2 exp( 4- ,I

s

IDuI2 +2 4- ,I

u)

$2

with C2 < CI in (2).

Proof. 2(u - u) <

2 41r(u [Du-da)dVo

fs2

+ 41x

L

IVU12dVo

by using 2ab < a2 + P.

Notice the best constants CI in (2) is greater than 1, while the statement in Onofri's inequality indicates the best constant C2 in (4) is 1.

Proof of Onofri's inequality. We will use the following fact: Lemma 1. Let S[u] = I f S. I VuI2+2* fs2 u. Then S[u] is conformally invariant in the following sense: Given 0 a conformal transformation of S2, let T,u = v denote the induced conformal factor given by 4*(e2ugo) = e2°go, of more explicitly, T,u = (u o 0) + 1 log [J01. Then S[Tmu] = S[u].

Proof. Since log determinant is an intrinsic quantity of the metric it is invariant under conformal transformation, S[T,u] = S[u] as a consequence of Polyakov's formula.

Remark. This lemma also can be proved directly without applying Polykov's formula, but based on the following corollary.

Corollary. If u = 1 log [J0[ for some conformal transformation 0, then S[u] = 0.

Definition. We say u E S if fs2 e2nx, = 0 for j = 1, 2, 3. Lemma 2. Given a C' function u defined on 52, there exists a conformal transformation 0 such that T,,u E S. Proof. Consider the following family of conformal transformations of S2. Given

P E S2, t E [1, co), rotate the coordinates so that P corresponds to the north pole (0, 0, 1). Using the stereographic projection from P mapping the sphere to

the equatorial plane on which we have the complex coordinate z, we denote the

transformation Opt(z) = tz. Observe that 4,p,1 = id,¢p,t = 4-pt-x, hence this

86

S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

set of conformal transformations can be parametrized by the unit ball B3 = S2 x [1, oo)/S2 x {1}. Notice that for 0 = Opt we may write with a change of variable

Jen't'xi = Je9(xi o 0P t). I e'°=3 - p, Define the center of mass of the mass distribution e2 C.M. j = 1, 2, 3. The center of mass map may be considered as a map from B3 to B3, it has a continuous extension to the boundary mapping P to -P. It follows from the Brouwer's fixed point theorem that there exists P E S2, t E [1, oo) for which C.M.(Op,t) = 0.

Lemma 3 (Aubin [Au]). If U E S, for all e > 0 there is a constant C, such that

C`exp((Z+E)4a

I

IVu12+2L. f u)

We will now finish the proof of Onofri's inequality based on Lemmas 1, 2 and 3.

Proof. Consider the functional F[u]

log

4-

I

e2D - S[u]

s2

defined for all u on W1.2 (S2). Apply Moser's inequality, we get

log 1 JS2

SJU+ CI

so F(u] < C1. Choose Uk E WI.2(S2) so that F[uk] , M = maxF(u], Apply Lemma 1, we know F[u] = F[T,uJ for any conformal transformation. Apply Lemma 2, we may assume w.l.o.g. that uk E S and f uk = 0. Apply Lemma 3, then

log4_f e2uk
M - e < F[uk] < C, - (12 _ `)

IvukI2+24

1

r s.

for all e < .1. i.e., 1

r

fs

zIVukl2
IVuk

12

LECTURE 3. MOSER-ONOFRI INEQUALITY AND APPLICATIONS

87

is bounded. Choose f Uk = 0. Thus Uk -+ u weakly in W 1.2 and f) VU12 < f e2u. So F[u] attains the maximum value

f IVukI2, F[u] > F[uk] and f e2u,, and u satisfies

Au+e2u = 1 with u E S. This implies that e2ugo has Gauss curvature 1, which is equivalent to u = 1 log for some conformal transformation 0, but among this class of functions only the constant function u = 0 belongs to the symmetric class S. I

Exercise. Prove that if

Wp.t =

log I j#", 1, then f Ioop t 12 ___+ +oo as t

co

2 and f Pp.t as t -, oo. While f IV b1,,t 12 + 2 f 10p,t = 0. As an application, we have the following result due to Osgood-Phillips-Sarnak:

Theorem [O-P-S-2). An isospectral family of metrics on a compact surface without boundary form a compact set in the COO- topology.

Sketch of Proof. We will only prove here the genus zero case, i.e., M = S2. In this case the theorem says given {e2u,,} with Ok = e-2uk/o having the same spectral data, then the conformal class [uk] of uk forms a C°°- compact family. Recall the heat kernel asymptotics: Tre-4"

-+ (47rt) - I (ao +a, t + a2 t2

+...) as t -+ 0+

with ao = f dV = f e2udVo, aI=3fm KdV

sX(M)=

,

a2 = iio f K2dV = -M 60 f (1 - Opu)2e'2udVo. For k > 3, ak = fm B2kdV where B2k are universal polynomials of weight 2k in K and A with K, A each counting as degree 2. ao is fixed while a2 gives some information of Aou. Thus we need some information about the 1st derivative of u (which is missing - which al does not provide); while leading term ak provides norm of u. some informations about The key idea is to use the log determinant formula to bound f IVuI2. log

det Au det Do =

1

3

S[u]

here f e2u = 4ir say. W.l.o.g., assume that u E S, then Aubin's inequality implies that f IVU12 < bdd. Remark. In the papers of [0-P-S-1], they have also treated the higher genus case, and also the class of simply connected plane domains. In the case of plane domain (via conformal map, the unit disc), the role played by Moser's inequality is replaced by Beurling's inequality, the role of Onofri's inequality is replaced by the LebedevMilin inequality and the analogue of the Ray-Singer-Polyakov formula is proved by Alvarez [Al].

88

S.-Y. CHANG, THE MOSER.TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Beurling's inequality. Let D be the unit disc and f be an analytic function on D. If f f DI f'(z)I2dxdy < rr, f(0) = 0, then J{B E OD : if (e'o)l > t}l <

2rrel-e2.

Lebedev-Milin's inequality.

r < r IVU12dxdy +2 / ud' log f e2udo 27r IT 2rr 8D

JD

8D

where u is harmonic in D.

References [A]

[AlJ

[Au] [C-C]

D.Adams, A sharp inequality of J.Moser for higher order derivatives, Annals of Math 128 (1988), 385-398. O.Alvarez, Theory of Strings with boundaries, fluctuations, topology and quantum geometry, Nuclear Physics B126 (1983), 125-184. T.Aubin, J. Funct. Anal. 32 (1979), 148-174. L.Carleson and S.Y.A.Chang, On the existence of an extremal function for an inequality of J.Moser, Bull. S.C. Math 2e series 110 (1986), 113127.

[C-Y-5]

S.Y.A.Chang and P.Yang, Cori formal deformation of metric on S2,

[M]

J.D.G. 27 (1987), 259-296. C.W.Hong, A best constant and the Gaussian curvature, P.A.M.S. 97 (1986), 737-747. J.Moser, A sharp form of an inequality by N. Thudinger, Indiana U. Math

[O]

J. 20 Noll (1971), 1077-1092. E.Onofri, On the positivity of the effective action in a theory of random

[H]

surfaces, Comm. Math. Physics. 86 (1982), 321--326. E.Onofri and M.Virasoro, On a formulation of Polyakov's string theory with regular classical solutions, Nuclear Physics B 201 (1982), 159-175. [0-P-S-1] B.Osgood, R.Phillips and P.Sarnak, Extremals of determinants of Laplacian, J.Funct. Anal. 80 (1988), 148-211. [0-P-S-2] B.Osgood, R.Phillips and P.Sarnak, Compact isospectral sets of surfaces, J. F11nct. Anal. 80 (1988), 212-234. N.S.Trudinger, On imbedding into Orlicz spaces and some applications, [T] J.Math.Mech 17 (1967), 473-483. [O-V]

LECTURE 4 Existence of Extremal Functions for Moser Inequality In this chapter, we will describe a proof to establish the existence of extremal functions for Moser-Trudinger inequality as stated in the previous lecture. The proof has appeared in [C-C], see also [St].

Theorem 1. Let Bn denote the unit ball in R", n ? 2, then

sup 'B,, exp(au1

) dx

is attained for functions u E Wo'"(Bn) with f13. I p uI" dx < 1, where a" _ n(wn_1) , wn_1 is the area of (n- 1) spheres. We first remark that the result is surprising in the following sense: Moser's inequality is the limiting case of the Sobolev Embedding Theorem Wr1'4(D) c LP(D) with .1 = .1 - n for 1 < q < n and domain D in R". In the limiting case for q = n, we have Wo'" (D) C exp nnl class, while the embedding with exponent an is not compact. In another well-known case of Sobolev Embedding Theorem W01,2 (D) C L--2-N, while the embedding is not compact, the extremal function for the embedding does not exist unless D = R". In the latter case the study of the noncompactness phenomenon leads to the solution of "Yamabe" problem ([A] [S1]), and many other interesting problems in non-linear PDE (cf [B-N]). In contrast,

in the case of the Moser functional, the extremal function is attained. We also want to point out that the proof we shall present below is an existence proof, and the explicit expression of the extremal function is unknown and remains an open question. Our second remark is that we have obtained the existence result only for D = Bn : the unit ball in R'. The reason for this constraint is that only in this case can we apply the symmetrization technique mentioned in chapter 3 to reduce Theorem 1 to the following equivalent form by a change of variables as in [M]. 89

90

S: Y. CHANG, THE MOSER'TRUDINGER INEQUALITY do CONFORMAL GEOMETRY

Theorem 2. Let K denote the class of C' functions w(t) defined on 0 < t < 00 satisfying 00

o^(t) dt _<1,

w(0) = 0, cw(t) >_ 0 and

c

0

then 00

sup

(1)

wEK 0

e"(t)-tdt

is attained by some function w' E K. In a recent paper by M. Flucher [F], using an ingenious method in complex analysis, he was able to generalize the existence part of Theorem 1 to general domains in R2. In another related development, T.L. Soong [Sol has proved the analogous results of Theorem 1 for functions u defined on S2 with fs. u = 0 and fs, I p u12 dx < 1, and some partial results for functions v defined on S4, with

fs. v = 0 and < Pv, v >< 1 where P = (-A)2 - 2A is the Panitz operator on S4. In the remaining part of this chapter, we will prove Theorem 2 for the special case n = 2. The proof for general n is only more complicated in the technical sense.

Proof of Theorem 2, for n=2. The existence Theorem was established by contradiction. This is accomplished through the following two steps: Step 1 Assume there is no extremal function attaining the supremum in (1), then sup W

100

EK

sup wE K

f

e

2(t)-tdt<_1+ e.

oo

e"'2(t)-t dt > 1 + e.

The proofs of both step 1 and step 2 are motivated by the existence of a sequence of "broken line" functions {wa}a)o defined as follows: w1(t) = t when 0 < t 5 1, wi(t) = 1 when t > 1, and wa(t) = f wI (Q ). This is the sequence of functions which has been used by Moser (M) in establishing that 02 = 4w is the best exponent in the embedding Wo'2 C L2. Notice that for this sequence of func-

tions {wa}, wa E K, wa(x) - 0 as a --i oo uniformly in compact subsets, and I(wa) = fo-eeO(t)-t dt -r 1(0) = 1. The intuitive idea in the proof of the theorem is to prove that if the extremal of I (w) for w E K does not exist, the maximal sequence would behave quite like the sequence {wa}.

LECTURE 4. EXTREMAL FUNCTIONS FOR MOSER INEQUALITY

91

Proof of step 2 Define

when 0t _ 2, (t-1)i when 2
w(t)=

when t > e2 + 1

e

ee

then

1

I(w)=2fo eel-2edt + e ds+e>2.11+e>1+e,

2 f'e8

based on lower Riemann sum estimate of the integral fo a°3 ds. Proof of step 1

Choose a sequence w,n E K with I(w,n) -' M = Sup,,,EK 1(w). Assume that there does not exist any w E K attaining the supremum M, we claim that the sequence w,,, has the following properties:

(a) For each A>0, fA(wm)2dt-'0asm-00, (b) Let an be the first in (1, oo) with w»,(an,) = an - 2loga,n (if such a,,, exists).

Then an -4ooas

e20 (t)-t dt = 1, fm

(c)

(d) limsup,,,-. f f e-'- (0-I dt 5 e. (a) (b) can be established via argument by contradiction and elementary calculus.

To prove (c), we notice that it follows from (a) that w,n 0 uniformly on any compact subset of (0, oo). Thus for each e > 0, A and m large with w2,,,(t) < a for

t < A, using the property that an is the first point with wn,(t) >_ t - 2log+ t we have e"2(e)_e dt

I0 !A

< ec /

r

0

A

,,

e-e dt +

JO

r

= IA + f nm e-2 log+ a dt

fA'

=e`(1-e-A)+(A-a1-n,)<=

aam

as e -. 0, A -' oo. On the other hand

fo-

dt >

a-e dt

= 1- e-°m,

0

which tends to one as a,,, - oo, which finishes the proof of (c). To prove (d), we first establish a lemma.

92

S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Lemma 1. Let K6 = { : CI functions defined on 0 < t < oo, 0(0) = 0, f'OO 02 dt < b), then for each c > 0 we have

sup

(2)

.OEK6

o

Also when c2b -+ oo, the inequality in (2) tends asymptotically to an equality.

Proof of Lemma 1. We first remark that since fnr ea(t)-t dt is uniformly small for all 0 E K6 as N - oo, it is easy to verify that the extremal function for SUPOEKb fo ecO(t)-t dt exists.

Suppose 0 E K6 is such an extremal function, then via variational method, satisfies the following differential equation:

ecO(t)-t = A ', for some constant A.

(3)

Let k(t) = ca(t) - t, we may rewrite (3) into ek(t) = A k(t),

(4)

with k(t) satisfying

(k+1)2dt=c2b,k(0) = 0;

(5)

hence k(oo) = -1, k(oo) _ -oo. Multiply (4) by k and integrate, we get (6)

ek(t) = A(2 (k + 1)2(t) - (k + 1)(t) + C).

Letting t -+ oo and using (5), we find C = 0. Compare (4) and (6) we get (7)

k(t) =

2(k +

1)2(t) - (k + 1)(t).

After integrating (7), we have

1 + k(t) =

2

1 + Be"

where the constant B is determined by the equality (8)

c26=fx(k+1)2dt=4(logIBB-1+B)

LECTURE 4. EXTREMAL FUNCTIONS FOR MOSER INEQUALITY

93

Evaluating (6) at t = 0 we get 2

2

1 =A((1+B)2

(9)

1+B)'

On the other hand, integrating (4) directly we get from (9)

J=

1+B

2A

fedt = Ak(t)I o = - 1+B

B

Since 1 + k(t) = ca(t) ? 0 we have 1 + B >_ 0. Thus B >= 0. It follows from (9) that

B - 0 when c2b - oo and J:5 exp(.lc2b + 1) with the inequality approaching an equality when c2b oo as claimed in the statement of this lemma. We may now apply the above lemma to finish the proof of (d) as follows: Let x = t - am, Wm = Wm(x + am) - Wm(am), then W,,,(t) =

+Wm(am) for all

x>0, Wm (t) = Wm (am) + 2wm (am)Om (x) + 0,2 (x).

Since 0,,,(0) = 0, fo o,2,, (x) dx = J'0 wm(t) dt = b,,,. Thus if we set y = (1 - b)x, c = 2Wm(am) and 0(y) = 4)m(x) we get (10)

(t)-t dt < ew_2 f.,,- o e"^_

1

f

dy.

Applying the above lemma, we get

rJ a..,

i - bm

,

where k,,, = Wm(a,,,) - am + I n W2 (am). Since Wm E K, Wm(am) 5 (1 - bm)am, 2Iog+ a. 0 as m -, 00 from this using the definition of am we can check b,,, and k,,, -- 0 as m -- oo. Thus it follows from (10) we have

e`"^(t)-` dt 5 e, limsup m-oo Z-00

Remark. The author has learned from B. Beckner that Lemma 3 is actually equivalent to the Onofri's inequality on S2 as stated in Lecture 3. This fact is not an obvious one.

94

S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

References T. Aubin, The Scalar Curvature, Differential Geometry and Relations, Reider publishing Co, 1976. [B-N] H. Brezis and L. Nirenberg, Positive Solutions of nonlinear elliptic equations involving critical exponents., Comm. Pure. Appl. Math 36 (1983), 437-477. [C-C] L. Carleson and S.Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser., Bull. Sc. Math 110 (1986), 113-127. M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 di[F] mensions, Comm Math Hely 67 (1992), 471-497. J. Moser, as in Lecture 3. [M] M. Struwe, Critical points of embeddings of H01'2 into Orliz Spaces, Ann Inst [St] H Poincare Anal Nonlieaire 5 (1988), 425-464. [Al

[S-1]

R. Schoen, Conformal deformation of a Riemannian metric to constant

[So]

Scalar Curvature, JDG 20 (1984), 479-495. Tien-Lun Soong, Extremal functions for the Moser inequality on S2 and S4, Ph.D Thesis (1991).

LECTURE 5 Beckner-Adams Inequalities and Extremal Log-Determinants in 4-D In this chapter, we will discuss some attempts to generalize the results of Onofri and [O-P-S) to higher-dimensional manifolds. Recall that Polyakov's formula depends highly on the invariant property of the Laplacian operator, i.e. if g = e2wgo, then JM IvuI2dV9 = JM IVOUI2dV0

for every u defined on a compact closed surface M. Rewrite IM IVU12dVg = -

fAM(Au)udV.

We see that this invariant property follows from the relation that Ag = e-2wOgo

Definition. We call an operator A a conformally covariant operator if 9 = e2w9o implies that A = e-b"Aoe°" with b - a = 2.

Example 1. A = A then b = 2, a = 0 in the case dim M=2. Example 2. M is a closed manifold of dimension n and let

4(n - 1)

where

n-2

R is the scalar curvature of M. L is called the conformal Laplacian. Then L has the following conformal covariance property

"Lo(e

L,,(m) = efor every 0 E COO (M). 95

"b)

96

S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Exercise. Prove the conformally covariant property of the conformal Laplacian.

Hint: Let u denote e T

,

then u satisfies the equation Lu =Rum where R

is the scalar curvature of g.

In [B-0-1] (see also (P-R]), it was proved that for the conformal Laplacian operator L, the 2-coefficient al of the heat kernel for L is invariant under conformal

change of metrics. And when n is odd, the log determinant of L is an invariant quantity. Restricting our attention to n = 4, for a compact 4-manifold, the a2 coefficient of the trace of the kernel for the conformal Laplacian operator is conformally invariant. The work of [O-P-S 1,21 thus suggests that the role of a2 should be replaced by log det L in the study of isospectral compactness problem. In this chapter, we shall discuss some aspects in this direction. First we recall the work of Branson-Orsted [B-0-2J, where they obtained some generalized form of the Ray-Singer-Polyakov formula for log-determinants of conformally covariant operators in four dimension on compact locally symmetric Einstein manifolds. A special case of their formula is when the covariant operator is the conformal Laplacian L. Suppose V (M, g) = fm e4i'dVo = vo, then

F(w)

det L,,

log j

et Lo

= - 2{ 4 k(M, go, L) log o JM M

JMJ(ow)2Ro

e4("'-W)

(,e,)2)

IV wI2}-4{RQJMIvw12- 1

where k(M, go, L) ivoc2 + 16ir2X, and c2 denotes the square-norm of the Weyl conformal curvature of go. Indeed, compactness information similar to those in two dimension can be obtained for metric e2"go with F(w) bounded. This is formulated as the following result.

Theorem 1 [B-C-Y]. If k(M, go, L) < 327r2, and with normalized volume fm e'dVo = vo, then 11w112,2 is bounded by a constant depending only on F(w).

Examples of manifolds with k(M, go, L) < 32ir2: any manifold M which is compact and locally symmetric, Einstein but not (S4,go), or a hyperbolic space form. k(S4, go, L) = 32x2. But RP4, CP2, S2 x S2, T4 and compact quotients of a polydisc with standard metric all have k(M) < 32ir2. Theorem 1'. The conclusion of Theorem 1 holds on (S4,go) with w replaced by Tow for some 4, a conformal transformation of S4. (TO(w) = wo,0+14 In IJmI, a 27'"go

= 4,' (e2"go)) The role 32ir2 plays here is the same as that of 4,r in Moser's inequality. Actually there is a higher dimensional analogue of Moser's inequality when n > 2 due to Adams, which we will now describe below. Recall by the classical Sobolev embedding theorem, we have W°,a C LI', when

= 1 - n if q > 1 and aq < n. The following result deals with the limiting case

aq=n.

LECTURE S. BECKNER-ADAMS INEQUALITIES & EXTREMAL LOG-DETERMINANTS 97

Theorem 2 (Adams). Suppose m < n are positive integers, 1 is a bounded domain in R1. Then there are constants Co = C(m, n), /io = /3(m, n) such that if u E Wm'Q of compact support in f) with II Vmullq < 1, where qm = n, then for,35 Qo,

L where q' =

and ,6o is the best constant for the inequality to hold.

When n = 4, q = m = q' = 2 we have t3(2,4)=321r2. L. Fontana [F] has generalized the above inequality to compact n-manifolds without boundary. In particular we have

Theorem 2'. Suppose that (M, g) is a compact 4-manifold without boundary, then there exists a constant Co such that for every u E C2(M)

t eXp(327r2( - )2 )dV9 < CoVol(M). fm(Au)2

JM r

Corollary. If M is a compact 4-manifold without boundary, then for every w E C2(M), log

Vol(M)

JexP(4(w - w)) 5 log Co + gns JM(Ow)2, where w =

Vol(M) /

w.

Suppose k(M, go, L) < 32,r2, then when FA(w) is bounded, we may apply the corollary above to show fM (d,,)2+fM IVWI4

is bounded. The proof is in the same spirit as the proof of Onofri's inequality that if u E S then S[u] is bounded implies that fM IVuI2 is bounded. Details of the argument are contained in [B-C-Y]. Restricting our attention to (S4, go), where go is the standard metric and vo = Vol(S4), we have

F(w) = constant log

= [log

1f vp

Sa

det L. det L,,

1 1 f (AW)2 _ 2 1 f IV l21 3 vo

3 vo

sa

J

Sa

1fsat\ }=I+?II. 2

+514-vo?fS.IDwI2-

vo

e`"

3

98

8. Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Theorem 3 [B-C-Y). On (54, go), F(w) < 0, and F(w) = 0 if and only if e2,,90 = 0* (go) for some conformal transformation on S4. That is, on S4 the log determinant L is extremal if and only if g is isometric to the standard metric. It is interesting to see that the proof of the Theorem 3 can be formulated into two extremal inequalities.

Lemma I. I < 0 on S4 and I = 0 if e2i"go = 0*(go)

Lemma 2. II < 0 on S4 and 1I = 0 ife2u'go = q'(go). Lemma 1 is a special case of an inequality of Beckner which holds for general n. It is a linearized version of the Adams inequality just as Onofri's inequality is a linearized version of Moser's inequality.

Theorem 4 (Beckner [B]). If f E C°°(S") has an expansion Ek o Yk in spherical harmonic functions Yk, then

log

I e(f-t) <

V(S") Js

2n B(n'k)V(S") k=1

J,,

IYkI2

where B(n, k) = I'(n+k)/r(n)r(k) and equality holds if e2ff"go = b'(go) for some conformal transformation 0 of S". The proof of the Beckner inequality is quite delicate and depends on the Fourier analysis method (Lieb-Young inequality).

Proof of Lemma 1. Take n = 4 in Beckner's inequality, in this case, B(4, k) =

r(3 + k)

- k(k + 1)(k + 2)(k + 3)

r(3)r(k)

6

Recall that the k-th eigenvalue of A on S4 is Ak = k(k+3). Thus if Yk is a harmonic polynomial of degree k, then AYk + k(k + 3)Yk = 0. Letting f = e4u, we get :

109

V(S4)

fs e''

S6

k(k + 1)(k + 2)(k + 3)IYkI2 k=o

00

0

= 3 Is'

+ 2)Yk, Yk)

+ 2)w),

= 3(f 4(Ow)2+2 f IVwI2). Thus Lemma 1 is equivalent to Beckner's inequality on S4.

LECTURE 5. BECKNER-ADAMS INEQUALITIES & EXTREMAL LOG-DETERMINANTS 99

Remark. Beckner's proof also indicates that S4(w) =

f()2

+ 2IVwl2 + 4CO

is a conformal invariant and S4(w) > 0 if f e4" = Vol(S4). In particular we have w = 1 log Vol

Paw + 4!e4ri = 4! on S4

w here the operator P4(w) = -°(-° + 2)w, P4 has appeared in the literature previously and is called Paneitz's operator.

Proof of the Lemma 2. We will present a geometric proof here. Recall that Lemma 2 is equivalent to the inequality:

4f IVWI2 < with "=" holding iff e 2w go

f

(°w)z

(4go)

Let u = e°'. Then u satisfies the Yamabe equation

60u + Ru3 = Rou = 12u where R is the scalar curvature of the metric e2" g. Thus

° =0u=2- su2

We also observe that

°,W _ °w + IVwI2, and hence

f (°) = f lVwI

2

Denote V = Vol(S4), we have

-" =

i

(°;-)2 - V f IVW12

=V f (2- su2)2-4V J(2 - sue)=-4+ V,l Recall the Yamabe functional defined as: Q(u) = Q(S4,9o,u)

= VI/2 f IVUI2 + 2u2 (f u4)I/2 '

z Sa

36u4.

100 8: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

The infimum of Q(u) is achieved only by conformal factors u of the form u2go =

¢'go where ¢ is a conformal transformation of S'. We observe that inequality II is scale invariant, thus we may assume w.l.o.g that f u' = V. Hence applying Schwartz inequality and the Sobolev inequality in the Yamabe functional above, we get

r

rr

12

-I1=-4+V J 36 u'>-4+2IJ 6u'J>0, with equality holding if and only if u2go = 4*go for some conformal transformation . This finishes the proof of lemma 2 and hence the proof of Theorem 3. Beckner's inequality highly suggests that the results above for 4-manifolds holds

for all even dimensional manifolds. Yet so far there is difficulty to compute the precise logdeterminant formula for higher-dimensional manifold.

Remark added after the lecture: More recently, results in this section have been extended substantially in many directions. In [B-G], Polyakov-Ray-Singer type formula for functional determinant has been worked out for 4-manifolds with boundary. In [Br], Theorem 3 in this chapter has been extended to (Se, go). And in [C-Y], the zeta functional determinant F[w] as defined in this chapter has been studied for general compact 4-manifolds. Under conditions similar to that of Theorem 1 (i.e. k < 32712), existence for the extremal metric for F[w] has been established and proved to satisfy some sharp Moser-Trudinger type inequality. Also in [C-Y], a different proof of Beckner's inequality (Theorem 4 in this chapter) was given,the proof relies on the conformal invariant property of the Paneitz operator and Adam's inequality instead of the sharp Lieb-Young inequality.

References [A] [B]

[Br]

D.Adams, as in Lecture 3. W.Beckner, Sharp Sobolev inequalities on the sphere and the Moser1-udinger inequality, Annals of Math. T.Branson, Sharp inequalities, the functional determinant,and the complimentary series, preprint, 1992.

[B-G)

T. Branson and P. Gilkey, The functional determinant of a four-

[B-O-1]

dimensional boundary value problem, to appear TAMS. T.Branson and B.Orsted, Conformal indices of Riemannian manifolds, Compositio Math. 60, 261-293.

[B-O-2]

T.Branson and B.Orsted, Explicit functional determinants in 4-

dimensions, Proc. AMS. 113 (1991), 669-682. T.Branson, S.Y.A.Chang and P.Yang, Estimates and Ehtremals for Zeta Function Determinants, Comm. Math. Physics 149 (1992), 241-262. S.Y.A.Chang and P.Yang, Extremal metrics for zeta-function determi1C-Y] nant on 4-manifolds preprint (1993, to appear in the Annals of Math.). (O-P-S-1] R.Osgood, R.Phillips and P.Sarnak, as in Lecture 3. [B-C-Y]

[O-P-S-2] as in Lecture 3. [P-R] T. Parker and S.Rosenberg, Computation of heat-kernel for conformal

covariant operators, J.D.G. (1987).

LECTURE 6 Isospectral Compactness on 3-manifolds and Relation to the Yamabe Problem In Lecture 3 and 5, we have discussed the compactness results for isospectral families of conformal metrics for compact, closed manifolds of dimension 2 and 4 ([O-P-S-1] and [B-C-Y]) and the extremal metric of the log-determinant of the (conformal) Laplacian operator on S2 and S4 (10] [B-C-Y]). In this chapter, we will briefly discuss some progress which has been made for these problems on 3manifolds.

Recall that we have mentioned in the previous chapter that [P-R) when n is odd, L9 = conformal Laplacian operator w.r.t. the metric g, then log det L9 is a conformal invariant quantity when one changes the metric g in the same conformal class. Thus in particular, F[w] = log dee" for g,,, = e2`''g does not carry any information on w. Recall also that in the case of dimension n = 2, to achieve the C°° compactness for isospectral family of conformal metrics {g., = e2iog} on compact surfaces, one

of the chief strategies in [0-P-S-1] is to use the spectral information in F[w] = log to control the W1'2 norm of w, then use the information in the k-th a coeffient ak of the heat kernel expansion e'At ti Ek 0aktk-t to control the Wk,2 norm of for k 22. It turns in the case of dimension 2 when restricting metrics to a fixed conformal class, one can also give an alternative argument (using \1 to replace the log determinant of the Laplacian) to show that isospectral conformal metrics on

d°t,

compact surfaces form a compact set in the C°°-topology

Theorem 1

Suppose (M, go) is a compact surface, {e2' } is a sequence of conformal factors on M with (1) f e2ut dvo

= ao

(2) f KK e2i t dVo = a2 < oo where K., = Gaussian curvature of the metric e2' g0.

And assume in addition that, the first eigenvalue Al of the Laplacian w.r.t. to the metrics e2sj go are bounded from below by A > 0, i. e.

101

102 s.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

(3) For each function 0 defined on M dVo <

(r

2Vo/

M

IM

2

/

( JM

dyo) +

1

A JM

IDo phi12 dVo.

Then either (a) and (b): in case Ko (Gaussian curvature of the metric go) is < 0, or = 0 respectively; {ui} forms a bounded family in W2' (i.e. sup,, fey IVu,I2dVo is finite) or (c): in case Ko = 1 and (M, go) = (S2, go) with go = surface measure on S2, then the isometry class of ui forms a bounded family in W1.2. In the setting of Theorem 1 one can generalize the compactness result to compact manifolds of dimension 3 ([C-Y-2]).

Theorem 2

Let gi = u,'go be a sequence of conformal metrics satisfying the

following conditions:

(i) Vol(M,gi) = ao for some positive constant ao. (ii) f R2 (gi) + I P(gi )12 dV < 02i for some positive constant a2 where R(gi) is the scalar curvature of gi and p is the Ricci tensor of gi, and dVi = u. dVo, (iii) A,(gi), the lowest eigenvalue of the Laplacian of the metric gi, has a positive lower bound A, (g,) > A > 0; i.e. For each 0 defined on M, we have

(JM0 dVJ)2/\JMdV,)+'rJMIV

CfMo2dvj)

I2 d14.

< Then there exists constants c1, c2 so that

(a)

c1 <_ u(x) <

(b)

I lui 112.2 <_ c2;

1 cl

,

in the case where (M, go) is the standard 3-sphere. isometry class of ui satisfy (a) and (b). except

In the latter case,

the

Theorem 2 was proved in the special case when Ro is negative by [B-C-Y] and when (M, go) = (S', go) in [C-Y-1] [C-Y-31, see also Theorem 3 below. Notice that in dimension n = 3 (actually for 3:5 n:5 6, cf [K], page 3), one can make a explicit computation in the formula of McKean and Singer (M-S] in Lecture 2, and obtain on (M, go)

ao = volume of

constant

g=

J

RdVo

a2 = A2 f R2dVo + B2 f IPI2dVo A2i B2 > 0 Furthermore [8] for k >_ 3

ak = Ak f Iok-2RI2dVo + Bk f lVk-2 pl2dVo + lower order terms,

LECTURE 6. ISOSPECTRAL COMPACTNESS AND THE YAMABE PROBLEM

103

with Ak, Bk > 0. Thus the condition (i) (ii) (iii) in Theorem 2 all are spectral in formation, and once one controls W2,2-norm of the conformal factor {u3 } for an isospectral sequence of metrics gj = u,go, one can gain the control of Wk,2-norm of {uj} through the information in ak, and we get as a corollary of Theorem 2.

Corollary 3

An isospectral set of conformal metrics on a compact 3-manifolds is compact in the C°°-topology.

Since conformal transformations preserv the spectral information another immediate corollary of Theorem 2 is the following result of Obata for the special case

n=3.

Corollary 4

The confonnal group of a compact 9-manifold is non-compact if and only if it is comfonnally equivalent to the standard 3-sphere. Before we discuss the ideas in the proof of Theorem 2, we would like to mention

a result of Gursky [G] which generalize Theorem 2 to general dimensions. First we remark that, in the statement of Theorem 1 (for compact surface), it is wellknown that the "local information" (i.e. those computed using local coordinates in the metric g) like that in conditions (1) and (2) in Theorem 1 are not enough to conclude the W1.2 compactness of {wj} for the metrics {e2"'igo}. This can be seen through the famous "Dumbell" surface with the neck getting thinner and thinner. Notice that for such surfaces, using Rayleigh quotients, it is easy to see that Al -+ 0. Thus it comes as a surprise that when n >_ 3, some "local information" (e.g. size of curvature tensor of the metric) suffice to provide the compactness of the metrics. This is the content of the following theorem of Gursky [G].

Theorem 3

Let (M, go) be a compact manifold without boundary. Suppose that

9j = u go is a sequence of metrics satisfying (1)

f u" dVo = Vol(9j) < ao

fIRm(gj)IPdVj 2, where Rm(gj) is the full curvature tensor of gj. Then there exist c1, c2 > 0 so that -CL < uj c1 and II uj II w2., < c2, unless (M, go) = (S", go), in that case the (2)

conclusion holds in the isometry class of uj.

One would like to point out that the exponent p > 2 is necessary in Theorem 3. In a recent article [C-G-W], two examples have been constructed to indicate that Theorem 3 fails when p = 2 .

Gursky's proof of Theorem 3 is quite ingenious. One of the key idea is to apply condition (2) to Bochner's formula to start Nash-Moser iteration process and to establish "Harnack type" inequality for sequence {U3 }. In this lecture, instead of proving the general result [8], we give a complete proof some special cases of Theorem 2 to convey the idea how to use the condition Al ? A > 0 in the compactness result. We start with a definition.

104 S: Y. CHANG, THE MOSER..TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Definition

We say a sequence of positive function {uj} satisfies condition (*), if there exist 1o, ro > 0 so that for all j

(*)

J ufxZra)

Theorem 2' (1)

(2)'

dVo2!

toJM duo

Suppose {uj} is a sequence on (M3, go) satisfying

vol(gj) = f u,duo = ao fey Rti, <_ a2 R; = scalar curvature w.r.t. 9j

A > 0 where .11 is the first eigenvalue of the Laplacian operator. And if in addition, {uj} satisfies condition (*) then there exist some co > 0 and a constant Co depending only on the data ao, a2, A, lo, ro so that (3)

(4)

IMtL+(OC0

Remark The underlying analysis of the compactness results in Theorem 2 and Theorem 2' is the optimal Sobolev inequality: 1.

1/3

Q(M) (IM uduo)

< 81 II+

Rout duM

IM

The optimal constant Q(M) is an invariant of the conformal class of M. For a conformal metric 9 = u4go, its scalar curvature R is given by the equation (5)

8Au + Rus = Rou

on

M.

Thus the Sobolev x0 quotient Q[ul =

f (8IVu12 + Reu2) dVo (f u6duo)1/3

is exactly given by f Ru6dVo if the volume is held to be 1, (i.e. f u6duo = 1). The celebrated recent solution of Yamabe's problem ([A], [SI) asserts that (a) Q(M) < Q(S3) unless M is conformally S3 and (b) a minimizing sequence for Q[u] is compact if Q(M) < Q(S3). Thus in our compactness assertion, we have substituted an L2 bound for the curvature in place of the condition Q[u j] < Q(S3), and substituted the condition a1(gj) > A > 0 in place of the minimizing property for Q[u]. 2. As we have mentioned in Lecture 4, one fact which has a key role in Yamabe problem is that the embedding W1"2 C L is not compact. On (S3,go), if we adopt coordinates on S3 through its stereographic projection mapping with pole of

LECTURE 6. ISOSPECTRAL COMPACTNESS AND THE YAMABE PROBLEM

105

S3 to 0 = (0, 0, 0) in R3, then in this coordinates system, volume form dVo on S3 is defined by

dVo=(I+Ix12)3dx For each a > 0, consider the function IX12

ua(x) _

1012

Ixl2

+

then {ua} is a family of function with

6(21r2) = Q(S3)(f uddVo) = 8 / 1Vu0I2dVo +6J uQd1/o,

,3

S

s9

where Q(S3) = 6(2x2)'1. Since ua(x) - 0 uniformly on compact subset off x = 0, {ua} is a typical example of family of functions which indicate the failure of compactness of the inclusion W1.2 C L6 on S3. Notice that {ua} does not satisfy comdition (*) in our Theorem 2', and also for all c > 0, f y., uQ+`dVo -, oo as a - 0. Thus it is convincing that a sequence {uj} satisfies condition (1), (2), (3), (4) in Theorem 2' is a sequence which is bounded in the sense of the conclusions (a) and (b) in Theorem 2; i.e. there exist constants c1, c2 so that c1 < uj <_ and IIuj1I2,2 < c2. Indeed, using an iteration argument (Lemma 3.4 in [C-Y-11) one can verify this latter fact. Thus Theorem 2' is a special case of Theorem 2. Examples of (M, go) for which conditiopn (*) automaticallyfollows from as3. sumptions (1), (2) and (3). (a) When Ro < 0 (we may assume w.r.l.g. Ro is a constant), then

8Auj 5 fxt (

uj

=J (a')ZdVo-16R0J '3'dVo+RoJ M uj

M Bj

M

Since fm ° dVo = 3 fm IVuj12ui 4dVo 5 0, we have

f Rj2ugdVo < a2, so Ro I

u'-2dVo

a2.

M

And it is easy to check that for {u, } satisfies fM u; 2dVo < /9, {u,} satisfies condition

H.

(b) On (S3,go), given a sequence {uj} satisfying (1), (2), (3) we can find v, in the same isometry class as uj with v., satisfying (1), (2) ,(3) and condition (*). To see this, given uj, we apply Lemma 2 in Lecture 3 to obtain vj = TO, uj for some

conformal transformation 4j of S3 and with vj E S, then A1(t4go) = \I (vjgo), and {vj} satisfies (1), (2), (3) with the same constants ao,a2 To see {vj} satisfies

106 S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY 3c CONFORMAL GEOMETRY

condition (*) we apply and get dVj = vjdVo (6)

JS9

= zk, k = 1,2,3,4 in the Rayleigh quotient of A1(vjgo)

xkdvj

IvJZkl2dVJ

IVQXkvjdV0

< dVj =

A

J 3 IVozkvj2 dVo

Sum (6) over k = 1,2,3,4 we get

ao
j

vdVo

Thus {vj} satisfies condition (*). Proof of Theorem 2' Denote u = uj, we will show that u satisfies (4) with constant depending only on a0a2, At lo, ro

Multiplying the equation (5) by uQ (Q > 1 to be chosen later) we have (denote

f = fm dVo) for w = u'V 40

J

8(1

(7)

IVw12+RDJ w2 = (Ru4w2

We will now apply our assumptions (1), (2), (3) to estimate the term I = f Ru4w2. Taking a suitably large number b (again to be chosen later) on the region IRI ? b we have b2

J IRI>_b

u2dV0

< JR1_b R2u6dV0 < a2

Thus

(8)

(r fictb RU4w2 = \ J <

U)1/6(Jw6) 1/3

1/2

6

R2U6)

( J IRj2b

)1`6 ( r w6)1/3

For the remaining part of the proof, we will apply condition (*) in the statement of Theorem 2'. For dV = v6dVo, we have from the Rayleigh-Ritz characterization for Al,

(9)

.6dV) 2/(J dV)+A1IMIV r 1 t2dV < (J M

M

4P12CW

LECTURE 6. ISOSPECTRAL COMPACTNESS AND THE YAMABE PROBLEM

107

where I VufI2dV = I V4I2u2dVo. We will denote E.y = {x E M, u(x) > 7} and IE,I = fE, dVo. By assumption (*), there exist some 70,10 > 0 so that J-1,1 >_ loI fey dVoI. Applying (9) and (3) to 4' = u` with $ = 1 + 2e and a small, we have

us+o/lzl(r u6dVJl + (r (J \J

f us+2 dvo

(10)

JpvuI2u2dvo.

For simplicity, we will now normalize u and assume that ao = f u6dVo = 1. We f64( dVo as may then estimate the term r us+EdVo

J

= /u6+'dvo + =f

r

u6+Edyo

JE4

fE.0

(u6 - 7o)u`dVo + fE,0

udVo +

Eo

f

u6dVo ;ro

1/2

1/2

JE,o

(u6 -

,yo)u2 dVo

/

\f E (u6 -

70)

0/

+ C('Yo),

where C(7o) is a constant depending only on 7o and f dVo. Thus, for each 11 > 0 we have (11)

(fu6+)2(1+)(f,o

(u6 - ,y06)u2

dVo)

(fE,o (u6

+ I+

- 706)0)

)C2(7o) u6+2edVo)

(1+n)(1-706 IEtiol)( f

I

+ 1 1+ C2(7o) 17

(we may assume w.1.o.g. that 7o is small and 7oIE,, I « 1). Since by our assumption on IE,0I we have $IE.yoI _> -foflo > 0, we may choose vi so that (1 + t)(i - IEymI) < 1 - 6 for some positive 6, 6 = 6(70, lo) and obtain from (10), (11) (12)

6 f u6+2 dVo < C(7o, lo)

+ 1 f I Vu` I2u2dVo,

where again C(7o, lo) is a constant depending only on 70, lo. From this point on, we may estimate the term f IVuEI2dVo as follows: f IVudI2u2dVo = (1+e)2

f

Ioul+cl2dV0

108 S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

and notice that for fl = 1 +2E, w = u4fi = ul+'. Thus, combining (7) and (12) we have

fu

6+21

(13)

dVo

E2

<

(1 + )a

=

where

I = r Ru4w2 and L = 0(!! b° r ue+2`dv0) + 1 C(-to, 10) Combining (11), (13) with (8), we find

(14) 1< (1) \\!/ 1/3

JRi4w2

1/3


O2

b

I

1/3

/ w6dVo I

I

1/3

(fw6dvo)

+ b / u4w2dVo bet

+ 8A I + bL

so that 1/3

a2

(15)

8A)C \ b)

(1

(Jw61v0)

1/3

+bL.

Now choosing b sufficiently large so that (1) < Q, and then choosing e sufficient z small, we get

r

1/3

4Q(J w6dVo

U

1 + IRoI

J

w2dVo

\ 1/3

< 3Q(J w6dVo)

+ 3 U + IRaI

J

wa dV0.

Recall w = u1+E, hence

(f u6+6EdVo)

l 1/3

1/3

= (f w6dVo)

`J

///

< 16 bL + 12 IRoI r u2+2f dVo \ (2+2E)/6

< C(b, IR0I) (f w6dVo 1

=Co
rJ dVo)

(4-2e)/6

LECTURE 6. ISOSPECTRAL COMPACTNESS AND THE YAMABE PROBLEM

109

References T. Aubin, The Scalar Curvature, Differential Geometry and Relations, Reider Publishing Co., 1976. [B-C-Y] T. Branson, S.Y.A.Chang and P. Yang, as in Lecture 5. [C-M-W] S.Y.A.Chang, M.Gursky and T.Wolff, Lack of Compactness in Conformal Metrics with L"/2 Curvature, preprint. S.Y.A.Chang and P.Yang, Compactness of Isospectral Conformal Metrics (C-Y-1] on S3, Corn. Math. Hely. 64 (1989), 363-374. [C-Y-2] S.Y.A.Chang and P.Yang, Isospectral Conformal Metrics on 3-manifolds, JAMS 3 (1990), 117-146. [C-Y-3] S.Y.A.Chang and P.Yang, The Conformal Deformation Equation and Isospectral Set of Conformal Metrics, Contemp. Math 101 (1989), 165[A]

178.

P. Kilkey, Leading Terms in the Asymptotics of the Heat Equation, preprint (1988). [C] M. Gursky, Compactness of Conformal Metrics with Integral Bounds on Curvature, Duke Math. J. [M-S] H.McKean and I. Singer, as in Lecture 1. E.Onofri, as in Lecture 3. [O] [0-P-S-1] B.Osgood, R.Phillips and P.Sarnak, as in Lecture 3. [O-P-S-2] B.Osgood, R.Phillips and P.Sarnak, as in Lecture 3. [P-S] Parker and Rosenberg, as in Lecture 5. R. Schoen, as in Lecture 4. [S] [K]

LECTURE 7 Prescribing Curvature Function on S" In this chapter we will describe an application of the Sobolev-Thidinger-Moser type inequalities to a problem in prescribing curvature functions on the n-sphere. We will start with a question originally proposed by L. Nirenberg: what function is allowed to be the Gaussian curvature function on the 2-sphere S2? Applying the uniformization theorem, the question can be described in analytic terms in the following way. Denote the metric with Gaussian curvature K as g = e2wgo, then K and w satisfy the following equation (1)

Ow + Ke2w = 1 on S2

where 6 denote the Laplacian operator related to the standard metric. The question is: For which functions K does there exist a solution to (1) ? In this chapter we will discuss some basic facts, techniques and progress made in answering the above question on the 2-sphere. Before we do that, we will mention some other related questions of prescribing curvature functions. It turns out many of the techniques for the S2 problem, which we will mention below, can be applied to these problems. We do not intend to give a survey of the progress made in this fast developing field, and will mention only a few relevant results related to the techniques of the 2-sphere problem discussed here. The reader is referred to [S-2] and [C-Y-G] for more complete and up to date references on the subject. On general compact manifolds M' (n > 3) without boundary, an analogous question to (1) is that of prescribing scalar curvature problem. In this case, when restricting metrics conformal to a fixed one go, denote the metric with scalar curvature R as g = uhgo, then R and u satisfy the following equation (2)

c A u + ROM = Rou

where c = 4n-3 , A denote the Laplacian operator w.r.t. the metric go and Ro is the scalar curvature of go. The question is: Which function R can be prescribed

so that equation (2) has a solution? When R = constant, this is the Yamabe problem mentioned earlier, and the question has been solved in the affirmative way by [A],[S-1].

III

112 S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Another analogous question, whose precise geometric meaning is well-fomulated only on 4-manifolds at this moment, is the problem of prescribing "curvature" for the Paneitz operator. The Paneitz operator was introduced by Paneitz ([P), see also [Br]) on 4-manifolds, it is a fourth order differential operator P. defined as p9

= (-0)2 + 5(3R9si -

where R,j is the Ricci tensor, b is the divergence, d is the differential w.r.t. the metric g = (g j). This operator enjoys the following good properties: (a) It is "conformally invariant", in the sense that if g,,, = e2i'g is a metric conformal to g, then P9,, = e-"P,. (b) Let Q denote the "curvature" function

Q = 6 (-3IPI2 + R2 - AR) where IpI2 = square norm of Ricci tensor, the P. and Q9 are related as

-P9w +

(3)

Q9 on M.

It is in the context of equation (3) when comparing equation (1) and (2) that a natural question to propose is that of prescribing "curvature" Q as defined by (3). When the manifold M is of dimension not equal to four, in general the existence and precise form of the Paneitz operator with properties (a) and (b) is unknown. But in a recent article (G-J-M-S), it is proved that when n is even, an operator of order n with conformal invariant property as described in (a) exists; also when M = R" with the Euclidean metric, the operator P = (-is) f . Using stereographical projection pulling S" to R", one can then compute the form of P on S". It turns out that this is precisely the operator which has appeared in Branson [Br, p 231] and Beckner [B] (which we have discussed in Lecture 5). And the precise form of

P" on S"is z

P"_ fl(- A +k(n - k - 1))

whenniseven

k=0

+(n )2). H (- A +k(n - k - 1)) whennisodd. k=0

In view of (1), (2) and (3), an analogous question to ask is that of prescribing Q in the following equation (4)

-P"w + Qe""' = (n - 1)!

on

S.

LECTURE 7. PRESCRIBING CURVATURE FUNCTION ON S"

113

It turns out that equation (4) is a more natural extension of equation (1) than that of equation (3) of the problem of prescribing scalar curvature. This point of view is explained in a recent article [C-Y1.

For the rest of this chapter, we will return to equation (1) -the problem of prescribing Gaussian curvature on S2. We first remark that integrating equation (1) over S2, we get (we denote f = 41 fs, dVo )

t Ke2w=1 Thus K > 0 somewhere is a necessary condition for (1) to be solvable. To explore other properties of K, we will use a variational approach. Consider the variation functional FK associated with (1): FK [w] = log f Ke2w - S[w]

where S[w] = f [VwI2 + 2 f w as defined in Lecture 2. Basic facts about FK [w]

FK[w]
(i)

< log max K,

via Onofri's inequality in Lecture 3. (ii)

When K > 0,

sup FK(w] is attained if K =- constant. WEW t.3

Proof. When u = it log [J,,[ for some conformal transformation .0 of S2, then u log c is a satisfies Au + e2u = 1 and K = 1, thus, when K = c > 0, u = log a nontrivial solution of (1) for which Fc[u] = logc. In general, suppose u is a local maximum of the functional FK(u], a direct computation yields: 0>

d22

FK[u + tv]

Lo

2

Ke2nv2 - (I Ke2uv)2] - I [Vv12

for all V E W1'2. This implies that the first non-zero eigenvalue Al of the Laplacian of the metric Ke2ugo satisfies Al > 2. While the result of Hersch in Lecture 1 says Al < 2 with equality if and only if the metric Ke2ugo has constant curvature 1. This coupled with the assumption that K, u satisfy (1) is equivalent to K =- constant. Thus the only way to solve equation (1) using the variational approach is to locate saddle point of the functional FK [w].

114 S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

(iii)

(5)

Kazdan-Warner condition ([K-W]): if K, u satisfy (1) then

fe2u=0 for j=1,2,3.

Proof. ([O-Y-4]) Recall Lemma 1 in Lecture 3, S[w] is a conformally invariant quantity in the sense that S[TT(u)] = S[u], where v = T#(u) denote the induced conformal factor given by 0'(e2ugo) = e2ogo, and Qi is a conformal transformation of S2. Suppose u is a critical point of FK [u], hence

d FK[T0'j

Lo = 0

where P E S2, t > 1, i0p t is the conformal transformation on S2 as defined in Lemma 3 of Lecture 3. But FK[TOP.,(u')J = log f

= log

J

Ke2Tm,,.<
Ke27"°P.t

- S[TTP.,(u)]

- S[u)

= log Ke2uoo"., det I J0P., I

= log

- S[u]

(K o 0pi)e2u - sm.

Thus e2u.

P't t=1

t=1

Now a simple calculation shows that

Ko4kP'=,

if x3=.P.

This gives the desired condition. (iv) K, u satisfy equation (1) if and only if K o 4, T4 (u) satisfy equation (1) for any conformal transformation 0 of S2. From the geometric interpretation of K, this is an obvious fact. It can also be checked directly.

(Moser [M-2]) When K is an even function (i.e. K(t) = K(-£) for all £ E S2), (1) has an even solution. (v)

LECTURE 7. PRESCRIBING CURVATURE FUNCTION ON S"

115

Proof. Recall that for even functions u defined on S2, the best constant in Moser's inequality is a = 8ir (Remark 3 in Lecture 3). Thus following the argument as in Lecture 3 after Remark 3, we have for u even,

(6)

log J e2u < 2 -/ I VU12 + 21 u + C2,

for some constant C2. From (6), the same argument as in the derivation of Onofri's inequality as in Lecture 3 gives an even solution for equation (1) in case K is an even function. We now describe an a priori estimate for solutions of (1), which we think is a key step to the prescribing curvature problem.

Theorem 1. Suppose K,w satisfy equation (1). (a) Assume that 0 < m < K < M, then S[w] < C(m, M).

(b) Assume further that K is a nondegenerate function in the following sense (nd)

LK(Q) 34 0

whenever OK(Q) = 0

then there exists a constant C = C(m, M, (nd)) > 0 so that

w
IIwIIwI.2
Remarks. 1. Considering the fact (iv), bounds for S[w] is the best which one can hope for (instead of the energy bound f IVwI2 ) for a given K. For the sequence wt = 5 log I Jm p,, I for some fixed P E S2, K = 1, S[wt] __ 0 while f I Vwe 12 -+

30 as t-40. 2. Part (a) of Theorem 1 has appeared in [C-Y-6], [Ch] part (b) is a special case of a result in [C-Y-G], where a similar a priori estimate for the prescribing scalar curvature equation on S3 was also obtained. One should also mention,

using a different approach (i.e. local estimate), the result analogous to Theorem 1 on S3 has been proved in [Z]. 3. Result with a similar statement as in part (a) also holds for the Paneitz equation (4) on S", with S[w] replaced by a suitably defined conformal invariant term S"[w] ([C-Y]).

Proof of part (a). What we actually will establish is: (a)' If w E S and satisfies the equation (1), then there exists a constant C = C(m, M) with

C_<w
and

fIvwP2C.

116 S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

In view of Lemma 2 in Lecture 3 and (iv) above, statements (a) and (a)' are equivalent. To begin the proof of (a)' we will first state the following result, which is a modified form of Aubin's result (Lemma 3 in Lecture 3) (cf also [Ch]).

Lemma 2. Given w E S, there exists a constant C(K) such that

(7)

where c is a real number, (w, K) satisfies equation (1).

Proof of (a)'. Aubin's Lemma shows that for all e > 0

(8)

for some constant C(e). But when one examines the proof of Lemma 3 in Lecture 3 ([A]), one notices that the reason the a-term e f IVwI2 appeared in (8) is to absorb the term f IVwj. Thus (7) is a consequence of (8) along with the observation that when w, K satisfies (1) one can easily verify f jVwI < C(K).

We multiply equation (1) by 2w and integrate and apply Jensen's inequality (recall 1Ke2"' = 1)

(9)

21 I VwI2 +2 -/ w=2 JKe21 w 11

< log -} Ke2w e2w C=4))
IVwI2+4f w

(by (7) with

hence - f w < C(K). From this we conclude w is bounded below by using the Green's identity:

-w(y) + i w = j(Iw)(x)G(Yx)dtt(x) = J(i - Ke2w)(x)(G(y,x) + constant)dp(x) (since K > 0) < f G(y, x)dµ(x) + constant.

LECTURE 7. PRESCRIBING CURVATURE FUNCTION ON S^

117

But once w is bounded from below, we may modify (9) as

2JIVwI2=2JKe2w(w-i) = 2 1(Ke2hJ1 - 6)(w - w)

6) log f

(Ke2w - 6) e2(w-W)

1-6

5(1-6) [logmaxK+2j'1vw12+2 f w-log(1-6) +logC(K)1 where 6 = minKe2w. From this, we conclude that f IVw12 < C(K). To obtain a pointwise bound of w, we notice that exp

\ J w) 5

f e2w

- m J Ke2w = m

hence

I Jw

(10)

C (m, M)

Notice that for any p > 1, we maythen apply Onofri's inequality to conclude

I ep"' < exp(42 /IVwI2 + P J w) < c(+n, M, P), hence

I - w(Q) + f wl = I fV G(Q, P) A w(P)dVo(P)I

o(P))J(f (Ke2w-1)2)+} <(f IG(Q,P)I2dV
Proof of (b). We will prove the result by contradiction. Given K > 0 satisfying the non-degeneracy condition (nd), suppose the statement of part (b) does not hold. Then there exists a sequence wk satisfying £wk + Ke2w"` = 1 on S2

118 S: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

with maxg. wk - +oo. Applying Lemma 2 in Lecture 3, we get a sequence of conformal transformations Ok = Opk,tk, with e2vkgo = -0k(e2v,k9o), vk E $ satisfying (12)

£vk + K o OPk,tk e2vk = 1 on S2.

Applying Lemma 2, we have 1Ivkjj, < c(m, M) and JIVvkI2 < c(m,M). We may then conclude that some subsequence of tk - +oo. For if not, i.e., tk < to for all k, for some to, then wk o dk = vk - log det d4k is uniformly bounded, which contra dicts our assumption that maxg22 wk -. +oo. Thus, after passing to a subsequence

we may assume that tk -. +oo, Pk - P E S2, and vk - v in C""° for some a E (0,1); the last fact following from the pointwise estimates on vk and (12) along with the Sobolev imbedding. Notice that K o mK -4 K(P) uniformly on compact

subsets of S2 - {-P}, hence v satisfies (13)

Av + K(P)e2v°° = 1,

at least weakly on S2 - {-P}. But after applying standard arguments from elliptic theory one sees that in fact v satisfies (13) on all of S2. By the uniqueness of solutions of (13) belonging to S we conclude that vo = - log K(P). Normalizing z we may assume that vk (by rotating Pk to P and adding a suitable constant), K(P) = 1 and that vk satisfies (14)

ivk + (K o 41k) e2vk = 1

with Ok = cbp,t,,. Also, by our work above we have (15)

(16)

lIvkIIoo = o(1) ask --* oo

IIVvkll.=o(1) Applying the Kazdan-Warner condition to (14) we have

(17)

J(VKok,Vzj)e2t k = 0, 3=1,2,3

where Ok = OP.t,, and tk -4 oo. We now claim that (17) implies (18)

VK(P) = 0 and AK(P) = 0,

thus contradicting our assumption that K is a non-degenerate function and finishing the proof of part (b).

LECTURE 7. PRESCRIBING CURVATURE FUNCTION ON S"

119

To verify (18) we will begin to compare the expression in (17) to f (VK o ck, Vx,), and then compute the asymptotic behavior of the latter term as tk -+ oo. To this end we define

Ak=J'e

Bk= J =2 J(KoOk-1)i. Lemma 3. Under the assumptions (15), (16) with 4k = WP,tk, tk -+ oo, K(P) = 1 we have

o(1) if VK(P) # 0, (19)

Ak = Bk + Ck with jCkl =

o(t2 log tk) if VK(P) = 0. k

Let Bk') denote the i-th component of Bk, 1 < i < 3, then

I Bk') = cl a, t +0( t2 ), i=1,2; k

k

(20)

Bk3)=c2(b,I+b22)

tk

logtk+O(tk.)

where c1, c2 are dimensional constants. Clearly (18) is a consequence of (17), (19) and (20).

Proof of Lemma 3. To prove (2.12) we first write Ak =

J

< VK o Ok, V x > e2vk

=J e2tk

_-

J(K0Ok-1)Axe2"k- J(K1)

=2J(KoIk-1)V +2 J(Ko-Ok-1)x(e2o"-1) 2I(KoOk-1)e29". Thus Ak = Bk + Ck where

Ck=2 r(Ko.0k-1) '(e2""-1)-2J(Ko¢k-1)e2"".

120 S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

We will see that the desired estimate of Ck follows from the same asymptotic computation of Bk as in (20) below together with the assumption that IIvklloo = o(1), IIVvklloc = 0(1)

To verify (20), we will use the stereographic projection coordinates of S2 to

compute Bk in terms of the Taylor series expansion of K. To do this, denote Q = (XI, x2, x3) E S2 and let y = (yI, y2) be the stereographic projection from S2 to the equatorial plane R2 sending the north pole No = (0, 0,1) to oo. We can also w.l.o.g. identify the point P as the north pole N. Thus xi = +Iv for 1 < i < 2,

and x3 = y+i We assume the Taylor series expansion of K around N is given by 2

(21)

K(xl, x2i x3) = K(x1, x2) = K(N) +

E i=1

2

aixi +

bi.'xixj

E

2

+ o(> x, )

ij=1

1=1

and (2.14) holds in a neighborhood D = {y E R2, lyl > M} of N, for some M > 0 large. Notice in this notation 0j: (y) = tky. Denote Dk = {y E R2, Iyi > M }, then (bk(Dk) = D. To estimate Bk, let dA(y) = * 1+[VI v2 dB denote the area form, then (22)

dA(y) = 2 J JD°

M2

dJyJ2 k

R

= tkM2 = O(k) as tk --+ oo.

(1 + Iy12)2

Thus

Bk=2 /(Kocbk-1)r =2 f (Kook-1)a+O(1). tk

11 Dk

Next we notice that by circular symmetry,

Lb

Bk')

=

fDk

(t(y)dA(y) = 0

if i

aixi(tky)xi(y)dA(y) + 2 f

Dk

j; 1 < i, j

3.

bjexj(tky)xt(tky)

xi(y)dA(y)

j,t=1

+Ek+O(iy) fori=1,2,3, where

k(

( ID,,

ItkItkylyll3 xi(y)IdA(y)

\1+

2l

, i=1,2,3,

121

LECTURE 7. PRESCRIBING CURVATURE FUNCTION ON S"

and x

Bk3)

=2J

j.ts1

x3(y)dA(y) + Ek3) + O(t2 )

bjtxj (tkY)Xt(tkY)

k

Similarly we define Ck`) for i = 1, 2, 3 to be the components of Ck, and B,() for i = 1, 2, 3 to be the components of Bk. (19) and (20) will be consequences of the following lemma which can be verified by direct computation.

Lemma 4.

as k -00

JDk xi (tky)x(y)dA(y)

(23)

1 < i,j,t < 2;

0,

(24)

J

0, i=3, jot;

xj(tky)xt(tky)xi(U)dA(y) = "

x1

logtk, i=3, 1<j=t<2.

k

IEUI=O(tk), i = 1,2,3. We conclude from (23), (24) that Bk') =Clat k

+ O(t ),

i = 1, 2,

k (25)

BM

=C2(b11 + b22)

tk

log tk + O( 2 ), tk

with c1 > 0, c2 > 0 dimensional constants. By a similar argument we can prove (26)

Ck(i) = o(lal)tk

109 tk k

+O(ta ) k

where lal = E Tail, Ibl = E Ibijl. Then (19), (20) are direct consequences of (25), i.j

(26). We have thus finished the proof of part (b) of Theorem 1. In the rest of this chapter, we will describe some sufficient conditions on K (n = 2) or R (n > 3) for equations (1) and (2) to have solutions.

122 S.-Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

Recall that in the proof of Kazdan-Warner condition (iii) above, we have, for a solution w of (1),

FK [TmP , (w)]

I

= 0, for any P E S=, and

c=1

I

d FK [Tmr.: (w)] dt c=1

_ J < VK, 0(x - P) > e2ru = 0

A relevant observation here is that if e2w is close to the constant 1, then the integral is comparable to

2jKV PdA. A parallel argument works also for the higher dimensional setting. Thus given a potential curvature function R (or K) we form the following map G : B Rn+1 given by (27)

G

t

1P) t

1".

(R o Opt)

x.

It is natural to investigate the asymptotic behavior of the map G : (P, t) 1--4 f (R o Opt) x as the parameter t - oo. We choose coordinates x1, ... , xn+1 so that P is the south pole P = (0, ... 0, -1) and let yl ... y be the stereographic coordinates

(x11+y ,

!l

1
+_ 11

xn = v (In the coordinate system y, mt (y) = l y.) Expand R in Taylor series in y coordinates around P: oc

R(y) = R(P) +

Rk(y) k=1

where Rn is a homogenous polynomial in y1, ... , yn of degree k, so that 00

(R o Op.e){y) = R(P) + E Rk(y)t-k. k=1

We denote by R(°)(y) = R(P) + Ek=1 Rk(y), the truncated Taylor polynomial of order a.

Definition. We say R is non-degenerate at P of order a if (i) the truncated Taylor polynomial of order a has the form R(°) (y) = R(P) + R°(y),

LECTURE 7. PRESCRIBING CURVATURE FUNCTION ON S"

(ii)

123

for t sufficiently large, the map G(P, t) = J(R(a) o Opt) x

satisfies a lower bound:

IG(P,t)l >

cl a
At any point P E S", one can examine this notion of non-degeneracy with some simple computations (as in introduction of [C-Y-6]), we can see for example: (1)

At a point P where VR(P) 0 0, it is non-degenerate of order 1.

(2)

At a point P where VR(P) = 0 but tR(P) 4 0, it is non-degenerate of

order 2.

We observe that for a function R uniformly non-degenerate of order a < n, the map G(P, t) is non-zero for t > to , hence the restriction of G to the sphere {(P, t), P E S"} are mutually homotopic fort > to, and the degree deg(G, B, 0) is well defined.

Theorem 2. There exists constants e(n) such that if K or R is a smooth function satisfying:

(i) IIK - 111. < e(2) or IIR - Roll. < e(n); (ii) K or R is a uniformly nondegenerate function of order a, where a < n when n is even and a < n - 1 when n is odd, i.e. IG(P, t) I -t

(*)

when a < n, C t, if a = n t' togs

t. ,

for t > to, uniformly in P; for all P in S". (iii) deg(G, B, 0)

0;

then the equation (1) has a solution.

Remarks. 1. In the case a = 2, uniformity in (*) is a consequence of the nondegenerate assumption on the critical points of the function K or R. In the general case, the uniformity requirement does not follow from the nondegenerate requirement on the critical points of K or R alone. In principle, it is possible to reduce this requirement to algebraic criteria on the Taylor coefficients of the function R at its critical points. For example we determine necessary and sufficient conditions on the Taylor coefficients of R at its critical points when a = 3 in [C-Y-6a]. 2. In dimension 2 (and 3), result in Theorem 2 is optimal in the sense that if the differential equation (1) admits a solution then it is necessarily captured

124 s: Y. CHANG, THE MOSER-TRUDINGER INEQUALITY & CONFORMAL GEOMETRY

by the variational scheme used in the proof. This amounts to an a priori estimate for solutions of (1) when I]K - III is sufficiently small It turns out in the case of n = 2 and 3, one can drop the smallness assumptionin the statement of Theorem 2.

Theorem 3 [C-Y-GJ. On S2 (S3), suppose K > 0 (R > 0) is a smooth function satisying the non-degeneracy condition that VK(Q) = 0 implies tK(Q) # 0 (respectively for R) and deg(GIt_ea0) # 0, then the equation (1) (respectively (2)) has a solution. Theorem 2 generalizes previous existence results of Chang-Yang ([C-Y-4J [C-Y5] and [H]) on S2 and Bahri-Coron ([B-C]) and Schoen-Zhang ([S-Z]) on S3 where K (respectively R) is assumed positive, having only isolated non-degenerate critical points and in addition satisfying OK(Q) # 0 at critical points, and the index count condition:

r`

(28)

(-1)ind(Q) # (_1)n_

Q cr1t cal,4K(Q)<0

In the appendix of [C-Y-G], it was shown that the index count condition (28) implies deg(G, 0) # 0. On the other hand, the nondegeneracy condition (nd) allows K (or R) to have non-isolated critical point sets. In dimension 4, there is an interesting example of Bianchi-Egnell ([B-E]) which indicates that further complications

arise and it will be necessary to understand the interactions of more than one concentrated masses. The validity of Theorem 3 remains open for n > 5. Theorem 3 is a consequence of a topological degree argument. The earlier perturbation result (Theorem 2) provides the initial step of a continuity argument. One then verifies that deg(G, 0) gives in fact the Leray Schauder degree of a nonlinear map whose zeroes correspond to solutions of the differential equations when R satisfies the close to constant condition. Then the a priori estimates as in Theorem 1 provide the continuity argument needed to verify the invariance of the Leray Schauder degree as one moves along the parameter in the continuity scheme. We refer interested reader to [C-Y-G] for details.

References [A]

[B]

[Br]

[B-C)

Aubin, T., Meilleures constantes dons le theforeme d inclusion de Sobolev et un theoreme de FPv4holm non lin6aire pour la transformation conforme de la courbure scalaire, J. of Funct. Anal. 32 (1979), 149-179. Beclmer, W. As in Lecture 5. Branson T., Group representations arising from Lorentz conformal geometry, JFA 74 (1987), 199-293. Bahri, A. and Coron, J.M., The scalar curvature problem on the standard three-dimensional sphere, Jour. Functional Analysis 95 (1991).

[B-EJ

Bianchi, G. and Egnell, H., An ODE approach to the equation Au + Ku = 0 in Rn, Math. Z.210 (1992), 137-179.

[C-Y]

as in Lecture 5.

LECTURE T. PRESCRIBING CURVATURE FUNCTION ON S"

[C-Y-4] [C-Y-5] [C-Y-61

[C-Y-6a] [Ch]

[Ch-D]

125

Chang, S.Y.A. & Yang, P.C., Prescribing Gaussian curvature on S2, Acta Math. 159 (1987), 215-259. ,

Conformal deformation of metric on S2, J. of Diff. Geo. 27

(1988), 259-296. , A perturbation result in prescribing scalar curvature on S", Duke

Math. Jour. 64 (1991), 27-69. , Addendum to "A pertubation result in perscribing scalar curvature on S"", Duke Math. J. (1993). Cheng, S.Y, Eigenvalue comparison for comformally related metrics, preprint. Chen, W.-X., & Ding, W-Y., Scalar curvature on S2, Trans. AMS 303

(1987), 369-382. [C-Y-G] S.Y.A. Chang, P. Yang and M. Gursky, The scalar curvature equation on 2 and 3-sphere., Calculus and Variation and PDE, June issue (1993). [GJMS] C.R. Graham, R. Jenne, L. Mason and G. Sparling, Conformal invariant power of the Laplacian I. existence., to appear in J. London Math. Soc.. [H] Han, Z-C., Prescribing Gaussian curvature on S2, Duke Math. Journal 61 (1990), 679-703. [K-W] Kazdan, J. and Warner, F., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. of Math. 101 (1975), 317-331. [M-2] [O]

[On] [P]

Moser, J., On a non-linear problem in Diff. Geometry Dynamical Systems (M. Peixoto ed), Acad. Press., N.Y., 1973. Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, Jour. Math. Society of Japan 14 (1962), 333-340. Onofri, E., On the positivity of the effective action in a theory of random surfaces, Comm. Math. Physics 86 (1982), 321-326. S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint 1983..

[S-1]

as in Lecture 4.

[S-2]

R. Schoen, A report on some recent progresses on non-linear problems in geometry., JDG No.1 (1991), 201-242. Schoen, R., and Zhang, D to appear.. Zhang, D., New result on geometric variational problems, Thesis, Stanford University (1990).

[S-Z] [Z]

The Effect of Curvature on the Behavior of Harmonic Functions and Mappings Richard Schoen

lAS/Park City Mathematim Series Volume 2, 1996

The Effect of Curvature on the Behavior of Harmonic Functions and Mappings Richard Schoen

Introduction The purpose of this lecture series is to explain the role that curvature plays in the analysis of solutions of geometric partial differential equations, especially those

which arise from variational principles. We do this by considering two concrete situations. The first of these is the study of harmonic functions on Riemannian manifolds. This subject occupies the first three lectures. The second problem we consider is the harmonic mapping system which arises from the variational problem of extremizing the L2 energy for maps between Riemannian manifolds. Although we consider two specific problems here, much of the theory we describe has been generalized to other problems, and the general ideas involved have broad application to the study of geometric partial differential equations of elliptic type. In this introduction we describe in general terms the content of the various lectures, and give references for various extensions of the theory which have been obtained. The first two lectures give the proof of the gradient estimate and Harnack inequality for harmonic functions on Riemannian manifolds with constants depending only on a lower bound of the Ricci curvature of the manifold, the dimension, and

the radius of a ball on which the function is defined. This result in global form; that is, for positive harmonic functions defined on complete manifolds, is due to S. T. Yau[Y]. The local version is due to Cheng and Yau[CY]. An important point of this estimate, is that it does not require a global coordinate system, nor a bound on the injectivity radius of the manifold. Standard Harnack inequalities in PDE require bounds on the coefficients of the operator in some fixed coordinate chart, so these are not suitable for our purpose. There are two basic ingredients in the proof of the gradient estimate. The first is the use of comparison theorems for the Riemannian distance function. It is important in many PDE estimates to have an appropriate exhaustion function for your manifold or domain. The basic properties of the Riemannian distance function which are needed for the Harnack inequality 'Mathematics Department, Stanford University, Stanford, CA 94305

E-mail address: echoen4gause.stanford.edu Q 1996 American Mathematical Society

129

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R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

are that it is uniformly Lipschitz (true on any metric space), and that its Laplacian has an upper bound depending on the lower bound of the Ricci curvature. This lower bound of the Laplacian holds in the distributional sense beyond the injectivity radius, and so does not require the distance function to be smooth. Geometrically, the Laplacian of the distance function equals the mean curvature of the geodesic sphere, so this comparison result says that the mean curvature of the geodesic sphere may be bounded from above in terms of a lower bound of the Ricci

curvature of the manifold. The second main ingredient is the Bochner formula which yields a lower bound of the Laplacian of IDul2 for a harmonic function u in terms of a lower bound of the Ricci curvature. The proof then relies on a clever application of the maximum principle. The outline of this proof may be used to obtain important results for other problems as well. For eigenfunctions, a modification of this proof was used to obtain sharp lower bounds for the first eigenvalue of a manifold by P. Li[Li], and more generally by Li and Yau[LY1]. Li and Yau[LY2] have obtained an analogous estimate for the heat equation. S. Y. Cheng[C1] used a similar argument to obtain a gradient estimate for harmonic mappings into simply connected manifolds of nonpositive curvature in terms of a bound on the oscillation of the map. This result was improved by H. I. Choi[Cho] to handle mappings whose image falls in a convex coordinate chart. The reader may also see the recent book [SY] for more details of these results. In the third lecture we prove an existence theorem for harmonic functions with prescribed boundary data on complete, simply connected, manifolds with bounded negative curvature; that is, all sectional curvatures bounded between two negative constants. Such manifolds have a geometrically natural compactification as a closed ball. This is given by adding a sphere at infinity, which may be thought of as the sphere of directions of geodesics from a point. This boundary is referred to as the geometric boundary. We then give an elementary proof of the unique solvability

of the Dirichlet problem for continuous boundary data on such manifolds. This theorem is due to Anderson[A] and Sullivan[Su], and the proof given here is due to the author, and appears in [AS]. The basic idea is that the theory of harmonic functions on manifolds of this type is similar to that on bounded domains in euclidean space. The condition of bounded negative curvature corresponds to a very weak smoothness assumption on the boundary, so one should imagine doing analysis with a very singular boundary. The result of Anderson and the author[AS] is much more general than the solvability of the Dirichlet problem because it constructs with estimates the Poisson kernel functions of such a manifold, showing that there is a unique kernel function for each point of the geometric boundary, and that these kernel functions are Holder continuous on the boundary. The main ingredient which is established in [AS] is a Harnack inequality at infinity. These results were generalized and improved by A. Ancona[An]. Results for the special case of coverings of compact manifolds which weaken the curvature assumption were proven by Lyons and Sullivan[LS], and by W. Ballman[Ba]. A result of Cheng[C2] establishes solvability of the Dirichlet problem under the assumption of pointwise bounded negative curvature; that is, the curvature is less than a negative constant,

and the ratio of sectional curvatures at any given point is bounded. This asumption removes the lower bound in two dimensions, and allows the curvature to be unbounded in higher dimensions. It is not known whether the Dirichlet problem is

INTRODUCTION

131

solvable without the assumption of a lower curvature bound. The reader may see [SY] for more discussion, and questions in this direction. The last six lectures deal with questions involving harmonic mappings. Here the key geometric ingredient is the curvature of the target space. There have been recent developments which extend much of the existing harmonic mapping theory to singular target spaces for which the curvature is bounded from above in the sense of triangle comparison hypotheses. We discuss here only the first part of this theory which was developed by Gromov and the author[GS]. This theory has been extended substantially by N. Korevaar and the author[KS], and many existence results were also treated from a different point of view by J. Jost[J]. These new

results handle arbitrary metric space targets, and in case the target space is a length space of nonpositive curvature, the results of [KS] prove that the harmonic mappings are Lipschitz continuous in the interior. T. Serbinowski[Se] has shown

that harmonic maps are C° for any a < 1 provided the boundary map is CO. In a recent joint work with Korevaar and Serbinowski we have also extended the regularity theory of Uhlenbeck and the author[SU] to work under the hypothesis that the target be a length space with curvature bounded from above by a constant. This result implies that minimizing mappings are Lipschitz continuous on an open set whose complement has Hausdorff codimension at least three. In these lectures, we deal only with the case considered in [GS] where the target is a Riemannian simplicial complex of nonpositive curvature. The results of Lectures 4,5, and 6 extend to harmonic mappings to arbitrary length spaces of nonpositive curvature. Much of this extension appears in [KS). The major new ingredient which is needed is a detailed construction of the nonlinear Sobolev space of maps with finite energy given in [KS]. Lecture 4 gives the definition of metric spaces of nonpositive curvature, together

with examples and motivation. One of the basic motivational issues is that the classical theory developed by Eells and Sampson[ES] makes use, in a crucial way, of a Bochner formula (analagous to the one used above for harmonic functions). It is necessary to find a replacement for this formula in the case of singular targets in order to prove the Lipschitz bound which is critical for the theory. Lecture 6 derives a monotonicity inequality which is crucial for the theory of [GS] concerning the structure of harmonic maps to euclidean buildings. This inequality extends to harmonic maps to arbitrary length spaces of nonpositive curvature; however this extension is not carried out in [KS], as the Lipschitz result is derived in [KS] by a different method, closer to the Eells-Sampson method. In [GS], the Lipschitz bound was derived from the monotonicity inequality. The extension of the monotonicity inequality for maps to general targets, and its consequences will be discussed in a future paper of Korevaar and the author. Lectures 7 and 8 describe the detailed results of (GS] for harmonic mappings to euclidean buildings. This theory uses special properties of the target space to derive very special structure of harmonic mappings to such spaces. Finally Lecture 9 describes the application [GS] to proving

the arithmeticity of lattices in rank 1 Lie groups. This involves the use of the structure theorem to prove a vanishing theorem for harmonic mappings to euclidean buildings.

Finally we mention that in light of the direction which is pursued here of studying harmonic mappings to singular spaces, it seems to be natural to propose

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a similar direction for the material in the first three lectures. Since the gradient estimate of Lecture 1 depends only on the dimension and a lower bound on the curvature, one might guess that it is true for harmonic functions on length spaces with curvature bounded from below. This is not known in full generality, but some interesting partial results have been obtained by J. Chen[Chj, and recently by J. Chen and H. Pei. The main issue is similar to that for harmonic mappings, namely making sense of a Bochner formula (and proving it). The author expresses his sincere thanks to Dan Pollack for taking excellent notes during the RGI lectures, and writing them up clearly. We thank Bob Hardt and Chang You Wang for providing us with a typed copy of the body of these notes.

Without their assistance, this paper would not exist. Finally we thank Bob Hardt and Mike Wolf for their efforts in organizing the RGI, and for their persistance in getting the present volume published.

LECTURE 1 Gradient Estimate and Comparison Theorems

Let (Mg) be a n-dimensional complete Riemannian manifold. We will be interested in several higher order quantities. Consider first the Riemannian curvature tensor Riem(g). In terms of a local orthonormal basis {ej} 1 for the tangent bundle of M, the coefficients of Riem(g) are written Rj,ki. Using these, we may describe the following important geometric components: 1. Sectional Curvature, K(ir). Given the 2-dimensional subspace it g TZM spanned by el, e2, we get the sectional curvature K(7r) = R1212. This is equal to the Gauss curvature at x of the two-dimensional image of 7r under the exponential map expx. 2. Ricci Curvature, Ric(g). A diagonal component Rjj Rikjk = sum of the sectional curvatures of the coordinate 2-planes containing ej. The off-diagonal components Rj may be obtained by polarization. 3. Scalar Curvature, R(g). R = E'i=1 Rj,jj = sum of the sectional curvatures of all the coordinate planes. Among the interesting geometric operators on (M, g), we have the most fundamental operator, the Laplace-Bertrami operator A on (M, ds2), which is defined in local coordinates by

Q_ 1 8 C fgji 09

8 BxJ J

where ds2 = gijdx'dx&, (9'i) = (9j.,)-1, g = det(g;3). This operator is associated to the Dirichlet energy E(f) = JM IVf I2dp where f E and dp the volume form of M; in local coordinates, dp = det (gj9 )dx. The relationship between harmonic functions, curvature, and spectral theory is quite interesting. Basic intuition for this comes from 2-dimensional case. Here we have that a Riemannian surface is a Riemann surface and we can invoke uniformization. Suppose (M2, g) is simply connected so that the possible conformal types are S2, C, D. Moreover, from the point of view of harmonic functions, a surface of nonnegative curvature behaves like C, and a surface of nonpositive curvature behaves like D. Therefore, one might expect that, in higher dimensions, manifolds 133

134

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

of sectional curvature > 0 behave like R" and manifolds of sectional curvature < 0 behave like H". Unfortunately, there are few theorems like this. Here is one:

Theorem 1.1 (Gradient Estimate). If u is a positive harmonic function on a ball BB(x) C M, k > 0, and Ric > -(n -1)k2, then sup B./z(x)

I D-I U

< Cn

a \ I ka)

Corollary 1.2. Suppose RicM > 0. Then any positive harmonic function u on m is constant. In order to prove the gradient estimate theorem, we first need to review several important comparison theorems.

Recall first, for x E M, the exponential map exp.: TM - M. For X E TxM with IXI = 1, y(t) = exp.(tX) is the unique unit speed geodesic that starts from x and goes in the direction X. When t is small, y is the unique minimal unit speed geodesic joining x and exp.(tX). Note that the differential dexp. It X : Tex(T.M) - T.y(t)M is a linear isomorphism. Let to = sup{t > 0: y be the unique minimal geodesic joining p and y(t)}. If to < +oo, then y(to) is called a cut point of x. Let Cut(x) = set of all cut points of x. Observe that the distance function x), while only Lipschitz on M, is actually smooth on M \ C(x). In particular, we may take derivatives, which will involve various curvatures of M.

Theorem 1.3 (Hessian Comparison). Let M1 and M2 be two n-dimensional complete Riemannian manifolds. Assume that -yi : [0, a] - Mi (i = 1, 2) are two geodesics parametrized by arc length and that -y, doesn't intersect the cut locus of denote a 'yi(0). Let pi be the distance function from yi(0) on Mi, and let sectional curvature of Mi. Assume that 0 < t < a and that at -Y1 (t) and 72 (t), we have K1

(x1-.) 1 K2'r Xs, 82)

where Xi is any unit vector in Tx,(t)Mi perpendicular to

.

Then

H(p1)(X1,X1) <_ H(p2)(X2,X2)

where Xi E T.y,(a)Mi With (XiKa > (yi(a)) = 0, and H(pi) denotes the Riemannian Hessian of the distance function pi.

Proof. We start with the definition H(f)(X,Y) = JI'C(Y(f )) - (OXY) f where f E C2(M), X, Y E TM are two vectors, and X (or k) are two extended vector fields of X (or Y) near x, and V denotes the Riemannian connection of M. Since -yi f1 Cut(yi (0)) = 0, we may extend Xi E T., (.)Mi) with (*, Xi) = 0 to a Jacobi

field Xi along -yi satisfying Xi(yi(0)) = 0, X1(yi(a)) = Xi, and [Xi, -] = 0. Since

LECTURE 1. GRADIENT ESTIMATE AND COMPARISON THEOREMS

136

grad ps = 4 , we have that H(p2)(Xi,X1) = XiXiPs - (°X"Xi) Pi

= X; (ki,

_ (X;, °

f

d

o

di

,/

- (v.i1, vr) = 0)

Xi) (because [ki, J

=J Since X; is a Jacobi field, it satisfies

and we get

H(pi)(Xi, Xi) =

fO

I°

Xi I

2

- (R,

,

Xi, 5;

X;

Ti

We may now compare these two Hessians as follows: Let E',,..., E,, be orthonormal. ) (-y;(0)) = 0, we Since (Xi, parallel vector fields along yj such that En =

A'(t)EE. We may assume have (Xi, -) = 0 at each point of 'yi. Set X2 that X1(a) A (a)EJ (-yl(a)). Define now a vector field Z along; by

Z= n-1 E aj(t)E;. j=1

Then Z(0) = X1(0), Z(a) = X1(a). Moreover IZI = IX2I and n-I

n-1

j-1

j=1

A'

(t) E.,2

V g2- 12

136

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Since a Jacobi field minimizes the index form among all vector fields along the same geodesic with the same boundary data, we have

H(Pl)(X1,X1)=

f

n

2

(ki

I VkXI

8

ayl)'rT1,X1

2

1r°

< To I v&'T Z

ja

V, .

=

(z) 5;i1F z)

- (RI X2

Jon I V

Z'tyrl 8y,'Z

R1

(R, X2r

''2) '_J /

= H(p2) (X2, X2).

Theorem 1.4 (Laplacian Comparison). Let M be an n-dimensionsl complete Riemannian manifold with Ric(M) > -(n - 1)k2. Let 6) be the distance on M from a fixed point 0. If x E M and is smooth at x, then

£PM(.,6) Ix S

0){1+kpM{x,0)).

Proof. Let N be a space form of constant curvature -k2, and 0 be a fixed point on N. Let of N. We start to compute the Hessian of distance function

with Xly(p). For y be a geodesic in N starting at 0, p > 0, and X E 0 < t < p, let X (t) be the parallel displacement of X along y. Then the Jacobi field Y(t) along y with Y(0) = 0, Y(p) = X has the form Y(t) = f(t)X(t) where f (t) satisfies

df(t) - k2f(t) = 0 , f (0) = 01 f (P) =1.

It is easy to see that f (t) _kp it.. Now, if {

, X 1, ... , Xn- I } is an orthonormal

basis of T,(,)M and ii(t) = f (t)Xi(t) is the Jacobi field with X,(p) = Xi, then we have n-1

APN(',0) = > H(P)(XX, X,) S=1

= (n - 1)

12

dtf (t) = (n - 1)k coth kp.

J a P I`

+k2 f2(t)] dt

Applying the Hessian Comparison Theorem, we have that APM(',6)

I=<_ APN(',0) 1

v

LECTURE 1. GRADIENT ESTIMATE AND COMPARISON THEOREMS

137

where y is point on N with pN (y, 0) = pM (x, 0). Therefore 6)

APM

I < (n -1)k coth(kpM(x, 0)).

It is easy to check that kpcoth kp < 1 + kp , so

OPM(',6)

PM(x,0)(1+kpM(x,0)).

Ix

Remark 1.5. F4om the Laplacian Comparison Theorem, we see that if Ric(M) >

-(n - 1)k2 and y E M \ Cut(x), then OPM(',x) Iy<

n-1 PM(y,x)(1+kPM{y, x))

Moreover, we can show that a corresponding global inequality holds in a distribution sense, i.e.,

(*) fM PM(', 0)I 4O < JM Pn

p)

(1+kpM(', 0))V

V 'P E Co (M, R) and cp > 0.

1.6 Proof of (*). Let Sl = M\Cut(O), and v E 0000(M) with cp > 0. Since Cut(0) has Hausdorff H" measure zero, we have

f PM (', 0)0'P = jPM(.U).so. M

Defining 11E C it as 1l, = {x E it : d(x, Cut(0)) > e}, we know ilf converges to it as e . 0, and hence, applying Stokes formula and the Green formula gives JMPM('AAW

M

VPAf (', 0)VV

= lim Jf VPM(.,d)V'P coo fly

_+limfn. 1PM(',0)cP--lim < lim fo, n - 1 (1 + E-.o

=f n-1 M

f

as2

V avf lim fan, E-0

OPAf av

W°PM

t-0 otL

av

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

138

We also note that fl is star-shaped and hence so is ft.. In particular, > 0. Therefore, fan,
fMPm (-, O)Ap

:1M

0 and

Per(

Now, we return to the proof of the Gradient Estimate Theorem. First, we calculate the Laplacian of the energy density of u, namely 12IVU12/ J = IVVU12 + VuV(Au) + Ric(Vu, Vu)

\

> IVVU12 - (n - 1)k21Vu12.

Since, by hypothesis, Ric(M) > -(n - 1)k2 and Du = 0, we see that

1

(ivui2)

=1VUTAIVUl + Ivlvu112.

In particular, we have (1.1)

IVulAIVUI + (n -1)k21VU12 > IVVu12 - lvIVu112.

Now, we want to calculate the right hand side in some detail. Since the calculation

can be taken locally, we fixed a point p E M and chose a suitable orthonormal frame near p such that (p) = 0 for i > 2 and iii; = I Vul (p) then at p V,(lVul) = V2(

u, 1,.; Ivul = U1,i

(ui,u,)) _

and

VIVUI VIVui = j:u1.,IVVUl2 - IVIVu112 2 u,a -

is

u2

j

l,i

FU.

i>2, i

> Eu1, + i>2

u,a. i>2, j>2

LECTURE 1. GRADIENT ESTIMATE AND COMPARISON THEOREMS

139

2

Since Du = 0, we have u1 1 = (Ei>2 ui i) . So by the Cauchy-Schwartz inequality, we have 2

IVVu12-IVIVU112>-Eu 1+nl 1

i>2

Eu2l i>2

1

) 2

+ n-1u1,1

1

2

i>1

=

n

1 1

IVIVu112.

Thus, combining (1.1) and (1.2), we get (1.3)

IVukkIVul >

n

1 1IVIVu1I2 - (n -1)k2IVuI2.

LECTURE 2 Gradient Estimate Proof and Corollaries We continue our proof of the gradient estimate theorem. Using inequality (1.3) from last lecture, we introduce the function V 11To use the maximum principle, we compute

VV =

vlvul U

-

- Ivulyu U2

DIVui

2Vlyul Vu

u

u2

IVulAu u2

DIVui _ 2ViVui Vu u

+

u2

DIVul _

+

2Iyu13 u3

21Vu13

u3

2yV Vu u

u

1IVIVU1I2 - (n - 1)k2Ivui2)

(n

Ivuiu

IVIVU1I21 - (n (n - 1)Ivulu

-1)k2Ivul

-

u

- 2V V Vu

Vu u

On the other hand,

vlvuI = uvV +

Iyulyu U

IOZI4

IVIVU1I2 = u21VV12 + so that

IVIVu1I2

Ivulu

Iv3I3

>

u

+ 21VulVu va ,

+2vV

Vu u

Therefore, (2.1)

AV >

1

(n

1) V'

- (n - 1 ) k 141

- 2- n 2 1) ,,V

vu

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

142

Note that if w attains its maximum inside B.(x), say at x0, then we know that AV(X0) < 0, (VO(xo) = 0, and hence, 1

1)`p3(xo) - (n - 1)k2V(xo) 5 0

or

'7(xo) 5 (n - 1)k2.

(n -

Unfortunately, V can also attain its maximum on 8BQ(x). We introduce a new function for B. (x) as F(y) = V(a2 - p2(y)), where p(y) = d(y, x), and note that F(OBQ(x)) = 0 and F > 0 on BQ(x). We may assume that there exists x0 E BQ(x) such that IVul(xo) 0 0 (otherwise, the conclusion holds automatically). Suppose that x1 E BQ(x) is a maximum point of F on BQ(x). Case I: x1 f Cut(x). In this case, F E C2 on BQ(x). Therefore, VF(x1) = 0, AF(x1) < 0. At x1i this implies V0

Vp2 a2 - p2

App

2Vp2VV

Ape

a2-p2

V

(a2-p2)p- 0'

hence, Dip

21Vp212

- a2Opt - p2

(Q2 - p2)2 -

0 .

Since R.ic(M) > -(n - 1)k2, we have that Opt = 2pAp + 21 VpI2 = 2 + 2pAp

<2+2pnp 1(1+kp) = 2 + 2(n - 1)(1 + kp).

Substituting this into the previous inequality, we get

2+2(n- 1)(1+kp)

AV

a2 1p

8p2

-

0.

(a2 - p2)2

- p2

We can also plug the inequality (2.1) for App into it 0

¢2 (n - 1)

-(n- 1)k - (22

2

8p2

n - 1

Ou

a2 - p2

At x1, we have, by (2.2),

VO Vu u

2pVp Vu (a2 - p2)u

2pw a2 - p2

(a2 -

p2)2.

LECTURE 2. GRADIENT ESTIMATE PROOF AND COROLLARIES

143

8o 2

0 > n

4(n - 2)

1 - (n - 1)k2 -

p

n-1 az - p2

_

c(1 + kp)

8p2

a2 - p2

(a2 - p2)2

or

0> nF21 nF >

1

-(n-1)k2(a2-p2)-c(l+kp)(a2-p2)-8p2-4n-1pF - (n - 1)k2a° - 8a2 - 2c1aF - c(1 + kp)a2

(since a2 - p2 < a2)

>

nF F2

1 - 2c1aF - c2(1 + ka)2a2.

Therefore F(x1) = supB.(z) F:5 cna(1 + ka) restricted on Bf (x), we will have 3a2 sup 4

I-I < Cna(1 + ka)

B3(x) u

sup -D-- < Cn (1 +aka)

B3(x)

Case II: X1 E Cut(x). In this case, even though the inequality of Ap holds distributionally, we don't have a pointwise estimate of Ap at x1. In order to overcome this difficulty, we employ a "support function". Let ry denote a minimal geodesic joining x and x1, and choose t on ry which is very close to x. Then x1 0 Cut(2). If we denote e = d(x, x), then by triangular inequality, we have

e + ft, 2) >- d(y, x) V y E NN, a small neighborhood of x, e + d(xl, z) = d(xi, x).

We consider F = (a2 - (e + p2)2)¢. Then F:5 F near x1 and F(xl) = F(zo) so that x1 is maximum point of F. We can apply the above argument to and then let e -, 0. We finally get the same conclusion. This method, or modification of it, have been used to prove many results in geometry. For example: 1. (Li-Yau [LY1],[LY2]) Obtain lower bounds on eigenvalues of compact manifolds. Gradient estimate for a positive solution of the heat equation on a manifold with a lower Ricci curvature bound. The heat equation then gives information on the eigenvalues.

144

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

3. (S.Y. Cheng [Cl] and H.I. Choi [Cho]) Used a similar method to obtain gradient bounds for harmonic mappings to special target manifolds in terms of a bound on the oscillation of the mapping. Now, we draw some direct corollaries of the Gradient Estimate Theorem.

Corollary 2.1. Suppose u is harmonic on BQ(x) C M, where Ric(M) > -(n - 1)k2. Then sup IVul < c (l +aka ) sup Jul. Proof. Let A = supBQ Jul. Then v = u + A + e > 0 on B.. Applying the theorem to v, we get

sup Vu 1 = sup Vv I_ c

B.12

l

1 + ka

l

B,12


a

sup(u+A+E) BI

12 pJul+e

Letting e 10, we get sup I VuI < c

1 + ka sup

B,/2

IuI

a

Corollary 2.2 (Harnack inequality). Let M be a complete Riemannian manifold with Ric(M) > -(n -- 1)k2. Suppose that u is a positive harmonic function on the geodesic ball BQ C M. Then

sup u < c(n, a, k) inf u. B,/2

Ba/2

Proof. By the gradient estimate theorem, we have supB.,3 Ivul < c(n, a, k). Let x1, x2 E Ba be such that SUPB./2 u(x) = u(xl), infB,,, u(x) = u(x2). Let -y be the minimal geodesic joining x1 and x2. Then

/

Y

I VuI ds < c(n, a, k)

JY

ds < c(n, a, k)2a.

On the other hand log

u(xl) _

-

y

U(X2)

r

dlogu(-y(s)) ds ds

IDu1 ds

u

< 2ac(n, a, k) .

Therefore, u(x1) < e2ac(n.a.k)u(x2). The following Liouville theorem now follows readily.

LECTURE 2. GRADIENT ESTIMATE PROOF AND COROLLARIES

146

Corollary 2.3. Suppose M is a complete Riemannian manifold with Ric(M) > 0, then any positive harmonic function is constant. Proof. Applying the theorem with k = 0, we have sup B.

IVul u

<

c a

Letting a go to infinity, we have that IVul - 0, hence, u - const.

Remark 2.4. Here is an example which indicates when the Harnack inequality goes bad. Consider the catenoids in R3, MM = {(x,y,z) E R3 : x = ecosh(!)cos9, y = ecos(z)sinO, z E R, 0 E (0, 21r]}. e

E

Truncate these catenoids by the planes z = ±1 to obtain the catenoidal regions M. Then, as e becomes smaller, ME looks more like a pair of discs. The harmonic function solving the Dirichlet problem with u 1 i90 = 1 and u 1 01 = 0 will have wide oscillation on the neck. The reason is that the curvature KK of ME approaches oo

as e-40. Now, we have seen that M with Ric > -(n -1)k2 looks like R" from the point of view of harmonic function. Finally we mention the

Conjecture 2.5 (Yau, 1974). Let (M, g) be complete with Ric(M) > 0, then, for

each 1>0 dim{u : Au = 0, sup Jul < cR[} < +oo. BR

LECTURE 3 Harmonic Functions on Negatively Curved Manifolds Let (M, g) be Cartan-Hadamand manifold, i.e., M is simply connected, complete, and all sectional curvatures are nonpositive. By the Cartan-Hadamand Theorem the exponential map expp : TPM M is, for every p E M, a diffeomorphism. We will now further assume that every sectional curvature KZ(M) satisfies the strict inequalities -b2 < Ks(7r) < -a2 < 0 for some constants 0 < a < b. Question: Do such manifolds behave like H"? The answer is roughly yes. In order to see this, we wish to compactify M by adding a "visibility boundary" 8M to M. In other words, we will add an end point to each ray so that we may identify asymptotic directions. Here a my from p is a geodesic

'y : (0, oo) - M, -f (t) = expp tv

corresponding to some v E TPM. We say that two rays ryi, rye are equivalent provided that suptdist,y(y1(t),ry2(t)) < oo, and we define 8M = {equivalence classes of rays).

How should we think of this? Consider the spheres Sp(M) _ (v E Tp(M) (vI = 1}.

For each p E M, a vector v E Sp(M) gives a unique representative in 8M, and we have an identification OM -- SS(M). Using a different point q E M deSq(M), and hence a homeomorphism termines similarly an identification OM Vbp.q : Sp(M) --+ Sq(M). Some distance estimates using the Rauch and Toponogov Comparison Theorems shows that 1bp,q is actually Holder continuous of order

alb [SY, Prop. 11.1.3]. Thus 8M has a natural Holder structure. For real hyperbolic space H", f/ip,q is a conformal transformation. For R" (where one would have

a = b = 0) Op,q is induced by a translation of R". In general, the transformation ,bp,q is badly behaved. For example, for complex hyperbolic space CH", it is not even quasi-conformal.

The existence of many non-trivial bounded harmonic functions on M is guaranteed by the following solution of the Dirichlet problem at oo. 147

148

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Theorem 3.1 [A], [SuJ. For any V E C°(M), there exists a unique u E CO°(M) U C°(M) satisfying

onM

Au=O

u='

on 8M.

Proof. We give a simplified argument which appears in [AS]. Fix p E M and write W = So(f) for t E Sp(M). We first assume W is Lipschitz in f. We may use polar coordinates (p,t) on TpM to describe the corresponding points on M, via the diffeomorphism expp. Thus p(x) = pp(x) = dist(x,p)

for x E M.

Consider now the unit ball B1(x), centered at x. From the comparison theorem we find that BI (x), when viewed from p, subtends an angle no more than ce-*P(') Thus the oscillation of as y = expp(po) varies in B1 (x) satisfies ce-aP(x)

Bl(x)(A = (pmax -'Pmin 5

Here V(t) may be viewed as a function on M \ {p} extending gyp. To obtain suitable decay also on the derivatives we need to cut-off and mollify this function. This will use the lower bound on curvature, which implies that unit balls are basically the same everywhere on M. Consider the expression X(dist2(x, y)) where X E Co (R, [0,11)

is a fixed function that is identically one near 0. We define

V(x) = fM fiu X(dist2(x,y))dµ(y) Then,

]Vxdist2(x,y)] 5 C, I'&zp2(x,y)] S C, we have the desired decay OSCB,(x) 0 < Ce'aP(x)

IV (x)]

<_

I AO(x)I <

Ce-aP(x) Ce-0P(x).

LECTURE S. HARMONIC FUNCTIONS ON NEGATIVELY CURVED MANIFOLDS

149

Recall that in H", the distance function p = pp is strictly convex, the Hessian of p is strictly positive definite, and the Laplacian of p is bounded away from 0. Similarly here in M we find that

Op> n

p

(l+ap)?co>0,

hence,

De-6p = -be-6p(Op - bIVpi2) -ICiIbe-6p

because IVpl = 1. Thus A(rp + ce-6p) < 0,

0(0 - c'e-6p) > 0, and we have super- and sub-harmonic functions which agree with 9 on 8M. We may now use either the standard Perron process or solve a suitable Dirichlet problem on

the bounded domain BR(p) and let R t +oo. We find that there exists a u : M - R satisfying

Du=0 and 0-c'e-6p
elwj - 'Pal

Sup luj - UkI = Max M

Thus the {uj} are uniformly Cauchy in C°(?) and converge uniformly to u E C°(M) with u I OM = gyp. Standard local representation formulas show that u is harmonic and smooth. To conclude this chapter we will mention a few related results and questions. First a more refined result on the Dirichlet problem involving Poisson kernel functions is given in the paper [AS). W. Ballman [B] discusses the Dirichlet problem for the universal covering of a compact manifold M with nonpositive sectional curvature. He also assumes that M is rank one, where here the rank of M is the largest dimension of a flat totally geodesic submanifold of M, the universal cover of M. His proof involves a certain amount of dynamical systems theory. Some interesting problems are to extend results to finite volume manifolds or higher rank manifolds. S.Y. Cheng [C2J showed that the Dirichlet problem is solvable if every sectional curvature K,,(ir) <- -a2 and, for each x E M,

max(-K=(ar)) < Cmin(-KK(rr)). n

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

150

This allows cases where infxEM min,, KK(a) = -oo. The DeGiorgi-Nash-Moser machinery for regularity of nonlinear elliptic equa-

tions may be treated in a Riemannian manifold (M, g) in a coordinate free way. Think of M possibly having a boundary. There are three crucial hypotheses involving positive constants cl, c2, c3 depending only on M. H1. Isoperimetric inequality. vol(fl)"-1I' < Cl vol(C

)

for any 1lccM. H2. Volume bound.

vol(B,(x))
inf aER

J o13(z) I f - c 12dµ <- c3a2 J e(=) I Vf I2dt .

Note that these are all lower order quantities. They involve no curvature hypotheses and may make sense if the metric is not smooth or even if M is not a manifold.

These hypotheses have some useful consequences. The Harnack inequality states that a positive harmonic function on a ball Bo C M satisfies sup u < c(n, cl, c2, c3) inf u. ft./2

Bo/2

The mean-value inequality, which is implied by H1, says that a nonnegative subso-

lution (i.e., Au > 0) on B, satisfies

sup u' < c(n, p, cl )a-" J ul'dfa for any p > 0. Be/2

Be

More generally, a solution of Au + fu = 0 with f Be If IE+9 du :5 r satisfies

sup u' < c(n, r, p, cl)a-" J Bei2

u1'dµ

for any p> 0.

B

To understand hypothesis H3, consider a family of catenoids with shrinking necks. Here H1 and H2 continue to hold with fixed c1, c2, the Poincare inequality H3 eventually fails for any fixed c3. That is, the first eigenvalue of a small ball centered on the neck approaches 0.

LECTURE 4 Harmonic Mapping into Singular Spaces 4.1 Motivation

The motivation for us to establish a theory of harmonic mappings into singular spaces is some new application of harmonic mapping in proving "Rigidity Theorems!'. These applications, which will be described in the next chapter, rely heavily on both existence and vanishing results, and the so-called Bochner Method. This has two points. (i) Analysis. We seek to represent a topological object by a geometric one. By analogy, in Hodge theory, we represent a cohomology class by a differential form that is harmonic. The general issue is that we need to establish the existence and regularity for a "canonical representative" of a topological object. (ii) Vanishing Theorems. Under certain conditions, we try to prove that some quantities vanish by using a Bochner formula for these quantities. This has a more algebraic flavor.

Roughly speaking, the combination of an analysis and an algebraic result will imply a geometric result. To illustrate this idea, we'll examine a theorem due to Preissmann [P].

Theorem. If X is a compact manifold with negative sectional curvature, then any abelian subgroup of the fundamental group 7r1(X) is cyclic. Proof. Assume that a and b are two commuting elements in 7r1(X ), i.e., aba-1 b-1 aba-1b-1 and 1 will give a smooth N 1. Then the homotopy mapping between mapping 0 of the torus T2 = S' x S' to X. Note that X is compact with negative

sectional curvature. We claim that 0 is homotopic to a mapping ' whose image lies in a closed geodesic of X. In order to prove the claim, we need the following analytic contributions. (i) The Eells-Sampson theorem: (see [ES]). Any w : M - X is homotopic to a harmonic mapping, provided that X is compact with sectional curvature Kx < 0.

151

152

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

(ii) The vanishing result: Suppose u : T2 - X is the harmonic map homotopic to 0. Then we have the Bochner formula (see [ES])

o(IVul2) =

Rfc77(Vau, VQu)

IV2uI2 +

0=1 2

- E (R X(Vnu, Vpu)VQu, Vpu) 04=1

where 0 denotes the laplace operator on T2, V2u the Hessian of u, RicT2 the Ricci operator on T2, and RX the curvature tensor of X. Since T2 is endowed with flat metric, Ricv = 0; Kx < 0, and the third term has a positive sign. Therefore I A(IVu12) >

IV2uI2 > 0

i.e., IVUI2 is subharmonic and hence constant by the maximum principle. Moreover,

IV2uI = 0, that is, u is a totally geodesic mapping. By using the strict negativity of Kx, we know that the rank dim(u.(TTT2)) < 1

dzET2.

This implies Im(u) is contained in some closed geodesic. Therefore a and b are homotopic to multiples of some closed geodesic.

4.2 Nonpositively curved metric spaces In order to study more complicated group actions by the theory of harmonic maps, we want to allow the image space X to be singular. In fact, we will consider a singular metric space X with nonpositive curvature. Suppose X C Rk is a closed set, and fl is a bounded, smooth domain in M. We want to find energy minimizing maps from ft to X with specified boundary data cp : Oft -- X.

Let HI(ft,X) = {v E HI(ft,Rk)

:

v(x) E X a.e. x E ft} where HI(fl,Rk)

is the Hilbert space of Rk-vector-valued L2 functions on fl with first derivatives in L2. Recall that the energy of u E HI (ft, X) is given by

E(u) =

In

IVu12 dp

where d IA is the volume element of M. The relation u = cp on 80 is to be understood

in the HI (ft, X) trace sense.

Lemma. Suppose V : Oft -' X occurs as the trace of some map in HI(ft,X). Then there exists u E HI (ft, X) such that u = V on On, and E(u) < E(v) for all v E H1(Sl, X) with v = ,p on On.

LECTURE 4. HARMONIC MAPPING INTO SINGULAR SPACES

153

Proof. (direct method) Let {ui } be a minimizing sequence of maps in H' (S2, X) with ui = V on BSI. Since a bounded subset of H' (ft, X) is weakly compact, there is a subsequence again denoted {ui } which converges weakly to a map u E H' (fl, X). On the other hand, E is sequentially weakly lower-semicontinuous. Therefore

E(u) = inf{E(v)

:

v E H' (fl, X),

v = ip on BSI}.

Note that if every pair of points in X can be joined by at least one Lipschitz curve, then Lemma 4.3 implies that any pair of points x, y in X can be joined by an energy minimizing curve y parametrized by arclength. Using the infemum of the length of such curves -y one obtains an intrinsic metric on the space X. This should not be confused with the topologically equivalent,but smaller extrinsic distance Ix - y1. Next, we explain the generalized notion of nonpositive curvature which we use. We assume the X is metrized as above so that any two points x, y in X may be joined by a unique curve in X whose length in Rk is dist(x, y). Consider a geodesic triangle in X, wxy and a corresponding triangle in lit2, Oxy, with the same side lengths. Let y(s) denote the minimizing curve parametrized by arclength between

x and y. Then the squared distance D(s) = dist2(y(s),w) corresponds to D(s) = 189 + (1- 8)212 = s2 + as + Q which is the unique quadratic polynomial determined by boundary conditions D(0) = 1x12 = Id(x,w)12 and 13(t) = 1912 = d(y,w)12. Note that U "(s) = 2. The nonpositive curvature condition of X is that the corresponding inequality D"(s) > 2 holds, in a weak sense. Since D(s) is only Lipschitz, x = y(0),

and y = y(t), the condition D"(s) >_ 2 means that f t S"(s)D(8)d8 >_ 2 f t S(s)ds

V S E C,,00 (0, e), s> 0.

Definition. X has nonpositive curvature if, for any three points w, x, y E X, the inequality D"(s) > 2 holds. There are two elementary examples of non-positively curved singular spaces: (1) 1lcees. A tree is a connected and simply connected graph. One determines an intrinsic metric on a tree by assigning a length to each edge. (2) Surfaces with cone metric: On the disk, the metric is smooth and flat away from the origin. The sign of the curvature at 0 is determined by the cone angle a at 0; namely,

1. a < 27r b positive curvature at 0. 2. a = 27r zero curvature at 0. 3. a > 21r negative curvature at 0. One can often understand these two examples by approximating them by a sequence of smooth surfaces {Xi}, and hope that some properties of harmonic mappings into X, can be inherited. In fact, here is a question due to M. Wolf [W]. X Degeneration question: Consider a sequence of smooth manifolds X. where convergence is in the sense of the distance functions. If of : Cl - Xj are harmonic maps with bounded energy, E(u,) < c, what do the limits of the uj's look like?

154

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Suppose each Xi has nonpositive curvature. As before, by virtue of the Bochner formula,

2o(IVuuI2) > Rico(Vui, Vui) > -clVuil2. I Vu, I2 is thus a subsolution for the operator i + 2C. We can apply the DeGiorgiNash-Moser mean-value inequality ([GT]) to conclude that

-

Sup IVUj 12
4.3 A two-dimensional result Consider a map u from disk D2 into X. If u is smooth, we may define its Hopf differential 0 = Odz2 by letting .0

= Iu=I2 -

IU. I2 - 2i(uz, u1)

If u is also harmonic, then d is holomorphic. If u is simply energy minimizing, not necessarily smooth (e.g. X isn't smooth), then 4 is nevertheless still holomorphic and hence smooth. Here, we sketch a proof of the latter (see [S]).

Proof. Consider a variation of domain Ft = D -+ D for Iti small such that F0 = identity, Ft = identity near 8D. Then, by virtue of minimality of u,

E(u o Ft) > E(u)

V t.

On the other hand, the map t -+ E(u o Ft) is 01 in t. To see this, let y = F, (z) and perform a change of variables in the integral to get

E(uoFt)= fD

C?u)det(c9xy)dY axiax (8u 8xi exi pp,,' ' J yi Wk eyQ J i,i ,k

Therefore, A It=0 E(u o Ft) = 0. Now, for t (x, y) and ((x, y) any smooth functions with compact support in D2, consider the variation given by

Ft(x,y) = (x+t((x,y),y) Gt(x, y) = (x, y + tn(x, y))

The first variation of these shows that 0 satisfies the Cauchy-Riemannian equations weakly. Linear regularity theory (e.g., Weyl's lemma) shows that 4, is thus holomorphic and smooth.

LECTURE 4. HARMONIC MAPPING INTO SINGULAR SPACES

155

Harmonic maps from surfaces to an R -tree X were recently studied by M. Wolf [W]. By involving the appropriate definition of harmonic mapping, M. Wolf used holomorphic quadratic forms 4' on R, whose profile can be viewed as the Hopf differential of some harmonic map. In particular, such 4' are used to construct

harmonic maps into trees. To indicate his idea, we suppose u : D2 -' R is a )2 harmonic function. Then its Hopf differential 4' = ( dz2. If b(0) 0 0, we can do a change of coordinates so that u is a projection map, i.e., u(x, y) = x in a suitable coordinate. Changing coordinates so that 0 - 1 near 0, the map u contracts (or expands) maximally on regions where 4' < 0 (or 4' > 0). e.g., If 4' = zdz2, then the map which projects, in a suitable way, the regions between the maximal negative curves onto the maximal positive curves defines a harmonic map from the disk to the tree. See Figure 1.

Figure 1

0>0 Suppose X is a surface with a negatively curved cone metric. e.g., We may obtain such an X in R3 by choosing any curve (embedded) in S2 of length 0 > 27r

and taking a cone over it with vertex at the origin. This gives a cone angle 0. Let g be a cone metric on X, u*g is a real symmetric (2, 0) tensor on D2 and 0 + F + Fdzdz. Here 4' = ip(z)dz2 is the Hopf differential of u, and F is the energy density of u. We can view Fdzdz as a conformal metric; its sketch factor is constant in all directions. On the other hand, 4'(v v) < 0 along the direction v of minimal sketch, and 4'(w w) > 0 along the direction w of maximal sketch. Moreover, vlw. The direction fields v, w, define the positive and negative foliations of 4' away from the discrete set I0+-' {0}. Recent work of E. Kuwert[K] analyzes the possible collapse of u along the negative foliation of 4'.

Theorem ([K]). Assume that the boundary map

156

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Thus, in general, u-1 (0) is 1-dimensional, e.g., if we place a large number of cone points on a Riemannian surface of genus g, E9 and consider a harmonic map u homotopic to the identity, where u : E9 - E9 cone. Then the number of zeros of 0 Is controlled by the genus of E. Most preimages of cone points are arcs on the negative foliation of 0. In particular, u isn't in general quasiconformal.

LECTURE 5 Energy Convexity of Maps to an NPC Metric Space We want to deduce the convexity of energy for a map into an NPC metric space X. First, we assume that X is smooth NPC. Let ut [0, be a smooth map with F Inx {t}= ut. Then we have dtz E(ut)

=2 f {itVevII2 - (Rx(v,F"ea)v,F'"eo)+(VeV vF*e,)1dp \

where v = F * (Ft is the velocity of the variation, let,..., e } is an orthonormal basis for T x Cl, V a is the acceleration vector, and V' is the connection on u"TX gotten by pulling back the Levi-Civita connection on Cl. For details on this formula, one can refer to [ELI.

Since we are assuming nonpositive curvature, we may disregard the second term. We also want to disregard the third term. In order to do that, let uo, u 1 be two given maps, and ut be a homotopy from ua to ul. This is a homotopy ut such that for each x ut(x) is a constant speed geodesic in X parametrized on [0,11. This is done by choosing {ut(x) : t E [0,11) to be the unique geodesic between uo(x) and ul(x). For such a geodesic homotopy, the acceleration term vanishes; thus d2 E(ut) > 2 fn

IV'vI2dµ

Now, we need to modify this argument so that it works when X is NPC Riemannian simplicial complex. This is a simplicial complex whose faces are endowed with Riemannian metrics which extend smoothly to the closure in such a way that the lower-dimensional faces have the metric induced from the higher dimensional

faces. We may also assume X C RN in such a way that each face has the metric induced from RN. The following properties can be derived for a NPC metric space X. First, any two points in X may be joined by a unique length-minimizing path. Secondly, if po, pl and 0o, 01 are two pairs of points in X, and if we parametrize the 157

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

158

geodesic paths from po to p1 and from Bo to 81 by p(t), 0(t) for t E [0, 11, then the function d(p(t), 0(t)) is a convex function of t. The second property implies that geodesics from a point spread apart more quickly than in Euclidean space, since we may take po = Oo and conclude that d(p(t), 0(t)) > 8d(p(s), 0(s))

for a< t.

By virtue of the convexity of distance function, we have the following proposition concerning energy convexity.

Proposition 5.1. Suppose X is a Riemannian simplicial complex of nonpositive curvature, and uo and u1 are Lipschitz maps from Il to X. Then if ut is the geodesic homotopy from uo to ul we have

2E(ut) > 2 fn IVd('uo,u1)t2dµ weakly on [0,1], i.e.,

fo

1

E(ut)("(t)dt > 2fa IVd(uo, u1)12 f 1 C(t)dt `d S E C ([0,1], R) C > 0. 0

Proof. (See [GS], also). First consider the one-dimensional case in which we have Lipschitz curves 7o, -ti : (-6,6) --+ X and a geodesic homotopy yt for 0 < t < 1. Assume that 6 = 0 is a point of differentiability for -yo, -y1 and 7e for a.e. t E (0,1). We fix s = 0 and calculate Iaso We can replace -yo, 71 by the corresponding constant speed geodesics. Let e(s) be the length of the curve t -, -yt(s) and observe that t(s) = d(ryo(s), y1(s)) is a Lipschitz function of s, and reparametrize the homotopy by setting ryr(s) = ry(rs/t(s) for r E [0,e(s)]. Thus r H 7y, (s) is now a unit speed geodesic. For any h the function r ,- d2(ryr(h),7y.(0)) is convex because X has nonpositive curvature. At any r for which d -y,(0) exists we have

lim h-2d2 (7r(h),7r(0)) =I

h-.0

12. dyr (0) ds

Since d (0) exists at r = 0, and r = 1(0), it follows that there is a sequence h. tending to zero such that r ,- hi 2d2 (ry . (h,), ryr (0)) converges uniformly on [0,1(0)] In to a convex function which agrees a.e. with the function r H I2 (0).

particular, we can assume that I t I2 (0) is convex in r. Now by the chain rule we have

(o) =

-re(o)-2Z(0)r&(0)(0)+

Wary.,/1(0)(0)

or in terms of t d ds

7t (0) =

dit (0) + ds

te(o)-1

de(o) d1ft (0). ds dt

LECTURE 5. ENERGY CONVEXITY OF MAPS TO AN NPC METRIC SPACE

159

For any r1, r2 E (0,1(0)) with r1 <,r2 we have d(ry (s), ry (a)) = r2 - r1. Differentiating with respect to a, we then conclude that

djr 8i ds 8r

d70r ds O

r=Ti

i ra

Therefore, for a.e. t E [0,1], d-tt ds

(0) = V(t) + I a+tt(0)-1de(0)

I

`(0).

at

For a constant a where V(t) L(0) = 0 for a.e. t and

t

(°)12 = IV(t) 12+ I a + tt-' (O)' (0) 1' It(0)12

I

IV(t)12 = I

(0)

1'+ a2t2(0)

12

dt2

IV(t)I2 + 2 I

d8t

I

>2I

d(°)

I2

'(0)12

= 2 (4-d(io(s)7i(s))

2 I

s=

since IV(t)12 is convex.

Now to prove the result in higher dimensions, observe that the map (x, t) ut(x) is Lipschitz. Thus, for almost every line parallel to the t-axis, it is differentiable at a.e. point of the line. At such points of differentiability the previous results tell us 2

dt2IVu`I2 > 21Vd(uo,u1)I2

in the weak sense. Thus, if (E Co (0,1) and C > 0 we have, for a.e. x E M

f

I

IVutl2(x)C""(t)dt > 2J IVd(uo(x),ul(x))I2C(t)dt.

o 0

o

Integrating w, t, and x we get f' IVutI2S"(t)dµdt > 2 r1I

n

I.

IVd(uo,ul)I2d, I

(t)dt.

o

Corollary S.Z. Assume X C It" is the same as the above proposition. Suppose uo and ul are Lipschitz and e > 0 is small enough so that E(u1) < Eo + e for i = 0, 1 where Eo = inf {E(u) : u E H' (fl, X), u = rp on &Z}. Then fn d2(uo, ul) < ce with a constant c depending only on Poincare constant fl.

R. SCHOEN. BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

160

Proof. Apply the proposition and the Poincare inequality (since d(uo, u1) = 0 on ice) to get 2E(ut) 2! cind2(uo,u1)dµ weakly on [0,1]. By the convexity of the energy, `da E 10, 1]

Eo < E(ua) < aE(uo) + (1 - a)E(u1) < Eo + e. Now, use the fundamental theorem of calculus and an appropriate test function to get the required result.

Corollary 5.3. Suppose X is simply connected NPC. There is a unique energy minimizing map u : 0 -, X wsth given Lipschitz boundary data.

Proof. Suppose uo and u1 are two minimizers with the same boundary data W. Applying Corollary 5.2 with e = 0, we can conclude that fey ld(uo, u1)12dp = 0. It follows that uo = u1. This completes the proof.

5.4 Monotonicity We may choose a smooth variation of the domain of the form u, = uoFT, where

.,(x) = (1 + r£(x))x, where r is small and l; is a smooth, compactly supported approximation to the characteristic function of a ball B,(0). Since u is a minimizer, we can deduce from a first variation argument, as in L. Simon [Si], that

(5.1)

(2 - n)

IVuI2dµ + a f Be (o)

z

IVu12dE = B (0)

2o. f

Be

I

car

for a.e. a. One can integrate the identity w.r.t a to get the usual monotonicity formula for the normalized energy. Th derive the monotonicity of another useful quantity, we will combine (5.1) with an inequality having to do with the convexity of the distance function in the target. For example, if X is a smooth NPC space and u is a smooth harmonic map, then we have, by the chain rule, for Q E X, fixed Ad2 (u(x), Q) = tr Vdd2(u(x), Q) (Vu, Vu) R

_

Hessd2(u,Q)(Ve,,u,

21VuI2.

a=1

Since Hessp d2(p. Q)(v, v) > 2]v12 for all v E TpN i.e., d2(u(x), 0) is a strongly subharmonic function. For NPC metric space X, we need to make this proof variational since this is the only tool available.

LECTURE 5. ENERGY CONVEXITY OF MAPS TO AN NPC METRIC SPACE

161

Fix Q E X. Given p E X. There exist a unique geodesic y(t) parametrized with constant speed on [0,11 such that y(0) = Q, and y(1) = p. Define R,\,Q : X -+ X by RA,Q(p) = ry(A). Then Ra,Q is a Lipschitz and

contracting map. In fact d(Ra,Q(1h),Ra,Q(p2)) 5 Ad(p1,p2)

R1,Q = Identity RO,Q(p) = 0 V P E X. So R,\,Q is a retraction map of X onto Q with finite Lipschitz constant.

Proposition 5.5. Suppose X is a NPC metric space. If u E H' (St, X) is minimizing and u(fl) is a compact subset of X, then Od2(u(x),Q) > 21Vu12 weakly.

Proof. Fort E Co (St) with t > 0, consider the deformation u.(x) = R1_Tf(S),Q (u(x)) where r > 0. Then u, = u near Oft, and E(u,) has a minimum at r = 0. Therefore

d dt

E(uT) > 0 .

IT.o

On the other hand,

8 = D8 R1-=((x),Q(u(x)) - Ti 8

11

12

tai

I2 -I D &u (x)R1-*(,Q(u) a R1-rt.Q(u(x))+T2 FX

Lt

- 2TOx DOxt (x)R1-*(,Q(u(x)) 2

2

ax i) 18 R1-*((x),Q(u(x)) I

Note that RA,Q(p) = d(Ra,Q(p),Q)y1(Ra,Q(p)) where y is the unit speed geodesic from Q to p. We have that D. RA,Q (P) . 811 R,\,Q (P) = Dvd2 (gA,Q (P), Q). 2

Using the contracting property of RA,Q, we also have I D 8x; (x)R1-T(Q(u) I< (1-

Therefore

E(u1) <

f (1 - TC)2jVuj2 L

i

8x;

12.

r`)2 I

e

(d2(R1-T(,Q(u)+Q)J dµ+O(72)

162

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Letting r j 0 and using the minimality of u, we have

05-2 J

II

tIVuI2dii+J Otd2(u(x),Q)dµ

in

I [(AC)` (u(x), Q) - 2IVuI2] dµ > 0. Remark. Since the function involved is only Lipschitz, our derivative is taken almost everywhere.

LECTURE 6 The Order Function We define the order function for u E H'(11, X) as follows: Let X E f Z and 0 < o < dist(x,BfZ). If u $ Q on 8B, (x), then o fB.(x) IDuI2d,L

ord(x, o, Q) =

Q)dE.

fea, (x)d2(u,

Proposition 6.1. Suppose X is a NPC space and u E Hl (Il, X) is a locally minimizing map. Then, for any x E ft, either u = Q near x, or ord(x, o, Q) is monotonically increasing.

Proof. For x E ft given, we assume that ord(x, o, Q) is defined for o small; otherwise, from Proposition 5.4 it follows that d2(u(y), Q) is subharmonic, and u Q near x. Define E(o) = fB,,(x) JVuI2dp, 1(c) = foB,(.) d2(u, 8)dE. From the monotonicity formula (5.1), we compute the logarithmic derivative of E,

E(s)

o

E(a) JoBe(x)

or

Recall from Proposition 5.4 we have that Od2(u, Q) > 21Vu12. Integrating this inequality over B,(x) using a smooth approximation of the characteristic function of B,(x), we get that d2 21Bo(=) IVu12d z < JOB,,.(z) (u(),Q)dE 8r

2E(a) <

e J aB e(x) ) Or 163

d2(u(.),Q)dE.

164

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

The logarithmic derivative of I(o) is

n-1

I (o)

+ I(a) OB,(.) or d2(u( ),

Q)dE.

Combining (6.1) with (6.2), we compute d

do log

&E(o)

_

I(a)

8u 2

2

E(o) 'oBe(=)

_

Or

8 d2(u(.)

-

1

T 01)

Or

,

Q)dE.

Noting that I 'd(u, Q) 1:51 r I, we use the Schwartz inequality and (5.1) to verify that the lefthand side is nonnegative. Thus

ord(x, o, Q) _ ) is monotonically increasing. Remark. The monotonicity of ord(x, or, Q) was also studied by F. H. Lin [L] and R. Hardt [HL] for energy-minimizing harmonic maps into round cones. Recall that d2(u,Q) is a convex function of Q. Consider Q 1-4 f8Be(z) d2(u,Q) dE. Then there is a unique maximum point Q.,,. Let ord(x, o) = ord(x, o, Then ord(x, a) is still monotonically increasing in or.

Definition. ord(x) = lim.1o ord(x, o). Note that the function x . ord(x) is upper semicontinuous because it is a decreasing limit of continuous functions.

Proposition 6.2. If X is a Riemannian simplical complex of nonpositive curvature, u E HI (fl, X) is a locally minimizing map (with compact image), then u is locally Lipschitz and moreover (6.3)

sup IVu12 < c(SZ,111) in IVul2dp n,

n

for any fl, CC Cl. Remark. For harmonic maps into a smooth NPC, the conclusion of Proposition 6.2 is a well-known result of Eells and Sampson [ES]. The usual proof of this is based on the Bochner formula for IVuI2 which relies heavily on the smoothness of X.

Proof. First, let's recall a measure theoretic property of HI (Cl, R') function u (see [Z]); namely that u is approximately differentiable a.e. in Cl. This means that at a.e. x0 c Cl there exists a linear map L(x) = A(x - x0) + B which approximates u

LECTURE 6. THE ORDER FUNCTION

165

in the sense that lii o

Ju 2_' 4.

IVu-VLI2dp =0.

Iu-LI2dµ+o-- J

B.(zo)

(xo)

An asymptotic calculation (see Lemma 1.3 in [GSJ) then show that if A g& 0, then ord(x) = 1 and thus by upper-semicontinuity of ord(x), ord(x) > 1 whenever xo is in the closure of the set of points at which u has a nonzero approximate derivative. Let a = ord(xo) > 1 and fix oo > 0 so that Boo(xo) C A. Let o1 E (0, oo) and note that the monotonicity of the order implies o

f

IVuI2dµ ? a f

d2(u,Q1)dE

8B. (xo)

B, (xo)

for all or E [a,,oo) where Q1 = Qxo,o,. Combining this with (6.2) yields al(o) <_

< This implies

I'(a) I(o)

laf 2

d d2(u(x),Ql)dE

8B,(xo)

2(01'(0) - (n - 2)1(o)).

n-1+2a o

for a E [01,00).

Integrating from of and oo and fixing ao, we obtain a1'

' I (o1) <_ cai°I(oo) 5 c(oo)o1

1

The subharmonic property of d2(u,Q1) on fl implies sup d2(u(x),Q,) S ca?' B-° (xo)

Then, for all x E B

a (xo), choose o1 = 21xo - xo) and by triangle inequality

d(u(x), u(xo)) 5 d(u(x), Q1) + d(u(xo), Q1) <_ cjx - xoI °

since a > 1, this implies that for any xo at which approximate derivative exists and is non-zero we have I N-4 1:5 c. Thus { N7 1:5 c ae., and u is locally Lipschitz. We can also apply the above argument to show the interior estimate (6.3). See Theorem 2.4 in [GSJ for details. Estimate (4) can actually lead to the following Eells-Sampson type of Existence theorem.

166

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Theorem 6.3. Let M be a compact Riemannian manifold without boundary, and let X be a Riemannian simplicial complex and its universal covering X has nonpositive curvature. Then any Lipschitz map W : M -a X is freely homotopic to a Lipschitz map u : M -, X which minimizes energy in the sense that

E(u) = inf{E(v)

:

v : M -+ X Lipschitz, v homotopic top)

Proof. We use a local replacement idea. Let Eo = inf{E(o) : v

: M -' X

Lipschitz, v homotopic to co}. Let {ui} be a sequence of Lipschitz maps homotopic

to

ball B. We lift a map u1 : B - X to X. Denote this lift by ui : B - X. Let vi be a minimizing map from B -, X which is equal to us on 8B. We then defined a replaced map ui by rr(v.(x))

ui(x) =

1u-(X)

xEB

xEM\B

where rr is the projection of X onto X. Since E(ui) < E(ui), {ui} is again a minimizing sequence. Applying Corollary 5.3 in B we have, for all e > 0 and i sufficiently large,

fd2(us,u)d1i < ce. On the other hand, ui is uniformly Lipschitz on compact subsets interior to B. Thus a subsequence of {fii} converges uniformly near x0 to a Lipschitz map which must

be u by uniqueness of limit. In order to prove u is homotopic to ,p, we note that geodesic homotopy vi,t with vi,o = gyp, vi,1 = fii converges to a geodesic homotopy

vt with vo = rp, v1 = u near xo. Then, by a covering argument, we have a global geodesic homotopy from V to u.

LECTURE 7 Approximation and Smoothness Results for Harmonic Maps To apply harmonic map theory to singular spaces, we need to improve its regularity in certain cases. First, we consider approximation of harmonic map by homogeneous minimizing (tangent) maps and note that the linearity of the approximation will indicate differentiability in a strong sense. Let X be a geometric cone in R" (i.e., AX C X, VA E R+), and suppose X has nonpositive curvature. We say a minimizing map u : B1(0) - X is intrinsically homogeneous of degree a (a > 1) if u maps each ray through origin onto a geodesic in X parametrized in such a way that d(u(tt), u(0)) = t°d(u(t), u(0)). Then the following lemma gives us a criterion of intrinsically homogeneous map. Lemma 7.1. If u : Bl (0) -+ X is a minimizing map such that ord(0, a, Qoa) = a for all or E (0,1) with a fixed a > 1, then u is intrinsically homogeneous.

Proof. First, we note that ord(0, or, Qoa) a implies that ord(0, a, u(0)) = a. Then all inequalities in the proof of monotonicity of order must be equalities. It follows that drd(u,

u(0)) =I dr

a.e., I

and there is a function h : (0,1) -+ R such that

d d(u,u(0)) = h(a)d(u,u(0)) on OBo(0). Integrating the first equality along ray ry : (a, 1) --+ Bi given by -f(r) = r£ for some { E OB1(0), we have

I,1

I

d

I ds = d(u(e), u(0)) - d(u(a{), u(0)) S d(u(;), u(aC)). 167

168

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

In particular it follows that u(y) is a geodesic path in X. We also have

E(a) =

f

Be(e)

d(u, u(O)) d d(u, u(0))dE = h(o)I(a)

so we conclude from E(o) = as-1I(a) that h(a) = as-. Then integrate along the ray from of to t to obtain d(u(of ), u(0)) = a°d(u(t), u(0))

V

f E 8B1(0).

Remark. (1) If 0 E Il and u is a harmonic function, then u(x) - u(0) = p°(x) + 0(IxI°+1) where p0(x) is a spherical harmonic of degree a with

a = lim I(or)

and Q = u(0).

(2) The converse of the above lemma is also true. In fact, if u : B1 (0) X is intrinsically homogeneous of degree a, then it follows immediately that ord(O, a, u(0)) is constant. Moreover, the constant is a and ord(O) = a. On the other hand, homogeneity of u implies that ord(O, o, Qoo) is independent of o, i.e., ord(O, or, Qoa) = a. An intrinsically homogeneous map of degree 1 is very special. Roughly speak-

ing, it looks like a map of afne type. More precisely, let (: R' -' X be an isometric and totally geodesic embedding. This means that d(((x), ((y)) = Ix - yi, and the straight line between x and y is mapped to a unique geodesic between ((z) and

((y). Take v : R" - Rr given by a harmonic function of degree a. We consider u = ( o v : R" - X and note that u is a minimizing homogeneous map of degree a. We call this u a regular homogeneous map.

Proposition 7.2. Any homogeneous minimizing map u : R" -i X is regular if it is of degree 1.

Proof. (Sketch). We divide the proof into several steps. Step 1. ord(0) = 1 and ord(0, a, Qoa) = 1. This follows from (6.1) of the above Remark.

Step 2. ord(x) = 1 for all x E R". In fact, x r-4 ord(z) is upper semicontinuous and the degree 1 homogeneity of u implies that ord(x) is homogeneous of degree 0. Therefore ord(x) = 1 bx E R".

Step 3. ord(x, a, u(x)) = 1 for all z E R" and all or > 0. To verify this, note that ord(x, a, u(x)) = ord(Ax, aa, Au(x)) VA > 0. So we take A = a- I and conclude that

lim ord(x, a, u(x)) = 1. 0-00

On the other hand, limo_.o ord(x, or, u(x)) = ord(x) = 1. It follows that for all

xER" and all o>0we have ord(x,o,u(x))=1.

LECTURE 7. SMOOTHNESS RESULTS FOR HARMONIC MAPS

169

Step 4. From Lemma 7.1, we conclude that u is an intrinsically homogeneous of degree one about every point. It follows that the restriction of u to any line parametrizes a geodesic in X with constant speed and from the equality Ozd2(u(x),u(xo)) = 2IVu12 and the degree 2 homogeneity about xo of the function x '- d2(u(x),u(xo)) we also conclude that IVuI2 is homogeneous of degree 0. In particular IVuI2 is constant and d2(u(x), u(x0)) is a quadratic polynomial with positive value. Thus a linear algebraic computation implies that u can factor through an isometric, totally geodesic embedding of R', for some t, into X. Now we return to the general situation of a minimizing map u : ft -' X. Given xo E ft we approximate u near xo by homogeneous map. We choose coordinates

so that xo = 0 = u(xo). We define a rescaled map of u, ua,K : Ba lco - r.-'X by ua,u(x) = K-1u(A,x). Note that is-1X has nonpositive curvature since distances are multiplied by a constant factor. We have, by a change of variables,

f Ivua.Kl2d, = µ-2A2-" f Be

JVul2dµ

Bae (0)

d_,

0)dE = µ 2A1-"

u

aB ,

raB,e d2X (u, O)dE.

In particular, ord u,\,,, (0, o, 0) = ord u(0, Aa, 0) for o E (0,A- lao). For small A > 0, let ,c = (A1'"I(A))1/2, so that we have

f

d,X(u,O)dE = 1 B,

Since ord ua,,,(0) --> ord u(0) = a as A - 0. We thus have, for A small IVua.Kl2dlz S 2a.

Thus ua,K has uniformly bounded energy and then by Proposition 6.2. has uniformly bounded Lipschitz constant on compact subsets of B1(0). We can extract a subsequence {A,} tending to zero, so that the corresponding u; converges uniformly to a Lipschitz limit u.: 111 (0) -, TOX where TOX denotes the tangent cone of X

at 0.

Proposition 7.3. The map u. is a nonconstant energy minimizer that is homogeneous of degree a.

Proof. See [GS], Proposition 3.3. For simplicity, we will focus on the smoothness of harmonic map into an Fconnected space. This is a higher dimensional generalization of a tree space. Definition. We say that a nonpositively curved Riemannian simplical complex X is F-connected if any two adjacent simplices are contained in the image of a totally geodesic embedding of a Euclidean space into X. We will assume that X is finite dimensional and locally compact.

170

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Notice that (1-dimensional) trees are F-connected. The most important higher dimensional F-connected complexes are the Euclidean buildings of Bruhat and Tits (see [Br)). The importance of F-connected complexes is that there are lots of interesting groups which act isometrically on these spaces. Definition. A k flat F of X is the image of a isometric totally geodesic embedding of Rk into X. A point xo E SZ is called a regular point of u if there is an e > 0 such that u(BE(xo)) lies in a k-flat. Let R(u) _ {x E D : x is a regular point of u}. The singular set is the complement

8(u) = Cl \ R(u). Notice that 11(u) is an open subset of Cl by definition, and that 8(u) is a relatively closed set. The main smoothness theorem concerning minimizing map into F-connected complexes is the following.

Theorem 7.4. Suppose X is an F-connected complex. If u : Cl -+ X a locally minimizing map, then: 1. dim% S(u) < n - 2 where dim7j denotes the Hausdorff dimension. 2. For any S21 CC Cl, there exists a sequence of Lipschitz cut-off function {lfii} such that iii - 0 in a neighborhood of s(u) f1511, 0 < 9l' < 1, t/r,(x) -> 1 for

all x E 01 \ S(u), and

lim J IVujIVdp = 0. 1

The idea is that we want to show that almost all points x in Cl has ord(x) = 1 and that, at such an x, the degree 1 homogeneous map u. has rank equal to dim X; moreover, such an x is a regular point of u. In order to carry out the details we need several preliminary results. First, we generalize the notion of homogeneous map without the cone structure of the target. Let x1,. .. , x" be a normal coordinate system centered at xo, and

let r = Jxj,

= >sf denote polar coordinates in Br.(x0). We will say that a

Lipschitz map l : B.ro(xo) - X is essentially homogeneous of degree 1 if there is a nonnegative function A : S"-1 - R and an assignment 'y to each E Sii-1 of unit speed geodesic in X with 'yy(0) = 8(0), y (A(l;)'y) = £('y) for x = ryt; E B.ro(xo). In other words, a map is essentially homogeneous of degree 1 if the restriction of u to each ray is a constant speed geodesic in X. For xo E l and o > 0 such that Bo(xo) CC II, we consider the error with which u can be approximated by degree 1 essentially homogeneous maps. Define

R(xo, o) = inf

sup

d(u(x), e(x)) : l is essentially homogeneous of degree 1

zEBe(xo)

Note that R(xo, o) < supXEB,(xo) d(u(x), u(xo)) < Lip(u)v.

LECTURE 7. SMOOTHNESS RESULTS FOR HARMONIC MAPS

171

Definition. A minimizing map u : fl -' X is intrinsically differentiable on a compact subset K C St provided there exist ro > 0, c > 0 and 0 E (0, 1] such that R(x, o) C cot+13R(x, ro) for all x E K and o E (0, ro). The constants c, 0, ro depend only on K, A, X and the total energy of u.

Definition. A subset S C X is essentially regular if for any minimizing map u : A - X with u(fl) C S, the restriction of u to any compact subset of it is intrinsically differentiable.

Suppose that X0 is a totally geodesic subcomplex of X and e : R" -, Xo is an essentially homogeneous degree 1 map, we have the following concept.

Definition. I is said to be effectively contained in Xo if t-1(XI) is of codimension at least one in R" where X1 is the subcomplex of Xo consisting of simplices which

are faces of a simplex in X but not in X0. Notice that X1 C X0 is of at least codimension one. We are ready to state the following theorem:

Theorem 7.5. Let u : A -> X be a minimizing map. Let xo E fl and ro > 0 be such that B,.o(xo) CC A. Let Xo C X be a totally geodesic subcomplex, and let e : B,.o(xo) - Xo be an essentially homogeneous degree I map. Assume that Xo is essentially regular near p = £(xo). There exists bo > 0 depending only on Q, 12, X, Xo such that if l is effectively contained in Xo and SupB., (xo) d(u(x), e(x)) < bo, then u is intrinsically differentiable near xo, and there exist o < bo < ro such that u(Bs, (xo)) S Xo. The proof is rather technical. We omit it, and refer to [GS].

LECTURE 8 Order 1 Points and Partial Regularity The next result convinces us that a point x E ft where the order of u at x, ord°(x) = 1 and where its approximating map u. has rank k = dimX is a regular point, provided that a little regularity condition holds on X. Roughly, this regularity condition means that there exist an isometric totally geodesic embedding i : B, (x) C ft - X corresponding to that of u.. Notice that if X is an F-connected complex, then this condition holds.

Theorem 8.1. If X is an F-connected complex, and u : ft X is a minimizing map, then any point x E ft where ord" = 1 and where u, has rank k = dim X is a regular point. We also omit the proof and refer to [GS].

Suppose that X is F-connected, denote Xp as the tangent cone of X at po which is also F-connected. Let J : R'" -+ Xpo be an isometric totally geodesic embedding for 1 < m < k. It can be easily seen that J(Rm) is contained in at least one k-flat. We need the following geometric result.

Theorem 8.2. Let X k be F-connected, and let X0 be the union of all k-dimensional flats in Xp which contains J(Rm). Then the subcomplex Xo is totally geodesic and is isometric to R'" x X1 where X 1 is F-connected of dimension k - n. Moreover J is effectively contained in Xo. Combining these theorems we can then prove the simplification result.

Theorem 8.3. Let X be F-connected. Then the following properties hold:

1. For any positive integer u and a compact set Ko C X there exist eo = eo(Ko, u) > 0 such that for any minimizing map u : fZ" -' X with u(f2) K0, we have either ord(x) = 1 or ord(x) > 1 + co for all x E ft.

2. Let u : 11 -, X be a minimizing map, and let x0 E with ord(xo) = 1. There exists a totally geodesic subcomplex X0 of X, (xo) which is isometric to R'" X Xj for some 1 < m < min{n, k} and some F-connected X1 of

dimension k - m such that u(B,(xo)) C X0 for some o > 0. Moreover, if we write u = (ul,u2) = B,,,(xo) - R""' x X1, then ui is harmonic with rank m at xo, and ord"2(xo) > 1 3. X is essentially regular. 173

174

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Proof. (Sketch) (i) Denote k = dim X. If k = 1, i.e., X is a tree, we will show explicitly that co only depends on n. In fact, if ord"(xo) > 1 and u(xo) is not a vertex of X, then u is smooth near xo and thus ord"(xo) > 2 since ord"(xo) is integer. If u(xo) is a vertex with p edges emanating from po = u(xo), then p > 2 by the maximum principle. If p = 2, then, again, ord" (xo) > 2. We assume that p > 3, and consider every homogeneous approximating map u, : R" -+ XP. If we choose an edge e emanating from po and introduce an arc length parameter 8 along e, then on Oe = u.-1{e} the function he = s(u.) is a homogeneous harmonic function of degree a = ord"(xo). Of course, Oe is the cone over a region De C S"-1. It follows that heIDe is a first eigenfunction of De and the corresponding first eigenvalue AI (De) = a(a + n - 2). We thus decompose S"-1 into p disjoint regions all with the same eigenvalues. We can choose one such region D. that vol(De) < " S"-1 P Standard results about eigenvalues then imply that there exist bn > 0 such that ,\I (De) = a(a + n - 2) > n - 1 + bn. In particular, a > 1 + co with co = co(n) > 0. Property 2 is an easy consequence of Theorem 8.1 and Theorem 8.2. To establish 3 we work by induction on k. For k = 1 the result follows easily. Assume that k > 2 and all F-connected complexes of dimension less than k are essentially regular. By a compactness argument, it suffices to prove the following result for any xo E 13. There exists ro > 0 such that for or E (0, ro] R(xo,or) < ar1+BR(xo,ro)

for constants c, ro, A depending on x0, E(u), Il, Xo. There are two cases to consider.

First suppose ord"(xo) > 1. Then from 1 we know that ord"(xo) > 1 + Co. This implies, by lecture 6, that sup d(u(x),u(xo)) 5 cal+eo sup d(u(x),u(xo)) XEB., (;o) Bro (xo)

for some constant c and ro > 0. Therefore, the desired decay on R(xo, o) follows. In the remaining case ord(xo) = 1, the result follows immediately from 2, and the inductive assumption.

8.4 Proof of Singular Set Estimate Now, we start to prove the main Theorem 7.4. The estimate on the Hausdorff dimension of S(u) is an application of the basic argument of Federer dimensional reduction [F2]. For any subset E C fZ and any real number a E (0, n), we recall the definition of Hausdorff (outer) measure p{" (E) = inf

{r: : U B,&o) E a

o0

{-1

i=1

and of the Hausdorff dimension

dime E = inf{s : 7{"(E) = 0}.

LECTURE 8. ORDER 1 POINTS AND PARTIAL REGULARITY

175

We observe that S(u) = So U . . . U Sk where ko = min{n, k -1} and S. consists

of those singular points having rank j, where the rank at xo is the rank of the approximating map it. if ord(xo) = 1, and the rank is zero if ord(xo) > 1. We will first show dime So < n - 2. In fact, if we define So = {x E 0, ord"(x) > 1}. Then we actually show that dim,{ So < n - 2. Notice that $o C So. In order to do this, we need the following lemma whose proof is very standard.

Lemma 8.5. If {ui} is a sequence of minimizing maps from BI to X with E(uj) and Image (ui) uniformly bounded, then a subsequence of {ui} converges uniformly

on compact subsets of B1(0) to a minimizing map u : BI - X, and N8(So(u) n B*(0)) > lim f'(So(ui) n B,.(0))

for all r E (0, 1). In particular, dimw(So(u)) > lim dimw(So(u)). We now show that dim So (u) < n - 2. Suppose s E [0, n] with 7 i° (So(u)) > 0. Then by [F1] we may find x0 E fl such that

l a- w*(So(u) n B.,(x)) > 2-'. 0-0

Let u.: R" -

be a homogeneous approximating map for u at xo. Let a = ord"(xo) so that it, is of degree a. Since xo E So(u), we have a > 1 + co. We may apply Lemma 8.5 to suitable rescalings {ui} of it near xo to conclude that ?1 (So(u.)) > 0. Since So(u.) is a cone, it follows that there is x E S"-1 fl So(u.) such that

11'm o-w(So(u.) fl B,(xl)) > 2 °. 0-0 Let ul be a homogeneous approximating map for it, at x1. Then ord"1(x1) > 1+eo. It is easy to we that derivative of u1 is zero along ray t "- tx1. If we choose

coordinates in which x1 = (0,..., 0, 1), then Rk = 0. Therefore the restriction of u1 i denoted u1, to

R"-1 is a

homogeneous map of degree a > 1 + co. We then have So(ul) = So(ul) x R

and thus ?V-I (So('01)) > 0. Ifs > n - 2, we may repeat this argument inductively and produce finally an go > 0 and a minimizing map v : R2 homogeneous of degree a > 1 + Eo such that xa-("-2)(So(v)) > 0. Thus repeating the argument again will produce a geodesic w with degree > 1 and contradict with ord'(xo) = 1 for all xo.

We now show by induction on k = dim X that dim S(u) < n - 2. For k = 1 we have S = So, and we have established this case. Assume that k > 2 that the conclusion is true for F-connected complexes of dimension less than k. Let

176

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

xo E Sm -So for a minimizing map u : f2 --+ X1. We then have ord"(xo) = 1, and by 2 in Theorem 8.3 there is a bo > 0 such that u(B6. (xo)) C Xo = Rm x X1 with X 1 an F-connected of dimension k - 7n. Thus we have u = (u1, u2) where ul : BQO (xo)

R"', U2 : Boo(xo) --+ Xl are both minimizing. S,,,(u) n Bo(xo) S S(u2) n B,,(xo) n B,0(xo)) < n - 2 and thus By inductive assumption we then have

dime S(u) < n - 2. To prove 2 we use induction on k = dim X again. Fbr k = 1, we have S = So and dim% S < n - 2. Let e > 0 and d > n - 2. Let 122 be a fixed domain of D with 121 CC 122 cc f2, and choose a finite covering {Br, (xj)}j_1 of So n f21 satisfying xj E So and Erg < e, and B4r, (xj) C 122. Let on 0 \ B2r, (xj) 1 a:5 Sj < 1 on B2r,(xj) \ Br,(xj) . 0

Brj (xj )

Then IVS, I < 2r; 1. Define +p = min{cp, : j = 1... P} and observe that W = 0 near so n S21 and Cpl = 1 on f2 \ Ujt=1 B2r, (xj). Now let Wo = e and observe

f IVVuIIV boldp =

2f

pIVVuIIVVldp 1/2

52(f

Ra.,(x,)

jvlVVuIIVul-102dj4

(f IVuIIV'l2dp ` Uj_, R2.' (-j)

1/2

by Schwartz inequality. On the other hand, a result for harmonic maps (see [ES]) implies that on regular set we have 20IVu12 > IVVU12 - cIVu12.

1

For j = 1, ... , e let p3

on B2r, (xj)

a < pj < 1 on Bar, (x.) \ B2r, (xj) with 12\B4r,(xj) 0

I V Pj 1 5

. Define

p = max{pj, j

= 1, ... , t}

and observe that p = 1 on Uj=I B2,., (x,), and p = 0 outside Uj=1 Bar, (xj). We therefore have

f

IVVuI2loul-'92dp < f lVV I2IVul-1,p2p2dp. B2,,(x,}

0

LECTURE 8. ORDER I POINTS AND PARTIAL REGULARITY

177

An equality in [SY( implies that on regular set we have (1- e)IVVuI2 > IVIIVUI2 for some e,, > 0. Therefore we have AIVuI >

cIVUI

on fl \ So. Using integration by parts, we have En

f

n

.5-2

f

inn

n

This implies fn IVVUI2IVuI-1p2,p2di <

cfn IVuI(w2IVPI2 + IVVI2p2 + p2,P2)dµ.

Combining this with the above estimates, we have

t

f IVVulIVGoIdµ n

< c>r;2 j_I

fa..j(;j)

t IVuldµ <

rj _2

j=1

sup IVUI. B3,3(S1)

On the other hand since xj E So we have ord°(xj) > 1+eo and therefore SUPB3r (ii) IVul < crjeo. Thus we have

I

ff1

IVVuIIVa'oldµ 5

cF,rJ -2+e0 < e j=1

provided n - 2 < d < n - 2 + co. Now we can assume that 2 holds for maps into Fconnected complexes of dimension less than k. We cover (S - t=I Br,(x;)) n S21 with balls {B,., (yy) : p = 1,... } such that in B,, (yy) the map can be written u = (u1, u2) as in (ii) of Theorem 8.3. By the inductive assumption, there exists a function Op vanishing near V n B,., (yy) and identically one outside a slightly larger neighborhood with

fn IVVuIIV+GpIdu5 2-Pe. We finally set 0 = m1n{,0o, v/1 i ... , . 0.) and conclude

in IVVuIIV,PIdp 5 E fn IVVuIIV &Idp < 2E. P=Q

LECTURE 9 Rigidity Results via Harmonic Maps In this chapter we prove some rigidity theorems for discrete groups with the help of the theory developed in the previous lectures. In particular, we prove our p-adic superrigidity results for lattices in some groups of rank one.

First, let's review briefly the history of rigidity. We assume that (Mg) is complete (compact or non-compact with finite volume) locally symmetric space of

non-compact type, i.e., in a neighborhood of any point, M is isometric to k, a simply-connected globally symmetric space with nonpositive sectional curvature.

Note that the isometry group of k is a semi-simple Lie group of non-compact type. A typical example of non-compact type locally symmetric space is hyperbolic space X (i.e., KX =_ -1, and consequently X is locally isometric to the hyperbolic

disc (H",gH)). For M a given smooth manifold, we consider Mo = {g : (M,g) is locally symmetric} and define an equivalence relation in Mo by saying that, for g', 92 E Mo, 9i ^' 92 if and only if there exist a diffeomorphism F : M -, M such that gl = F'g2. We thus consider the moduli space Mo/ Diff(M). For example, if dim M = 2, then it is well-known that ,Mo/ Diff (M) is a (6g - 6) dimensional space where g = genus of M. The situation is different for locally symmetric manifolds of dimension larger than two whose universal cover is irreducible. In 1960 Calabi-Vesentini ([CV]) proved the local rigidity result (in the Kahler case) which says that there is no nontrivial curve in Mo. In 1970, Mostow ([M]) proved the rigidity theorem (for the compact case) which says any two locally symmetric structures in Mo are equivalent. In particular, the moduli space Mo/ Diff(M) is a point. In the 1970's, G. Marqulis ([Ma]) proved his celebrated "superrigidity" for lattices in groups of real rank at least two. One analytic approach to prove Mostow rigidity theorem is to use harmonic map theory. In fact, in 1979, Y.T. Siu ([Siu]) used harmonic maps to prove rigidity in the Hermitian locally symmetric case; in the 1980's, Sampson ([Sa]) proved a vanishing theorem of harmonic maps from Kahler manifold into an arbitrary manifold with a certain negative curvature assummption. Now, we formulate both Archimedian superrigidity and p-adic superrigidity in our setting. One can refer to [Ma] or [Zim] for details on superrigidity.

179

180

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

(i) Archimedian Case: Let M be locally symmetric, N be symmetric space of non-compact type. Denote the universal cover of M as M so that M = M/I' with r = ir1(M). Let p : r - isom(N) be a homomorphism. Consider a p-equivariant map W : M -' N i.e., .p o ry = p(y) o p Vy E F. Then Archimedian superrigidity concerns whether there exists a totally geodesic p-equivariant map u : M - N. (ii) p-adic Case. Let M be as above. We replace N by an F-connected complex X such that isom(X) = p-adic Lie group. (Notice that there exists a procedure due to Bruhat-Tits (see [Br}) for starting with a p-adic Lie group G and constructing a building X such that GC- isom(X)). Then p-adic superrigidity concerns whether there exists a constant p-equivariant map, i.e., whether p(r) lies in the isotropy subgroup of a point.

Theorem 9.1 (Margulis). If rank (M) > 2, then both superrigidity properties stated in (i) and (ii) are true. Note that rank (M) is the dimension of a maximal flat in M, i.e., a totally geodesic submanifold isometric to Rk. Therefore, H2 has rank 1; H2 x H2 has rank 2, SL(n, R)/SO(n, R) has rank n - 1. Margulis also showed that (i) and (ii) imply the arithmeticity of r (see [Ma] or [Zim]). Margulis's result left questions unanswered for rank (M) = 1. In fact, it is known that superrigidity fails for lattices in the isometry groups of real and complex hyperbolic space(see Introduction in [GS]). A result of K. Corlette ([C]) showed that Archimedian superrigidity (i) holds for quaternionic hyperbolic HQ and Cayley hyperbolic spaces HCB. Here, we show that p-adic superrigidity (ii) holds for H$ and He s

Theorem 9.2 (Gromov-Schoen). In the case of M = HQ or HC"., Corlett's vanishing theorem can be derived to prove p-adic superrigidity (ii). Consequently the corresponding r are arithmetic. We will sketch the proof of this theorem in the remainder of this lecture. First, we need to prove the existence of a finite energy equivariant harmonic map. Let X be a Euclidean building associated to a p-adic Lie group H(=- isom(X)). Then X has a compactification X = X UOX such that any h E H acting isometrically on X extends as a homeomorphism to X. Moreover, if {Ps} is a sequence from X with lim,-. P; = P E OX, and if {Q;} is another sequence from X with d(P;,Q;) < c independent of i, then lim; Q; = P. Finally, the isotropy group of P E OX is a proper algebraic subgroup of H (see [Br]). Let M be a complete Riemannian manifold, M be its universal covering manifold such that m = M/r. Suppose we have a homomorphism p : r - H. Then we have the following existence result.

Theorem 9.3. Suppose p(l') is Zariski dense in H (i.e., p(T) is not contained in a proper algebraic subgroup of H), and suppose there exists a Lipschitz p-equivariant map v : M - X with finite energy. Then there is a Lipschitz equivariant map u of least energy and the restriction of u to a small ball about any point is minimizing. Proof. See Theorem 7.1 in [GS].

LECTURE 9. RIGIDITY RESULTS VIA HARMONIC MAPS

181

Using the result of our main theorem, we now prove the following extension of Corlette's vanishing theorem [Cl.

Theorem 9.4. Let w be a parallel p -form on M, and assume that u is a finite energy equivariant harmonic map into an F-connected complex X. In a neighborhood of any regular point of u, the form w A du satisfies d* (w A du) = 0.

Proof. Suppose xo E M is regular point for u. Then there exists 60 > 0 such that u(B6a (xo)) C k-flat F. The calculation of [C] then implies dd' (w A du) = 0 in B6o (xo). Note that sets R(u) and S(u) are 17-invariant, and we define Ro =

R(u)/t, so = S(u)/t. We then have from Theorem 7.4 that dimSo < n - 2, and for any compact subdomain III C M there exists a sequence of nonnegative Lipschitz functions {t/ii} which vanish in a neighborhood of So n NI and tend to 1 on M \ (So n S21) such that

ilim f 00 m

0.

Let p be a nonnegative Lipschitz function which is one on BR(xo) and zero outside B2R(xo) with Iopl < 2R-1. We then apply Stoke's theorem on M using the identity r/iip2 (w A du, dd-(w A du)) = 0. Thus we obtain

1' p2lld(A du)II2dp = ± fM (*d(ip2) A *(w A du), d`(w A du)) dµ. IM

This implies

fM

A du)II2du < CfM (,Oip(Vpl + IV ',Ip2) IVullld'(w A du)II

Using Young's inequality, we then have

f t,b p2lld'(w Adu)II2du < C M

fM"IVpI2IVuI2dii+C f p2I0llvuldµlvVul. M

Therefore, r, IId*(w A du)II2d/A < cR-2E(u) + e(R)

IJV0ijjVVujdjA.

M

By first choosing R large and then taking i to infinity, we finally have that d' (w A du) __ 0 on R(u). We now derive a consequence of this result.

182

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

Theorem 9.5. A finite energy equivariant harmonic map from either I or KCn. into an F-connected complex is constant. See Theorem 7.4 in [GS] for the proofs. Let M be 1I (or HC,), so that isom(M) is (n, 1) (or F4 2D). We then have Lemma 9.6. There exists a finite energy Lipschitz equivariant map. See Lemma 8.1 in [GS] for the proofs. Combining all these results will give Theorem 9.2.

References A. Ancona, Negatively curved manifolds, elliptic operators, and Martin boundary, Ann of Math 125 (1987), 495-536. M. Anderson, The Dirichlet problem at infinity for manifolds of negative [A] curvature, J. Diff Geom 18 (1983), 701-721. [AS] M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann of Math 121 (1985), 429-461. W. Ballman, On the Dirichlet problem at infinity for manifolds of nonpositive [B] curvature, Forum Math 1 (1989), 201-213. [Br] K. Brown, Buildings, Springer-Verlag, Heidelberg and New York, 1988. [Ch] J. Chen, On energy minimizing mappings between and into singular spaces, preprint (1992). [Cl] S. Y. Cheng, Liouville theorem for harmonic maps, Proc Symp Pure Math 36 (1980), 147-151. [An]

, The Dirichlet problem at infinity for nonpositively curved manifolds, Comm Anal Geom. 1 (1993), 101-112. [CY] S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm Pure Appl Math 28 (1975), 333-354.

[C2]

[Cho] H. I. Choi, On the Liouville theorem for harmonic maps, Proc AMS 85 (1982), 91-94. K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann of Math 135 (1992), 165-182. [CV) E. Calabi and A. Vesentini, On compact, locally symmetric Kdhler manifolds, Ann of Math 71 (1960), 472-507. [C]

[EL] [ES]

J. Fells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. J. Fells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 88 (1964), 109-160.

[Fl]

H. Federer, Geometric Measure Theory, Springer-Verlag, Heidelberg and New York, 1969.

[F2]

, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing fiat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767-771. M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, I. E. H. S. Publications.

(GS)

LECTURE 9. RIGIDITY RESULTS VIA HARMONIC MAPS

183

(GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Heidelberg and New York, 1977. [HL] R. Hardt and F. H. Lin, Harmonic maps into round cones and singularities of nematic liquid crystals, Mat. Zeit. 213 (1993), 575-593. [J] J. Jost, Equilibrium maps between metric spaces, Calc of Var and PDE 2 (1994), 173-204. (KS]

N. Korevaar and R. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm Anal Geom 1 (1993), 561-659. [K] E. Kuwert, Harmonic maps between flat surfaces with conical singularities, To appear, Mat. Zeit. (1995). [Li] P. Li, A lower bound for the first eigenvalue of the Laplacian on a compact Riemannian manifold, Indiana Math J 28 (1979), 1013-1019. [LY1] P. Li and S. T. Yau, Estimates of eigenvalues of a compact Riemannian manifold, Proc Symp Pure Math 36 (1980), 205-239. [LY2] , On the parabolic kernel of the Schrodinger operator, Acta Math 156 (1986), 153-201.

F. H. Lin, On nematic liiquid crystals with variable degree of orientations., Comm. Pure Appl. Math. 44 (1991), 453-468. [LS] T. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J Diff Geom 19 (1984), 299-323. [Ma] G. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Heidelberg and New York, 1989. [M] Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Studies 78, Princeton University Press, Princeton, 1973. [P] A. Preissman, Quelques proprigtes globales des espaces de Riemann, Comment Math Helv 15 (1942-43), 175-216. [Sa] J. H. Sampson, Applications of harmonic maps to Kiihler geometry, Contemp Math 49 (1986), 125-133. [S] R. Schoen, Analytic aspects of the harmonic map problem, Seminar in Nonlinear Partial Differential Equations M. S. R. I. vol.2 (S. S. Chern, eds.), Springer, Heidelberg and New York, 1985. [SU) R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, 3 Diff Geom 17 (1982), 307-335. [SY] R. Schoen and S. T. Yau, Lectures on Differential Geometry, International Press, Boston, 1994. [Se] T. Serbinowski, Boundary regularity of harmonic maps to nonpositively curved metric spaces, Comm Anal Geom 2 (1994), 139-154. [Sim] L. Simon, Lectures on Geometric Measure Theory, Centre for Mathematical Analysis, Australian National University, 1984. [Siu] Y. T. Siu, the complex analyticity of hasrmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. 112 (1980), 73-112. [Su] D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, J Diff Geom 18 (1983), 723-732. [W] M. Wolf, Harmonic maps from a surface to R-trees, Math. Z. 218 (1994), [L]

577-593. [Y]

S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm Pure Appl Math 28 (1975), 201-228.

184

[Z]

R. SCHOEN, BEHAVIOR OF HARMONIC FUNCTIONS AND MAPPINGS

W. Ziemer, Weakly Differentiable Rnctions, Springer-Verlag, New York, 1989.

[Zim] R. Zimmer, Ergodic Theory and Semisimple Groups, Birkhauser, Boston/ Basel/Stuttgart, 1984.

Singularities of Geometric Variational Problems Leon Simon

IAS/Park City Mathematics Series Volume 2, 1996

Singularities of Geometric Varational Problems Leon Simon LECTURE 1 Basic Introductory Material Introductory Remarks These lectures are intended as a brief introduction, at graduate level, to the techniques (principally analytic and measure-theoretic) needed in the study of regularity and singularity of minimal surfaces and energy minimizing maps (sometimes loosely referred to as harmonic maps-see the discussion of terminology in 1.1 and 1.3 below).

Since it is technically simpler, we concentrate almost exclusively on energy minimizing maps, but the reader should keep in mind that essentially all the results discussed in these lectures have very close analogues for minimal surfaces. The first 3 lectures are meant to be essentially self-contained, assuming no prior knowledge about harmonic maps; the main analytic tool used in the first 3 lectures is the Schoen-Uhienbeck regularity theorem. We defer the proof of this until the second series of lectures (given in week 3 of the RGI). The last lecture in this present series touches on more recent work. This is too lengthy to be covered in any detail in the available time, but we do state the main results in a self-contained way, and prove a few things which give at least some hint of the kinds of techniques which are involved.

1.1 Definition of Energy Minimizing Map Assume that fl is an open subset of R", n > 2, and that N is a smooth compact Riemannian manifold of dimension p > 2 which is isometrically embedded in some

Euclidean space RP. We look at maps u of fl into N; such a map will always be thought of as a map u = (ul, ... , uP) : fl - RP with the additional property that u(fl) C N. Consider such a map u = (u1, ... , uP). We do not assume that u is 'Mathematics Department, Stanford University, Stanford, CA 94305

E-mail address: lasCgauss. stanford. edn ® 1996 American Mathematical Society 187

188

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

smooth-in fact we make only the minimal assumption necessary to ensure that the energy of u is well-defined. Thus we assume only that Du E Li (SZ), and then the energy EBo(y) (u) of u in a ball BP(Y) - {X : IX - YI < p} with BP(Y) C ft is defined by

EB,(Y)(u) = B,(Y)

IDuI2.

Notice that here Du means the n x p matrix with entries D;u'(- 80/8x'), and I2 = E: .l Ej=1(D;u!)2. We study maps which minimize energy in St in the I

sense that, for each ball BP(Y) C ft, 8B,(Y)(U) < eB,(Y)(W),

for every w : BP(Y) -+ RP with Dw E L2(BP(Y)), with w(B,,(Y)) C N, and with w =- u in a neighbourhood of 8BP(Y). Such u will be called an energy minimizing map into N.

1.2 Definition of Regular and Singular Set Given an energy minimizing map u in the sense of 1.1 above, the regular set regu of u is defined simply as the set of points Y E ft such that u is smooth in some neighbouhood of Y; thus reg u is an open subset of fl by definition. The singular set sing u of u is then defined to be the complement of reg u in fZ. Thus

sing u=f\regu, and sing u is a closed subset of R.

1.3 The Variational Equations Suppose u is energy minimizing as in 1.1, suppose BP(Y) C Il, and suppose that for some 6 > 0 we have a 1-parameter family {u,},E(-6,6) of maps of BP(Y) into N such that Du, E L2(fl) and u, _- u in a neighbourhood of 8BP(Y) for each a E (-6, 6), and uo = u. Then by definition of minimizing we have £B,,(y)(u,) takes a minimum at a = 0, and hence

whenever the derivative on the left exists. The derivative on the left is called the first variation of £B,(y) relative to the given family; the family {u,} itself is called an (admissible) variation of u. There are two important kinds of variations of u: Class 1: Variations of the form (i)

U. =llo(u+SC),

LECTURE 1. BASIC INTRODUCTORY MATERIAL

189

where ( = ((1,... ,(P) with each (' E Cr(B,(Y)) where II is the nearest point projection onto N. (Here and subsequently denotes the COO functions with compact support in BP(Y)).) Notice that this nearest point projection onto N is well-defined and smooth in some tubular neighbourhood {x E RP :

dist(X, N) < ao} for some ae > 0, and hence ua defined in (i) is an admissible variation for 1al < vo. We recall the general facts that the induced linear map dIIy gives orthogonal projection of RP onto the tangent space of N at Y E N, and the Hessian Hess fly has the properties that v1 Hess IIy (v2i v3) is a symmetric function of vi, v2, v3 E RP and is related to the second fundamental form of N via the identity v1 Hess IIy (v2, v3) = -1 E vo, Ay (vT,, v1 where the sum is over all permutations al, a2, 03 of the integers 1,2,3 and where vT means orthogonal projection onto the tangent space of N at Y. On the other hand by using a Taylor series expansion for II it is straightforward to check that Diu, = Diu+s((Di()T +( Hess fu(Diu,.))+O(82), where ( )T means orthogonal projection into the tangent space at the image point u(X), and hence for such a variation * implies the integral identity

1.3(i)

J0 i=1

(Diu DDS - ( Au(D;u, D,u) = 0

for any ( as above. Notice that if u is C2 we can integrate by parts here and use the fact that ( is an arbitrary C°° function in order to deduce the equation

6U+

n

E Au(Diu, Diu) = 0, i=1

where Au means simply (tut, ... , tuP). The identity 1.3(i) is called the weak form of the equation 1.3(i)'; of course if u is not C2 the equation 1.3(i)' makes no sense classically, and must be interpreted in the weak sense 1.3(i). It is worth noting (although we make no specific use of it here), that, in case u E C2, 1.3(i) says simply

(Du)T = 0

at a given point X E B,(Y), where (Au)' means orthogonal projection of Lu(X) onto the tangent space Tu(X)N of N at the image point u(X). Class 2: Variations of the form

u,(X) = u(X + s((X)), where ( = ((1,... ,(n) with each (i E C°°(B,,(Y)). Then D,u,(X) = F,'=, Diu(X + s() + sD;('DJu(X + a(), and hence after making the change of variable ( = X + s( (which gives a C°° diffeomorphism of

190

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

BB(Y) onto itself in case Ia1 is small enough) in this case * implies

1.3(ii)

JB(Y) i,7=1

(IDuI26i, -

2Diu DJu)D,C' = 0.

The identities 1.3(i), (ii) are of great importance in the study of energy minimizing maps. Notice that if u E C2 we can integrate by parts in 1.3(ii) in order to deduce that 1.3(i) implies 1.3(ii); it is however false that 1.3(i) implies 1.3(ii) in case Du

is merely in L2 (and there are simple examples to illustrate this). One calls a map u into N which satisfies 1.3(i) a "weakly harmonic map", while a map which satisfies both 1.3(i) and 1.3(ii) is usually referred to as a "stationary harmonic map". Thus the above discussion thus proves that energy minimizing implies stationary harmonic. We shall not here discuss weakly harmonic maps, but we do mention that such maps admit far worse singularities (see e.g. [RT1, 2]) than the energy minimizing maps . (Except in the case n = 2 when there are no singularities at all-we show this below in the case of minimizing maps, and refer to recent work of F. H61ein [HF] for the general case of weakly harmonic maps.)

1.4 The Monotonicity Formula An important consequence of the variational identity 1.3(ii) is the "monotonicity identity" 1.4(i)

f IDuI2 =2f p2'" IB P(Y) IDuI2 - 02-"B.,(Y) ,(Y)\B.(Y)

R2_"I ft I2

VR

'

valid for any 0 < o, < p < po, provided B ft (Y) c 0, where R = IX - Y J and 8/8R means directional derivative in the radial direction IX - Y I -1(X - Y). Since it is a key tool in the study of energy minimizing maps, we give the proof of this identity.

Proof. First recall a general fact from analysis-Viz. if a. are L' functions on BA(Y) and if fBn°(Y) Eni a'D,,S = 0 for each C which is C°° with compact support in BPo (Y), then, for almost all p E (01 po), fs,(Y) E, -j a,D,C = fes,(Y) n' a( for any C E C°°(-ffP(Y)), where a = (al,... , a") and r)(=- p-1(X - Y)) is the outward pointing unit normal of 8B,,(Y). (This fact is easily checked by approximating the characteristic function of the ball Bp(Y) by CO° functions with compact support.) Using this in the identity 1.3(ii), we obtain (for almost all p E (O,po))

that

Io(Y) is=1

(IDuI26i, - 2Diu Dju)Di(' _ (IDuI26i,, - 2Diu Dju)p 1(X `

IOB(Y) i)=1

- Y')(-'.

LECTURE 1. BASIC INTRODUCTORY MATERIAL

191

In this identity we choose (3(X) = Xj - Yj, so DCj = b;j and we obtain

(n - 2)

JB9(Y) IDuI2 = P-I j Bv(Y)

(IDu2 - 210u/8R12).

Now by multiplying through by the factor pl_n and noting that LB,, f = for almost all p, we obtain the differential identity d

p2_n

JBp(y) IDuI2

dP C

I

=2d

r

T p-

R2-n I R

fB, f

12

for almost all p E (0,po). Since JB0 f is an absolutely continuous function of p (for any LI-function f), we can now integrate to give the required monotonicity identity. Notice that since the right side of 1.4(i) is non-negative, we have in particular

that 1.4(ii)

p2-n f

IDuI2 is an increasing function of p for p E (0, po), P(Y)

and hence that the limit as p - 0 of p2_n ',,(y) IDuI2 exists.

1.5 The Density Function We define the density function eu of u on 11 by

eu(Y) = limp2-n P1o

IDuI2.

Bo(Y)

(As we mentioned above, this limit always exists at each point of f1 for a minimizing map u.) We shall give a geometric interpretation of this below. For the moment, notice that the density eu is upper semi-continous on Cl; that is

1.5(ii)

Yj - Y E 0

eu(Y) > limsupeu(Yj). j-.oo

Proof. Let e > 0, p > 0 with p + e < dist(Y, 851). By the monotonicity 1.4(ii) we have $, (1'j) <_ P2-n fe,(y1) IDUI2 for j sufficiently large to ensure p < dist(Y,,BSt).

Since BP(Yj) C B.4.E(Y) for all sufficiently large j, we then have the inequality

eu(1) < p2-n fB,+.(y) IDuI2 for all sufficiently large j, so limsupj_ eu(Yj) < p2-n fB,+.(Y) IDuI2. Letting e 10, we then conclude that limsupj_ eu(Yj) < p2'n fa,(y) IDuI2, and the required inequality follows by taking the limit as p 10.

192

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

1.6 The Regularity Theorem Here we want to state the regularity theorem (due to Schoen-Uhlenbeck [SU]). This result is fundamental to the subsequent discussion in these lectures, but its proof is rather lengthy, so we defer discussion of that until our our later lectures (see [SL1]). The theorem is quite easy to state: it says (for any dimension n > 2) that if A is any given fixed positive constant, then there is e > 0, depending only on n, N, A, such that if u is an energy minimizing map into N as in 1.1, if BP(Y) C f2, if p2-n fB,(Y) IDuI2 < A, and if

min p

of

AERP

lu-AI2 <e2,

BO(Y)

then Y E reg u, and furthermore for each j > 0

sup p'ID'ul S Ce,

B,/2(Y)

where C depends only on n, N, j. Thus, roughly speaking, and subject to a fixed bound on the energy of u on the ball BB(Y), the theorem says that if the mean square deviation of u away from some constant vector is sufficiently small then Y is a regular point of u and all the derivatives of u are controlled in the ball of radius p/2 and center Y. We mention here that there are analogous theorems for various classes of minimal surfaces, due to De Giorgi, Reifenberg, Allard, Ahngren, and Schoen-Simon. Discussion of these, and abundant references to related work, can be found for example in the references [DeG], [R], [AW], [A], [G], [SS].

1.7 Corollaries of the Regularity Theorem Corollary 1. There exists e > 0, depending only on n, N such that if Bp(Y) C Il and if p2-n fB (Y) IDuI2 < e, then Y E regu and supR,/,(Y) prlDlul < C for each j > 1, where 6 depends only on j, n, N. Proof. The Poincare inequality for arbitrary functions f on BB(Y) with Df E L2 asserts that inf., fBo(y) If _'\12 < Cp2 fB,(Y) IDul2, where C depends only on the dimension n. (See e.g. [GTJ for a proof.) Since this is trivially also valid for RP-valued functions f = (f I , ... , f P) we can apply it to u, thus giving inf.,ERP p'n fB,(Y) lu Al2 < Cp2-n fB(Y) IDul2 < Cc, and hence Corollary 1 is a direct consequence of the regularity theorem 1.7. A second corollary gives a nice characterization of the regular set:

Corollary 2. 9 (Y) = 0 4=a Y E reg u. Proof. "G" follows trivially from the fact that u smooth near Y implies IDul is bounded near Y, while

follows directly from Corollary 1.

LECTURE 1. BASIC INTRODUCTORY MATERIAL

193

The third corollary shows that the singular set of the energy-minimizing map u is actually quite small:

Corollary S. N"-2 (sing u) = 0. (In particular sing u = 0 in case n = 2.) Remark. Here Itl is the j-dimensional Hausdorff measure; recall that ?{1 (A) = 0, for a given subset A C R", means that for each rl > 0 there is a covering of A by a countable collection B,,, (Y;) of balls such that Ei pp < r,. Proof of Corollary 3. Let K be a compact subset of Cl, 6o E (0, dist(K, &1)) and 6 E (0, 6e). For Y E sing u fl K we ]mow by Corollary 2 that IDuI2 > Epn-2

(1)

JB(Y)

for all p < 6. Choose a maximal pairwise disjoint collection {B6/2(Yj)},j=I...Q of balls with Y, E K fl sing u. (Here maximal means that we take such a cover with maximum Q.) Then the balls B6 (Y,) coverall of K f l sing u and by summing over j in (1) with YJ in place of Y and 6/2 in place of p we get (2)

Q(6/2)n-2 < e-1 Ls12(yj) IDuI2 < E-1

IDuI2, f(Kr"Wng u)6

where (K fl sing u)6 = {X : dist(X, K n sing u) < 6}. This evidently implies that IDuI2,

Qwn6" < 2nwnb2E-1 l(Kreing u)60

where Wn is the volume of the unit ball in Rn. Since B6(Yj) cover all of sing ual

K, and since we can let 6 1 0, this last inequality shows that singu fl K has Lebesgue measure zero. But then f(ainguflK)6 IDuI2 - 0 as 610 by the dominated convergence theorem, and hence (2) implies that h--2 (sing u fl K) = 0. Since K was an arbitrary compact subset of Cl, this completes the proof of Corollary 3. There is also a nice compactness theorem for energy minimizing maps, as follows:

Lemma 1. If {u,} is a sequence of energy minimizing maps from Cl into N with sup, fB,(Y) IDu,I2 < oo for each ball Bp(Y) with Bp(Y) C Cl, then there is a subsequence {u,,) and a minimizing harmonic map u of Cl into N such that u j,, Duy converge in L2 locally on St to u, Du respectively.

Remarks. (1) In particular the energy fB,(Y) IDu,,'I2 converges to f8,(Y) IDuI2 for each ball BP(Y) C Cl. (2) Notice that, by the Rellich compactness theorem for sequences of functions with

gradient bounded in the L2 norm (see e.g. [GT]), there is an LL(fl) function u with Du E L a (Cl) such that up converges in L2 to u on compact subsets of Cl and

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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

Du,' converges locally weakly in L2 to Du in f2. Of course then u maps into N (in the sense that u(X) E N a.e. X E 0) because a subsequence of the subsequence u,' converges pointwise a.e. to u. Thus the content of Lemma 1 is that Du.,, converges in L2 and that u is minimizing. To give the proof of these facts we shall need the following lemma, which is a consequence of the main result of Luckhaus [Luckl]. (The fact that Du,, converges in L2 locally on 11 is originally due to Schoen-Uhlenbeck, who used the regularity theorem to establish it. This approach however does not establish the fact that u is energy minimizing; this was not proved in full generality until the paper [Luckl].)

Lemma 2. Let A > 0 be given. There are constants co = eo(n, N, A) > 0 and C = C(n, N, A) such that the following holds for any p > 0, e E (0, 1]: If B(1+E)p(Y) C fl, u,v : B(l+E)P(Y)\Bp(Y) - BY, if u, v have L2 gradients Du, Dv with p2-" fB{1+.)o(Y)\Bo(Y)(IDui2 + IDvJ2) < A, u(X),v(X) E N for X E B(1+.)p(Y))\Bp(Y), and if a-2np-" fB(1+.)v(Y)\B,(Y) Iu - VI2 < e0, then there is w on B(1+,)p(Y)\BB(Y) such that w = u in a neighbourhood of 8Bp(Y), w = v in a neighbourhood of 8B(1+i)p(Y), w(B(1+1)p(Y)\Bp(Y)) c N, and IDw12

IB(I+.)(Y)\fi(Y)

<

C

(IDuI2 + IDvl2) + Ce-2 /

Iu - vt2.

B(1+.)o(Y)\Bp(Y)

B(1+<)0(Y)\Bp(Y)

We shall give the proof of Lemma 2 in the second series of lectures-see Lecture 3 of [SL1].

Proof of Lemma 1. Suppose R.,(Y) C ft and u = lim, uj, in the L2 norm on Bpo(Y) as in the Remark (2) above. Also let 6 E (0, and 9 E (2, 1) be given. (We shall eventually choose 6 small and 9 close to 1.) The4)main step in the proof (and the step which uses Lemma 2) is to prove that we can find p E (9po, po), e E (0, po/p-1),

a subsequence j C {j'} and a sequence w, : B(l+E)p(Y)\Bp(Y) -4 N with L2 gradients such that w, = u on 8Bp(Y) w, = u3 on 8B(l+f)p(Y) and (1)

limsupJ j-00

IDw,I2 < C6, C = C(n, N, A).

B(1+.)o(Y)\B,,(Y)

We proceed to prove this. (Once we have (1) the remainder of the proof is quite straightforward.) First choose A such that sup, pp-" fBno(Y) IDuj12 < A and note that if e E (0, 6(1 - 9)/2(1 + A)) then there must be an integer t E (2,2(1 + A)/b) such that IDu, I2 < b

(2) f8(e+j.)p0 (Y)\B(e+(#-1).)po (Y)

LECTURE 1. BASIC INTRODUCTORY MATERIAL

195

for infinitely many j, since otherwise by summation of the reverse inequality over integers e E (2,2(l +A)/6) we would get fB,,(Y) IDu,12 > A for sufficiently large j, thus contradicting the definition of A. Thus select such an 8 and let p = (0 + (I 1)e)po. Since (0 + le)po = p + epo > (1 + e)p, we then have from (2) p2rn

(3)

IDuI 12
for some subsequence (3) of {j'}, and of course then also (by lower semicontinuity of fB(1+.)P(Y)\8,(Y) IDu,12 with respect to weak convergence)

I

(4)

Du12 < (1+.)v(Y)\Bo(Y)

Since fB(1+.)P(Y)\B,(Y) Iu - u;12 -' 0, we can then apply Lemma 2 with uJ in place of v to deduce that there are w1 : B(1+E)p(Y)\BP(Y) -+ RP such that wi(B(1+E)P(Y)\B,(Y)) C N, w,I8B(1+E)v(Y) = up w,I8Bp(Y) = u, and

p2-"J

IDw, 12


( 1+.)P\Bv(Y)

(IDul2+1,12

+ e`2p-2 IB(l+,)p(Y)\Bp(Y) u-u,12)

for sufficiently large j by (3) and (4), thus establishing (1) as claimed. Now let v : BPo(Y) -+ RP with v(BPo(Y)) C N, Dv E L2, and v = u in a neighbourhood of 8B. (Y). Assuming 0 is close enough to 1 to ensure that v = u in BPo(Y)\Be.(Y) we then let i, be defined by u3 on B,, (Y)\B(1+,)v(Y)

u=

(4)

w., on B(1+,,)v(Y)\B,(Y) v on BP(Y)-

Then by (1) and the minimizing property of u1 we have

IDU;12 <(5)

fs".(Y)

f

fPoIDui12 (Y)

=

IDu'12+

fB(,+-).-(yABP(y)

BIDw'12+

f0(Y)

IDv12,

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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

and hence (7)

liminf j.

(l+<),(Y)

a

OO

IDu;I2 < r

JDvI2+C6,

C=C(n,N,A).

B,Y

Since 6 > 0 was arbitrary and fB(,+<),(Y) JDu12 < lim info .OO fB(,+<), (Y) JDu, I2,

this shows that u

J ,,o(Y) IDuI2 < J

JDvI2, ,o(Y)

so that (letting 0 j 1) we conclude that u is energy minimizing on Finally to prove that the convergence is strong we note that if we use (7) with v = u and (1) then we have li}minf

Ja,(Y)

IDU,12 < rBP

IDuI2 + C6. (Y)

Thus using the weak convergence of Dud to Du we have

Iiminf / ..//B ,(Y)

IDu., - Du12 - liminf

j

(IDu,, 12 - IDuI2) < C6.

!B ,(Y)

The subsequence {j} depends on 6, but in any case (by a diagonal process) we can prove that this last inequality implies that there is a new subsequence j such that Du; converges in L2 to Du on Bsp,,(Y). In view of the arbitrariness of 0 and po this gives the desired conclusion.

1.8 A Further Remark on Upper Semicontinuity of the Density Notice that if we use the result of Lemma 1 above, a very minor modification of the argument used in 1.5 to prove the upper-semicontinuity of 9u(Y) as a funcis actually upper-semicontinuous with respect to the tion of Y shows that

joint variables u, Y in the sense that if Y, - Y E S2 and if u., are a sequence of energy minimizing maps from Cl into N with locally bounded energy in the sense of Lemma 1 converging In L2 locally to u, then e (Y) > lim sups -,,,, O i (Y,) .

LECTURE 2 Tangent Maps Here u continues to denote an energy minimizing map of fl into N, with H an open subset of R1.

2.1 Definition of Tangent Map Take any ball BP0(Y) with BP0(Y) C St and for any p > 0 consider the scaled function uy,, defined by uY,P(X) = u(Y + pX ).

Notice that uy,, is well-defined on the ball BP11P(0); furthermore, if or > 0 is arbitrary and p < po/Q, we have (using Duy,(X) = p(Du)(Y + pX), and making a change of variable k = Y + pX in the energy integral for uy,P) 2.1(1)

p.2-n

DuY.PI2

J B(O)

I

= (ap)2-n J

<po_n

IDuI2 ev(Y)

f

,.(Y)

I Du12,

where in the last inequality we used the monotonicity 1.4(i) from Lecture 1. Thus if pj 1 0 then lim 8upf-,o f B, (o) I Duy Pf I2 < oo for each or > 0, and hence by the compactness theorem (Lemma 1 of 1.7) there is a subsequence pi, such that

u;,,, - V locally in Rn both with respect to the L2-norm and in energy, where 9 : R" -' N is a minimizing harmonic map (in the sense of 1.1 with 12 = Rn). en mma of u at Y. In general it Any tp which is obtained in this way is called a t is not true that such tangent maps need be unique (see [WB])-that is, if we choose different sequences pj (or different subsequences py) then we may get a different limit map. We discuss this further below.

2.2 Key Properties of Tangent Maps First note that by the equation 2.1(i) above, with p,, in place of p (keeping in mind that uy,,, , converges in energy to gyp) we have, after taking limits on each side

o.2-n J e (0) IDpI2 = eu(Y), 197

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

198

where we used the definition eu(Y) = limplo p2-" fB,,(y) IDuI2. Thus in particular o.2-" fBB(o) IDVI2 is a constant function of or, and since by definition e,o(O) _ limolo o,2-" fB,(o) ID(pI2, we thus have

eu(Y) = e,,(0)

2.2(i)

a2-"

J

Vo > 0. -(0)

Thus any tangent map of u at Y has scaled energy constant and equal to the density of u at Y; this is also a nice interpretation of the density of u at Y. Furthermore if we apply the monotonicity formula 1.4(i) to V then we get the identity

0 = v2-"

JB.,(0)

T2-" r I

JB.(O)

R2-n I O1112,

IDwp 12 =

JB,(O)\B(O)

OR

so that Oco/OR = 0 a.e., and since W has L2 gradient it is correct to conclude from this, by integration along rays, that

W(AX) - W(X) VA > 0, X E R".

2.2(ii)

This is a key property of tangent maps, and enables us to use the further properties of homogeneous degree zero minimizers. (See 2.3 below.) We conclude this section with another nice characterization of the regular set of u: 2.2(iii)

Y E reg u b 3 a constant tangent map V of u at Y

(and in this case of course the tangent map is unique). To prove 2.2(iii), note that by Corollary 2 of 1.7 we have Y E regu . . 9 (Y) = 0, but e (Y) = 0 e- 'P const. by 2.2(i).

2.3 Properties of Homogeneous Degree Zero Minimizers Suppose o : R" - N is a homogeneous degree zero minimizer (e.g. a tangent map of u at some point Y); thus W(AX) - V(X) for all A > 0, X E R". We first observe that the density ep(Y) is maximum at Y = 0; in fact by the monotonicity formula 1.4(i), for each p > 0 and each Y E R",

2

f

P(y)

R2-n I _ RV 12

+

p2-"

JBP(Y)

LECTURE 2. TANGENT MAPS

199

where Ry(X) - IX - YI and 8/8Ry = IX - YI-1(X - Y) D. Now BB(Y) C Bp+l yl (0), so that P2_n f

B(y)

P2_"

IDWI2 <-

_ (1 +

j

IDWI2 n+,Y1(O)

IYI)"-2((P+

P

IDWI2) = (1 + IYI )n-2e,(o),

IYD2-n j

p +IYI(O)

because cp is homogeneous of degree zero (which guarantees that

6,p(0)). Thus letting p j oo, we get 2

P

r2-n

,,,(O)

+0,(y) s @,v(0), fn Rn Ry nI 8Ry 12

which establishes the required inequality 2.3(i)

ew(Y) < e9(0).

Notice also that this shows that equality can hold in 2.3(i) only if 8cp/8Ry = 0 a.e., that is only if W(Y + AX) = cp(Y + X) for each A > 0. Since we also have (by assumption) W(X) _- W(AX) we can then compute for any A > 0 and X E R" that W(X) = W(AX) = W(Y + (AX - Y)) = W(Y + A-2(AX - Y)) +,\-2(,\X - Y))) = V(X + tY), = W(A(Y

where t = A - A-1 is an arbitrary real number. So let S(cp) be defined by 2.3(ii)

S(W) = {Y E R" : Op(Y) = e,p(0)}.

Then we have shown that W(X) _ V(X +tY) for all X E R", t E R, and Y E S(W). Combining this with the fact that if Z E R" and ,p(X + Z) - W(X) for all X E R", then e,,(Z) = e,,(0) (and hence then Z E S(W) by definition of S(cp)), we conclude

S((p) is a linear subspace of R" and V(X + Y) = W(X), X E R", Y E S(W).

(Thus cp is invariant under composition with translation by elements of S(W).) Notice of course that 2.3(iii)

dim S(p) = n

W = const.

Also a homogeneous degree zero map which is not constant clearly cannot be continuous at 0, so we always have 0 E sing W if p is non-constant, and hence, since p(X + Z) _- W(X) for any Z E S(W), we have 2.3(iv)

S(W) C singcp

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L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

for any non-constant homogeneous degree zero minimizer V.

2.4 Further Properties of sing u For any Y E Sl and any tangent map V of u at Y we shall let S(V) be the linear subspace of points Y such that $ ,(Y) = e,(0), as discussed in the previous section. Notice that then by 2.2(iii) we have 2.4(1)

Y E sing u b dim S(V) < n - 1 for every tangent map W of u at Y.

Now for each j = 0,1, ... , n - 1 we define Si = {Y E sing u : dim S(W) < j for all tangent maps W of u at Y}. Then we have 2.4(11)

S0 C SI C ... C Sn-3 = Sn-2 = Sn-I = Sing u.

To see this first note that Sn_I = singu is just 2.4(i), and the inclusion Si_I C S3 is true by definition. Also, if Sn_3 is not equal to both Sn_2 and S,,- 1, then we can find Y E sing u at which there is a tangent map yo with n - 2 < dim S((p) < n - 1; by 2.3(iv)) we have but then 7{n-2(S(W)) = oo and hence (since S(W) C 7{n-2 (sing gyp) = 0 by Corollary 3 7{n-2 (sing gyp) = oo, contradicting the fact that of 1.7.

The subsets S, are mainly important because of the following lemma, due to F. Almgren [A]; the lemma can be thought of as a refinement of the "dimension reducing" argument of Federer (for this see the discussion in the appendix of [SL3]):

Lemma 1. For each j = 0,.. . , n - 3, dim S, < j, and, for each a > 0, so fl {X eu(X) = a} is a discrete set. Remark. Here "dim" means Hausdorff dimension; thus dim S. < j means simply that Nf+,, (Sj) = 0 for each e > 0. Before we give the proof of this lemma, we note the following corollary.

Corollary. dim sing u < n - 3. Proof. By 2.4(ii), sing u = Sn-3, hence the lemma with j = n - 3 gives precisely dim sing u < n - 3 as claimed.

Proof of Lemma 1. We first prove that So fl {X : eu(X) = a} is a discrete set for each a > 0. Suppose this fails for some a > 0. Then there are Y, Yf E So rl {X :

eu(X) = a} such that Yj 0 Y for each j, and Yj --+ Y. Let pf = IY1 - Yj and consider the scaled maps uy, p, . By the discussion of 2.1 there is a subsequence pj,

such that uy p,, -# tp, where V is (by definition) a tangent map of u at Y; also,

by 2.2 we have e,p(0)=eu(Y)=a. Let i = JY, -Y1-1(Yt - Y)(E Sn-1). We can suppose that the subsequence f is such that E,, converges to some t E S"-1. Also (since the transformation

LECTURE 2. TANGENT MAPS

201

X *-4 Y + p, X takes Yj to E,) O, (Y;) = Ou, of (f j) = a for each j, hence by the upper semi-continuity of the density (as in 1.8) we have Op(t) > a. Thus since O,,(X) has maximum value at 0 (by 2.3(i)), we have Op(1) = 010(0) = a, and hence t E S(ip), contradicting the fact that S(W) = {0} by virtue of the assumption

that Y E& Before we give the proof of the the fact that dim S; < j, we need a preliminary lemma, which is of some independent interest. In this lemma and subsequently we use ly P to be the map of R" which translates Y to the origin and homotheties by

the factor p-'; thus

IIY,P(X)=p 1(X-Y). Lemma 2. For each Y E S and each 6 > 0 there is an e > 0 (depending on u, Y, b) such that for each p E (0, eJ

GYP{X E BP(Y) : 9u(X) > 9u(Y) - e} C the 6-neighbourhood of Lyp for some j-dimensional subspace Ly,P of R".

Proof. If this is false, then there exists b > 0 and Y E S, and sequences pk 10, ek 10 such that (1)

{X E B1(0) : 9u,.,pk (X) > Ou(Y) - ek} ¢ the b-neighbourhood of L

for every j-dimensional subspace L of R". But uy,P,, -, gyp, a tangent map of u at Y, and 0. (Y) = O,P(0). Since Y E S,, we have dimS(V) < j, so (since S(W) is the set of points where 9,o takes its maximum value 9,,(0)), there is a j-dimensional subspace Lo D S(W) (Lo = S(ip) in case j) and an a > 0 such that (2)

O,,(X) < O,p(0) - a

for all X E B1(0) with dist{X, Lo} > 6.

Then we must have, for all sufficiently large k', that (3)

OuY,k, (X) < 0,x(0) - a b'X E B1(0) with dist{X, Lo} > 6.

Because otherwise we would have a subsequence {k} C {k'} with Ou,,,,. (Xk) > 010(0) - a for some sequence Xk E B1 (0) with dist{Xk, Lo} > b. Taking another subsequence if necessary and using the upper semi-continuity result of 1.8, we get Xk --# X with 9,,(X) > 8,(0) - a, contradicting (2). Thus (3) is established. But (3) says precisely that, for all sufficiently large k',

{X E B1(0) : 9u, ok (X) > O,,(0) - a) C the b-neighbourhood of Lo, thus contradicting (1).

Completion of the proof of Lemma 1: We decompose S. into subsets S.,t, i E {1,2,. .. }, defined to be the set of points Y in S1 such that the conclusion of

202

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

Lemma 2 above holds with e = i-1. Then, by Lemma 2, Sj = Ui>1Sj,i. Next, for each integer q > 1 we let 5,,,,q = {X E Sj,, : 9,,(X) E

and note that 8j = Ui,gSj,i,q. For any Y E Sj,, we have trivially that

Sj,i,g C {X : 9u(X) > 6,(Y) - 1/i}, and hence, by Lemma 2 (with e = i-1), for each p< i-1 r)Y.p(Sj,i,q n B4(Y)) c the 6-neighbourhood of Ly,

for some j-dimensional subspace Ly,,, of R. Thus each of the sets A = Sj,i,q has the "6-approximation property" that there is po (= i-1 in the present case) such that, for each Y E A and for each p E (0, p0], *

i?y,, (AnBP(Y)) C the 6-neighbourhood of Ly,p

for some j-dimensional subspace Ly,, of R". In view of the arbitrariness of 6 the proof is now completed by virtue of the following lemma:

Lemma 3. There is a function /3 : (0, oo) - (0, oo) with limt jo /3(t) = 0 such that if 6 > 0 and if A is an arbitrary subset of R" having the property * above, then gj+P(6)(A) =0. This lemma is quite easy to prove, using the fact that there is a fixed constant C" such that for each or E (0,1) we can cover the closed unit ball B1 (0) of Rj with a finite collection of balls {B,(Yk)}k=1,...,q in Rj with radii o and centers Yk E B1(0) such that Qoj < C,,. In view of the arbitrariness of o, it then follows that for each f3 > 0 we can find o = 0(/3) E (0,1) such that there is a cover of RIM by balls {B,,(Yk)}k=1,...,Q such that Qoj+A < 2. More generally, if L is any j-dimensional subspace of R" and 6 E (0,1/8) there is /3(6) (depending only on n, 6) with /3(6) 1.0 as 6 j 0 such that the 26-neighbourhood of L n B1(0) can be covered

by balls B,(Yk), k = 1,... ,Q with centers in L n B1(0) and with Qoj+00) < 12' By scaling this means that for each R > 0 a 26R-neighbourhood of L n BR(0) can be covered by balls B,R(Yk) with centers Yk E L n BR(0), k = 1, ... , Q such that Q(oR).+0(6) < !Ri+0(6). The above lemma follows easily from this general fact by using successively finer covers of A by balls. The details are as follows: Supposing without loss of generality that A is bounded, we first take an intial cover of A by balls BPo/2(Yk) with A n BPo/2(Yk) # 0, k = 1,... Q, and let For each k pick Zk E An B,/2(Yk). Then by * with p = po To = Q(po/2)?+p(6).

there is a j-dimensional affine space Lk such that A n BPo (Zk) is contained in the 6-neighbourhood of Lk. Notice that LknB,./2(Yk) is a j-disk of radius < po/2, and so by the above discussion we can cover its bpo-neighbourhood by balls B,poi2(Zj,t), P = 1,... , P, such that P(opo/2)j+0(6) < 1(po/2)j+0(6) Thus A can be covered

LECTURE 2. TANGENT MAPS

203

by balls B,,.12(Wt), k = 1, ... , M, such that M(opo/2)j+0(6) < !To. Proceeding iteratively we can thus for each q find a cover by balls B,1,12(Wk), k = 1, ... , Rq, such that Rp(ogpo/2)J+#(6) < 2-qTo.

LECTURE 3 The Top-Dimensional Part of Sing u Here u continues to denote an energy minimizing map from 11 C R" into N C RP; the discussion is mainly only relevant when there are actually genuine "(n - 3)dimensional parts" of singular set in the sense that there are points Y E sing u at

which there are tangent maps V with dim S(,p) = n - 3. But the reader should keep in mind that all the discussion here carries over with an integer m < n - 4 in place of n - 3 if the target manifold N happens to be such that all tangent maps V of energy minimizing maps into N have dim S(w) = m < n - 4 (one such case is in fact mentioned later in this lecture, when dimN = 2 and N has genus > 1). Since the discussion is essentially identical in this case, there is no conceptual loss of generality in adopting the definition of top dimensional part in the following section.

3.1 Definition of Top-dimensional Part of the Singular Set We define the top dimensional part sing. u to be the set of points Y E sing u such that some tangent map V of u at Y has dim S(V) = n - 3. Notice that then by definition we have sing u\sing.u C and hence by Lemma I of the last lecture we have 3.1(i)

dim(sing u\sing. u) < n - 4.

To study sing. u further, we first examine the properties of homogeneous degree zero minimizers W : R" -' N with dim S(W) = n - 3.

3.2 Homogeneous Degree Zero ap with dim S(V) = n - 3 Let V : R" -+ N be any homogeneous degree zero minimizer with dim S(q) _ n-3. Then, modulo a rotation of the X-variables which takes S(W) to {0} x R"--3, we have 3.2(i)

Ox, y) = -POW, 205

L. SIMON. SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

206

where we use the notation X = (x, y), x E R3, V E Rs-3, and where Wo is a homogeneous degree zero map from R3 into N. We in fact claim that 3.2(u)

sing,po = {0}

and hence (PoIS2 E C°°,

so that e,oIS2 is a smooth harmonic map of S2 into N. To see this, first note that sing Wo D {0}, otherwise go, and hence V, would be constant, thus contradicting the hypothesis dim S(V) = n - 3. On the other hand if a 0 0 with t E sing coo, then by homogeneity of Wo we would have {A : A > 0) C sing o, and hence {(AC Y)

: A > 0, y E R"-3} C singp.

But the left side here is a half-space of dimension (n - 2), and hence this would give fl' 2 (sing cp) = oo, thus contradicting the fact that 7f'2(sing) = 0 by Corollary 3 of Lecture 1. Thus 3.2(ii) is established.

We also note that if W (j) is any sequence of homogeneous degree zero minimizers

with coU)(x,y) - po')(x) for each j, and if limsup1.O° fB,(0) IDp(')I2 < oo, then lim sup sup lDtWo) I< 00 j-.oo S2

for each e > 0. Indeed this follows easily from the compactness theorem (Lemma 1 of 1.7) and from the fact that all singular sets of minimizers have dimension < n - 3; the details are left as an exercise. It then follows that for a particular co with

fs2IDvoI2
sup ID'cpol <- C, S2

where C depends only on t, N, A. Finally we note the important fact that the set of densities of such homogeneous degree zero minimizers cp with dim S(cp) = n-3 is a discrete subset of (0, 00) if either the target manifold N is real-analytic or if dim N = 2 or if a certain integrability condition (see 3.2(vi) below) holds. We first check the case dim N = 2. In this case we use the fact that all non-constant smooth harmonic maps 10 from S2 (into any target) are conformal (except at finitely many points where I D'I = 0). (See e.g. [JJ].) Then for such a i/i, we have ID4I2 == +J or -- -J, where J is the (signed) Jacobian. Therefore by z the area formula (see e.g. [SL3, §8]) the energy of 0 is exactly 2d(O) area N, where d(pi) is the topological degree of i/i : S2 - N. Since this all applies to cpo(x), where co(x, y) - Wo(x) is any tangent map of u with dim S(,p) = n - 3, this implies that the set of densities 9,0 (0) of such cp forms a discrete set; but this set of densities is exactly the set of densities Y E sing. u} by definition of sing. u. This completes the discussion for the case dim N = 2. For dim N > 3 and N real analytic we can also establish the discreteness as follows. Let M(O) by the harmonic map operator on S2. Thus M(O) = AO + E3 A*(V.*, V:'+G), where Atp = (4tp ...... A 9) and V,ti I = (VjPI,... ,V,iiD), 2IN12

LECTURE 3. THE TOP-DIMENSIONAL PART OF SING u

207

With Q &119011g thl 1aplacian on $2, and with V1I such that (V1(1 Vif1 V3() = the S2-gradient off for any C' scalar function f on S2; thus lp is a smooth solution

of MM = 0 on S2 if and only if the homogeneous degree zero extension of 1p to R3\{0} is a smooth solution of the variational equation 1.3(i) of Lecture 1, and in fact M(+y) = 0 is exactly the variational equation (analogous to 1.3(i)) corresponding to the the energy functional es2(0) = fs, IVV)12, Where [V0[2 =

E IIV I2.

Now let 00 be any smooth solution of M(tk) = 0 on S2 and v E T, where T

denotes set of smooth RP-valued function on S2 with the property v(x) E T,yo(y)N for each x E S2. (Thus T is a the set of smooth sections of the pull-back by too of the tangent bundle of N.) The linearized operator Gvov of M(+') at tb = +Go is defined by

zvo(v) = ZM('0s))1'=0, where 0, is any 1-parameter family of smooth maps of 52 into N with ili, (x) varying smoothly in the joint variables (x, s) E S2 x (-e, e) for some e > 0, and with v = j t/isl,=o. (Of course such a family always exists for any given v and the derivative on the right is independent of which particular family is used, so long as ids ti, 1=o = v.) The linearized operator G,yo always has non-trivial kernel;

in fact if 0, is any 1-parameter family of smooth harmonic maps of S2 into N (that is, M(P,) = 0 Vs) with ip,(x) varying smoothly in the joint variables (x, s) E then v is automatically such a S2 x (-e, e) for some e > 0, and if v = solution, by definition of to.. In particular by using 0, = e'Atf10, where A is any skew-symmetric transformation of R3, we get a linear space of solutions spanned by the special solutions 3.2(iv)

v(x) _- x`DJ,po(x) - x'Djt,bo(x),

i, j = 1, 2, 3

(In computing D,00 we assume 1(,a is extended as homogeneous degree zero to R3\{0}.) Similarly by considering the homotheties of S2 (which are conformal and hence preserve harmonicity), we get a family generated by the special solutions 3.2(v)

Dit/io(x),

i = 1, 2, 3.

Now if K denotes the L2 projection of T onto the kernel of C,r,o, then the operator N(v) = M(II(Oo + v)) + K(v), for v E T with [vI < b where b > 0 is small enough to ensure that the nearest point projection n onto N is smooth in the b-neighbourhood of N, then the linearization of .A( at 0 is just Gee + K, which has trivial kernel. Then using the implicit function theorem (applied on the appropriate Holder spaces-we refer to the the discussion of [SL2, pp.537--5401 for the details)

together with the fact that, by smoothness of N, there is 61 E (0, b) such that any smooth map t' : S2 -, N with 10 - 001 < bI can be uniquely represented in the form +P = II(Oo + v) for v E T, we can find a real analytic embedding W of a ball B,,(0) C ker G,y into T such that all solutions of M (tk) = 0 are contained in

208

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

P(B,(0)), and the set of such solutions is precisely W applied to the set of points f E B. (0) such that V f (f) = 0, where f is the real analytic function on Bs (0) defined by f(O) - Cs. (*(0)) for 0 E B,(0). (In fact' is just the restriction to B, (0) of the local inverse A(-', which exists by the inverse function theorem.) Then since real analytic functions f have the property that f is constant on the connected components of the set where V f = 0, and since there is d < a such that at most one connected component of the set of points where V f = 0 intersects the ball B. (0), we deduce that the set of energies of smooth harmonic maps : S2 -' N is discrete. We actually note here that the above construction of the embedding W and the map f makes sense in the smooth case, and shows that the solutions near 00 are contained in the manifold W(B0(0)) which has dimension equal to the dimension of the kernel ker 4,,0. We shall want to refer to the following integrability condition:

f = const.

3.2(vi)

Notice that by the above discussion this is equivalent to saying that all of the manifold 'Y(B,(0)) corresponds to solutions of .M(II(tyo ++1')) = 0 and the energy

E is constant on'I(B,(0)). That is, it is equivalent to the requirement that there is a > 0 such that the set of solutions of M(v) = 0 with (v - OoIca < a forms a manifold of dimension equal to the dimension of ker£,yo. (This explains why 3.2(vi) is called an integrability condition.) Notice that by the Schauder theory for elliptic equations the condition Iv-001c3 < or is equivalent to the condition Iv - tbo J', < a modulo a fixed multiplicative constant. Using the definition of sing, u, the above discussion in particular implies 3.2(vii)

{eu(Y) : Y E sing, u} is discrete

whenever either dim N = 2 or N is real-analytic or when the integrability condition 3.2(vi) holds.

3.3 The Geometric Picture Near Points of sing. u Let K be a compact subset of fl and Y E sing, u f1 K and let W be a tangent map of u at Y with dim S(W) = n - 3. As in 3.2, we can assume without loss of generality (after making an orthogonal transformation of the X variables which takes S(W) to {0} x Rn-3), that 3.3(i)

cV(x, y) ° Vo(x),

X E R3, U E Rn-3.

By definition of sing, u, there is a sequence pj 10 such that 3.3(ii)

lim pn

3-00

J

Iu -W12 = 0, P (Y)

LECTURE 3. THE TOP-DIMENSIONAL PART OF SING u

209

so for p = p3 with j sufficiently large we can make the scaled L2-norm p-" fB0(y) Iu- WI2 as small as we wish. On the other hand we claim that for any homogeneous degree zero minimizing maps 'p : R" -. N as in 3.3(i) and any ball B,(Y) with BPo (Y) C fl we have the estimate 3.3(iii) sing u fl BP12(Y) C {X : dist(X, (Y + {0} x R"3)) < b(p)p}

`dp < po,

b(p)=C(p"f p Iu-'pl2)I11", where C depends only on n, N, A with A any upper bound for

p02_" fB,, (y)

IDuI2. In

view of 3.3(ii), this perhaps suggests that the possibility that the top dimensional

part of the singular set is contained in a C' manifold (or at least a Lipschitz manifold) of dimension n-3. But there is a problem in that 3.3(11) only guarantees

that b(p) is small when p is proportionally close to one of the pj, and, without further input, we cannot conclude very much about the structure of singe u from this-see the discussion in 3.4 below. We conclude this section with the simple proof of 3.3(iii). We assume Y = 0.

Proof of 3.3(111). Let p < po and Z = ({, n) E sing. u r Bp/2(0). Take o = Qolti, with 60 < 2 to be chosen. By the Schoen-Uhlenbeck regularity theorem there is co = eo(n, N, A) > 0 such that

f0:5 O-" f.,(Z) lu - w(Z)I2 B

(1)

<20-" 4.

IU _ p12 + 2a" f B,(Z)

(Z)

I'p-'(Z)l2.

By virtue of 3.2(iii) we know that IDWo(x)I < CIxI-I, where C depends only on N, A, and hence IV(X) -,p(Z)j:5 CI{I-lo < COO for X E B0(Z), where C depends only on N, A. Then (1) gives

c0:5 2f3 "ItI-" f

Iu -'Pl2 +

C'002.

Bn(0)

Then selecting C#02:5 Zeo and multiplying through by ItI", we have

Itl" < C (p"

j

Iu -

I2

) p" ,

P(0)

with C depending only on n, N, A. Taking nth roots of each side, we then get the required inequality.

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

210

3.4 Consequences of Uniqueness of Tangent Maps We want to show here that the geometric picture established in the previous section does give good information about the structure of the top dimensional part

sing, u if the tangent map at each point Y E sing, u is unique; because then for each 6 > 0 and each Y E sing, u there is W as above, an orthogonal transformation Q of R" and a py,6 > 0 such that 3.3(iii) holds for all p:5 py,6i with Q independent of p. We claim that such a property implies that sing, u is contained in a countable union of (n - 3)-dimensional Lipschitz graphs: To be precise, we could apply the case j = n - 3 of the following lemma:

Lemma. Let j E (1,... , n - 1). Suppose 6 E (0, ] and A is a subset of R" such that at each point Y E A there is a j-dimensional2 subspace Ly of R" and py > 0 such that *

AnBB(Y)c{X : dist(AnB,(Y),Y+Ly))<Sp}

dp
then A C U°_IE where each Ei is the finite union of graphs of Lipschitz functions, each the graph over some j-dimensional subspace (in the sense that there is an open subset Ui of some j-dimensional subspace Li C R" and an L, -valued Lipschitz

function fi on Ui such that Ei = {X + fi(X) : X E Ui}). Remark. In standard terminology, this says that A is countably j-rectifiable. Proof. We decompose A = U901Ai, where Ai+1 C Ai is the set of points Y E A

such that * holds with py = i-1. Notice that then * holds for all Y E Ai with py = i-1, and Ai satisfies a uniform cone condition-in the sense that Ai n Bi-1(Y) c Ky,

V Y E Ai,

where Ky is the cone given by Ky = {X : dist(X, Y+Ly) <5X-Y}. Now select j-dimensional subspaces L1i ... , Lq of R" such that for each j-dimensional subspace L C R" there is one of the L, such that IIL, - LIP = SUPXELnsn-I dist(x, L, n Sn-') < (1 - 6)/2. Then we can decompose A, = UQ where

Ai,,={YEA1 : IILy - L,II < (1-b)/2}. Then each Ai,, has the uniform cone property that

A1.,nBi-1(Y)CY+K,,

VYEAij,

where K. = {X : dist(X,L,) < (1 +6)IXI/2}. It is standard that such a uniform cone condition implies that, for each given Y E Ai,,, Ai,, nBi-, (Y) is contained the graph of a Lipschitz function with domain Bi-, (Y')nL,, where Y' is the orthogonal projection of Y on L,. (See e.g. [SL3,§5].) The lemma is thus proved. Notice also that the argument above actually shows that the following is true:

LECTURE 3. THE TOP-DIMENSIONAL PART OF SING u

211

Corollary. If A satisfies the same hypotheses as in the lemma, except that we can choose py m pa, with pa > 0 independent of Y, then Aft $,,(Y) is contained in the finite union of j-dimensional Lipschitz graphs for each Y E A. Notice that, by 3.3(iii), such a uniform choice of py can be made for sing, unK, K any compact subset of f2, provided that for each 6 > 0 and each compact K C 5l, there is p(6, K) E (0, dist(K, O t)) and a homogeneous degree zero spy : R" -+ N with sing cpy = Ly (= some (n - 3)-dimensional linear subspace) such that **

p"I

B,(Y)

Iu-cpy12<6, p
In fact by 3.3(iii) if ** holds with suitably small 6 = 6(n, N, K, A) > 0 (A is any upper bound for sup d2-" fB4(y) IDul2 over all Y E K with d E (0,dist(K,85t))), then we have the hypotheses of the above corollary with 6 = (for example) . Notice also that in this case it follows automatically from **, 3.3(iii), and thez triangle inequality that the hypotheses of the corollary hold not only with py independent of Y by also with Ly independent of Y, and hence we deduce the following: Corollary. There is 6 = 6(n, N, K, A) such that if for each Y E sing* u n K there is ipy such that ** holds, then there is p > 0 such that sing. u n BB(Y) is contained in an (n - 3)-dimensional Lipschitz graph for each Y E sing. u n K. We see in the next lecture that there are stronger conditions on the L2-norm which guarantee much stronger results in certain cases.

LECTURE 4 Recent Results Concerning sing u Recall that in the previous lecture we identified a "top dimensional part" sing. u of the singular set sing u. Here we further refine this set by defining

sing.u={YEsing. U : Notice that by 3.2(vii) of the previous lecture we know that if either dim N = 2 or if N is real-analytic then for each compact K C fI there is is a finite set F = {al, ... , aq} such that

K n sing. u= 0, a g .F. Thus

singu n K = (uQ 1sing., u) u ro,

with dim ro < n - 4; our aim is to give conditions which ensure that each sing, u has closure which is (n - 3)-rectifiable in a neighbourhood of each of its points.

4.1 Statement of Main Known Results If dim N = 2, we can show that each of the sets closure sing. u is locally (n - 3)rectifiable (i.e. countably (n - 3)-rectifiable with locally finite (n - 3)-dimensional Hausdorff measure (so that for each Y E closure sing. u there is o > 0 such that 7in-3(B,(Y)nclosuresing. u) < oo). As a matter of fact, a bit more can be proved:

Theorem 1. If dim N = 2, then sing u is countably (n - 3)-rectifiable, and, for each a > 0, S. is (n - 3)-rectifiable in a neighbourhood of each of its points, where

S. = {X E sing u

:

0.(X) > a} (D closure sing. u). (S. is closed by upper

semicontinuity 1.5(ii).) Furthermore, if N = S2 with its standard metric, or if N is S2 with a metric which is sufficiently close to the standard metric of S2 in the C3 sense, then sing u can be written as the disjoint union of a properly embedded (n - 3)-dimensional C'4'-

manifold and a closed set S with dim S < n - 4. If n = 4 then S is discrete and 213

214

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

the CI.µ curves making up the rest of the singular set have locally finite length in compact subsets of ft.

Remark. In case n = 4 and N = S2, Hardt & Lin [HL] have proved, by different methods than those to be described in these lectures, that the singular set is a union of arcs with endpoints forming a discrete set. In case dim N > 3 we unfortunately can only get information about the part of the set sing. u consisting of points Y E sing. u such that all tangent maps ap of u at Y with dim S(W) = n - 3 have the following "integrability property", in which we assume that we have made an orthogonal transformation of the X variables to ensure, as in 3.2(i), that cp(x, y) = apo(x). Then we require:

t

the condition 3.2(vi) holds with apo in place of 'o.

Unfortunately this integrability condition is not always satisfied in case dim N > 3, so the following theorem in general fails to establish rectifiability of the entire singular set, even if N is real analytic.2

Theorem 2. Suppose dim N > 3 and suppose that the integrability property t holds for all tangent maps ap of u with dim S(ap) = n - 3. Then sing u is countably

(n - 3)-rectifiable, and the set S. = {X : Au(X) > a} has finite measure in a neighbourhood of each of its points for each a > 0. We want to give some brief indications of the kinds of techniques which are needed to prove such results. Without explaining the terminology (for which we refer to [SL4]), we want to mention here again that there are analogous results (with analogous techniques of proof) for minimal surfaces. For example, the following theorem about mod 2 minimizing surfaces is proved in [SL4]:

Theorem. If M is an n-dimensional mod 2 minimizing current in an open subset f2 of some (n + k)-dimensional smooth Riemannian manifold, and if fI has zero mod 2 boundary in 0, then the singular set sing M of M is (n - 2)-rectifiable, and can be decomposed sing M = u, oS,, where W 2(So) = 0 and where each Sj, j ? 1 has locally finite 9{"-2-measure.

4.2 Preliminary Remarks on the Method of Proof "Blowing Up" We initially suppose 0 E singu and work in balls B9(0). To begin, we note that the argument used in 3.3 to prove that singu n Bp/2(0)) is contained in a (6p)-neighbourhood of {0} x R"-3, where 6 = C(p-" fBPio? [u - apl2)1/", assuming

that S(W) = {0} x R"-3, actually gives more information. Namely, we can use that argument together with the regularity theorem 1.6 and the inequality 3.2(iii) in order to deduce that for any eo small enough (depending only on n, N, A) we have for each t > 0 4.2(i)

ptIDt(u - ap)I < Cr eo in B,/2 (0)\{X : dist(X, {0} x R"-3) < bp}

21n the meantime, this integrability condition $ has been shown to be unnecessary in case the target N is real analytic; see (SL6]

LECTURE 4. RECENT RESULTS CONCERNING SING u

215

W > 0, where b = C(eo'p " fBp(o) lu - w12)I/n, with C depending only on n, N, A, provided 6 is small enough depending on n, N, A. Thus W smoothly approximates u away (at least distance bp) from the singular axis {0} x Rn-3

Now analogous to the discussion of the linearized operator L. on S2 (in the previous lecture), we can also discuss linearizing the harmonic map operator M(u) _ Au + E" I Au(Diu, Diu) at gyp: 4.2(ii)

whenever 41/, is a family of smooth maps of BR, (0)\{X : dist(X, {0} x Rn-3) < 6,), varying smoothly in both X, s, where R, j oo and 6, 10 as IsI 10, with tfio = cp, and where v = To p. 1,=0. Notice that then v(X) E Tp(X)N for each X E R"\{0} x Rn. Then in view of the definition 4.2(ii) and the inequalities 4.2(i) (with B = 0, 1, 2)

it is clear that the difference u - W satisfies, in the "good" region B.,2(0)\{X dist(X, {0} x R"-3) < by}, an equation of the form C,p(u - w)T = E,

4.2(iii)

where vT(X) means orthogonal projection of v onto T,(X)N, where IEI < C(ID(u(p)I2+p 2Iu-WI2) < Ceo on {X E BP/2(0) : dist(X, {0} x Rn-3) 2!6p), and where C depends only on n, N, A. In view of the fact that IEI is small relative to u - gyp, this suggests that we should try to approximate u -,p by solutions I(i of the linear equation t, = 0; this is essentially the idea of "blowing up", going back to De Giorgi (in his work on oriented boundaries of least area).

In fact our initial aim is to show that such an approximation can be made provided we have the following "no 6-gaps at radius p" hypothesis for the part S+ = {X E singu : 9u(X) > 9,p(0)} of the singular set: 4.2(iv)

({0} x Rn-3) n Bp(0) C UZES+B6p(Z)

notice in particular that if this hypothesis fails (for a given 6 > 0), then there is a point Yo = (0, yo) E ({0} x R"-3) n Bp(0) such that S+ n B6p(Yo) = 0. Thus, if IYoI < p/2 and if p -n f8P(o) Iu - w12 < e2 with a small enough (depending only on 6, n, N, A), then by 3.3(iii) we have 4.2(v)

S+ n p I(B6P(Yo)) n BP/2(0) = 0,

where p is the orthogonal projection of R" onto {0} x R"-3. This explains the terminology "no 6-gaps at radius p". (We explain how to handle the alternative when there are 6-gaps at radius p as in 4.2(v) later.) Subject to the above no 6-gaps hypothesis 4.2(iv), it is in fact possible to prove the approximation 4.2(vi)

(Bp)-"

I

B, (o)

Iu -'P -

I2

4p n fBP(o) Ju -'PI2,

216

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

where 0 is fixed E (0, 2), and where 10 is a solution of L = 0 of very explicit (and controlled) type-we show that such an inequality holds with i/' of the special form n-3 2

V,(x, y) = E E aijgp'Diwwo(x) +'d'o(t), j=1 i=1

where ail are constants, w = IxI-1x E S2, where i

is a smooth solution on 52 of

the equation G,,sk = 0 as in 3.2, and where Ia,sl+supS2 I1GoI <- Cp-n fB,(o) An outline of the methods needed to establish such an approximation are described below.

Iu-W12_

In any case, notice first that the information in 4.2(i), (ii), (iii) could not possibly be sufficient to establish such an approximation, because 4.2(i), (ii), (iii) say nothing about the behaviour of u - cp in the "bad" region {X E Bp12(0) dist(X, {0} x R11-3) < by). So we need information about the behaviour of u - cp in this region. For example, we at least need to know that the contribution to the quantity p-n fBo(o) Iu - cdI2 from the bad region is relatively small. This motivates the discussion of L2-estimates in the next section.

4.3 L2 Estimates We continue to assume that cp is a homogeneous degree zero energy minimizing map from Rn to N with S(cp) = {0} x Rn-3, and u : fZ -+ N is energy minimizing. Later we assume that in fact SP (modulo composition with a rotation of the X-

variables) is one of the tangent maps of u at a point Y E sing. u, but for the moment ,p, u have no particular connection, and Y is an arbitrary point of 12. We d e f i n e no to be a bounded subset of R n s u c h t h a t ? Q C fl, and A, d are positive constants such that d < dist(53o, Of) with d2-n ffl. IDuI2 < A. (Notice that then by the monotonicity formula we have p2-n fB,(y) IDuI2 < A for any Y E Sto and any p E (0, d).) The main L2-estimate is the following:

Lemma 1. If ao(Z) C no, if 0 E (0,1), and if e (Z) > ea(o), then p2-n

f

BOP(Z)

12

IDvuI2 + J e v(Z)

R22 n 18 < Cp n 1

Bp(z)

lu - wz I2,

where WZ(X) =W(X+Z), X E Rn, Rz = IX-ZI, a/ORz = and C depends only on n, N, A. We give the proof of this lemma at the end of this section, but for the moment we record some corollaries.

Corollary 1. Under the same hypotheses, for any a E (0,1), Iu - pzI2 4.,.(Z)

W°

< cpn+° /

JBo(Z)

I.u - PzI2,

LECTURE 4. RECENT RESULTS CONCERNING SING u

217

where C depends only on n, N, A, a.

This corollary is a direct consequence Lemma 1 together with the calculus inequality P

J

R°-1 f2(R) dR< 0°

R°-1f2(R) dR+C° rP RI+°(f'(R))2 dR, I/2

0

valid for any bounded CI function on (0, p). We apply this to f (R) = u(Z+Rw) (pz(w), where w = IX - ZI-'(X - Z) E Sn-1.

Corollary 2. Under the same hypotheses, if Z = ({,7l) E B0 , and if Bp(0) C (Zo we have

p-2If,I2+p n r B, (Z)

Iu-wZl2
where C depends only on n, N, 0, A. This corollary is is easy to prove once we note that, by 3.2(iii) of Lecture 3 and a compactness argument, we have a fixed constant C, depending only on n, N, A,

such that frER3.I=1<1 IDap012 > CIaI2, for all a E R3. For the details we refer to JSL5J.

Corollary 3. There is bo E (0,1), depending only on n, N, such that if 6 E (0, 65J, i f BP(0) C 1 1 0 i and i f p n fBo(0) lu - W12 < 64, and if there are no 6-gaps at radius p (i. e., if 4.2(iv) above holds), then for each 0, a E (0,1)

p-n+1-°

Iu I1,12(0)

1

l2

< Cp-° J

r6P-Q

J B,,(0)

Iu -,pl2

where C depends only on n, N, A.

Remark. Notice that Corollary 3 implies that

p -n f

Iu - WI2 < Cal-°p`n /

J((x.y)EB1,2(0) : IZI
lu - cpl2,

B,(0)

for each v > 6, and hence (if we choose 6 small) we see that very little is contributed to the quantity p -n fB,(o) lu - wl2 from the region B,,/2(0) f) {(x, y) : I xl < 6p}.

Proof of Corollary 3: For o E (6p, p/4), we choose points Z1,... , Zq in Bp12(0) with eu(ZJ) > $,,(0) and such that any given point X ties in no more than C,, of the balls B,(Z,,), and such that (1)

{(x, y) E BP,2(0) : Ixl < a/2} C u,Q IB,(Z.,).

By virtue of the no 6-gaps hypothesis at radius p, this can evidently be done with (2)

Q5 C(P/a)n-3,

218

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

where C depends only on n. Now by virtue of Corollary 1 we have for each Z = Z.,

1Bev(z)

Iu - cpZI2 < R"-a

Cp-n

it, - SPZI2,

JBy/2(Z)

and hence by Corollary 2

(3) o-n+a JB.

Iu-VI2.

IU_-pZI2 < Cp n+a Js,2(Z) IU_,PZI2 < Cp n+a Bp(0)

(Z)

Notice also that

Iu-wPzI2S21u-wit+2IV-cPzl2,

(4)

and, by the same computation (based on the inequality 3.2(iii)) that we used in the proof of 3.3(iii), we know that

XEB,(Z),

ISP(X)-,Pz(X)I S CIxi-1o,

where C depends only on N, A, and hence (4) together with another application of Corollary 2 gives

lu - wz I2 < 2Iu _ WI2 + CIxI-2pn 1

Iu _
BP(0)

Therefore, by integrating this over B,(Z), we see that (3) yields o2-n+a

Iu -

w12 <

Cp2-n+a

Be(Z)

r

in - (PI2,

Bv(O)

where C depends only on N, A. Now summing over Z E { Z1, ... , Zq } and using (1)

and (2), we obtain

p n+l-aQ-I+a r

Iu - w12 < Cp n

lu - w12. BP(o)

t(x,y)EBp/2 (0) : lzl
This evidently implies the required inequality.

Notice that at one point in the above proof we "gave up" two powers of o; by being a little more careful, and noting that if Z = (r;, r}) then (by 3.2(iii))

=

3

f'DjVo(x) + E,

w(z, v) - wPz(x, v) = PO(x) - ,Po(x + t) j=1

with IEI < CIxI-2ICI2, we could have proved a more precise inequality. In fact (see again [SL5] for the detailed discussion):

LECTURE 4. RECENT RESULTS CONCERNING SING u

219

Corollary 4. Under the same hypotheses as Corollary 3,

- -

Jr

r6.

JB,,12 (o)

where Ia3 I <

y)D, o(x)I2

C'P-n fe,(a) Iu

-

Bo(o)

- WI2.

There is one final L2 estimate, which follows from the variational equation 1.3(ii) and from the part of Lemma 1 which bounds the L2-norm of IDyuI2, as follows:

Corollary 5. For each i = 1, 2, 3 let v, (r, y) = r f52 D,W(w) (u - rp)(rw, y). Then for each smooth function C E C°O({(r,y) E R x Rn-3 : r2 + IyI2 < p2}) with support in {(r, y) : r2 + IyI2 < 82p2} and with OC/8r = 0 in a neighbourhood of the (n - 3)-dimensional space r = 0, there is a constant C = CP,(,e,n,N,A such that n-3

(DrvjDrDye( + E Dy,viDy, Dye() I 3 j=I < C( 4p(0) Iu - 0I2)1/2(JBv(o) ID(u - ,)I2)112,

foreacht=1,...,n-3. Remark. In view of the fact that it is not necessary here that ( vanish in a neighbourhood of the axis r = 0, this says that the even extension of (p -n f8,(o) Iu (pI2)-I/2Dyevi is close (in a certain sense-see below) to being harmonic with re, yn-3 in the region {(r, y) E R x Rn-3 : r2 +IyI2 < spect to the variables r, y1, p2).

4.4 Special Solutions of the Linearized Equation Now recall that C,, is the linearized harmonic map operator at cp, where ,p(x, y) _- po(x) is as in the previous section. Notice that then, in the coordinates (x, y) _- (rw, y) with w = IxI 'ix E S2 and r = IxI, 4,v has the form n-3

4.4(i)

r'2 (r2

v/

+ F Dy, v + r'2CPOv, j=1

where C,,. is the linearized harmonic map operator on S2 as discussed in 3.2 of Lecture 3. As we discussed briefly in 4.2, we want to show that u - cp is well-approximated by special solutions of this, provided p-' fB,(o) Iu - VI2 is small enough. To see this the crucial ingredient is the following lemma concerning solutions of the linear

equation Cpv = 0 which satisfy L2 bounds analogous to those satisfied by u - o in 4.3 above:

220

L. SIMON, SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

Lemma. If v E L2(BI (0)) f) C2 (BI (0)\{0} x Rr-3) satisfies the equation G pv =

0 on BI(0)\{0} x R"-3, if, for some a E (0,1), $ > 0, v also satisfies the L2 inequalities 3

r JB,12(0)

I- E a2(r,y)Dio(x)I2 <,0(J j=I

B1/s(0)

IvI2)2,

B1(0)

with sup jail -< 13(fs,(o) Iv12)1/2, and if each vi(r,y) _ fs

v(rw,y) dw

satisfies n-3

(D,.viD,Dye + E Dyj viDyj Dye() = 0 fmoo)xR-3

j=1

for any C°° -function C on {(t, y) E R x R"-3} with C having compact support in t2 + IyI2 < 1, then

= v(rw, y)

3 n-3

E E aij1/o D.,(po(x) +'IGo(w) + E, i=1 j=I

(o)1E12)1/2 < C(fB,(o) lvl2)1/2pµ for p E (0, 2), where p E (0, 1) dewhere (p-" f,,,(.) B JE12)1/2 pends on n, N, A, and where a,j are constants and ,00 is a solution of C,,Oo = 0 (as

in3.2)onS'2 with Iaijl+supsa1001


Here C=C(a,p,n,N,A).

Remark. Notice that vi automatically satisfies the strong form of * on {(r, y) : r2 + IyI2 < 1, r > 0}, as one easily checks by direct computation based on 4.4(i) and the facts that Dicpo(rw) = r-I Dicpo(w) and that DiVo(w) is a solution of G,,(v) = 0 on S2 (see 3.2(v)). The point about * is that C need not vanish along r = 0, so that the solution vi satisfies 82vi/OrOy' = 0 along r = 0 in the weak sense, hence in the classical sense.

4.5 Brief Sketch of the Proof of the Results of 4.1 We here only discuss the case when the integrability condition t holds at each point of sing. u (as for example in the case dim N = 2). Only minor modifications

are needed to prove the more general case stated in Theorem 2. By virtue of the L2-estimates of 4.3 and the above lemma, it is now quite easy to check the approximation result 4.2(vi), provided b is chosen sufficiently small (depending only on n, N, A) and provided p-" f B"(Y) lu - W12 < e2 with e sufficiently small compared with b (e.g. e < 63 with b sufficiently small); the details of this can be found in [SL5].

On the other hand the solutions V D=. , (x) are generated (analogous to the solutions in 3.2(iv), (v)) by the 1-parameter family W(e"A), where A is the skewsymmetric matrix such that esA is a rotation by angle s in the yI-xi-plane, while the integrablilty hypothesis t guarantees that there is a 1-parameter family gyp, of

LECTURE 4. RECENT RESULTS CONCERNING SING u

221

smooth harmonic maps of S2 into N with initial velocity tb . Then we deduce that corresponding to the linear solution 10 in 4.2(vi) there is such a

0, a E (0, 1) depend only on n, N, A, provided e is small enough, depending only on n, N, A, 8, 6. Then it follows from 4.2(vi) (assuming the integrability condition and assuming e, 6, 0 small enough) that 4.5(i)

(ep)-" fBe(Y) Iu -'vI2 <

2P_n

j

In o(Y)

Notice that by the triangle inequality in 4.5(i) we get III - WIIL2(B,(o)) < Cp ' jP(Y) Iu - pI2.

4.5(ii)

Provided there are no 6-gaps at the new radius Op, the inequality 4.5(i) can now be iterated.

Remark. Actually a technical point here is that the new p- need not be minimizing, although it is harmonic and has distance (in the L2-sense) less that Ce from the original gyp, and hence it can be shown that all the discussion of 3.2 and 4.3 applies with such a ip in place of the minimizer gyp; we should therefore duplicate the entire discussion above starting with a homogeneous harmonic w which lies in a suitably small neighbourhood of a minimizer and which (modulo a rotation) has W(x, y) cpo(x), with V homogeneous degree zero harmonic and sing cp = {0} x RI-3. Since the inequalities 4.5(i), (ii) ensure that we always remain close to the same minimizes, this point turns out to cause no serious difficulty. Notice in particular that the above argument only used the equation 1.3(i)' for v, and not 1.3(ii). Implementing such an iterative procedure, based on 4.5(i), 4.5(ii), then gives a homogeneous cp such that 4.5(iii)

a_n

fee (Y)

In - V.- 12 <

C(a/P)°P-" j P(Y) Iu - ,pI2, 0 < a < p

provided there are no 6-gaps at any radius a = B.'p, where IJW - W112 (B,(0)) < Cp ' fBo(y) In - cpl2. Notice that we continue to assume in this discussion that Y is an arbitrary point of Cl with W,(Y) C Cl; 4.5(iii) shows that we must encounter 6-gaps at some radius if either Y 0 sing,u or if eu(Y) 96 0,,(0), because 4.5(iii) guarantees in particular that W' is the unique tangent map of u at Y. Now at the points Y where 4.5(iii) fails, we must have a 6-gap in S+ at either the initial radius p, or else at some a = 8'p; notice that in the latter case we can select the smallest such j, so that 4.5(iii) still holds at least for 9)p < a < p.

L. SIMON. SINGULARITIES OF GEOMETRIC VARIATIONAL PROBLEMS

222

Keeping in mind that p-" fB p(y) Ju - ,,I2 < E2 implies that p-n

Iu -

w12 < CE2 for any Z E Bp12(Y), the rest of the proof involves using the above alter-

natives to show that (assuming the ball Bp(Y) is a sufficiently small neighbourhood of some initial point Zo at which u has a tangent map ,p(°) with dim S(,p(°)) = n - 3

and 9,p0)(0) = 9,,(0)) we can decompose S+ = B,(Y) fl {Z : eu(Z) > o m) into a part which is contained in an embedded CI," manifold and a part which can be covered by a countable collection of balls B,,, (Yk) with Ek °k 3 < (10),;n-3, and such that a similar decomposition can be made starting with any of the balls B,,, (A;) in place of Bp(Y). Here fl is a fixed constant E (0,1), and we need to take e small enough depending only on n, N, A. Thus after j iterations we get we get that S+ is contained in the union of j embedded CI.v manifolds together with a set which is covered by a family of balls B,,,(Yk) with Eko'k-3 < (1 - Q).pn-3. Thus S is contained in the countable union of C14 manifolds together with a set of (n - 3)-measure zero. This iterative procedure has the additional property that it controls the sum of the measures of the embedded manifolds, thus giving the local finiteness result stated in the theorems of 4.1. Finally to prove the additional conclusions of Theorem 1 in case N is S2 or metrically sufficiently close to S2, we need to use a result of Brezis, Coron, & Lieb [BCL] which asserts that a homogeneous minimizer ' from R3 into the standard S2 is such that 0192 is an orthogonal transformation. This result is easily seen (for topological reasons) to imply that there we can encounter no 6-gaps (at any radius) in the iterative argument described above, and hence we obtain an inequality like 4.5(iii) uniformly for non-isolated points Y of S+, assuming that Y E Bpo/2(Y°), where po" fB'.'(Y0) lu -,p(°)I2 is sufficiently small, where ,p(°) with dim S((°)) _ n - 3 as above. This evidently implies the stated Cl," property of S+.

The fact that the exceptional set sing u\sing. u is discrete in the case n = 4 involves a simple scaling and compactness argument together with several applica-

tions of the result in the previous paragraph. The details are given in (SL5] (see [HL]).

Acknowledgments. Partially supported by NSF grant DMS-9207704 at Stanford University. It is a pleasure to thank Tatiana Toro for her invaluable assistance in the preparation and correction of these notes.

References (AW] (A]

[BCL]

W. Allard, On the first variation of a varifold, Annals of Math. 95 (1972), 417-491. F. Almgren, Q-valued functions minimizing Dirichlet's integral and the regularity of of area minimizing rectifiable currents up to codimension two, Preprint. H. Brezis, J.-M. Coron, & E. Lieb, Harmonic maps with defects, Comm. Math. Physics 107 (1986), 82-100.

LECTURE 4. RECENT RESULTS CONCERNING SING u

223

[DeG]

E. De Giorgi, Frontiers orientate di misura minima, Sem. Mat. Scuola

[GT]

Norm. Sup. Pisa (1961), 1-56. D. Gilbarg & N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1983.

[G]

E. Giusti, Minimal surfaces and functions of bounded variation, Birk-

[HL]

[HF]

[JJJ

hauser, Basel, Boston, 1984. R. Hardt & F: H. Lin, The singular set of an energy minimizing harmonic map from B4 to S2, Mansucripta Math. 69 (1990), 275-287. F. H61ein, Reularite des applications faiblement harmoniques entre une surface et une varietee Riemannienne, C. R. Acad. Sci. Paris Ser. I Math. 312 (1991), 591-596.

J. Jost, Harmonic maps between Riemannian manifolds, Proceedings of the Centre for Mathematical Analysis, Australian National University 3 (1984).

[Luck1] S. Luckhaus, Partial Holder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988), 349-367. [Luck2] S. Luckhaus, Convergence of Minimizers for the p-Dirichlet Integral, To appear, 1991. [MCB] C. B. Morrey, Multiple integrals in the calculus of variations, Springer Verlag, 1966. [RJ

[RT1] [RT2] [SS]

[SU)

[SL1]

R. E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta. Math. 104 (1960), 1-92. T. Riviere, Everywhere disconinuous harmonic maps from B3 to S2, C. R. Acad. Sci. Paris Ser. I Math. 314 (1992), 719-723. T. Riviere, Axially symmetric harmonic maps from B3 to S2 having a line of singularities, C. R. Acad. Sci. Paris Ser. I Math. 313 (1991), 583-587. R. Schoen & L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), 741-797. R. Schoen & K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307-336. L. Simon, Proof of the basic regularity theorem for harmonic maps, this volume.

[SL2] [SL3] [SL4] [SL5] [SL6]

[WB]

L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals of Math. 118 (1983), 525-572. L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University 3 (1983). L. Simon, Cylindrical tangent cones and the singular set of minimal submanifolds, Journal of Differential Geom. 38 (1993), 585-652. L. Simon, On the singularities of harmonic maps, in preparation. L. Simon, Rectifiability of the singular set of energy minimizing maps, Calculus of Variations and PDE 3 (1995), 1-65. B. White, Preprint.

Proof of the Basic Regularity Theorem for Harmonic Maps Leon Simon

IAS/Park City Mathematics Series Volume 2, 1996

Proof of the Basic Regularity Theorem for Harmonic Maps Leon Simon LECTURE I Analytic Preliminaries This second series of lectures is meant as an elementary introduction to the basic regularity theorem for harmonic maps, which we used frequently in the first lecture series [SL1]. We deal here with energy-minimizing maps rather than minimal surfaces, because the basic regularity theory for these (while being very analogous to the theory for minimal surfaces) is technically simpler than the corresponding theorems for minimal surfaces. Interestingly enough though, the regularity theory for harmonic maps into compact Riemannian manifolds was not established until much later than the basic regularity theory for minimal surfaces. (The work of Giaquinta-Giusti and Schoen-Uhlenbeck was done in the early 1980's, whereas the regularity theorem for area minimizing hypersurfaces was proved by De Giorgi [DG] in the early 1960's.) The explanation for this (apart from the difficulty in proving the energy inequality discussed in Lecture 3 below) is perhaps that the close parallel between regularity theory for area-minimizing surfaces and regularity theory for energy minimizing maps (e.g. monotonicity and harmonic approximation as a key ingredients in the proof of the main regularity lemma) was not widely appreciated until the work of Schoen-Uhlenbeck [SU] and Giaquinta-Giusti [GG]. The plan of this series of 4 lectures is as follows: Lecture 1: Analytic Preliminaries. Lecture 2: Proof of the Schoen-Uhlenbeck theorem modulo a "reverse Poincare" inequality.

Lecture 3: Proof of the reverse Poincar6 inequality used in Lecture 2. Lecture 4: Higher regularity and other mopping-up operations. I Mathematics Department, Stanford University, Stanford, CA 94305

E-mail address: 1msigauss . stanf ord. edu © 1996 American hint hemaUral Sor,rty 227

228

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

1.1 Holder Continuous Functions Recall that ifflC R" is open and if aE (0, 11, we saythatu :

fl -Ris

uniformly Holder continuous with exponent a on Nl (written u E C°,°(St ), if there

is a constant C such that Iu(X) - u(Y)l < CIX - YI° for every X,Y E fl. There are various reasons why Holder continuity turns out to be so important in geometric analysis and PDE. We mention two reasons here: (1) (Scaling.) Notice that if Iu(X) -u(Y)I < 1IX -YI° for every X,Y E fl and

if for given R > 0 we define the scaled function u(X) = R-°u(RX) for X E Sl a {R-'Y : Y E fl}, then II (X) -u(Y)I <,QIX -YI° for every X,Y E Q. (ii) (Dyadic decay implies Holder continuity.) If u : B2R(Xo) -, R and if there is a fixed 0 E (a, 1) such that oscBo/,(y) u < 00SCBP(y) u (oscA U = supx,yEA Iu(x) -

u(y)l) for every Y E BR(Xo) and every p < R, then u E C°"°(BR(Xo)) with a = 1og2(1/0), where 1092 means the base 2 logarithm. Indeed by iteration the given inequality implies oscB,,,i(y) U < 0' 08CBP(y) U (2-j)lo 2(1/s) 05CB,,(Y) U The following result of Campanato will be needed in the sequel:

Lemma (Campanato). Suppose u E L2(B2R(Xo)), a E (0,1), fi> 0 and *

inf p -n AER

r JB0(y)

Iu - AI2 <

a2(P/R)2a`

for every ball BP(Y) such that Y E BR(Xo) and p:5 R. Then there is a Holder continuous representative u for the L2-class of u with

Iz(X) -1(Y)I < C$(IX -YI/R)°, V X,Y E BR(Xo), where C depends only on n, a. Remarks: Notice that, since fBB(y) Iu-AI2 = fBo(y)(u2-2Au+A2), it is easy to check that the infmum on the left of * is attained when A has the average value AyP = (w"p")-1 fB,(Y) u, where w" is the volume of the unit ball in R". It will be convenient to use this fact from time to time in what follows.

Proof of the lemma. First note that (1)

(p/2) -n JP/z(Y) Iu- AyPI2 < 2np"JB'(Y) IU

- AyP12 < 2"32(P/R)24,

where Ay,,, is the average of u over BP(Y) as in the above remark. Using the given inequality with p/2 in place of p, we also have (P/2)_"

(2)

IB,/2(y) IU - AyP/212 < Q2(P/R)2°

LECTURE 1. ANALYTIC PRELIMINARIES

229

Adding (1) and (2) and using the squared triangle inequality Ia - b12 < 21a - c12 + 21b - c12, we conclude that (3)

I AYp - AYp/21 <- 2"+2Wn

i/2Q(P/R)a

provided that p!5 R and Y E BR(XO). Now for any integer v E {0, 1, 2... } we can choose p = 2-1R, whereupon (3) gives 2"+2Q2-"a

IAY,R/2"+i - AYR/2"I <

(4)

Since 2-11 is the with term of a geometric series, we see that the series a defined by

a = E o(AYR/2.+t - AYR/2..) is absolutely convergent. But the jth partial sum aj is just AY2-1R - AY,R, so we havelimv--AY,2-.R exists, and we denote this limit by Ay. Since AY,2-"R - AY = - Ejt "(AY,R/2l+i - 4R/2,) we see I1\Y2-'R - AYI 5 C,62-'

(5)

,

where C depends only on n, a. Then combining (1) (with p = 2-"R) with (5) and using the squared triangle inequality again, we conclude P

(6)

"I

lu - Ay12 < C/3(P/R)2a,

Bo(Y)

for p = 2-1R, v = 0,1, .... On the other hand for any p E (0, R] there is an integer v > 0 such that 2-"-'R < p < 2-"R, and it evidently follows (since B2-..R(Y) D Bp(Y) for such v) that (6) holds (with 2'C in place of C) for every

p
Now take any pair of points Y, Z E BR(Xo) with IY - ZI < R/4, and apply (6) with p = 21Y - ZI on balls with centers Y, Z and add the resultant inequalities. Since Bp12((Y + Z)/2) C BB(Y) n Bp(Z) this gives

p--

(Iu - Ay12 + Iu - AZ12) < 2CQ(P/R)2a By/2((Y+Z)/2)

Since 1Ay - AZ12 < 21u - Ay12 + 21u - AZI2, this gives (7)

IAy - AZI < 2C/3(P/R)' = 2CQ22+a(IY - ZI/R)a.

Now for any pair of points Y, Z E BR(Xo) we can pick points Y = Zi, ... , Zs = Z on the line segment joining Y, Z such that 1Z1- Zi+11 <- R/4, and applying (7) to each of the pairs Z1, Z;+I and adding, we deduce finally that (8)

IAY - AZ1 < CO (IY - ZI/R)a,

VY, Z E BR(XO)

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

230

On the other hand by letting p j 0 in (6) and using the Lebesgue lemma, we have Ay = u(Y) for almost all Y E BR(Xo), so the proof is complete (because then 11(Y) _- Ay is a representative for the L2 class of u which by (8) satisfies the required Holder estimate.)

1.2 Functions with L2 Gradient Recall that if u E L2(fl), then we say that Du = (DIu,...

,

is an L2

gradient for u in fl if Diu are L2(fl) and if 1.2(i)

L

where, here and subsequently, CEO denotes the set of COO functions with compact support in fl. Of course such functions D,u as in 1.2(i) are uniquely determined by u if they exist at all; further if u E C' (I2) then such an identity holds by integration by parts, so this notion of L2 gradient really does generalize the classical notion of partial derivatives.

So from now on, when we say Du E L2 we mean that u E L2 and that there are DIu,... , D,,u E L2 such that 1.2(i) holds. In this section we present some of the important facts about such functions. Lemma 1 (Rellich Compactness Lemma). Suppose uk is a sequence of L2 functions on a ball BR(Xo) with Duk E L2 for each k and

limsup k-C.0

(Juk12 + IDukI2) < cc. BR(X0)

Then there is a subsequence uk' and u E L2(BR(Xo)) with Du E L2(BR(Xo)) such that lim fBR(XO) Juk, - U12 = 0, Duk' Du (weak convergence in L2(BR(XO))), and fBR(XO) IDu12 < liminfk'-°°fBR(XQ) IDuk'I2. For a proof see e.g. [GT Chapter 7). We shall also need the following Poincare Inequality:

Lemma 2. If Du E L2(BR(Xo)), then R_n / /BR(Xo)

Iu - Aft < CR2-" f

IDu12,

BR(X0)

where A = fBR(xO) u and C is a constant depending only on n; here and subsequently w denotes the measure of the unit ball in R°. There are various proofs of the Poincare inequality-one nice way to prove it is to use the above Rellich compactness theorem. (By a translation and change

of scale, it suffices to prove the lemma with R = 1 and X0 = 0, subject to the assumption fB, (0) u = 0. Then if the theorem f a i l s with C = k, k = 1, 2, ... , there

LECTURE 1. ANALYTIC PRELIMINARIES

231

is a sequence uk with f B, (o) IDuk I2 -+ 0, f B1 (o) Uk = 0, and fB, (o) uk = 1 for each k; now apply Rellich to get a contradiction.) Finally we want to state and prove Morrey's lemma.

Lemma 3 (Morrey's Lemma). Suppose Du E L2(BR(Xo)), 0 > 0 is a constant, and p2-n /

B1(Y)

IDuI2 < #2(p/R)2.,

dY E BR/2(Xo), P E (0, R/2).

Then u E Co"(BR12(Xo)), and in fact VX,Y E BRI2(Xo)

Iu(X) - u(Y)I s C13(IX - YI/R)°,

Proof. Let Ay,,, = (wp")-1 fB,(y) u. The Poincare inequality gives

p -n f (Y)

Iu

_,\Y",12

Cp2(p/R)2o

<-

for each Y E BR/2(Xo) and each p E (0, R/2). Using the Campanato lemma of §1.1, we then have the required result.

1.3 Harmonic Functions Recall that a COO function u on a domain fl C R" is said to be harmonic if Au =

0, where Au a Ej=1 D,D.,u, with Dj = e ,, in classical notation. Thus we can write Du = div(Du), where Du = (Diu, ... , D"u) as usual denotes the gradient of u, and div 0 means E, D? 40 for any smooth vector function 0 = ($1, ... , 0"). If we choose a ball B = BR(Xo) with closure contained in St, and if we integrate

the identity div Du = 0 over B and apply the divergence theorem, then we can check rather easily (see e.g. [GT] for the details) that harmonic functions have the mean-value property 1.3(i)

u(Xo) _ (wnR")-i

U

B R(XO)

for any such ball BR(Xo). This property is quite fundamental; for example using it together with the fact that harmonicity of u trivially implies harmonicity of each partial derivative (of any order), one can easily check (see e.g. [GT] for the details)

that 1.3(ii)

sup BR/2(X0)

\ 1/2

R' ID'uI < C'+1 j! ( R-" J

\

where C is a constant depending only on n.

BR(XO)

u2/

,

V j = 0, 1, 2,... ,

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

232

1.4 Weakly Harmonic Functions If Du E L2(B), where B = BR(XO) is an arbitrary ball in R", we say that u is weakly harmonic on the ball if

L1=0

1.4(i)

°dVEC,,(B).

Notice this formally generalizes the notion of smooth harmonic, because if u is smooth harmonic then we could integrate by parts in the identity f Du 'p = 0, thus establishing 1.4(i), and, conversely, if u E C2(B), 1.4(i) evidently implies Au = 0 by virtue of the arbitrariness of ,p. In fact H. Weyl proved that the two notions (weakly harmonic and classical harmonic) are the same: Lemma (H. Weyl). Suppose Du E L2(B) and that 1.4(i) holds. Then the L2 class of u has a COO representative which is harmonic.

We shall not give the proof, but we do want to point out that the proof is elementary, based on smoothing and the mean-value property 1.3(i) of the previous section. (See e.g. [GT].)

1.5 Harmonic Approximation Lemma The following harmonic approximation (or "blow up") lemma will be of fundamental importance:

Lemma (Harmonic Approximation Lemma). Let B = B1(0), the open ball of ra-

dius 1 and center 0 in R". For each e > 0 there is 6 = 6(n, e) > 0 such that if Df E L2(B), fB IDf 12 < 1, and

If Df Dol <6sup IDcI,

V E C°°(B),

B

then there is a harmonic u on B such that fB IDuI2 < 1 and If-u12<e2. JB

Proof. If this fails for some e > 0, then there is a sequence fk such that Dfk E L2(B), fB I DfkI2 < 1, (1)

I fDfk . DPI < k-1 sup ID,I B

for each co E CO (B), and such that Ifk-u12>e2

LECTURE 1. ANALYTIC PRELIMINARIES

233

for every harmonic u on B with fB IDU12 < 1. Notice that since the same holds with fk = fk - Ak for any choice of constants

Ak, we can assume without loss of generality that fB fk = 0 for each k. But then by the Poincare inequality of 1.2 we conclude

limsup(IfkI2 + IDfkI2) < 00, k-oo

J

B

and hence by the Rellich compactness theorem of 1.2 we have a subsequence fk' and an f with Df E L2 on B such that (3)

lim I If - fk, I2 = 0 B

and Dfk' Df (weak convergence in L2(B)). But using this weak convergence in (1) we deduce that fB D f Dcp = 0 for each V E C,°(B), so that f is weakly harmonic on B, and Weyl's lemma (in 1.4 above) guarantees that f is smooth harmonic on B, and hence (since fB ID f I2 < lim inf fB ID/k' 12 < 1) we see that (3) contradicts (2).

LECTURE 2 A General Regularity Lemma Here we shall prove a general regularity lemma which establishes the SchoenUhlenbeck regularity theorem modulo the proof of a "reverse Poincar6" inequality, the proof of which we shall give in Lecture 3.

2.1. Statement of Main Regularity Lemma and Remarks Lemma i (Regularity lemma). Suppose u = (u',... , uP) : BR(Xo) -, RP, and a E (0,1), ,l3 > 1 are given, with Du E L2(BR(Xo)). Suppose further that u satisfies the equation

Au = F

*

weakly in BR(Xo),

where F is a measurable function satisfying

IFI < ,IDuI2 on BR(Xo),

(a)

and that (P/2)2_n

(b)

J

aP n f

IDuI2 n/2(Y)

Iu - ayl2 (Y)

whenever Y E BR12(Xo) and p< R/2. Then there is bo = bo(n, a,#) E (0,1) such that the inequality R_n fBR(XO) Iu ,xo,RI2 < 602 implies that u is Holder continuous with exponent a on BR/2(Xo) and

Iu(X)-u(Y)I < C(IX-YI/R)°(R-nf BR(XO)IU_AXO,RI2)'/2, where C is a constant depending only on n, a, P. 235

X,Y E BR/2(Xo),

236

L. SIMON. BASIC REGULARITY THEOREM FOR HARMONIC MAPS

Remarks: (1) Notice that Au = F weakly means that

J

Du.Dip=-J R(XO)

BR(XO)

(Here Du Dco = Es I E

=I D;uiD;90.) (2) If N is a smooth compact manifold embedded in RP, and if u : BR(XA) RP with u(BR(Xo)) C N is energy minimizing in the sense that fBR(Xo) IDuI2 <

fBR(XO) IDvI2 whenever v

:

BR(XO) - RP with v(BR)(Xo) C N and v = u in

a neighbourhood of OBR(Xo), then (see the discussion in Lecture 1 of [SL1]) u satisfies an equation of the form *, (a). Thus to apply the above lemma to energy minimizing maps it is enough to check that the "reverse Poincare inequality" (b) automatically holds for such maps. This we shall do in the next lecture. (3) Actually, as we show in Lecture 4 below, the hypotheses of the above lemma ensure that u E CI-* rather than in CO-1. Before we begin the proof of this lemma, we need to mention the appropriately scaled version of the harmonic approximation lemma 1.5.

2.2 Resealed Version of the Harmonic Approximation Lemma Lemma. For any given e > 0 there is 6 = b(n, e) > 0 such that if D f E L2(Bp(Y)), IP2_" fB,(Y) Df - DPI 5 SPSUPB,(Y) I DPI for every if P2-" fB,(Y) I Df I2 < 1, and if W E CC°(Bp(Y)), then there is a harmonic function u on Bp(Y) satisfying the inequalities p2-" fB'(Y) IDul2 < 1 and p" fB,(Y) Iu - AY,PI2 < e2. Notice that this easily follows from the unsealed version of 1.5; in fact one just checks that 1.5 applies to the resealed function fp(X) = f (Y + p(X - Y)).

2.3 Proof of the Regularity Lemma As in Remark (1) of 2.2 we have (1)

J R(XO)

Du Dip = - f

F,

BR(XO)

By using this identity with cp E C°°(Bp,2(Y)), where B,,(Y) C BR(Xo) is an arbitrary ball, and using also the hypotheses (a), (b), we have sup I'PI, I(p/2)2-" J P/2(y) Du DPI < Op-" J P(Y) Iu - )Y,PI2 B,/2(Y) and since supap/2(Y) I (PI 5 PSUPB1/2(Y) I DPI (by 1-dimensional calculus along line

segments in Bp/2(Y)), we thus have (2)

I(P/2)2-" JB,2(Y) Dv DI

(p" fa'(Y)

- AY.PI2)I/2p

suP Bo/2(Y)

DW,

LECTURE 2. A GENERAL REGULARITY LEMMA

237

where v = e-lu, with e = Q(p-" fBP(Y) Iu - AYPI2)1/2. Also, we have trivially by (b) and the definition of v that (P/2)2_n

(3)

fP12(Y)

IDvl2 < 1.

Let e > 0 be arbitrary. By (2), (3) we can apply the harmonic approximation lemma in 2.2 above in order to conclude that there is a harmonic function w on Bp12(Y) such that (p/2)2_n

(4)

J

IDwI2 < 1 and (p/2) -n j

B/2(Y)

Iv - WI2 < e2, P/2(Y)

in the harmonic assuming that p-n fBP(Y) Iu - AY'PI2 < 62, where b(n, e) is approximation lemma of 2.2 above. Now take 0 E (0, and note that by the squared triangle inequality a] (5)

(OP) n Bf ,P(Y)

Iv

- w(Y)I2 < 2(9 )-n(JBaP(Y)(I v - wI2 + Iw - W(y)12).

Now using 1-dimensional calculus along line segments with end-point at Y together

with the estimate 1.3(ii) with j = 0 (applied to D;w), we have C92p2_n

sup Iw - w(Y)I2 < (Op sup IDwI)2 < B#, (Y)

BeP Y

r

IDwI2.

BP/2 (Y)

Using this together with (4) in (5), we conclude that Iv - w(Y)I2 < 9-ne2 + C92,

(Op)-Beo(Y)

where C depends only on n. Recalling that v = e-Iu, we have

(,g,) -n

j

In - \12 < X32(9-ne2 + C02)P-n

jP (Y)

Iu - \Y.p12,

e P(y)

where A = tw(Y) is a fixed vector in R. Now we choose 9 and e: first select 9 E (0, a] so that C32#2 < 202"; notice that such 0 can be chosen to depend only on n, a,13. Having so chosen 9, now choose e > 0 such that f329-ne2 < 292 Therefore

(OP)` IBev(Y) In - Ay.Bpl2 <

92°rp-n

J BP(Y)

Iu - Ay.PI2.

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

238

Thus, to summarize, we have shown that if a E (0,1) if Bp(Y) C BR(Xo), and if p-" fBo(Y) Iu - u(Y)I2 < 602, with bo = bo(n,,6, a) E (0,1) sufficiently small, then

(op)-n J o (Y) Iu - AY,epI2 < 02ap n J '(Y) Iu - \Y.pI2 ,

(6)

with 8 = 0(n, a,,0) E (0, 4 ') Next, let Io = R-n fBR(XO) lu-AXo,R12 and note that if to < 2-nb02, then, with p = R/2, we have Bp(Y) C BR(Xo) and P -n

(7)

1

Iu - .XXo,R12 < 2- to < bo

V Y E BR12(Xo).

B1,(Y)

But notice that (6), (7) in particular imply that we have the "starting hypothesis" p -n fB,(Y) Iu - \y pit < by is satisfied with Op in place of p, and hence (6) holds with Op in place of p (p = R/2 still). Continuing inductively, we deduce that (8)

f (81R/2)-n J lu - \y,gjR/2I2 < 02-"(R/2) -n

B, R/2(Y)

J

ju - )1y,R/212 < 2'I0 R/2(Y)

for each j = 0, 1, 2 .... provided only that the inequality 2nlo < bo does hold. Now on the other h a n d if a E (0, R/2], there is a unique j E {0, 1, ... } such that 9j+IR/2 < o < B'R/2, and it is then easy to check that (8) actually implies

0

u-AY,012 B0(Y)

C 2nt2ao-n-2a(o,/R)2aI0

V Y E BR/2(Xo), o E (0, R/2],

provided 2nIo < bo. Then the required Holder continuity is established by virtue of the Campanato Lemma of 1.1.

LECTURE 3 The Reverse Poincare Inequality 3.1 Statement of Main Inequality Let N be a smooth compact Riemannian manifold (without boundary) isomet-

rically embedded in RD, let f c R" be open, and let u : 0 - R" with Du E L2 locally in f, be energy minimizing amongst maps into N in the sense that u(fl) C N and f B IDuI2 < fB IDvl2 whenever B is a ball with B C ft, v : B - RP, v(B) C N and v = u in a neighbourhood of OB. Then we have the following:

Lemma 1. If u is energy minimizing as described above, if A is a given constant, and if R2'" fBk(XO) IDu12 < A for some ball BR(Xo) with closure contained in 12, then p2_"

JB,/4(Y)

IDUI2 < C/! " f

lu - \Y.PI2 ,(Y)

for each Y E BR/2(Xo), p:5 R/2. Here C = C(n, N, A) > 0. Remark: Notice that, once we have established this, we will have completed the proof that u E C°'°(BR/4(Xo))-subject to the same hypotheses as in the above lemma-by virtue of the Remark (2) following the regularity lemma of 2.1.

3.2 A Lemma of Luckhaus, and Some Corollaries The following lemma is due to Luckhaus [Luckl] (see also [Luck2]), and extends the Lemma 4.3 of [Sin.

Lemma 2. Suppose N is an arbitrary compact subset of RA, n > 2 and u, v S"-1 -, RP with Vu, Vv E L2 and u(S"-1), v(S"'1) C N. Then for each e E (0,1) there is a w : S"-1 x [0, el RP such that Vw E L2, wIS"'1 x {0} _ U, wIS"-1 x {e} = v, IowJ2

-

< Ce

f

Sn-1

(IVuI2 + IVvI2) + Ce-1 J 239

n-I

lu - vl2

240

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

and

dist2(w(x, s), N) < CE'-II(J

lu - v12)1/2

IVul2 + IVvl2)'/2(J

+ Cc"

f

S^-'

lu - v12

for a e. (x,8) E Sn-1 x [0, E1. Here V is the gradient on S"`1 and V is the gradient on the product space S"-' x [0, e1. We give the proof of this in the next lecture, but for the moment we want to establish two useful corollaries:

Corollary 1. Suppose N is a smooth compact manifold embedded in RP and A > 0. There is Co = eo(n, N, A) > 0 such that the following holds:

If E E (0, 1], u : S"-' -' R' with Vu E L2, fs^_, lVul2 < A, u(S"-1) c N, and if there is A E RP such that F--2n fs^_, lu - Ale < 4, then there is a w Sn' x [0, E1-+ RP such that Vw E L2, w(Sn-' x [0, e]) C N, WIS"-' X {0} = u,

wIS"-1

J ^-' x fo.e) IVW12 < CE J3^-' IVul2 +Ce_1

x {E} _- const.,

Sn-'

lu - Alt,

C = C(n, N)

Proof. First note that, since u(S"-1) C N, dist2(A, N) < lu(x) _A 12 for each x E Sn-'. Thus, assuming for the moment that co E (0,11 is arbitrary, by integrating over S"-1 we obtain IS,,-11

dist2(A, N) <

lu - Alt < C2 04E

so that dist(A, N) < CEOE2" < CEO, where C depends only on n. Thus assuming

that CEO < a, where a > 0 is such that the nearest point projection II onto N is well-defined and smooth in Na =- {X E R}' : dist(X, N) < a), we can apply Lemma 2 with Nca in place of N and with v - A in order to deduce that there is x {e} = A, a Wo : S"-' -* BY such that woIS"-' x {0} = u, woISn-1

f

3^-' x (oeI

IVwol2 < CE

f

s^

IVul2 +CE-' f

s^

Iu - Ale

C(1 + A'/4)EO/2. Thus if C(1 + A'/4)e /2 < a, then we and dist(wo(x, s), can define w = IIowo. Since dw = dfwo(t,,) odw0 at each point (x, s) E Sn-' x [0, e], w is then a suitable function.

LECTURE 3. THE REVERSE POINCA" INEQUALITY

241

Corollary 2. With the same hypotheses as in Corollary 1, there is w : B, (0) -. RP with wlSn-I = u, w(BI) C N, and

J Bi(0)

lDwl2 < Ce r

I Vu12 + Ce-1

S*-1

JS^-1

lu - A12.

Proof. Let wo be the w from Corollary 1, let AO E N be the constant such that wolSn-1 x {e} __ Ao, and define w(rw)

wo(w,1-r) ifrE (1-c,1) = tl AoifrE(0,1-e), (

where w = 1X1-1X, r = IXl for X E B,(0)\{0}. Then w has the required properties.

There is a scaled version of Corollary 2, proved by applying Corollary 2 to the scaled function uo(X) = u(Y + pX) as follows:

Corollary 2 (Scaled version). If e E (0,1], if u : 8B,,(Y) -, RP with VTu E

L2 and with p3-n f0B,(Y) lVTU12 < A (VT the gradient on 8B,,(Y)), and if also e-2nP1-n foB,(Y) lu - A12 < 4, then there is w : Bp(Y) -+ RP with Dw E L2,

N, wl8B,(Y) = u, and

w(BB(Y)) C

p2-n

JB,(Y)

JDwl2 <

Cep3-n

r

I VTUl2 + CE-1p1-n

6Bp(Y)

LOW)

lu

12.

Finally we want to prove the lemma used in Lecture 1 of the first lecture series. For this we need a further corollary of Lemma 2 above. In the proof we shall need the following important general fact about slicing by the radial distance function. Suppose g > 0 is integrable on BB(Y). Since f B0(Y) g = fa (faB.(Y) g) da, for each O E (0, 1) we have 3.2(i)

I

B,(Y)

g

< 20-1p1 / B,(Y)

9

for all a E (p/2, p) with the exception of a set of measure Op/2. (Indeed otherwise the reverse inequality would hold on a set of measure > Op/2 and by integration this would give fB,(Y) g < fo (foB.(Y) g) Furthermore, if w a function with L2 gradient in fl and if ii is any representative for the L2 class of w, then for each ball Bp(Y) C 0

3.2(ii) vi(a) has gradient in

and

lDw(a)l2 = /BO(Y) lDwl2

a"-1

fS.-I

for a.e. a E (p/2, p). where w(a) is the homogeneous degree zero function (relative to origin at Y) on Rn defined by wi (a)(X) = iv-(Y + aw), w = IX - Yl-1(X - Y).

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

242

We can now state the third corollary. (This is Lemma 2 of Lecture 1 of the lecture series [SL1J).

Corollary 3. Let A > 0 be given. There are constants co = eo(n, N, A) > 0 and C = C(n, N, A) such that the following holds for any p > 0, e E (0, co):

If B(1+,)p(Y) C It, u,v

:

B(1+E),(Y)\Bp(Y) - RP, if u, v have L2 gradients

Du, Dv with p2-" fB(1+.)P(Y)\BP(Y)(IDwI2 + IDvI2) < A, u(X),v(X) E N for X E B(1+,,)p(Y))\BB(Y), and if a-2np-n fB(,+.)P(Y)\8P(Y) Iu - vI2 < eo, then there is w on B(1+,,)p(Y)\Bp(Y) such that w = u in a neighbourhood of 8Bp(Y), w = v in a neighbourhood of 8B(I+,,)p(Y), w(B(l+E)p(Y)\BB(Y)) C N, and

IDwI2 < fB(,+.)P(Y)\Bp(y)

C

(IDwI2 + IDvI2) + CE-2

Iu - v12

Bc1+.)P(Y)\BP(Y}

Proof. By scaling and translation, we may evidently assume that p = 1 and Y = 0. We abbreviate B,(0) = Bo. First note that by 3.2(i) with 0 = e/4 we have a set of o E (1,1 + e/4) of positive measure such that (1) 18B.

(IDuI2 +

IDvI2)+

J/

(IDu12 + IDvI2) < Ce-1

(IDuI2 + IDvI2)

J

and

It!JaBP J

(2)

Iu-v12
I2
Also, by 3.2(ii) we know that a.e. these o can be selected such that uIBB vIBB, have L2 gradients VTU, VTV on BB0. Now we can apply the Luckhaus lemma (with e/2 in place of e) to the functions

u(w) _- u(ow) and v(w) = v(aw), thus giving w on {0} = u, x {e/2} = v and

S"-1

x [0,e/2] with tulS"-I x

t`VIS"-1

J

IomI2 < Ce I ^ - ' x [o.e/21

dist2(t (w, s), N) < C(

(IVuI2 + IVuI2) + Ce-1

S^-1

fn-I

(IVuI2 + Ivvt2))1/2(E2-2n

JS"-1

f

Iu -vI2 Iu - 4/12)1/2

+Ce-nfS.-I Iu-vI2
243

LECTURE 3. THE REVERSE POINCAR$ INEQUALITY

and in view of (1) and (2) we get

j

°-1 X (O,c/2I

I wl< C f

- v12

(IDul2 + IDvl2) + CE-2 j

8,+. \ Bl

i +. \B,

(IDul2 +

dist2(w(w, s), N) < C(f

j- v12)1/2

lDvi2))112(E-2n

B,+\Bi

,+.\Bi

+ CE-n-I 1

Iu - v12 < CEO

Bi+.\B,

with C = C(n, N, A). Taking co (depending on n, N, A) small enough so that CEO < o02, where ro is the width of a tubular neighbourhood of N in which the nearest point projection 11 is smooth, we see that now the lemma is established if we define w as follows:

v(W(IXI)X),

w(X) =

w(IXI-IX,

11 o

u(X),

(1 + E/2)v < Ixi < (1 + E) IXI/o -1), v < IXI < (1 +c/2)Q

1
where p : [(1 + E/2)o,1 + E] --' 11 + E/2,1 + E] is a CI function such that V(t) - a for t close to (1 + e/2)o, W(t) - 1 + e for t close to 1 + e, and 0 < W'(t) < 9 for all t. (Notice that (1 + E/2)a < 1 + 7E/8 because a < 1 + E/4, and hence there exists such a function gyp.)

3.3 Proof of the Reverse Poincard Inequality For any p:5 R/2 and Y E BR12(Xo), we then have by the monotonicity (see 1.4 of the lectures [SL1J) that (1) p2-n

j

IDuI2 < (R/2)2j

IDu12 < 22R-n R/4(y)

O(Y)

f

IDu12 < 2nA.

BR(XO)

Take a fixed p E (0, R/2]. We want to prove p2-n fBP(y) IDti2 < Cp n fB,(y) lu Ay,P12. By (1), there is no loss of generality in assuming that

(2)

p-nf

lu-Ay,PI2 <6p,

Bv(Y)

where 6o is to be chosen depending only on n, N, A. (Because if the reverse inequality to (2) holds, then by (1) we would trivially have the reverse Poincare inequality with constant C = 2nA/602.)

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

244

pJ

Applying the general fact 3.2(i) above with g = lu - Ay,P12 and g = IDuI2, in view of (2) we then have

c7"

J8Be(Y)

+1L-AY,PI2
(3)

IDuI2 <

Cp2-n

J

8e(Y)

lu-)1Y,PI2
(Y)

for all o E (p/2, p) except for a set of measure p/4, where C depends only on n. Now, assuming or E (p/2, p) is as in (3) and that bo/Ceo < 1, where eo is as in Corollary 2, we can apply the scaled version of Corollary 2 with A = Ay,,, and e = (6o/Ceo)I/2n

to give w : B,(Y) -, RD with wIBB,(Y) = ul8B,(Y), w(B,(Y)) C N, and (4)

a2-n f

IDWI2 < Cep2-n f

Bo(y)

IDuI2 +Ce-Ipn f

lu - Ay,P12.

BP(Y)

Ba(y)

But since u is energy minimizing we have fa,(y) IDuI2 <- fB.(Y) IDwl2, and hence, since o > P/2, (4) gives (5)

p2 -n

IDuI2 <

CEp2-n

4,(Y)

4P/2(Y)

IDuI2 + Cc-Ip n 4"

Iu - AyPl2. (Y)

Thus in particular (6)

p2-n

f

C = C(n, N, A).

IDuI2 < CeA + Ce-Ibo < Cb0/2'",

BP/2(Y)

In view of monotonicity (Cf. (1) above) we see that then also a2-n fB,(Z) IDuI2 < C 60112n provided B,(Z) C B,,12(Y). Then the ordinary Poincare inequality implies

then that for any such ball B., (Z) we have Q-n

fB.(Z)

Iu - UZ,,I2 <

Cb0/2n

provided only that (2) holds with suitably small bo, depending only on n, N, A. Hence if e E (0,1) is given and if we now choose bo = bo(n, N, A, e) small enough in (2) then we also have (5) (with this e) uniformly for such balls B,(Z). That is, a2

JBe/2(Z)

IDuI2 < CE0,2 J IDuI2 + Ce-I f .(Z) r,(Z)

(7)

Ceo-2

JB.(Z)

IDuI2 + CE-III

u-

LECTURE 3. THE REVERSE POINCARE INEQUALITY

245

whenever B,(Z) C B 12(Y), where It = fB,(Y) Iu - AY,PI2. (Here we used fB.(Z) Notice that c is still to be fB.(Z) Iu - AY,PI2 < fap(y) Iu IU chosen to depend only on n, N, A; for the moment we have (7) for arbitrary e E (0,1) provided only that (2) holds for suitable bo = bo(n, N, A, E). Now in view of the arbitrariness of the balls B,(Z) we claim that (7) implies the required reverse Poincare inequality. To see this, first define

Q=

sup

{B,(Z): B2,(Z)CBo/2(y)} a2

f ,(Z) IDuI2,

and then take an arbitrary ball B,(Z) with B2o(Z) C BP12(Y). Notice that such a ball can be covered by balls B,/2(Zi), i = 1,... , S, with Z, E B,(Z) and with B. A) C B 12(Y); further we can evidently bound the number S by a fixed constant depending only on n. Now using (7) with Zi in place of Z and summing over I we have 02

J B,(Z)

IDuI2 < CESQ + CSE-III.

Taking sup on the left we thus have

Q:5 CESQ+CSE-II1i whereupon choosing c (depending on n, N, A) such that CES < 1, we have o2

/

B,(Z)

IDu12 < CII

for each ball B,(Z) with B2o(Z) C BP/2(Y), where C depends only on n,N,A. Taking Z = Y and o = p/4 we thus have the reverse Poincare p2_n f a,,4(y) IDul2 < Cp n fBP(Y) Iu _.\Y,.12 for any ball BP(Y) with p:5 R/2 and Y E BR/2(Xo). On

the other hand for any Y E BR/2(Xo) and p < R/2 we can cover BP/2(Y) by a collection of Q = Q(n) balls BP18(Y,,) with Yj E B.12(Y). Since the above shows that p2-^ fBo/8(Y IDuI2 < Cp "fB,,2(YY) Iu - Ay,,P/2I2 < Cpn f,,(Y) Iu AY,PI2 for each j = 1... Q, the required reverse Poincarb p2_n feo/2(y) IDuI2 < Cp n fB,(Y) Iu - AYPI2 follows by summing over j.

LECTURE 4 Completion of the Regularity Proof Here we want to tidy up the loose ends. Specifically, we want first to prove the Luckhaus lemma stated in §3.2 of the previous lecture (this will then complete the proof that an energy minimizer u is CO- in B,/2(Y) assuming that it satisfies p -n f ,,(Y) Iu - Ay,PI2 < 802 with 6o sufficiently small), then we want to show that u E C1'11 under the same hypotheses; standard linear PDE then shows that u E C°O. We also want to point out that the modifications which are needed to establish the

regularity theorem in case the domain Sl is equipped with an arbitrary smooth Riemannian metric are very minor.

4.1 A Further Property of Functions with L2 Gradient x We note an important property for L2 functions 0 on a cube Q = [al, b1] x [a,,, b"] with DO E L2: there is a representative for the L2 class of tj, such that, for each j = I,- , n, xj-1, xj, xj+1, ... , x") is an absolutely continuous x"). Here of function of x! f o r almost all fixed values of (xl, ,x'-l,xj+l, course "almost all" is with respect to (n -1)-dimensional Lebesgue measure on the (n - 1)-dimensional cube [a,, bl] x ... x [a._1, b?-11 x [a.,+1, 6,+1] x ... x [an, b,].

Furthermore, the classical partial derivatives D,i (defined in the usual way by D.V;(X) = lint-1(+/(X + tel) - i(X)) whenever this exists) agree a.e. with the L2 derivatives Djo. A discussion of these properties can be found in e.g. [GT] or [MCB]. We here make one further point: one procedure for constructing suchho a representative (see e.g. [MCBI) is to define ii(X) = Ax at all points where

there exists a Ax such that 1imp1op" fs,(y) (ax - u(Y) I dY = 0, and to define arbitrarily (e.g. a/i(X) = 0) at points where the limit does not exist. Thus in particular if 0 = (401,... ,.OP) : Q -+ RP, and if N is any closed subset of R1' with the property that ii(X) E N for almost all X E fl, then we can select the representative /i to have the property ii(X) E N forevery X E Sl, in addition to the absolute continuity properties mentioned above. 247

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

248

4.2 Proof of Luckhaua' Lemma In the case n = 2 the functions u, v have an L2 gradient on Sl and hence have absolutely continuous representatives U, U such that VU = Vu, Vv- = Vv a.e., where V denotes the gradient on S1. Furthermore, by 1-dimensional calculus on S' and by the Cauchy-Schwarz inequality we have

-Ss

Ilii-U12

ssuipl--vl2<J 1IvIu-UI2I+(2n)-'J < C(J

1

IV(UU - U)I2)1/2(J

`

lu - vl2)1/2 t C J 1 lu - vla,

and hence (1)

sup iu - 1 <

(JS' IV(U _ U)12)'14( ISI I*l - U12)1/4 +,C( JS` ju - UI2)1/2,

If we now define

w(w, 8) = U(w) + (U(w) - U(w)),

then, letting Vw denote the gradient of w on S' x [0, e], we have

IVwI < IV-uI+IV(U-u)I+ EIv-uI, and hence

(2)

IVwI2 < 8(IVUI2 + IV1I2) + 2e -21;U - UI2.

Notice that, since U(S1) C N, (1) implies that for each w E S', s E [0, e],

dist(w(w, s), N) < C(/ IV(U - U)I2)1/4(I Iu - UI2)1/4 + C( j' IU - vI2)1/2, Is s 1

1

and by integrating (2) over S' x [0, e] the remaining inequality also follows. This completes the proof in the case n = 2, so from now on assume n > 3, and again choose absolutely continuous representatives U, v such that Cu = Vu, vv = Vv a.e. on Sn-1 Without changing notation, we extend U, U to give homogeneous degree zero

functions on Rn; thus u(rw) _= u(w) for r > 0 and w E S"-', and Du = Vu on Sn-1, DU = Vv on Sn-1. Notice also that DU(X) = IXI-1Du(w), with w = IXI -' X, and hence (since n 2 3) we have in particular that

(3)

(IDiI2 + IDUI2) < C JS^-1(Ivul2

+

IvvI2),

LECTURE 4. COMPLETION OF THE REGULARITY PROOF

249

and also of course

Ill_ UI2
(4)

J f-1,11"

Js"-1 [u -v[2.

Now fore E (0, and i = (i1,... in) E Z", we let Qi,E denote the cube [i1e, (iI + 1)e] x . . . x [ine, (in + 1)E], and for a given non-negative measurable function f [-1, 1]" -+ R we let j : Qi,E R be defined by

f (X) _

f (X + e i),

X E Qi,E.

{i:Q...cl-4,j"} Then

JQo..

f(X)dX < fl-

j(X) dX = J

111"

j(X)dX,

and hence for any K > 1 we have

f(X)dX

f(X +ei) < K

e"

{i:Q:.,c[-,#I"}

J-1.11"

for all X E Qo. with the exception of a set of measure < Cell 1K, where C = C(n). Similarly, since by Fubini's theorem

f(X)dX <e / Qo..

QO.,

[-1.1j"

we have

f (dX

en-1 -1.11"

1 1 Zjn}

for all X E Qo,,with the exception of a set of measure < CE"/K, and generally,

foranyeE{0,...,n}, F(t)

f(X +Y+ei)dH'(Y) <_ K L11. f(X)dX

for all a-faces F<<} of Qo,, and all X E Qo,, with the exception of a set of measure < CE"/K. Notice that this last inequality implies (5) e11-t

E

F

J,: Q... e[- J.J]" l f-faces F") of X+Q,..

J Fcn

f (Y) dH'(Y) < K

j-1.1j,.

f (X) dX

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

250

for all l E (0,... , n} and all X E Qo, with the exception of a set of measure < Ce"/K. Now by the absolute continuity properties of 4.1, we can select representatives u, U, Du, DU, for the L2 classes of u, v, Du, Dv such that for almost all X E Qo,E all of the functions u, U, Du, Du are defined ?{t-a.e. on each of the 1-dimensional faces Ft of each of the cubes X + Qi,c with Qi,E C [- 2, 2]n for each e = 0,... , n, and such that furthermore, on each such 1-dimensional face, i, U have L2-gradients which coincide xt-a.e. with the tangential parts of Du, DU. Now by applying (5) with f = Iu _:U12 and f = IDu12 + IDi12 we see that we can select X = a E Qo,, such that the above properties hold and also such that (6)

,"-t

E

{i: Qt.

(_1,I)n

J(c) f (Y)

dit(Y)

f (X) dX with f = Iu - vI2 or f =

IDUI2,

for each P E f0,--- , n}, where C = C(n).

Next, let Q be any one of the cubes a + Q, , with Qi,, C [-2, 2]", and we proceed to define a W1,2 function w = w{i.0 on Q x [0, e] which agrees with U on Q x {0} at all points of Q where u exists, agrees with v on Q x {e} at all points of Q where v exists, and which is such that n-1

(7)

4IO,c)

IDwI2 <- C E e"-'+1 ally-fac, F(J)

Q fp(I) I

+Ce"-2

(IDuI2 .. IDI2)

I:

all1-facesF(') IF(')

(-

v12

(8) diet2(w(X), N) <

C

max

1-facesF(')ofQ

((f

F(i)

I D(u - 1/)12)1/2(1

Iu - VI2)1/2 + CE-1 JFM Itt - VI2).

Let E be any one of the edges (i.e. 1-dimensional faces) of Q. By 1-dimensional calculus along the line segment E, we have (since the length of E is e) (9)

Ep[u-VI2<11.1DIu-V121+e-1fEI1-l-V12.

Hence by using the Cauchy-Schwarz inequality we obtain (10)

sup [u - U12 < C(JF ID(u - 1)I2)1/2(Je ju

where C is a constant depending only on n.

- UI2)1/2

+CE-1 Jiu

-

1112,

LECTURE 4. COMPLETION OF THE REGULARITY PROOF

251

Then we can define an R"-valued function w on Q x (0, e) by the following inductive procedure. We first define w on Q x {O} and Q x {e} by

w(X,0) = u(X),

(11)

w(X,e) = U(X),

X E Q.

Next we extend w to each F(1) x [0, e], where FM is any 1-dimensional face (i.e. edge) of Q, by defining

w(X, s) = (1- s)u(X) + 8v(X ),

X E F(1), a E [0,C].

E

C

Notice that then by (10) and the fact that u(R") C N we have (12)

dist2(w(X, a), N) < C

maX 1-facenF(i)ofQ

mFa(x

1 faces

ufQ

Sup IV - u12 <

((fF0) ID(u--)12)1/2(fF(')

Iu-y12)1/2+CcI f (1)

r-v12).

Also notice that by direct computation (13)

sup IDw(X, 8)I2 < 8(IDU.(X)12 + IDv(X)I2) + - Iu(X) - v(X)I2 8E(O,eJ

at any X in any edge F(1) of Q (where D = ( hence (14)

J

I

(11x[0,e]

wI2 < Ce

J

(IDul2

),

ax =

+ IDUI2) + CC-1

(1)

f

gradient on F(1) ),

F(')

lu - V12.

For t > 2 we now proceed inductively by homogeneous extension into faces of larger and larger dimension. More precisely, assume t > 2, and that w is already defined

(with L2 gradient) on all F(I-1) x (0,e] and w(X,0) _- u(X), w(X,e) _- U(X) on F&). Since 8(F(e) x [0, e]) is the union of F«-1) x [0, e] (over the t-1 faces F(1-1) of F(t)) together with F(t) x {0} and F(t) x {e}, we then have that w is already well defined fle-a.e. on 8(Fip) x [0, e]). We can thus use homogeneous degree zero extension of wI8(FV) x [0, e]) into Fit x [0, e] with origin at the point (q, a/2), where q is the center point of F(t . Then by direct computation we have (15)

Lt)x( O,eJ

FallF-1)

IwI2,

IDwI2 < CC f (IDUI2 + IDUI2) + CC

18111)x(O,E}

where D = (a , a ), a = gradient on F(t) on the left and on F<<-1) on the right. So by mathematical induction based on (15) we conclude that, for all t E (2,... , n},

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

252

to can be extended to all of F(t) x [0, E] (F(t) = any t-face of Q) such that to has L2 gradient Dw on all F(t) x 10, E] with (16)

alit

E

IDw[2 < CJ-1

J

-facF)ofQfF«)X1O,

l

IDwl2

JF(')xjO,]

Z

FM allj-facesF(f)ofQ FU)

j =l

Furthermore notice that the homogeneous degree zero extension preserves the bound (12). Thus by (12), (14), (16) (with I = n) we conclude the existence of w : Q - RP as in (5) and (6). Since Q = a + Qi,,, we should write w = w(i,`). Since the construction of w(i,c) is such as to ensure that w(i,E) = w(j,E) 7{t+l-a e. on F(t) x [0, E] for any common a-face F(t) of two different cubes a+Q,, and a+Qj,E, we can then define a W1,2 function to on [-4, 4]n by setting w(X, s) = w0") (X, s) for (X, 8) E (a + Qi,,) x [0, J. (We are assuming E E (0, and a E Qo,E, hence

we automatically have that [-4, 4]n C U{Qi,e a + Q,, C [-2, 1jn.) Notice that then by summing for i in (7) and also using (6) and (8) we get (17)

IDwI2
J

I-#,1l"Xfo,Ej

(IDuI2+IDvI2)+CE-1 J

[-1 11^

1-1,11^

Iu-v12

and (18)

dist2(w(X), N) < Ciracmax)EF(J F(1)

< `i`E1-n(J

ID(u-v)I2)1/2(Iu-uI2)1/2+CE-IJ JFM

(ILU12 + IDfJI2))1/2(J

I4L-vI2)

F(')

Iu - vI2)1/2 + C, ,-n J

Iu - v12,

where F denotes the collection of all 1-faces of cubes in the collection {a + Qi,, C [- 2, 21n). Define ii = w]([-4, x [0,E], and using 4.1 choose p E [g, A] such that to has L2-gradient on 8([-p, p]n) x [0, e] and such that

I

I

(I-p,pl^)XIo,E)

wI2


f

IwI2.

I-},V^X(o.f)

Then finally let 'P be the radial map from 0 taking Sn-I to 8([-p, p]n) (notice that this is a Lipschitz piecewise C' map with a Lipschitz piecewise C' inverse). Thus we s), and one then readily checks can define ii) on Sn-I x 10, e] by w(w, s) = that this map w has the properties claimed for w in the statement of the Luckhaus

lemma. (In particular, since w(X,0) - u(X) for X E 0([-p, pin) and since u is homogeneous of degree zero in Rn, we then have by definition that w(w, 0) - u(w) v(w) a.e. on Sn-1 ) a.e. on Sn-I. Similarly

LECTURE 4. COMPLETION OF THE REGULARITY PROOF

253

This completes the proof of the Luckhaus lemma, and hence the proof that u is C°"°.

4.3 Proof of CI41 and Higher Regularity Here we assume the hypotheses are as in the abstract regularity lemma 2.1. First let BP(Y) C BR/2(Xo) be arbitrary. We assume without loss of generality that R = 1, so that the Holder continuity for u proved above gives Iu(XI) - u(X2)I < CP°,

(1)

X1, X2 E Wp(Y), 0 < p < 1/2.

Since U E W1"2(Bp(Y)) n C°(B,(Y)), we know (see e.g. ICT]) that there is a V E C2(Bp(Y); RP) n C°(Bp(Y); RP) n WI,2(Bp(Y)) which is harmonic on Bp(Xo) and which agrees with u on 8Bp(Y). Of course u satisfies the weak form of Laplace's equation on BB(Y); that is,

r

E DjvDjV= 0, 1o E Cc°(Bp(Y); R").

JB,,(Y) j_1

Taking the difference of this equation and the weak form of the equation for u, we

thus get that

f

Dj(u - v)Dj4p = P(Y)j=1

JBP(Y) c F,

Now since u, v agree on the boundary it is easy to check that this is also valid with the choice W = v-u; here we use the general fact that if a C°(Bp(Y))nW I"2(BP(Y))

function is zero on 8Bp(Y), then it is the limit in the WI.2(Bp(Y)) norm of a sequence of C'°(BP(Y)) functions. Thus we obtain

(2)

f

BID(u - v)12 = f(Y) (u - v) F (Y)


IDuI2
IDuI2,

B.(Y)

where we used the fact that supB,(y) lu - u(X°)I < Cp° (by (1)) for any Yo E W ,(Y) and that, for Yo E 8Bp(Y), supBP(y) Iv - u(Y°)l SUPBP(Y) Iv - v(Y°)I < psuP8BP(y) lu - u(X°)I < Cp° by applying the maximum principle to each component W of v = (v', ... , vP). Now on the other hand we have, by using the reverse Poincare inequality on the right of (2) and again using (1), (3)

p-' fBP(Y) ID(u - v)I2 =

Cp3°-2,

P E (0, 1/4)

L. SIMON. BASIC REGULARITY THEOREM FOR HARMONIC MAPS

254

Now let us agree that a was chosen in the first place so that 3a > 2, and that hence

3a=2+2p for some p>0. Thus we have

p-n j

B"M

ID(u - v)12 = Cp2'`,

p E (0,1/4)

with 2p = 3a - 2 depending only on n,,Q, co, and this is all valid uniformly for Y E B114(Xo). Thus we have

a-" j e (Y) IDu - Dv(Y)12 2a_n /Bo(Y)

<

ID(u - v)12 +

2a-n jBe(y) I Dv - Dv(Y)12

< C(pl a)np2' + 202 sup ID2vI2. B,(Y)

Now by the estimates 1.3(ii) for harmonic functions we have

2a2 sup ID2vI2 < 2a2 sup ID2vI2 < Ca2p-2 B,/2(y)

Be(Y)

sup

IDv - Ay,,I2

B3,/4(Y)

< Ca2p n-2 )BAY) IDv - Ayp12,

with Ay,,, = IBp(Y)1-1 fB,(Y) Du. (Note that the last estimates would be valid with any constant A In place of Ay,p.) Now on the other hand

pn

j

B(Y)

ID(u - v)12 + 2p j

IN - Ay,, 12 < 2p'n f (Y)

I Du "(Y)

<- Cp2µ + Cp n j

IDU12,

B,.(Y)

and by the reverse Poincare and the Holder estimate for u we have

p2-n j

IDuI2 < Cp-n j o(Y)

Iu- AY,2p12 < Cp2a. 3,(Y)

Thus the above inequalities combine to give

I, <

C(p/a)"p3a-2 +

C(a/p)2p 2+2a,

where to = a-n fB0(Y) IDu-Ay,,,I2, and this is valid for all a < p < 1/4. Choosing O. = p" with r. = 1 + a/(n + 2), we then have

I, < Ca27

LECTURE 4. COMPLETION OF THE REGULARITY PROOF

255

for any y < 2/(rc(n+2)), provided a is chosen sufficiently close to 1, and this holds for any o,:5 R/4 and any Y E B1 J4(Xo), with C depending on a, n and the constant a. Therefore the Campanato lemma of 1.1 can be applied to give Du E C°"'r on B114(X0). In particular this gives the boundedness of Du on B1/4(X0), and hence the original equation has the form Au = f, with f bounded on B114(X0). But then the standard C1,°-Schauder estimates (see e.g. [GTJ) give that u is C1,° for each a < 1, as required, on BR14(X0) But now we can "bootstrap"; we have shown that u is a C""° weak solution of the equation n s

Du = - E

Dju)

j=1

in BR14(X0). But since u E Cl-, it is clear that the right side here is in CO,*

on BR/4(Xo) and hence the equation has the form /u = f with E C". But the standard Schauder theory (see [GTJ) for equations Au = f now implies that on BR/4(X0), and then the right side in * is of class C1'°, so now u UE satisfies an equation of the form Au = f with f E C1'Q, and the standard Schauder theory this time gives u E C3.°. Continuing in this way we deduce inductively that u E Ck,° on BR14(X0) for each integer k > 1. We here used the qualitative part of the Schauder theory (that if Du = f in a domain Sl with f E Ck.°(S2), then u E C2.°(S2)); of course we can also use the quantitative part of the Schauder theory (i.e. the relevant Schauder estimates) at each stage in order to deduce the bounds (4)

R3

sup

ID'u1 < Cc

BR,e (XO )

(assuming R2'n fBR(xo) IDuI2 < A and R-n fBR(xo) Iu - AXO,R12 < e(n,N,A)2). This completes the proof of the regularity theory claimed in 1.6 of the first lecture series, except that we have the required estimates only over BR/$(Xo) rather that BR/2(Xo) as originally claimed. But this evidently follows, because the original hypotheses R2-n'BR(xO) IDuI2 < A and R-n fBR(xO) Iu - AxO.RI2 < e(n, N, A)2 imply that (R/2)2'n fBRI,(Z) IDuI2 < 2n-2A and (R/2)'n fBR12(z) Iu - AZ.R/212 < 21 2 uniformly for Z E BR/2(Xo), and so the original hypotheses on BR(XO) with new e(= 2'"e(n, N, 2n-2A)) gives us the required hypotheses on each of the balls BR/2(Z), and hence (4) holds with Z, R/4 in place of Xo, R:

R'

sup

IDrul < Cc.

Bit/16(z)

Since BR12(X°) can be covered by a finite collection (with cardinality depending only on n) this gives inequalities like (4) on BR/2(XO) as required.

256

L. SIMON, BASIC REGULARITY THEOREM FOR HARMONIC MAPS

Acknowledgments. Partially supported by NSF grant DMS-9207704 at Stanford University. It is a pleasure to thank Tatiana Toro for her invaluable assistance in the preparation and correction of these notes.

References [DeG]

[GT]

E. De Giorgi, F4 ontiere orientate di misura minima, Sem. Mat. Scuola Norm. Sup. Pisa (1961), 1-56. D. Gilbarg & N. Tlrudinger, Elliptic Partial Differential Equations of Sec-

ond Order (tom edition), Springer-Verlag, 1983. M. Giaquinta & E. Giusti, The singular set of the minima of certain quadratic functionals, Ann. Scuola Norm. Sup. Pisa 11 (1984), 45-55. R. Hardt & F: H. Lin, Mappings minimizing the LP norm of the gradient, [HL] Comm. Pure & Appl. Math. 40 (1987), 555-588. [Luckl] S. Luckhaus, Partial Holder continuity for minima of certain energies [GG]

among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988), 349-367. [Luck2] S. Luckhaus, Convergence of Minimizers for the p-Dirichlet Integral, Preprint (1991).

[MCB] C. B. Morrey, Multiple integrals in the calculus of variations, Springer [SU]

Verlag, 1966. R.. Schoen & K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307-336.

[SL1]

L. Simon, Singularities of geometric variational problems, this volume.

Geometric Evolution Problems Michael Struwe

IAS/Park City Mathematics Series Volume 2, 1996

Geometric Evolution Problems Michael Struwe

Introduction Variational principles play a fundamental role in geometry. Geodesics, minimal surfaces, or Einstein metrics can be characterized as stationary points of suitable variational integrals; moreover, extremals of variational problems for mappings between manifolds are tools for their differential-topological classification. A particular such class of maps is the family of harmonic maps. Due to their geometric nature, these variational problems in general give rise to non-linear partial differential equations which rarely can be attacked by standard

p.d.e. methods. Instead, it is more natural to use a variational approach. However, in many instances also the well established direct methods fail because the functional concerned is not (weakly) lower semi-continuous or because the space of admissible functions is not (weakly) closed. In addition, more sophisticated techniques derived from Morse theory or Ljusternik-Schnirelman theory cannot be applied because of lack of differentiability or failure of the Palais-Smale condition. The latter is a compactness condition, and its failure in the context of geometric variational problems quite often naturally arises from invariance of the problem with respect to a non-compact group of manifest or hidden symmetries like gauge-invariance, conformal invariance, etc. Hence, lack of compactness reflects an important geometric aspect of the problem and is not simply a technical shortcoming. Moreover, special solutions which exhibit that symmetry often reveal highly interesting features of the problem. Thus, while it is not possible to apply Morse or Ljusternik-Schnirelman theory in a standard way, it seems desirable to follow the ideas underlying these theories as closely as possible. Outstanding among these ideas is the concept of a gradientlike flow, deforming the sub-level sets of a given functional to its set of critical points and connecting orbits. In the sequel, we will investigate various geometric variational problems and the corresponding `gradient flows'. In contrast to the setting of a C2-functional on a Hilbert space V, where the gradient flow is defined

by an ordinary differential equation in V satisfying a Lipschitz condition, most I Mathemati, ETH-Zentrum 8092 Zurich Switzerland.

E-mail address: struvemath. ethz. ch. Q 1996 American flat

259

So. tcty

260

M. STRUWE. GEOMETRIC EVOLUTION PROBLEMS

of the time we will find ourselves in a non-smooth environment and our `gradient flows' will involve unbounded operators and possibly singular behavior. However, as in case of the heat equation, the unbounded operators that we encounter will often have a smoothing effect. Most remarkably, moreover, the singularities that we observe often can be related to the special `symmetric' solutions mentioned earlier by means of certain 'monotonicity estimates'. Thus, the flows we consider not only will provide us with existence results but will also provide some geometric understanding when and why existence results will fail to hold. We will pursue these questions in some detail in two model situations: the evolution of harmonic maps and the mean curvature flow. The latter is related to the Ricci-Hamilton flow, but can be stated in a simpler context better suited for the purpose of exposition as the analysis is less technical and more transparent. The evolution of harmonic maps in turn seems related to the evolution of Yang-Mills connections, and many results valid for the first have parallels in the second case. Finally, Ye's Yamabe flow [1831 is a beautiful example why an evolution approach might be preferred to a direct one. A different aspect of geometric evolution problems will be the subject of the third part of these notes. This is the vast field of hyperbolic equations and systems arising from variational problems on a Minkowski space-time. Here we can only briefly discuss recent progress and various open questions concerning harmonic maps of Minkowski space. Similar results should be expected to hold true for Yang-Mills fields on principal bundles over Minkowski space. Important progress towards an understanding of the 'hyperbolic' Einstein equations has recently been achieved by Christodoulou-Klainerman[26).

PART 1 The Evolution of Harmonic Maps

1.1. Harmonic maps Let MI be an m-dimensional R.iemannian manifold with metric y, N1 a compact !-dimensional manifold with a metric g, respectively. By Nash's embedding theorem we may assume that N C R" isometrically for some n. For a CI-map,

u=(u1,...,u"): M -'NCR" let e(u) =try" (x)uxau2n =: ZIVuiM be the energy density, written in local coordinates x = (xa)1
Repeated Greek indices tacitly will be summed from

1 to m, repeated Latin indices from 1 to n. Moreover, u=a = emu', etc. A Clvariation of u is a family (uE) of CI-maps of : M -+ N C R" smoothly depending on a parameter IEI < co, and such that ua = u. A variation (u,) of u is compactly supported if there exists a compact set Cl CC M such that uE = u on M \ Cl for all IEI < E0.

Definition I.I. A CI-map u: M - N C R" is harmonic if it is stationary for Dirichlet's energy

E(u) =

e(u) dvolM fM

with respect to compactly supported variations. Note that in local coordinates dvolM = I-y dx, where IryH = Idet(7ap)I. In order to derive the Euler-Lagrange equation satisfied by a harmonic map

u, let U C R" be a tubular neighborhood of N and irN : U - N the (smooth) nearest-neighbor projection. Denote TpN C TTR" the tangent space to N at a point p E N. Let 0 E Co (M, R") satisfy 4)(x) E Tu(x)N

for all x E M. 46 induces a Cl-variation u, = zrN . (u + E4)). 261

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

262

Since dtrN (p) I

Tn N = id for p E N, clearly we have d dE

U. 1E=6 = (dxN - u)O _ 0-

Suppose 0 has support in a single coordinate chart. Then

d E(u()I

- _

ryaA

[7[

8x° (Y°0

"'A u` )0' V 11 I dx

- Jm where OM =

ex, (y

[3 A. .)denotes the Laplace-Beltrami operator on M.

171

Thus, if u E C2 is harmonic, u satisfies

AMU 1

(1.1)

and conversely. To obtain a more explicit form of (1.1), let v1+1, ... , vn denote a local orthonormal frame for (TpN)l, the orthogonal complement of T9N in IIYn, near p = u(x) E N. Then, by (1.1) there exist scalar functions a1+I, ... , ,\n such

that -AMU =

Ak(vk

u).

k=1+1

Multiplying by vk (k fixed), since uxa vk(u) = 0 for all a, we obtain

ak = -AMU (vk . u)

div(Vu vk(u)) + 7apux° ' 8x (vk ° u)

= t Ak(u)(ux.,us), where Ak = dvk denotes the second fundamental form with respect to vk. Hence (1.1) is equivalent to

-AMU = A(u)(Vu,Vu)M,

(1.2)

where in local coordinates n

1*0Ak(u)(ux°,uxs)(vk

A(u)(VU, VU)M =

u).

k=1+1

Example 1.1. If M = T^' = R1/Z1, N = Sn C Rn+1, equation (1.2) simply becomes Au = IVU12U.

Harmonic maps generalize the concept of harmonic functions. Harmonic maps

S' -4 N uniquely correspond to closed geodesics on N. Important applications of harmonic maps are in Teichmiiller theory in understanding the Weil-Peterson metric on Teichmiiller space, see Earle-Eells [32], [33], Eells [35], Fisher-Tromba [47], Tromba [148], or in proving rigidity theorems for Kiihler manifolds (Mostow [113], Mostow-Siu [114], Siu [133]). A comprehensive survey of harmonic maps and their applications is given in Eells-Lemaire [36], [37]; see also Hildebrandt [73], [74], Jost [87], [88].

PART 1. THE EVOLUTION OF HARMONIC MAPS

263

In these notes for simplicity we shall mostly adopt the above parametric point of view. Intrinsically, the concept of harmonic map can also be defined as follows. A map u: M - N is harmonic if and only if its tension field

T(u) := trace(Vdu) = 0,

(1.3)

where V denotes the pull-back covariant derivative in the bundle T*M ® u-'TN. In local coordinates on M and N, the analogue of (1.2) then is _AMUk = ryas(x)r j(u(x))u uxo,

(1.4)

where (f ) are the Christoffel symbols on N. However, we prefer (1.2) to this intrinsic equation for its simple geometric interpretation (1.1).

Bochner identity A very useful identity is the differential equation satisfied by the energy density e(u) of a harmonic map u: M - N. Let R(M), Ric(M), RAN) denote the Riemann curvature tensor and Ricci curvature on M, respectively the Riemann curvature tensor on N. Given any point xo E M let MM) = Ric(M) = (Rap), R(N) = (Rikj1) be the coordinate representations of R(M), Ric(m), R(N) in normal coordinates x = (xa) about xo E M, respectively in normal coordinates u = (u') around uo = u(xo) E N. We use the notation of Jost [89[; (1.2.6), (1.2.7), (1.2.13).

Proposition I.I. If u E C3(M; N) is harmonic, then in local coordinates as above there holds (1.5)

-AMe(u) + IVdu12 + Rapu:.u:o =R.kjlul.uzu,ouL

or

where V denotes the covariant derivative on T'M 0 u-'TN. (See Jost [89]; p. 96f. for an equivalent expression in general coordinates and an invariant form of (1.5).) Proof. The proof relies on the following identities valid at xo E M in normal coordinates around x0 on M, respectively around u(xo) on N. Let rye' _ etc. Then we have It

(AMU)yw -

=

(-Y"O

IR Dz°x,.u:o = ryQ1µuzo +

27PP,oNU=°

1

ryap aµuya + 2ryPP,aµux° I 1

R°F = RaµB = rbR.µ - raµ,R = 2 (ryaK,Aa + ryRµ.aU) - 2 (7a3.µµ + ryµf,alf )

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

264

Hence, at xp,

©Me(u) = OM(271gij(u)'uzvus-) _ (AMU' .)U'. + ,-ly,a"aui,-U- +u=a=ruz°sµ + rlgij,kluuyµuJ=ru=ousa i

;

;

;

,

;

_ (AMU )xvuxµ + tAz.zHUz zM +7a,9,aIUxsuzU s ; U7 k - (7µu,aa + 7pp,ly)uxµuxr +2gij,k1(U)ux,, xµuxouxa {

1

I

_ (AMu )zPuzµ + U , xuuxoxµ + RapU=auA

+ (9,,,ki +gtl.lk -9I,{k)u=µi=µuzouxq Finally, note that by (1.4) we have (AMU')xruzn = -4'kl,j(u)u=.Ur-Ui, U'Z,.

at xo, since f (u(xo)) = 0 by our choice of coordinates. Moreover,

R;ljk = rkZ J - rjt,k, 1 rfl,k = 2 (9ij,kl + gil,jk - 911,1k), i

whence

AMe(u) = z°zMuZoxa +Raiu=ou=8 Since R;ljk = R;kfl the claim follows.

13

Note the following corollary as a consequence of (1.5); see e.g. [89j.

Proposition 1.2. If M is compact, RicM > 0, and if the sectional curvature of N is non-positive, then any harmonic map u E C°°(M; N) is totally geodesic in the sense that Vdu = 0; that is, du is parallel with respect to the pull-back covariant

derivative on T'M 0 u-ITN. Moreover, if RicM > 0 at a point of M, then u = cont. If the sectional curvature KN of N is negative, then u = const or u(M) is covered by a closed geodesic.

Proof. Integrate (1.5) over M to obtain IVduI2 = 0, Ric(M)(du, du) = 0, (R(N) (du, du)du, du) = 0 on M under the above assumptions.

0

For most of our purposes it suffices to note a weaker Bochner-type estimate.

Tb state this we return to the setting of maps u: M - N C RN. Differentiating (1.2) in direction xµ and taking the scalar product with ux.. with respect to y, on account of (1.1) we obtain (1.6)

-LMe(u) + IV2uI2 < IRicMle(u) +C(e(u))2,

where the second term on the right arises from estimating .1PV (Ac(u)(Vu,Vu)M(Vk ° u))xµ . u.-

= Ak(u)(Vu,Vu)M70'(dvk(u)ux- u..) = (A(u)(Vu, Vu)M)2 < C(e(u))2.

PART 1. THE EVOLUTION OF HARMONIC MAPS

265

Weakly harmonic maps Let

H1'2(M; N) = {u E H1.2(M; R" ); u(x) E N for almost everyx E M. where H1,2(M;R") is the standard Sobolev space of L2-mappings U: M - R" with distributional derivative Du E L2. That is, HI.2(M; N) is the space of maps u: M - N with finite energy E(u). It was observed by Schoen-Uhlenbeck (124] that in general HI.2(M; N) as defined above is larger than the weak closure of C' (M; N) in the H1"2-norm IIUIIHI.2 = j (IHH2 + IVu12) dvoiM, r

which in turn is larger than the strong closure of C°°(M; N) in H1,2(M; N). However, if m = dim M = 2 these spaces all coincide. By a result of Bethuel [7] the same is true if ir2(N) = 0. The respective relations between these spaces for general M and N were analyzed by Bethuel-Zheng (9] and Bethuel [7].

Definition 1.2. A map u E HI "2(M; N) is weakly harmonic if u satisfies (1.2) in the distribution sense.

Example 1.2. The map u : B1(0) C R' - S"`-1, given by u(x) = Izl belongs to H1'2(Bl(0);S"`-I) form > 3 and weakly solves (1.2), that is

-Du = IVuI2u.

Existence of harmonic maps As in Hodge theory, where one seeks to realize a de Rham cohomology class by a harmonic differential form, a basic existence problem for harmonic maps is the following:

Homotopy problem: Given a map uo : M -, N is there a harmonic map u homotopic to uo? This question, as we shall see below, has an affirmative answer if the sectional curvature KN of N is non-positive (Eells-Sampson [38]), or if m = 2 and v2(N) = 0 (Lemaire (102], Sacks-Uhlenbeck [121]). However, for N = S2 and m = 2, we have the following counterexamples:

Example 1.3. (Lemaire [102], Wente [151]): If u: B1(0) C R2 -' S2 is harmonic It., then u coast. and U 10H, (o) =

Example 1.4. (Eells-Wood [39]): If u: T2 - 52 is harmonic, then degu # ±1. In higher dimensions (m > 3), hardly any result is known for the homotopy

problem unless KN < 0. However, there are various existence results for the Dirichlet problem.

Dirichlet problem:Suppose OM 96 0 and let uo: M -+ N be given. Is there a harmonic map u : M - N such that u = uo on 8M? The Dirichlet problem can be attacked using variational methods. By minimizing E among the class H,I,a (M; N) = {u E HI.2(M; N); u = uo on OM}

266

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

one obtains a weakly harmonic map u satisfying the desired boundary condition. Similarly, one could attempt to solve the homotopy problem by minimizing E in a given homotopy class. However, Lemaire's example shows that in general homotopy classes are not weakly closed in H1"2(M; N). This is made explicit by the following example, whose construction relies on the fact that the conformal group on S4 acts non-compactly on HI2(S2, S2).

Example 1.5. The mappings ua = v, I ° D,,

irp : S2 . S2,

where ap : S2 \ {p} - R2 denotes stereographic projection and Dax = Ax is dilation by a factor A, are homotopic to the identity id = ul : S2 -, S2. But

ua-'u00(x)P (A-'oo), weakly in H1.2(S2; S2). (Incidentally, by conformal invariance of E in dimension m = 2, all ux are harmonic!)

Regularity The direct methods in general only yield weakly harmonic maps. In fact, the (singular) map x --' of Example 1.2 is minimizing from B1(0) C Rm into Sm-I for its boundary values, if m > 3 (Brezis-Coron-Lieb [13], Lin [105]). A partial regularity theory for energy minimizing maps was developed by Schoen-Uhlenbeck

[123], [1241 showing that in general an energy minimizing map u is smooth on an open set whose complement, the singular set Sing(u), has Hausdorff-dimension < m - 3 and is discrete if m = 3, which is best possible. Even if we regard the map u(x) = >zf as a map u: B1(0) C Rn -* Sm-I C N = Sm C R"1+1, for example, whence there is no topological reason for a singularity, u is still minimizing if m ? 7 (Jager-Kaul [82], Baldes [61). Moreover, Hardt-Lin-Poon [71] have constructed

examples of minimizing harmonic maps u: B1(0) C R3 --# S2 with cylindrical symmetry whose boundary data u l aB, (e) : 8B1(0) 95 S2 -' S2 have degree 0 and such that u possesses an arbitrarily large number of singular points on the axis of symmetry, and Rivii re [119] has exhibited weakly harmonic maps R3 -+ S2 with line singularities. By contrast, if m = 2, energy-minimizing harmonic maps are smooth (Morrey [110], [111], Giaquinta-Giusti [51]). Moreover, Grater [66] proved smoothness of

conformal weakly harmonic maps. This result was extended by Schoen [122] to harmonic maps which are stationary with respect to variations of parameters in the domain, hence possessing a holomorphic Hopf differential IU"12 - 2iux . uY)dz2 (au)2dz2 = (IuxI2 in suitable conformal parameters z = x + iy on M. Finally Halein [72] recently has shown regularity of weakly harmonic maps in general.

Theorem 1.1 (Helein [72]). Let m = 2 and let u E H1.2(MN) be weakly harmonic. Then u E C' (M, N). Proof. For N = Sn-1, M = Bl(0) C R2 his proof exploits the equivalence of (1.2) and

n

-Du' _ E Vuf (u`Vu& - Ad Vu`), i = 1, ... , n, j=1

PART 1. THE EVOLUTION OF HARMONIC MAPS

267

because 0 = V1u12 = 2 E uj Vu'. He then observes that for any I< i, j < n there holds (1.7)

div(u'Vu7 - uuVu') = u'DuJ - u3 Du' = 0.

Hence there is a potential a'j E H1.2 such that

rot a'; = u' Dug - u' Vu' and (1.2) takes the form n

-Au' = E(u=ay - uyai ), j=1

where the right hand side is the sum of Jacobians of H 1.2-mappings. Continuity of u (and hence smoothness) then follows from results of Wente [150] and Brezis-Coron [12].

Realizing that (1.7) is a consequence of Noether's theorem and the symmetries of Sn-1, Helein then generalized this simple and beautiful idea to arbitrary target manifolds by an ingenious choice of rotated frame fields on u-'TN. Inspired by Htlein's result, Evans [41] for m > 3 has obtained partial regularity results for "stationary", weakly harmonic maps into spheres.

1.2. The Eells-Sampson result By Examples 1.3, 1.4 and 1.5 above we know that it may be difficult (if not altogether impossible) to solve the homotopy problem for harmonic maps by direct variational methods. To overcome these difficulties, Eells - Sampson [38) proposed to study the evolution problem

u1 - 0Mu = A(u)(Vu, Vu)M on M x [0, oo[ with initial and boundary data (1.9) u = uo at t = 0 and on 8M x [0, oo[ (1.8)

for maps u: M x [0, oo[-+ N C IR", the idea behind this strategy of course being that a continuous deformation u(., t) of uo will remain in the given homotopy class. Moreover, the "energy inequality" (see Lemma 1.1 below) shows that (1.8) is the t) for t - 00 (L2)-gradient flow for E, whence one may hope that the solution will come to rest at a critical point of E; that is, a harmonic map. For suitable targets, this program is successful.

Theorem 1.2 (Eells-Sampson [38]). Suppose M is compact, 8M = 0 and that the sectional curvature KN of N is non-positive. Then for any uo E COO (M; N) the Cauchy problem (1.8), (1.9) admits a unique, global, smooth solution u: M x oo suitably, converges smoothly to a harmonic map [0,oo[-+ N which, as t u0, E COO (M; N) homotopic to uo.

The proof uses three ingredients.

Lemma 1.1 (Energy inequality). For a smooth solution u of (1.8), (1.9) and any T > 0 there holds rT

E(u(T)) +

J0

[utI2 dvolM dt < E(uo).

268

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

Proof. Recall that A(u)(Vu, Vu) 1

Hence, upon multiplying (1.8) by ut

and integrating by parts, we obtain

I

Iutl2 dvolM + d E(u(t)) = 0

for any t > 0, and the desired estimate (in fact, with equality) follows upon integrating in t.

Lemma 1.2 (Bochner inequality). If K^' < 0, then for any smooth solution u of (1.8) with energy density e(u) there holds (1.10)

with a constant C depending only on the Ricci curvature of M.

Proof. To derive this estimate we use the equivalent intrinsic form

ut - trace(Vdu) = 0

(1.11)

of (1.8), where now u = (u',. .. , u') denotes the representation of u in suitable local coordinates on N. As in deriving (1.5) from (1.4) in the stationary case, we then conclude that (in normal coordinates around xo on M)

- Am) e(u) + IVdul2 + Rpuyo'uyfl = Rkjluxauyouyfluyd C

at (xo, t), where V, RQp, R,kil, respectively, denote the pull-back covariant deriv-

ative on T'M ® u-I TN, the Ricci curvature on M, and the Riemann curvature tensor on N. Prom this identity, the claim follows.

Note that in the parametric setting of (1.8) and for general targets, upon differentiating (1.8) in direction x' , multiplying by uZ> and summing over 1 < a < m, by orthogonality A(u)(Vu, Vu) I TuN we obtain (1.12)

t& - AM e(u) + lV2ul2 < CMe(u) +CN(e(u))2,

and CM, CN, respectively, denote Constants where IV2ul2 = depending only on the Ricci curvature of M and the second fundamental form of N. The final ingredient is Moser's [112] sup-estimate for sub-solutions of parabolic equations. Let

Denote PR(zo) the cylinder

PR(zo)={z= (x, t); Ix - xol< R, to-R2 0 in local coordinates on M. Note that y019 E C°O(PR(zo)) satisfies the uniform ellipticity condition AItI2

for any f E R"' with constants 0 < A:5 A, uniformly in z E PR(to).

PART 1. THE EVOLUTION OF HARMONIC MAPS

269

Lemma 1.3. Suppose 0 < v E C°°(PR(zo)) satisfies Lv < C1v in PR(zo), where C1 E R, R < 1. Then for some constant C2 = rC2(A, A, C1) there holds v(zo) < C2R-(M+2) /

v dz.

Proof of Theorem 1.2. Local existence can be inferred from the a-priori estimates for uniformly parabolic equations (see Ladyzenskaya-Solonnikov-Uralceva [101]) and the (standard) implicit function theorem; see Hamilton [69]. Let u E C°°(M x [0,T[; N) solve (1.8). By Lemma 1.10 the energy density is a subsolution to the equation (8t - OM)e(u) < Ce(u).

Let tM be the convexity radius on M. Choose R < min{ 1, VT, tw } and apply Lemmas 1.1 and 1.3 to conclude that e(u)dz

(e(u))(zo) <- CR-(,+2) J R (xo)

< CR-(m+2) f'.-R2 E(u(t)) dt 0_R < CR-mE(uo) for any zo = (xo, to), to > R2, where C = C(M). Hence IVul is uniformly bounded on M x [0, T]. By a boot-strap regularity argument we obtain uniform bounds for all derivatives of u up to time T in terms of the initial data uo, T, and the manifolds M and N. The solution thus can be continued as a smooth solution to (1.8)) on M x [0, oo[. The preceding argument, applied with to > 1, R = min{ 1, &M), then gives uniform a-priori bounds for u and its derivatives on M x [1, oo[ depending only on M, N and E(uo). Hence, by Arzela-Ascoli's theorem, the flow (u(., t))t>1 is relatively compact in any Ck-Topology. Finally, by Lemma 1.1, for any sequence tk) -' 0 in L2(M), whence a sub-sequence (u(-, tk)) converges tk -+ oo we have tk) is homotopic to uo through the flow smoothly to a harmonic map u... As

for any k, so is u.. If KN < 0, the uniqueness of u,,, and hence t) - u°O follows from a maximum principle due to

convergence of the flow Jager-Kaul [81].

0

The Eells-Sampson result was extended to harmonic maps with boundary by Hamilton [69]. The condition KN < 0 can be replaced by the condition that the image of uo (and hence of u) should support a uniformly strictly convex function; see Jost [861, von Wahl [149]. However, none of these results can be applied in case uo: T2 -. S2 has degree 1 and, indeed, Example 1.4 above shows that the flow (1.8), (1.9) in this case cannot exist for all time and converge smoothly as t - oo.

1.3. Finite time blow-up The question remains whether in this case the heat flow (1.8) will develop singularities in finite or infinite time. Reversing the order of the historical developments, we first address this question and later present the existence results for weak solutions that prompted these investigations.

The heat flow (1.8) is a quasi-linear parabolic system and therefore hard to deal with explicitly. With enough symmetry, however, (1.8) can be reduced to a

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

270

scalar equation in only two variables. Consider equivariant maps

uo(x) = (f1sinho(r)ecoslzo(r)) of B1(0) C Rm into Sm C Rm+l, where r = lxi, ho(0)= 0, and let u: BI(O) x [0, T[ -+ Sm C Rm+I be the corresponding smooth solution of (1.8), (1.9), defined on a maximal time interval [0, T[. By uniqueness, also u is equivariant and can be

written (1.13)

u(x, t) = (fEsinhr,t,cosh(r,t)in

terms of a smooth map h: [0,1] x [0, T[- R satisfying the initial and boundary conditions

(1.14)

h(r,0) = ho(r)

for

0:5 r < 1,

h(0, t) = ho(0) = 0

for

0< t:5 T,

h(1, t) = ho(1)

for

0 < t < T.

b

In terms of h, the equation (1.8), that is Ut - AU = IVui2u, becomes

(1.15)

ht - hrr - m r l h,.

2r2h

=

+s

0.

If m > 3, it was shown by Coron-Ghidaglia [28] that for suitable he the solution h of (1.14), (1.15)-and hence the solution u of (1.8), (1.9)-cannot be smoothly continued beyond some finite time. We will later see a deeper reason for this; see Theorem 1.11. The case m = 2 is more interesting. In dimension m = 2 a family of stationary solutions h of (1.15) with h(O) = 0 is obtained by stereographic projection from the

south pole; see Example 1.5. In terms of r = lxJ and the polar angle 0 = 0(r) the stereographic projection is given by ros0 1

r - sing=-r; that is, 4(r) = arccos

l

r2

G+r2)

Composing with a dilation r -' r/A, we obtain the family

a (r) _ t rA)

/A2-r2/

= arccos A2 + r2

A > 0,

of stationary solutions of (1.15), with 4A(0) = 0.

Theorem 1.3 (Grayson-Hamilton [61], Chang-Ding [15]). Let m = 2. Suppose 1hol < ir. Then the solution h of (1.14), (1.15) exists for all time. Proof. The idea is to construct a barrier, preventing h from becoming discontinuous in finite time. Smoothness of h then follows from general results on quasilinear parabolic systems; see [101].

PART 1. THE EVOLUTION OF HARMONIC MAPS

271

First now that Lemma 1.1 translates into the uniform eneru bound lhr12rdr < E(u(t)) < E(uo).

Ir 0

Hence, by Sobolev's embedding theorem, h(., t) is locally Holder continuous on 10,1]

uniformly in t, and a singularity can only develop at the origin. We assume 0 < ho < 7r for simplicity. Let hl > ho satisfy 2 < h1 < 7r, h1(0) = zr, ht (a) < 7r for some a E10, 11 That is, hl maps into the (convex) lower hemi-sphere. Therefore, equation (1.8) and hence (1.15) with initial data h(., 0) = h1 possesses a global, smooth solution h. Moreover by the maximum

principle, h(r, t) < Ir for 0 < r < 1, t > 0. Choose a strictly increasing function A(t) such that 0a(o) > ho on 10, a] and 0a(t) > h(a, t) for all t > 0. Let h(r, t)

_ mf{0a(t)(r),h(r,t)}, h(r, t),

0:5 r < a a
be our barrier. Note that h is a supersolution to (1.15). Similarly, h e 0 is a subsolution. Hence 0 < h(r, t) < h(r, t) on [0,11 x [0, T], by the maximum principle, and h(., t) is continuous, hence smooth, at r = 0 for any t > 0. Quite surprisingly, Theorem 1.3 is sharp.

Theorem 1.4 (Chang-Ding-Ye[16]). Suppose Ib) > >r. (1.14), (1.15) blows up infinite time.

Then the solution h to

Proof. Let b > ir. We show the existence of a sub-solution f to (1.15) with f (0, t) = 0 < f < f (1, t) = b such that fr(0, t) - oo as t -+ T for some T < oo. Letting ho = f 0) and h the corresponding solution of (1.14), (1.15), the maximum principle then implies that h > f on [0,11 x [0, T[. Consequently, also h must blow up before time T. (For general initial data the proof is somewhat more complicated.) We make the following ansatz for f : f (r, t) = ox(t) (r) + Wp (r1+f ),

where e > 0, µ > 0 and A = A(t) will be suitably determined. Note that for any e > 0 we have

µ2 - r 2+2e (r 1+f ) =arc cos ( µ2 r2+2.) _' 0 (µ -b oo) + uniformly in r E [0,11. Hence, given e > 0, we can choose µ > 0 such that

cos0,(r1+f) > 1 + for r E [0, 1]. e Next observe that for any µ, e > 0 the function 0(r) _ 0µ(r1+f) satisfies

n20=0. -0rr--or+(1+22 r 2

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

272

Hence

r(f)

f, + rfr

2f

s2r2 _ (- sin 2(0a + 8) + sin 20,, + (1 + E)2 sin 20)/(2r2)

_ (sin((20a + 0) - 0) - sin((20a + 0) + 0) + (1 + E)2 sin 20) / (2r2) _ (-2cos(20a+0)sin8+2(1+E)2Cog 0sin0)/(2r2) > (1 + e) - cos(20a + 0) sin 9

r2 1+E

> r2 sing = T2 u2

+r2+2f

for EI = p2 + 1 > 0.

EIr`-1

On the other hand, we have d

+2r

etf (r, t) = N0alt)(r) =

A2

r2

Let a(t) solve ar,\ = -6A`, 6 > 0 to be determined; that is

Then fr(0, t) _ oo as t - T = i Eo Finally 0 } A2+r2 f -T(f) E

E 2-f

<-E1r`-I

= [ 2+r2

But by Young's inequality afr2-` < C(E)(A2 + r2),

and hence f Is a sub-solution of (1.15) for sufficiently small 6 = 6(e) > 0, as desired. Since e > 0 is arbitrary, supl f 0) - x] can be made as small as we please. 0

Similarly, Chang-Ding-Ye [16] construct solutions of (1.8) for M = S2 = N that blow up in finite time.

1.4. Global existence and uniqueness of partially regular weak solutions for m = 2 Theorem 1.4 shows that in general smooth, global solutions to (1.8), (1.9) do not exist. However, is there maybe a "weak" analogue of Theorem 1.2 that will still provide a satisfactory means towards deciding the homotopy problem? The following result is due to Struwe [138]; the result was extended to the case OM 36 0 by Chang [14].

Theorem 1.5. Suppose M is a compact Riemann surface, possibly with boundary, N C R' is compact. Then for any uo E H1'2(M; N) with smooth "trace" uojam = u° Ianf for some smooth function tb E C°° (M; N) there exists a global weak solution u: M x [0, oo[-. N of (1.8), (1.9) satisfying the energy inequality and of class C°O on Mx]0, oo[ away from finitely many points (2k, 1k), 1 < k < K. The solution u is unique in this class. At a singularity (2, ), a (non-constant) harmonic sphere u: S2 - N separates in the sense that for suitable sequences xk 2, tk / t, Rk \ 0 the rescaled maps uk(x) = u(xk + Rkx,1k): Dk C RZ --+ N

PART 1. THE EVOLUTION OF HARMONIC MAPS

273

(in a local conformal chart around x) converge in H (R2; N) to a non-constant harmonic limit u: R2 -+ N. it has finite energy and extends to a smooth harmonic

map ii: S2^='R2-+N. Finally, for a suitable sequence tk -+ oo, the sequence tk)) converges weakly in H12 (M; N) to a smooth harmonic map u.,,: M - N. The convergence is strong away from finitely many points x0°, 1 < 1 < L, where again harmonic spheres separate in the above sense, and there holds K + L < ea 1E(uo), where

co = inf {E(u); 6: S2 - N is a non-constant smooth harmonic map) > 0 is a constant depending only on N. The proof of this result is based on the following ingredients.

Lemma 1.4 (Ladyienskaya [100] Lemma 1, p.8). For any v E H01'2(R2) there holds v E L4 (R2) and IIvII

IIVvIIL4 <4

IIvII L2

La.

The same estimate holds for L2-functions v on a half-space with Vv E L2. (See also Friedman [48] or Struwe [144]; Lemma 3.5.7].) We now proceed with the proof of Theorem 1.5.

Positivity of eo: Applied to v = IVul4, where u solves (1.1) and 0 E C°°(S2) is a smooth cut-off function 0 < 0:5 1 with support in a coordinate neighborhood on S2, Lemma 1.4 gives (1.16)

f


fS In particular, if (Oj2) is a smooth partition of unity subordinate to a finite cover (Uj) of S2 by local coordinate charts U;,1 < j < 3, we obtain from (1.2), (1.16) and the Calder6n-Zygmund inequality that fS2IV2uI2 dvolss < C fV IouI2 dvolS2 + C L Iu12 dvols2


fs

2IVU14dvols2 +C

< CE(u)(f IV2uJ2 dvolss + E(u)) +C with constants C = C(N). Hence the number eo, defined in Theorem 1.5, is strictly

positive. Indeed, if E(uk) - 0 for a sequence of harmonic maps uk: S2 -+ N we obtain that Uk . conat in H2,2 (S2; N) `- Co(S2; N). In particular, uk(S2) lies in a convex geodesic ball on N for large k. Contradiction. L2-estimates for V2u: By (1.16) it is important to control the energy locally.

For ti c M denote E(u; fl) = f e(u) dvolM. n

Lemma 1.5. Let u: M x [0,T] - N be a smooth solution of (1.8) (1.9). Let xo E M and let BR(xo) be balls in a local conformal chart around xo, 0 < R < 2Ro. Then for R < RD there holds E (u(T ); BR(xo)) :5 E(uo; B2R(xo)) + CWy E(uo),

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

274

with C = C(M, N).

Proof. Let' E Co (B2R(xo)) satisfy 0:5 0:5

1, 0 =- 1 on BR(xo), IVOI 5 2-R, and

test (1.8) with ut42 to obtain

dx +at Um

IM

e(u02
Hence

e(u)02 dx) < CR-2E(u(t)) S CR-2E(uo),

dt (L, and the lemma follows by integration.

0

In particular, given el > 0, uo E H1'2(M; N) there exists a number TI > 0, depending only on a maximal number RI > 0 such that sup E(uo; B2Ri (x0)) < el

soEM

and the geometry of M and N, with the property that any (smooth) solution u of (1.8), (1.9) satisfies

sup

E(u(t); BR, (xo)) < 2el.

xoEM,0
Indeed, we may let TI = cE uo where C is the constant in Lemma 1.5. Similarly, given ei > 0 let RI > 0 be determined as above and let 4i be smooth cut-off functions subordinate to a cover of M by balls B2R, (xi) with finite overlap and such that 0 < 0; < 1, I00;1 < R, Ei ¢? = 1. Then by (1.16) we have , +

f

IVU14 dV01M

M

=

jIVuI4odvo1M

< Csup E(u(t); B2R, (xi))

M102U(t)12 dV01M + R1 2E(u0))

< Cc, (fu IV2u(t)I2 dvolM + R12E(uo)) for any t. Moreover, similar to our Bochner-type estimate (1.12), upon multiplying (1.8) by Amu and integrating by parts we obtain at (E(u(t))) + JM IOMUI2 dvOlM < C

Iou14 dvolM.

fM

Upon integrating over [0, TI I and combining the above estimates and the CalderdnZygmund inequality, we obtain that

1T11 _< CEI

f 0

C

M T,

f t Im

1+IIu011..=(M))

0

f 1V2u(t)12dvolll +C(1 f

+TIR-2 )E(uo)

PART 1. THE EVOLUTION OF HARMONIC MAPS

275

with constants C = C(M, N). Thus, for sufficiently small el > 0 we obtain an a-priori bound for u in the norm T V2

IIuII V

(M) =

osup

E(u(t)) +

IM (IV2u12 + 1utI2) dvolM dt

with T = T1, of the form IIUIIvr: (M)

< C (1 +

\

R I E(uo) + CT1II oII3p.1 1

Without proof we remark that VT (M) is a regularity class for (1.8) in the sense that, if u E VT(M) solves (1.8) with finite energy initial data uo E H1"2(M;N) and coinciding with a smooth function uo on OM, then u E C°O(MxJO, TJ; N). Local existence: R1 > 0 can be chosen uniformly for a set of initial data which is compact in {uo} + Ho'2(M; N). In particular, if uom E COO (M; N) converges to uo in H1'2(M;N), and if (um) is the corresponding sequence of local solutions (1.8) for initial and boundary data (uo,,), by the above a-priori estimate we have IIUrII12,TI(M) < C (E(uo) + 11411y..2) and the sequence (um) weakly accumulates at a function u E VT1(M). By Rellich's theorem, moreover, VUm --+ Vu in L2(M)

for almost every t, and it is easy to pass to the limit in equation (1.8); by the preceding remarks about regularity, in fact, u solves (1.8) classically in Mx]O,Ti]. Finally, by Lemma 1.1, u achieves its initial data continuously in H1 ,2(M; N). Uniqueness: The space of functions with bounded VT (M)-norm is a uniqueness

class. Indeed, if u, v E VT (M) weakly solve (1.8) with u(0) = uo = v(0), their difference w = u - v satisfies Iwt - AMWI < CIwI(IVu12 + IVvI2) + CIVWI(IVuI + IVvI).

Testing with w and integrating by parts, we obtain for almost every t > 0

2 JM

Iw(t)12 dvolM +

0G

r

J

fM

fM1w12(1Vu12

C ft

ff

ds

+ IVv12) dvolM ds

t

+C

0

(ft

IwIIvwl(IVul + IVvl) dvolM ds

M

1/2

(ft

lAS Iw14 dvolM d/

(ftrM

+C

1 4

1w14

dvolM

\ O//

ds/)

( pt

) f (1u1+ IVvI4) dvolM ds M

(o

t

)'/2

\Jo fu

IVwI2 dvolM ds 1/4

fM(IVu14 + LVvI4) dvolM ds)

)1/2

('

< C e(t) I (fo, fM

14 dvolM

+

L

< Ce(t)

[su p

f Iw(s)12 dvolM + f

t 0

fo,

fMIhh11201M ds I Vw12

fxf

dvolht ds0
J

276

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

0 as t -+ 0. Choosing t > 0 small enough

on account of Lemma 1.4, where e(t)

and such that IIw(t)IIL2(M) = sup0
Global continuation: The local solution u constructed above can be extended until the first time T = t1 such that limsup 1

supE(u(t);BR(xo))' > El

t1T \xo

for all R > 0. By Lemma 1.1, ut E L2(M x [0,71). Hence, the L2-limit ul = limtn u(t) exists. Let v be the local solution of (1.8) with initial data v = ul at time T and boundary data uo. The composed function

w(t) =

0
ru(t), v(t),

then is a weak solution of (1.8). By iteration we obtain a weak solution u on a maximal time interval [0,'P[. If T < oo, the above arguments again permit us to extend u beyond T, contradicting our assumption that T was maximal. Hence

T=oo. Finiteness of the singular set. Let t = l > 0 be the first singular time, and let Sing(l) = {xo E M;VR > 0: limsupE(u(t);BR(xo)) > e1}. t/i

Sing() is finite. Indeed, let x1, ... , xK E Sing(). Choose R > 0 such that B2R(xi)f1 B2R(x f) = 0 (i j), and fix r E [t - 2 °Eu , f[, where C is the constant in Lemma 1.5. Then by lemma 1.5 K

Ke1 <

lun sup E(u(t); BR(xi)) K

:5

i=1

(E(u(T ); B2R(xi)) + 2 /

E(uo) + ZE1

and K = K1 = #Sing(t1) < 2E(uo)ei 1. Moreover, for u1 = lime/=, u(t) we have E(u1) = Rlimm E

(ul;M\IJB2R(x4)) K,

< lim lim sup E I u(t); M \ U B2R (xi) R-»0

\

t/t,

i=1 Kt

E(u(t); B2R(xi))

< lim lim sup ( E(u(t)) R-.0

t/li \

i=1

Ki

< Rim (E(uo) -11 inf > E(u(t); B2R(xi)) 1=1

K,

< E(uo) -

lim limsupE(u(t);BR(xi))

R-.O t/l, < E(uo) - Kiel. i=1

PART 1. THE EVOLUTION OF HARMONIC MAPS

277

Similarly, let K2, K3, ... be the number of concentration points at consecutive

times t2 < t3 < ..., and let uj = limt,F, u(t) for j = 2,3,... Then by induction we obtain

E(uj) S E(u,-1)-Kje1 !5 ...

SE(uo)-(Kl+...+KK)ei,

and it follows that the total number K of concentration points, hence also the number of concentration times tj, is finite; in fact, K < E(uo)el 1 Smoothness: Let t = Ij for some j. To see that u is smooth up to time t away from Sing() we present an argument based on scaling, as proposed by Schoen [122] in the stationary case. By working in a local conformal chart we may assume that M is the unit disc B or half disc B+ = {(xI, x2) E B; x2 > 0};

moreover, by scaling we may assume t > 1. Finally, we shift time so that t = 0. The solution u then is defined on a domain containing M x [-1, 0]. For R > 0, Zo = (xo, to) denote

PR )(zo)_{z=(x,t);xEB(+),Ix-xol
N and r > 0, x E R2 we define E(u; B,- (x)) := E(u; BB(x) fl 0).

Proposition 1.3. Suppose U E C°°(P(+); N) solves (1.8) with smooth boundary data fo. There exist constants C, e2 > 0 depending only on N and rio such that, if sup E(u(t); B) < E2, -I
then there holds

supjDu(x,t)I
i

(and corresponding bounds for higher derivatives).

Proof. Choose 0:5 p < 1 such that (1 - p)2 Sup e(u) =

{(1 - a)2 Sup e(U)

and let zo E PP+) satisfy

(e(u))(zo) = sup e(u) =: eo. Then either eo(1 - p)2 < 4, in which case sup e(u) < 4(1 - p)2eo < 16, P(+)

or eo 4 < 1-o . In the latter case, the scaled function

v(x,t) = u(xo + eo 4x,to + eolt) is well defined on D = P, respectively on

D+={z= (x, t)EP;xo+eo'ix2>0}

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

278

with boundary data vo(x) = uo(xo + eo ix). Moreover, (e(v))(O) = 1 and

(1 - p)2 sup (+) e(u)

sup e(v) < eo sup e(u) < eo

=)22°

p(+)

D(+)

= 4.

( 2

ip

Thus, v satisfies a linear differential inequality

lvt - Ovl < CIVvI, and from the 17-theory for the heat equation, with p > 4 we get

D(+)np/ sup IVVI < C UDW IOvl pdz

1<

< C2 SUP (E (u(t); B(+))) t J

I/n + C (eo

l

1/p

+C (IIDvolILo + IIV2vollLo) + eo 1) J

;

see [101]. That is, eo < C if C2E2/p < 2. (If D = P, we can use Bochner's inequality 0 (1.12) and the sup-estimate Lemma 1.3 to achieve a slightly simpler proof.)

Blow-up of singularities: As in Struwe [138] we now use the scaling technique to analyze the singularities in more detail. Let (2,1) be a singularity of the solution u constructed above. Introduce a local conformal chart around 2 and shift time so

that (2,0 = (0,0). Moreover, after scaling we may assume that u E C- (P(+) \ {0}; N) where PR )(zo) is defined as above. Let (Rk) be a sequence of numbers Rk E]0,1[, Rk \ 0 (K -' oo). Let EI > 0 be as above and define sequences (xk), (tk) such that xk

2 while

E(u(tk); BRk (xk)) =

EI

sup

E (u(t); BRk (x)) = L (= t)E p(+); -1
where L is the number of unit discs needed to cover B2(0). We may assume tk 4R2k > -1. Scale Vk(x, t) = u (xk + Rkx, tk + Rkt) ,

vk : Dk+) = {(x, t) E R2 x [-4, 0]; (xk + Rkx, tk + Rkt) E P(+)) - N. Note that by Lemma 1.1 we have ftk

r lutl2 dvolM dt

I(vk)t12 dx dt < /tk

J

fD(+)

0

(k -. oo),

Rk JM

E(vk(t)) < E(uo),

-4 < t < 0,

k E N.

Moreover, we have

sup

E(vk(t);B2(x)) <_ L

(=.t)EDk+);

E(vk(t);BI(x))

sup (x.t)ED4+).

-4
-4

sup (x t)Ep(+);

E(u(t); BRb(x)) < LE (u(tk); BRk(xk)) = E1.

-I
PART 1. THE EVOLUTION OF HARMONIC MAPS

279

Him (R+ x [-1, 0); N) to a smooth solution v of (1.8) on R2 x [-1, 01 or R. x [-1, 0], respectively. Moreover, vi = 0, whence v(., t) =: u is, in fact, harmonic. Finally

E (u; B) = kym E (vk (0); B) =

k-00

E (u(tk); B,,,, (x)) >

e1

L

and u is non-constant. If u: R2 -+ N, by conformal equivalence R2 S2 \ {p}, u induces a weakly harmonic map u : S2 -+ N. By Helein's result u is smooth. If

u: R. -+ N, we also have filaR, = const. = uo(t). Since R+ is conformal to the unit disc B, u induces a non-constant harmonic map u: B -+ N with ulas = cont. But this is impossible by Lemaire's result, Example 1.3. Thus, singularities at small scales look like harmonic spheres. In particular, we obtain the estimate el > E. A similar analysis is possible at concentration points at "t = oo"; see Struwe [144), Lecture 111.5, for details. Moreover, it seems possible to iterate the above procedure and decompose u in the limit t / I into its weak limit u() and a finite sum of harmonic spheres u19 .... uL, similar to Struwe [137] for a related problem, and in such a way that energy is conserved; that is, L

E(u(t)) +

E(ul) = lim E(u(t)). 1=1

Applications As a first application we present a proof of the following theorem by Lemaire [102], and Sacks-Uhlenbeck [121].

Theorem 1.6. If dim M = m = 2 and if 7r2 (N) = 0, then any map uo: M -. N is homotopic to a smooth harmonic map.

Proof. Let to = inf{E(u); u: S2 --+ N is a non-constant, smooth harmonic map} > 0 as above. We may assume that uo E C°°(M; N) and

E(uo) < inf{E(u);u E C°O(M,N), u is homotopic to uo} + 4 Let u: M x [0, oo] N be the solution of (1.8), (1.9) constructed in Theorem 1.5 above. We claim that u is globally smooth and converges smoothly as t - oo to a harmonic limit u°Of solving the homotopy problem. Thus, assume by contradiction that u develops a first singularity at a point (x, 0, t < oo, and let Uk(X) = u(.t + Rkx,tk)

be a suitably scaled sequence for tk / d, Rk \ 0, defined on larger and larger balls in R2 and converging in Hoc (R2; N) as k -+ oo to a non-constant, smooth harmonic map f : R2 - S2 \ {p} -+ N. Replace the "large" part of Uk - u by the "small" part of u as follows. Fix p > 0 such that

E(u; R2 \ B,(0)) <

L<

2 < E(u* Bp(0)) = lim supE(Uk; Bp (0))

k-x

and choose ¢ E Ca (R2) such that 0 - 1 on B,(0). Then, as k -' oo we have

vk :=Ouk+(1 -0)u-ii

in

H2.2(R2;N)

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

280

and uniformly. Let ir: U6 -> N be the smooth nearest-neighbor projection of a tubular neighborhood Ua of N in R" and let wk = 7r vk. Then wk E C°°(R2; N) and wk - u in H' 2 (R'; N). Invert wk along the circle of radius p to obtain 4Dk(x) = Wk

(zj)

.

Then by conformal invariance of Dirichlet's integral

lim upE(zuk; B,(0)) = limsupE(wk; R2 \ Bp(0)) = E(ii;R2 \ B,(0)) < 4

k-00 Moreover

k-00

wk = wk = Uk

on BB,,

and upon replacing uk by wk on Bp(0) for some sufficiently large k we obtain a new

map v:M -Nsuchthat

E(u(tk)) - E(v) = E(uk; Bp(0)) - E(wk; B,,(0)) ?

4

That is,

E(v) < E(u(tk)) - 4 < E(uo) - 4 < inf {E(u); u N u0}. On the other hand, since a2(N) = 0 by assumption, u(tk) and v are homotopic. The contradiction shows that the flow cannot develop singularities in finite or infinite time. Hence the proof is complete. As a second application we present a key step in the proof of a result of MicallefMoore on a generalization of the Berger-Klingenberg-Rauch-Topogonov sphere theorem.

Theorem 1.7. Let N be a compact, simply-connected, n-dimensional Riemannian manifold and let k > 0 be the first integer such that ak(N) 96 0. Then there exists a non-constant harmonic sphere u: S2 - N having Morse index MI(u) < k - 2.

Recall that the Morse index of a non-degenerate critical point x of a C2functional f on a Hilbert space X is the maximal dimension of a linear sub-space

V c X such that d2f(x)I vxv < 0. Proof. Represent Sk = {

n) E R3 x Rk-2;

ItJ2

+ In12 = 1 }

S2 x Bk-2, where Bk-2 = BI (0; Rk-2) and S2 x {n} is collapsed for n E 8Bk-2. Let ho : Sk N represent a non-trivial homotopy class [ho] E irk(N). With respect to the above decomposition of Sk, for every n E Bk-2 this induces a map

uo(rl) = ho(.,rl): S2 - N. Note that u0(rl) - coast. for i E 8Bk-2. Let u(.;,1) be the corresponding solutions to the Cauchy problem (1.8) (1.9). First suppose supE(uo(rl)) < eo. n

PART 1. THE EVOLUTION OF HARMONIC MAPS

281

Then by Theorem 1.5 the flows q) are globally smooth and converge smoothly to constant maps as t - oo. The convergence is uniform in q. Indeed, for any t > 0 let µ(t) = SUPE(u(t;q)) > 0-

Note that the map t F-+ p(t) is non-increasing by Lemma 1.1. For a sequence tt - o 'I

select in E BI-' such that E(u(tt; qt)) = sup E(u(tt; q)). n

Then a sub-sequence qt -a any t < oo we have

and, by locally smooth dependence of u(.; q) on q, for

E(u(t;rh)) E(u(t;fl)) = tlim 0.0 For large I there holds t < ti, and Lemma 1.1 gives

E(u(t; qt)) >_ E(u(ti; qt)) = p(tt) Hence

slim µ(t) < Jim E(u(t; f )) = 0. 00

Thus E(u(t; q)) -+ 0 as t -+ oo, uniformly in q E BI-'. By Proposition 1.3 this implies smooth convergence. But then for large t the map ht : Sk ^-- S2 x Bk-2 9

(t, q) - u(e, t; q) E N is homotopic to a map h: (t;, q) '--+ u(q) E N where u E C°°(Bk'2, N). Since Bk-2 is contractible, u and therefore also ho is homotopic to a constant map, contradicting our assumption about he. Hence

ek = inf sup E(h(., tj)) J co > 0, h ho

17

where we take the infimum over all h homotopic to he. Consider the set ? t = {u : S2 -+ N;

u is non-constant, smooth and harmonic,

E(u) < 2ek}.

?{ is compact modulo separation of harmonic spheres u E W. Suppose any u E ?t has Morse index > k - 2. Then for each such u there exists a maximal sub-space V E H2.2(S2; u-1TN) of dimension > k - 2 such that d2E(u)lvxv < 0.

Moreover, there exist numbers po > 0, e > 0 with the following property: If a: U --+ N denotes nearest-neighbor projection in a tubular neighborhood U of N onto N, and if we denote

Vp = {tr(u+v);v E Bp(0;V)} C H2'2 (S2;N) then

sup E(u) =: µp uEBV1

is strictly decreasing for 0 < p < po and iz,,

<_ E(u) - e.

By the above compactness property of ?t, the numbers a and po can be chosen independent of u. Hence any path in H2.2(S2; N) of dimension < k - 2 that approaches an element u E ?t can be deformed within the corresponding set Va, to a path of strictly lower energy.

282

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

Now suppose ho is chosen such that sup E(ho(., n)) < Ek + 2 n

and let u(.; q) be the corresponding family of solutions to (1.8) with u(0; rl) _ rl). If for some fj E Bk-2 the flow u(.; fl) converges to a nontrivial limit, or if i1) develops a singularity at (z, 0 or some t < oo, a harmonic sphereu forms. By a deformation of the path n 1-4 u(t; il) fort < t close to t we achieve a path 17 -, u(t; n) homotopic to u(t; n), hence to ho, with E (u(t; ?J)) < Ek

for r) near il. By a covering argument we thus find a comparison path hl N ho such that

supE(hl(.,ri)) < Ek, 11EB

contradicting the definition of Ek.

Next Micallef-Moore establish that a strict point-wise 4-pinching condition

4x(p) < K(o) < c(p) for all two-plane sections v C TAN, p E N, where K denotes the sectional curvature, implies the lower bound MI (u) >_ 2

-2

for the index of any non-constant harmonic 2-sphere in N. In particular, if u is a harmonic sphere as constructed in Theorem 1.7, we obtain

k>2+ that is, Hk(N) = lrk(N) = 0 for 0 < k:5 R. By Poincare duality

Hk(N)=0,

0
Moreover, if k > 0 is the first integer such that 7rk(N) = 0, by the Hurewicz isomorphism theorem we also have 7rk(N) °-` Hk(N).

Hence 7rk(N) = 0 for all 0 < k < n, and N is a homotopy sphere. If n > 4, by the resolution of the generalized Poincare conjecture therefore N is homeomorphic to a sphere.

Extensions and generalizations Theorem 1.5 has been extended to target manifolds N with boundary by ChenMusina (201. The same technique can be used to study evolution problems related to other two-dimensional variational problems. For instance, in Struwe (1391, Rey [118] the evolution problem for surfaces of prescribed mean curvature is investigated; Ma Li (1031 has studied the evolution of harmonic maps with free boundaries.

PART 1. THE EVOLUTION OF HARMONIC MAPS

283

1.5. Existence of global, partially regular weak solutions for m > 3. Earlier we observed that singularities must be expected even for energy-minimizing weakly harmonic maps, if m > 3, and hence for the evolution problem (1.8). The following result was obtained by Chen-Struwe [21].

Theorem 1.8. Suppose M is a compact m-manifold, 8M = 0. For any uo E H1'2(M; N) there exists a distribution solution u: M x [0, oo[- N of (1.8),(1.9), satisfying the energy inequality and smooth away from a closed set E such that for each t the slice E(t) = E n (M x {t}) is of co-dimension > 2. As t -' oo suitably, u(t) converges weakly to a weakly harmonic limit u°° which is smooth away from a closed set E(oo) of co-dimension > 2.

Originally, the estimate on the co-dimension of E was obtained in space-time, the above improvement is due to X. Cheng [23]. For manifolds M with boundary OM 0 0 a similar existence and interior partial regularity result holds; see Chen [18]. Boundary regularity is open. The proof of Theorem 1.8 rests on two pillars: A penalty approximation scheme for (1.8), developed independently by Chen [17], Keller-Rubinstein-Sternberg [93] and Shatah [127], and a monotonicity estimate for (1.8), due to Struwe [140].

Penalty approximation Consider the case N = the Cauchy problem

S"-1

C R". Given u0 E H1"2(M;Sn-1), K E N consider

ut - OMu + Ku(IIu1I2 - 1) = 0,

(1.17)

(1.18)

ult=o = uo

for maps u: M x [0, oo[-+ R. That is, we "forget" the target constraint and regard all maps u: M - R" as admissible; however, we "penalize" violation of the constraint Iu12 = 1 more and more severely, as K -+ oo. (1.17) is the L2-gradient flow for the functional

EK(U) = E(u) + K r

JM

- 1)2 dvolM.

(Iu12 4

Indeed, we have

Lemma 1.6. If u E C°0(Mx]0,T];R") solves (1.17), (1.18), then there holds

EK(u(T)) + fT f I8ut2 dvolMdt = EK(uo) = E(uo); 0

M

in particular, u attains its initial data continuously in H'2(M; N)

Proof. Multiply (1.17) by ut and integrate to obtain the energy estimate. Since Btu E L2(M x [0,71), clearly u(t) -, uo in L2(M) and weakly in H'.2(M, N) as t -' 0. Since also limsoupE(u(t)) < limsoupEK(u(t)) < E(uo),

t-

t-

we, in fact, also have strong H1,2-convergence.

Moreover, we have an L°° a -priori bound.

O

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

284

Lemma 1.7. If u E C°°(Mx)0,7);R") solves (1.17),(1.18), then IIUIItoo <<< 1

Proof. Multiply (1.17) by u to obtain

(d - 0) Hz + IVu12 +K(Iui2 -1)Iui2 = 0; in particular, C d - A + 2Klul2) (Iu12 - 1) < 0. dt The claim now follows from the parabolic maximum principle, since Iu(0)I = luol <

0

1.

Thus for any K E N we have a unique, global, smooth solution UK of (1.17), (1.18). Moreover, II8tUKllL3(Mx(o.oo() < E(uo),

IUKI < 1,

supE(UK(t)) < E(uo),

SUPIIIUK(t)I2 -1IILz(M) <

4EK)

o.

Thus, passing to a sub-sequence, if necessary, we may assume that

a

UK` u in L (M x [0, oo[; R"), (1.19)

VUK -+ Vu weakly-* in L°°([O,oo[;L2(M)),

AUK - 8T weakly in L2 (M x [0, oo[), and Jul = 1. Relations (1.19) are not sufficient to pass to the limit K -+ oo in (1.17), directly. In case N = S"-I, however, a clever manipulation of (1.17) will do the trick.

Lemma 1.8. Suppose N = S"-i C R". Then (1.8) is equivalent to the relations (1.20)

Jul = 1

and (1.21)

utAu-div(VuAu)=0,

where we identify the vector fields u, ut, etc. with one forms in R" and where "A" denotes the exterior product.

Proof. For smooth u satisfying (1.20) equation (1.21) is equivalent to asserting

ut - Amu 1 which in turn is equivalent to (1.8); that is, ut-Au = JVuI2u. If u is a weak solution of (1.8), moreover, an approximation argument justifies testing (1.8) with u n ii, where 0 E Co (M x [0, oo[; n2(R")), whence we obtain (1.21). Conversely, suppose u weakly solves (1.20), (1.21). Note that by (1.20) for any r E Co (M x [0,oo[;R) the relation z d l (1.22) (Ut - 011 - IV1/I211) UT = { (dt - 0) 12 + IVuI2 r=0 IVu12IU12}

is automatically satisfied. Moreover, by (1.20) again, for any pair of vector fields v, 0 E C'° (Af x [0, oo(; A1(R")) there holds

v-0 =(vu)(u-,)-*[(vAu)s(uA)]

PART 1. THE EVOLUTION OF HARMONIC MAPS

285

almost everywhere on M x [0, oo[, where "*" denotes the Hodge star-operator and where again we identify vector fields and 1-forms. By an approximation argument,

ip = *(u A ¢) and r = u 0 are admissible as testing functions in (1.21), (1.22), respectively. RYom the resulting equations, we obtain the weak form of (1.8).

0

Now, taking the exterior product of (1.17) with UK, the nonlinear term from (1.17) drops out and we obtain (1.21). Because of the divergence structure of this equation and since by (1.19) we have

AUKAuK -AuAu VUK A UK -: Vu A u

weakly in Li (M x [0, oo[) we may pass to the limit K --+ oo and find that also u satisfies (1.21). That is, by Lemma 1.8, u is a weak solution of (1.8), (1.9). Note that the geometry of the sphere was used in an essential way. Moreover, the regularity of the approximating maps UK may be lost in the limit.

The monotonicity formula Monotonicity estimates first were introduced as a tool in regularity theory for minimal hypersurfaces. Schoen-Uhlenbeck [123] and Giaquinta-Giusti [52] observed that similar estimates hold for energy-minimi in harmonic maps and can be used to obtain partial regularity. Consider for instance a harmonic map u: B = B1(0; Rm) -> N.

Theorem 1.9. If u is energy-minimizing (for its boundary values), then for any 0 < p < r < 1 there holds p2-m

r J

BProof.

Ivu l2dx < r2m L..17u12

Note that the quantity (D(P) =

2p2-m j IVul2dx B,

is invariant under scaling u --+ uR(x) = u(Rx), which (in case M = B) also leaves (1.1) invariant. This observation allows us to give a simple proof of Theorem 1.9 for smooth harmonic maps, as follows. Note that ob(p) = 0(p;u) = t(l;up). Hence, for instance at p = 1, we have -

r Vu (1;u) = JB

f (x Vu) B

dp

'

(u)7

&u. d up dx.

udo B

WP

Since apup = x Vu E T"N, by (1.1) the last term vanishes and the boundary integral simply becomes dP

proving the theorem for smooth u.

LBI x Vul2dw > 0,

0

Moreover, since gy(p) scales as a dimension-less quantity, smallness of 4,(p) for some p > 0 yields a-priori bounds for u near the origin.

286

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

A similar result holds in the time-dependent setting. For simplicity we consider

smooth solutions u E CO°(Rm x [-1,0[;N) of (1.8) with E(u(t)) < Eo < 00 uniformly in -1 < t < 0. Denote

G(x,t)=

2

1

(

4 xt

)

'exp(-4itll, t<0,

the fundamental solution to the backward heat equation on R"' x R. Using G as a weight function, we define

,t(p)

= 1p2 f

Then we obtain:

Theorem 1.10 ([140]). For u as above and any 0 < p < r!5 1 there holds 4'(p) < fi(r). Proof. As in the stationary case, we use invariance of (1.8) under scaling u -. uR(x, t) = u(Rx, R2t). Note that t(p) = $(p; u) = 0(1; up). Hence, at p = 1 we compute

dpdp`g(P) = dpfi(1;up) = JRmx(_1} vu. v (dP) Gdx. Integrate by parts and use (1.8) and the relation VG(x, t) = ie G to obtain

dfi

dp (P) = JR*x{-l}

Du 2t

2tut + x Vu

(1.23)

dP

d dP P

R'^x{-1}

2 R^x{-1}

/

uGdx

uPGdx

12tut + x VU12Gdx > 0.

Since E(u(t)) < Eo < oo no boundary terms appear.

O

Remark 1.1. It was pointed out by R.Kohn that (1.23) is the energy inequality for (1.8) in similarity coordinates s = - logjtl, y = as introduced by Giga-Kohn [53] for a different problem. By a scaling argument as in the proof of Proposition 1.3, smallness of 4'(p) can be turned into an a-priori gradient bound for u.

Proposition 1.4 ([140]). There exists co = eo(m, N) > 0 such that for any solution u E C°°(Rm x [-1,0[;N) of (1.8) above, if fi(R) < to for some R > 0, then

supIvul <

CR

Pae

with constants 6 = 6(m, N, Eo) > 0 and C = C(m, N, Eo).

Proof. Scaling with R, we may assume R = 1. For 6 > 0 fix p E]0,6[, zo = (xo, to) E Pp such that

(6 - p)2 sup e(u) = omax6{(6 - v)2 sup e(u)}, Pp P. (e(u))(zo) = supe(u) = eo. PP

PART 1. THE EVOLUTION OF HARMONIC MAPS

287

First assume eo 1 < (L-2)' and scale v(x, t) = u(xo + e- 1 /2x, to + eo 1 t). Note that v E Coo (P; N) and sup e(v) = eo 1 P

e(u) < eo-1 sup e(u)

sup P.o _112(zo)

P§4.

< 4eo' sup e(u) = 4, Pv

while

(e(v))(0) = 1. By the Bochner inequality (1.12) therefore we have d

in P

dt

and Lemma 1.3 gives

r

1 = (e(v))(0) < C fP e(v)dz = Ceo

12

f

e(u) dx dt.

Denote G1 (z) = G(z - x) the fundamental solution with singularity at z = (x,1) and choose z = zo + (0, eo 1). Then

1
JP_j(zo)

e(u)G, dx dt,

0

and by applying Theorem 1.10 for each t E [to - eo ',to], the latter is


JR"x{-1}

e(u)G dx.

Now IzI < 6, Il < 62. Thus at t = -1 we may estimate

IGx-GI

<

4Er

Q1 Viit II42) -exp(-411+1/

< Cb,

and we obtain that

r 1 < Cb

JRmx{-1}

e(u) dx + C

JRmx{-1} e(u)Gdx < C16Eo + C1 4i(1)

with a uniform constant C1 = C, (m, N). Choosing 6 = 2C1 Eo , co = 21 , the above inequality will lead to a contradiction. Thus (b - p)2eo < 4, and we have

sup e(u) < 166-2. P4

The proof is complete. As an application we establish the following result from [140].

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

288

Proposition 1.5. Suppose uk E C°O(Rm x [-1,0[;N) is a sequence of solutions to (1.8) with E(uk(t)) < Eo < oo uniformly in t for k E N. Moreover, suppose uk(-1) -+ uo i n H , '. 2. (k -' oo) and u in L (Rm x [-1,0]),

uk

in L2 (R' x [-1,0]), Vuk -' Vu in L2J(Rm x [-1, 0J). Btu

Auk

Then u weakly solves (1.8),(1.9) and u is smooth away from a closed set E of codimension > 2; moreover, for any R > 0 we have

4i(R) + J 1R2 Jm 12tut 1t1 VU12Gdzdt < 41(-1).

(1.24)

Proof. Let E = n {z;liminf tz(R) > eo}, k-00

R>0

where 1

and 4

if 4' o(r) = r2 J to - r2 > -1, IVuk12Gz° dx, 2 "'x{t0-r3} e = k( 1 - Ito , else. E is relatively closed. Indeed, if 2 E E, let

zt E E, zt -. 2. By definition of E we have iminf lim inf i-oo

( 2R2 JRmx{t-R z} 1VukI2G=, dx > to

for any R > 0. Since G, -. Gr uniformly away from z and since E(uk(t)) < E0 < oo, the limits l - oo, k - oo may be interchanged for fixed R > 0, whence lim inf 4iz (R) > to,

k-oo for all R > 0; that is, z E E. Next observe that for zo f E there is a sequence (uk) and some R > 0 such that 0 u (R) < to. Proposition 1.4 implies that C

sup Mud <-

R

P6R(zo)

for 6 = S(m, N, E0) > 0, uniformly in k, and similar bounds for higher derivatives. Thus we may pass to the limit k -, oo in (1.8) and find that u is a smooth solution of (1.8) away from E. In order to be able to assert that u extends to a weak solution across E we need to estimate the "capacity" of E, respectively its m-dimensional Hausdorff measure with respect to the parabolic metric 6((x, t), (y, s)) = Ix - y1 + For a set S C Rm x R the latter is defined as

It --81-

inf`Fr;';SC UQr,(z;),z,ES,r;
11 m(S;S)=c(m)suP I`

I`

i

c

I

PART 1. THE EVOLUTION OF HARMONIC MAPS

289

where c(tn) is a normalizing constant and Qr(z0) = {z; Ix - xol < r, It - tot < r2}.

Fixa compact s e t Q C Rm x ] - 1, 0[ and l e t S= Q f1 E. Fix R > 0 and let Qr; (zi), ri < R, be a cover of S. Since S is compact, we may assume that the cover is finite. Moreover, a simple variant of Vitali's covering lemma shows that

there is a disjoint sub-family Qr;(zi), i E 9, such that S C UiEJQ5r;(zi). Let zi = yi + (0, r?), i E J. Since J is finite, there exists k E N such that t; -6'r; eo < 4 (6r2) < C f

t;-462r, fR-

:5

C(b)r, m

IVuk12G;, dxdt

I Vuk 12 dx dt + Cb-m exp

J

1) (- 3262

t62r; ftj -463r?

IVuk12G,; dxdt R'"

< C(b)r, m f

I Vuk I2 dx dt + Cb2-m exp

(__!_) Eo,

ri (ZI )

for all i E 9, where we used the fact that

G, < 6-m exp

(....i) Gx,

/

on Rm x [ti - 4b2ri , ti - b2ri ] \ Qr. (zi)

and Theorem 1.10 to derive the last inequality. If b > 0 is sufficiently small, we have

C62-m exp

- 32b2 1

Eo

e

<

2

and hence

ri-
IVukl2dxdt.

Qr, (zf )

Summing over i E 9, we obtain

ErQ1
iE7

=Cf

IVUk 12 dx dt

IVukl2dxdt
with constants independent of R > 0. That is, the m-dimensional Hausdorff measure of E is locally finite. In particular, for a suitable cover (Q,, (zi))iEJ of S, ri < R, we can achieve that Gm+1

I

`sEE

I Qr, (Zi) 7

0

as R -+ 0,

J

where,Cm+1 denotes Lebesgue measure on Rm x R. Now let 0 E C0 '10(Q2(0)) satisfy

, %410 E Co (Q2r,(zi)) Given 0 < 0:5 1,¢ = 1 on Q, (0) and scale 4i(z) = 4)( E Co (Q; R"), then r = ty infi(1 - Oi) is a Lipschitz function and r(z) - ty(z)

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

290

a.e. as R -+ 0. Testing (1.8) with r, we obtain

f

0

J

R JR"

(ut - Au - A(u) (Vu, Vu))irb dx dt


1 mIVuiiVinf(1-0;)Idxdt+o(1) o 1/ 2

< CIIVUIIL2(U,E,

Q.,(=,))

V

IV inf(1 Jetm

- 0;)12 dxdt+ 0(1)

1/2

r, 2dxdt)

<0(1) CSI

+o(1)

1/2

rr) f

= 0(1)

+ o(1) -* 0

(R - 0),

where o(1) -* 0 (R - 0), and u weakly solves (1.8). Finally, (1.24) follows from (1.23) and the fact that 45(R; u) < lim infk_ $(R; uk), t (l; u) = 0(1; uo) = limk-. 4 (1; uk). The proof of Theorem 1.8 uses the fact that results similar to Theorem 1.10 and Proposition 1.4 hold for solutions u to (1.8) on a compact manifold M, where 4i is defined with reference to a local coordinate chart V and where we truncate

the integrand with a smooth cut-off function r E Co (V). Moreover, if U6 is a 3b-tubular neighborhood on N in R" and if we define

EK(u) = E(u) + K fm X(dist2(u, N)) dx,

where(s)=8

for s< b, X'(s) > 0, X(s) - 2b for s > 36, for maps u : M -+ R", then the sequence of approximate solutions (UK) to (1.8) defined by the gradient flow of Ek again satisfies an analogue of Theorem 1.10 and Proposition 1.4. Similar to Proposition 1.5 we then establish that a sub-sequence (UK) converges weakly to a partially regular weak solution u of (1.8),(1.9). Moreover, inequality (1.24) holds. See Chen-Struwe [21] for details. Let us now turn to some further consequences of the monotonicity formula

Nonuniqueness Coron [27] observed that for certain weakly harmonic maps uo : B3 -+ S2 the stationary weak solution u(x,t) = uo(x) of (1.8) does not satisfy (1.24), hence must be different from the solution constructed in Theorem 1.8. Slightly modified, we repeat his construction. Suppose uo E Hja (R3; S2) is weakly harmonic, uo(x) = uo (), and consider u(x, t) = uo(x). Then u weakly solves (1.8) and -O2(p) =

f rIvuoi2exp (-fx 4p2 _ I2)

dx < co

2

for any z E R3 x R, any p /> 0. Suppose that u satisfies (1.24). This implies (1.25)

P JR3

IVuoI2exp I - Ix4p212 1 dz <

r r, IVuoi2exp (- Ix4r212) dx

PART 1. THE EVOLUTION OF HARMONIC MAPS

291

for any 0 < p < r < co. We show that (1.25) is violated for a suitable map uo. The map uo is obtained as follows. Let it : S2 \ {p} -. R2 S C be stereographic projection from the north pole (0, 0,1) of S2 and let g : C - C be a rational map. Composing the weakly harmonic map u : x - IXT from Example 1.2 with 7r and g we obtain a map

7-1 uo(x) =

((())).

) as a map uo : S2 -+ S2, by conformal invariance u0 is harmonic; hence uo : R3 S2 is weakly harmonic. By suitable choice of g (for instance, g(z) = .1z with A E R, A > 1), moreover, we can achieve that the center of mass Regarding uox) = uo (

q = f PVuo(x)I2xdv01s2 0 0. (Hence the map uo is not minimizing for its boundary values on B1(0) C R3; see Brezis-Coron-Lieb [13], Remark 7.6.) Denote

0(P, x) =

1 P

212 )

IVuot2 exp I - Ix -

dx

\\\

JJJ

for brevity. Note that

0(P,0) = foo US2 is independent of p > 0. Moreover, compute

o (P, 0) = =

j j

s

=9J

IVuot2 2p27 3

exp

I exl' (-4P2)

exp (-Q) do p

P

(_2)

(JIvHoI2dvs2). oo

= ao

ll

2p3

P sds

q

p

Hence for z = tq, 0 < p < r, if t > 0 is sufficiently small we obtain 0(p, x) = ao + t II PI2 + 0(t2) > 0(r, x) = ao + t 1!12 + 0(t2), contradicting (1.25). On the other hand, as in Theorem 1.8 we can construct weak solutions ii to (1.8) for initial data uo satisfying (1.24), showing that u # u and hence showing nonuniqueness in the energy class of weak solutions to (1.8), (1.9). Note that we have spontaneous symmetry breaking, since u cannot be of the

form u(x,t) = v (11,t). The latter map v would solve (1.8), (1.9) on S2 x [0,00[. Since uo : S2 - S2 is smooth and harmonic, by local unique solvability of (1.8),(1.9) on 52 x [0, oo[ for smooth data this would imply v(t) - uo. It remains an open problem to exhibit a class of functions within which (1.8), (1.9) possesses a unique solution. Certainly, the class of functions satisfying the strong monotonicity formula

ot(p) 5 qt(r) for all x and all 0 < p < r < OF is a likely candidate.

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

292

Development of singularities The most surprising aspect of the monotonicity formula is that it may be used to prove that (1.8), (1.9) in general will develop singularities in arbitrarily short time. The existence of singularities was first established by Coron-Ghidaglia [28]; see also Grayson-Hamilton [61]. These results were based on comparison principles for the reduced harmonic map evolution problem (1.15) in the equivariant setting. A deeper reason for the formation of singularities was worked out by Chen-Ding [19]. This is related to a result by White [152].

Theorem 1.11. Let M, N be compact Riemannian manifolds and consider a smooth map uo : M -+ N. Then inf{E(u);u E C°°(M,N),

u is homotopic to uo} > 0

if and only if the restriction of uo to a 2-skeleton of M is not homotopic to a constant.

Remark 1.2. In particular, there are examples of non-trivial homotopy classes of maps uo : M -+ N such that inf{E(u); u is homotopic to uo} = 0.

Example 1.6. Let u1 = id: S3

S3. Let 7r: S3 \ {p} - R3 be stereographic

projection, and let Da : 1R3 - R3, D,, (x) = Ax be dilation with A > 0. Then define U,\ = 7-1 . Da . 7C: S3 - S3.

Clearly, ua - u1 = id for all A > 0 and E(ua) - 0 (A -+ oo). The construction of Chen-Ding can be vastly simplified by combining Remark

1.10 with Proposition 1.4, as in Struwe [143] or [145]. Let M, N be compact manifolds, dim M = m > 3.

Theorem 1.12. For any T > 0 there exists a constant e = e(M, N, T) > 0 such that for any map uo: M -+ N which is not homotopic to a constant and satisfies E(uo) < e the solution u to (1.8), (1.9) must blow up before time 2T.

Proof. Suppose u E Coo (M x [0, 2T1; N) solves (1.8), (1.9). For z = (z, , T < t < 2T, R2 = T estimate $z (R) < CR2--E (u(t - R2)) 5 CR2-mE(uo) < co ife<`°T(m_X)12

whereC -CMNand) e= ( o = eo(M, N) > 0 is the constant in Proposition 1.4. Hence by Proposition 1.4 for z = (a, >, T < i:5 2T, we have the uniform a-priori bound

IVu(2)1< R =C(M,N,T). By the Bochner-type inequality (1.12) therefore we have d

AM) e(u) < Ce(u) on M x [T, 271

dt and Lemma 1.10 implies that

sup(e(u)) (x, 2T) < CE(uo) < Ce,

where C = C(M, N, T). Hence, if e = e(M, N, T) > 0 is sufficiently small, the image of u(2T) is contained in a convex, hence contractible, coordinate neighborhood on

293

PART 1. THE EVOLUTION OF HARMONIC MAPS

N and u(2T) is homotopic to a constant. But then also uo is homotopic to a constant map, a contradiction. Therefore u must blow up before time 2T.

0

Singularities of first and second kind Let u E C°°(Rm x [-1,0[; N) be a solution to (1.8) with an isolated singularity at the origin, and satisfying (1.24). If (1.26)

IVu(x,t)12 <- I'

the rescaled sequence

R>0

uR(x, t) = u(Rx, R2t),

satisfies the same estimate and hence a sub-sequence converges smoothly locally on Rm x ] - oo, 0[ to a smooth limit u as R -i 0. u satisfies (1.8). Moreover, u 0 const;

otherwise $(R; u) = (1.24) there holds

I

f

T isft m

+ 21t1

uR) < co for some R > 0 and u is regular at 0. Since by x. Qttl2

cdxdt <_ Rl O

ff r

I2tuRt + X DUR12

T

m

Z

r

R1

in-

21tj

cttxttt

I2tut 2ltl Du12Gdxl{t = U,

for any 0 < r < T < oo, it follows that u satisfies 2tut + xVu - 0; that is x u(x,t) = V

(

]ti

or

u(x,t)

(1!1x)

= w

We call singularities satisfying (1.26) of first kind. It is not ]mown whether self-similar solutions u(x, t) = v

actually may exist. All other singularities

are said to be of second kind. Since singularities in case m = 2 by Theorem 1.5 are related to time-independent harmonic maps u : S2 N of finite energy and since a non-constant, radially homogeneous map u(x) = w ( ) in m = 2 dimensions has infinite energy, Theorem 1.4 of Chang-Ding-Ye above shows that singularities of second kind, in fact, exist for the evolution problem (1.8).

Extensions and generalizations Further current developments include the evolution problem for harmonic maps on general complete, non-compact manifolds (Li-Zhm [1041) and for harmonic maps with symmetry (Grotowsky [65]).

PART 2 The Evolution of Hypersurfaces by Mean Curvature 2.1. Mean curvature flow Let M be a complete oriented m-dimensional manifold without boundary, Fo : M R' a smooth embedding into R", n = m + 1. Denote the scalar product in R". Let v be a smooth unit normal vector field on Mo = Fo(M) C R" and denote

H = E(e;,0",v) (m times) the mean curvature of Mo with respect to v, evaluated in a local orthonormal frame e1, ... , em on Mo. Also denote

H = -Hv the mean curvature vector. We say Mo = F0(M) is moved by mean curvature if there is a family t) of smooth embeddings t) : M -' R' with corresponding hypersurfaces Mt = F(M, t) such that (2.1)

0 F(p, t) = H(p, t)

for all p E M, t > 0,

0) = FO. Here we denote by ff(p, t) the mean curvature vector of Mt at a point x = F(p, t) E R". Equation (2.1) corresponds to the negative gradient flow for the volume of Mt. Indeed, if µt = t)'µ denotes the pull-back of the first fundamental form on Mt induced by the Euclidean metric µ , we have and

d

dtut =

-Haµt.

Recall that the first variation of volume Vol Mt =

JM=

r

dµ1

M

t

of the hypersurface Mt C R", when deformed in direction of a vector field t/i, is given by

f (9G, v) H d1im = Me

f4f,

('+G, L dpi"',

296

M. 8TRUWE, GEOMETRIC EVOLUTION PROBLEMS

where v denotes a unit normal vector field and H the corresponding mean curvature on Mt. Moreover, if A denotes the Laplacian in the pull-back metric At, we have

H=OF and (2.1) takes the form of a heat equation on M. The mean curvature flow was first investigated by Brakke [11], motivated by a study of grain boundaries in annealing metal, and later by Huisken [76]. While Brakke considered the problem in the general context of varifolds, Huisken approached the problem from the classical, parametric point of view (2.1). A weak form of (2.1) in terms of motion of level sets was later proposed by Osher-Sethian [116) and investigated in detail by Evans-Spruck [43], [44], [45], [46] and independently by Chen-Giga-Goto [22]. Finally, Ilmanen [79) has been able to relate the level-set flow and the Brakke motion of varifolds in the frame-work of geometric measure theory. We will trace a part of these developments. First we consider the parametric point of view. We will consider two model cases: the case when Mo is the boundary of a bounded region Uo C R" and the case when Mo is represented as an entire graph

Mo={(x',uo(x'));

x'ER"-II

Another special case is the case of plane curves or, more generally, curves on Riemannian surfaces. In the 80's Gage [49] and Gage-Hamilton [50] established that convex curves in the plane evolve smoothly to a nearly circular shape before they shrink to a 'round' point. Grayson [58], [59] extended these results to general closed embedded curves in the plane and on surfaces. In the latter case, another possible long-time behavior is the convergence towards a closed geodesic. AbreschLanger [1], Angenent [4) and Grayson [60] also studied immersed curves in the plane and obtained highly interesting examples of singular, self-similar behavior. However, in these notes we will study only the higher-dimensional case m > 2, where we will be able to observe singularities even if the original hypersurface is smoothly embedded, in contrast to the one-dimensional case.

2.2. Compact surfaces Let U0 be a compact set in R" with smooth boundary Mo and, say, outer unit normal v. If Uo is convex, then with our sign convention the mean curvature vector H is pointing inside Uo, and Mo, evolving by (2.1), is contracting.

Example 2.1. If Uo = BR(0), then Mt is a sphere of radius R(t), satisfying

=

-n-

R 1,

R(0) = Ro;

that is,

R(t) = R - 2(n - 1)t, and Mo shrinks to a point in finite time T = 2 "? I Example 2.2 (Angenent [5]). There exists a torus-like surface in R3 shrinking to a point by self-similar motion.

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

297

The evolution of spheres is in certain ways characteristic; moreover, their evolution gives nice "barriers" to control the evolution of more general hypersurfaces, due to the following

Theorem 2.1 (Comparison principle). If Mo and Mo are boundaries of smooth, relatively compact regions Uo C Uo CC R", respectively, and if Mo and Mo evolve

by mean curvature through families Mt = 8Ut, Mt = BUt, respectively, then for

t>0there holds UtCUt. IfUo0Uo,weeven have Ut000t. Heuristically, the comparison principle follows from the observation that if Mt were to "touch" Mt from "inside" at some point x, then the mean curvature of Mt at x would exceed the mean curvature of Mt at x, drawing Mt and Mt apart. A formal derivation, based on the maximum principle, will be given shortly. As a consequence of the comparison principle any compact hypersurface Mo = 8Uo evolving by mean curvature will become extinct in finite time T. Indeed, if

UoCBp,(xo)for some xo,Ro>0,we have T
Local existence Observe that (2.1) is only weakly parabolic, as the equation is invariant under diffeomorphisms of M. Local existence for (2.1) thus cannot be derived by the standard theory for parabolic systems; instead, more sophisticated methods, like the Nash-Moser scheme, have been used. However, as observed by Bollow [10], a simple alternative proof for local existence can be given based on a representation of the evolving surfaces Mt as graphs over Mo.

Since the general case is rather technical, let us derive this result only in the case that Mo is strictly star-shaped with respect to the origin, whence Mo may be represented as a graph

over the standard sphere.

Lemma 2.1. Let Mo be given as above with a smooth, strictly positive function uo : S"-1 -, R. Then (2.1) is equivalent to the evolution problem (2.2)

8u

at

-

u2 + U2

div

Vu u2 + IDUI2

n -1 u

on S"-1 x [0, oo[ with initial condition ult-o = uo, in the sense that for the solution of (2.1) there holds Mt = {u(e, t)l;; 1' E S"-1 }, where u solves (2.2). Conversely,

if u solves (2.2) then Mt, given as above, may be reparametrized by tangential difeomorphisms to achieve a solution F(-, t) of (2.1). Proof. Let C E Sn', and let et, 1 < i < n-1, be an orthonormal basis for TTSnand let x = u(1:,t)t: be a parametrization of Mt. Denote u., = e,(u) = Ve,u, etc., the derivative in direction e;. Then at the point £ we have the following expressions

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

298

for v, H and the first and second fundamental forms (9i j), (hj) on Mt, respectively:

x,i = u,, + uei, 9ij _ (x,i, xa) = u,iui + u26ij,

- ui

v

u2 + Dull' 2u,iu,, + u2bir - u,ij u2 + Vu12

H = trace(gi"h,k) = giihji

_

n-1

_1u d1V 1

u2 + lvul2

Vu u2 + 1Vu12

(To see the latter we may orient ej such that u,i = 0 for i 0 1.) Hence if Mt evolves according to (2.1), the radial component (gF)1 of its velocity satisfies u2T8 u u2+IVu12

-/&

Ff_(

\et

m)_-

Hu

u +lvu

U _-(n-1) u2+IVU12

1

+

u2+

1Vu12

div

Vu u2 + 1vu12

That is, u is a solution of (2.2). Conversely, if u solves (2.2) and Mt is the graph of u above S"'', then Mt evolves by mean curvature. 0

Now if uo > c > 0, then for u near uo the linearized operator (2.2) is uniformly parabolic and we obtain local existence and uniqueness of solutions of (2.2) and hence (2.1). Moreover, the comparison principle stated above for strictly starshaped hypersurfaces can be deduced from the strong maximum principle for uniformly parabolic equations. Finally, if Mo is convex, then Mt remains convex until Mo becomes extinct at a `round' point, as was proved by Huisken [761. Although one might expect a similar

result to hold for strictly star-shaped initial surfaces, the corresponding result is false in this case. To see this, consider the boundary of a fattened double-cone bounding two large spheres at the ends and circled by a self-similarly shrinking torus at the waist.

2.3. Entire graphs Suppose Mo can be represented as a graph Mo = { (x', uo(x')); x' E W-11,

with a smooth function uo : R"-' - R. By a result of Ecker and Huisken [34] then Mo generates a global smooth evolution by mean curvature. To see this it is useful to derive a non-parametric form of the evolution problem (2.1).

Lemma 2.2. Suppose Mo is given by the graph of a smooth function uo : R"-' R. Then (2.1) is equivalent to the evolution problem (2.3)

ju -

1 + IVu12 div (

1

Viyul2) = 0 M R"-' x [0, oo[,

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

299

with initial condition ult=o = uo, in the following sense. For any t > 0 the set Mi evolving by (2.1), starting from Mo at t = 0, is an entire graph

Mt = {(x',u(x',t));x' E Rn -t}, where u solves (2.3). Conversely, if u satisfies (2.3), then Mt, given as above, up to tangential dif'eomorphism evolves by (2.1).

Proof. Let x = (x', u(x', t)), e1,.. . , e the standard orthonormal basis for Rn. Then we have

x,i = ei + u,ien, gij = u,iu,i + bii, en - u,iei

v=

1+IVuP' (ei + u,ien, -u,iiej)

-

1 + [VuI2

-u.i3 1 + _VU 12

for 1
H = giihii = div

j_+_1 IV-U1 2

Hence if Mt = F(Mo,1) evolves according to (2.1), its vertical velocity satisfies Ttu

_

/a

\

1+IVuI2-{F,en)=-(Hv,en)=

1

Vu 1+JVI2

0

as claimed, and conversely.

Again, (2.3) is uniformly parabolic at smooth functions u satisfying a global Lipschitz condition. Hence, for smooth, globally Lipschitz initial data it is not

hard to show that (2.3) admits a smooth solution, which, in fact, is defined for all time. That is, an entire Lipschitz graph Mo evolves through smooth Lipschitz graphs Mt for all time. Surprisingly, Ecker and Huisken were able to show that a solution to (2.3) and hence to (2.1) exists globally even without any growth condition on uo near infinity. This result-somewhat reminiscent of results on removable singularities for the minimal surface equation-is obtained by first using spheres "above" and "below" Mo as barriers to control the "height" of Mt locally and then combining the local L°°-estimate thus obtained with the following interior gradient estimate for solutions of (2.3).

Theorem 2.2 ([34], Theorem 2.3 and Theorem 5.2). Suppose u is a smooth solution of (2.3) on BR(xo) x [0,71 with initial data uo. Then for all t E [0,7'] there holds

1 + [Vu(xo, t)[2 < C sup

1 + IVuo12

Bn(x,)

exp , CR-2

l

with constants C = C(n).

sup x1EBn(x0'),0<s,t
Iu(x', 8)

- u(xo,1)12

300

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

The proof of this result uses techniques of [98], adapted to a parabolic setting.

Theorem 2.3 ([341, Theorem 5.1). Let Mo = {(x',uo(x'));x' E R"-1} be a locally Lipschitz entire graph over RI-1. Then the mean curvature flow (2.1) has a unique smooth solution (Mt)t>o; moreover, each Mt is an entire graph Mt = {(x', u(x', t)); x' E R"-1} given by a smooth global solution u of (2.3).

Proof. Given xo E Rn-1, R > 0, T > 0 let Rp >R2 + 2(n - 1)T,

a= sup Iuol+Ro Blto (=o )

and choose balls Bo = BR. ((x'o, ±a)) "above" and "below" Mo as "barriers" to control u. Note that by time t the mean curvature flow will shrink the spheres

So = 8Bo to the concentric spheres St = 8Bt , where Bi = BR(t)((xo,±a)), with R(t)= Ro-2(n-1)t>RforO
IVu(xo,t)I
(

sup (Ivuol+ BR(z,)

IRl for 0 < t < T. Varying xo, we obtain a-priori gradient bounds on R"-1 x [0, T]. By a boot-strap argument similar a-priori bounds for higher derivatives of u may be obtained. Approximating uo by smooth functions u0k linear at infinity and such that 110k - uo uniformly on any compact subset 12 C Rn-1 with IVuokl bounded uniformly in 0, uniformly in k, the corresponding solutions uk to (2.3) for initial data uok converge smoothly to a solution u of (2.3), locally on Rn-I x]0, oo[.

Another use of the non-parametric form (2.3) of (2.1) is to give rigorous proof of the comparison principle Theorem 2.1: Suppose Mt. and Mto touch at a point zo E Rn. We may suppose 0 < to is minimal with that property and that xo lies on the boundary of Mt f1 Mt, relative to Mt. Represent Mt and Mt as graphs above their common tangent plan at (xo, to) and apply the maximum principle to equation (2.3) in a backward neighborhood of (xo, to).

2.4. Generalized motion by mean curvature Above we have seen that an initial surface near a double cone or, simpler, a long dumbbell-shaped surface develops a singularity before it becomes extinct under mean curvature flow. In order to be able to follow the evolution past the singularity, we now define

generalized motion by mean curvature as developed by Evans-Spruck [43], [44], [45], [46], and, independently, by Chen-Giga-Goto [22). The idea is to represent a hypersurface Mo in R" as the level-set {uo(x) = 0} for some function uo and

such that for a suitable family ut = u(., t), all level surfaces {u(x, t) = const}, wherever sufficiently regular, evolve by mean curvature. To derive the differential

equation that the family ut will have to satisfy in order for its level surfaces to evolve according to (2.1) we fix a point x E rZ = {u(x, t) = ry} and follow its path as r' is evolving by (2.1).

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

301

Note that we have v = OvV , H = div v; hence if we parametrize the particle

path of x by x(s), s > t, we have d

x(s) = -Hv =

x=

( IVVu

-

IVul

at the initial time s = t. On the other hand, the requirement u(x(s), s) = 'y for all s leads to 0=

d Z u(x(s),8)

Vu) +

au = -IVuI div (Dull + -u. 5i

Thus we say that the motion of level sets of u defines a generalized motion by mean curvature if u : R" x [0, oo[-. R satisfies (2.4)

69

u - (bij -

IVU12

/ U--,

= 0,

where i and j are now summed from 1 to n. Note that in the language of fluid mechanics, equation (2.4) corresponds to the Eulerian viewpoint while the parametric problem (2.1) corresponds to the Lagrangian one. In the present context, equation (2.4) seems to have first been proposed by Osher-Sethian [1161. For smooth functions u with Vu 34 0, equations (2.4) and (2.1) are in fact equivalent. However, if we want to model the motion of a closed hypersurface Mo bounding a compact region in R" by the evolving level sets rt = {u(x, t) = 0} of a solution of (2.4), it is clear that Vu must vanish at some point in the interior of Ft for each t and hence (2.4) cannot be interpreted classically on all of R" x (0, oo[. Moreover, by the geometric interpretation of (2.4), if u is a solution of (2.4), and if W is any continuous, one-to-one function, then %F(u) should be a solution of (2.4), as u and T(u) have the same level sets. To be able to include these cases a notion of weak solution is required. The appropriate notion of generalized solution is that of viscosity solution as in [29], [301, [40], [80], [83], and [106].

Definition 2.1. A function u E CO f1 LOO(R" x [0, oo[) is a weak sub- (super-) solution of (2.4) if the following holds: For each 0 E C°O(R"+a) and each (xo, to) E R" x]0, oo[ such that u - ¢ has a maximum (minimum) at (xo, to) we have (? 0) Ot - (bij - 7700-0-,-, < 0 for some 17 E R", InK < 1, and

n=

Vo(xo, to) IVOxo, to) I

if VO(xo, to)

0.

In particular, a weak sub-solution forces any smooth function whose graph "touches" the graph of u from "above" at (xo, to) to be a sub-solution at (x0, to), and similarly for super-solutions. In fact, for more general classes of equations one usually requires that ¢(xo, to) = u(xo, to) for comparison functions ¢. However, since translations u --+ u + e leaves (2.4) invariant, here this condition can be waived.

Often it may be convenient to represent admissible comparison functions as

0(x, t) = u(xo, to) + p(x - xo) + q(t - to) + 2 (z - zo)T R(z - zo) + o(Iz - zol2)

as z -+ zo, for p E R", q E R, R = (rij) a symmetric (n + 1) x (n + 1)-matrix, and z = (x, t).

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

302

Definition 2.2. A function u E co n L°°(R" x [0, ool) is a weak solution of (2.4) if u is simultaneously a weak sub- and super-solution. Let us now turn to some simple special cases.

Example 2.3. (The motion of spheres) Let uo(x) = Ro - 1x12. Then

u(x,t) = uo(x) - 2(n - 1)t satisfies (2.4) classically at all points where Vu 0 0, and the level surface

r t = {x E R' ; u(x, t) = 0}

is given by rt = BBR(i)(O)

with R(t) =

R-,0 - 2(n - 1)t as in the classical, parametric setting. To verify that u solves (2.4) also at xo = 0 it suffices to consider as comparison functions

0t (x, t) = u(x, t) f (z - zo)T R(z - zo), where z = (x, t), z o = (0, to), R = (ri,) >2 0. Then 0+ > u > r- near zo, V 0, and, letting rl denote an arbitrary unit vector in R', OP

(zo) _

- (bii - tli'Jj)WZx; = ut - (bii - 11070'u-,-, T- (bij -T(6,, - i7j%)rii

has the desired sign.

2.5. Uniqueness, comparison principles, global existence Thus, in a simple case we reobtain the classical motion by mean curvature. Note that in the example above we simultaneously recover the evolution of all spheres centered at the origin; moreover, the solution u remains smooth and is in fact globally defined even past the extinction time To = Ro/2(n - 1) of any individual sphere. However, it may not be obvious that this solution is unique; conceivably, after

passing through a "geometric" singularity a level surface might evolve in any one of several different ways, as is illustrated by examples of Brakke [11] for the mean curvature flow of varifolds. In fact, it is quite surprising that solutions to (2.4) are unique. Moreover, we shall see that the motion of a level surface 170 depends only on the surface, not on the function uo defining it. Finally, solutions persist for all time. Thus, (2.4) defines a unique global extension of the mean curvature flow of hypersurfaces to arbitrary level surfaces of continuous functions in R". We begin by the following observation. Theorem 2.4 (Naturality; [43], Theorem 2.8). If u E CO f1 LOO (R^ x [0, oo[) is a weak solution of (2.4) and if T : R --+ R is continuous, then v = W(u)

is a weak solution of (2.4).

Proof. First consider the case where T E COO with 'P' > 0 on R. C°°(R'+i) and suppose 0 = '+6(xo, to) - v(xo, to) <_ ,p(x, t) - v(x, t) near (xo, to). Then -0 = IF -I(a,) E C°O(Rn+I) and 0 = O(xo,to) - u(xo,to) <- O(x,t) - u(x,t)

near (x0, to),

Let u i E

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

303

whence for some il E R",1771/ < 1, we have

< 0 at (x0, to).

Ot - (bij -

Moreover, , = V,i/I V,iI = Vii/I VOI, evaluated at (xo, to), if Vq (xo, to) # 0. But then + W"(IV I2 - In . V012):5 0,

Wt -

and v is a weak sub-solution of (2.4). Similarly, v is a weak super-solution, which proves the theorem in case %F E C°° satisfies 91' > 0. The case 'Y' < 0 is similar. More generally, in order for 'I'(u) to be a weak solution of (2.4) near (xo, to) it suffices that T is monotone in a neighborhood of u(xo, to). Since an arbitrary continuous function can be uniformly approximated by functions which are locally monotone - that is, regions where ' is increasing or decreasing, respectively, are separated by intervals where W is constant - the assertion of the theorem for general 12 is a consequence of the next result.

Theorem 2.5 (Closedness of solution set; [43], Theorem 2.7). Let uk E CO n L' (R" x [0, oo[) weakly solve (2.4) and suppose uk u boundedly and locally uniformly on R" x [0, ool. Then u is a weak solution of (2.4).

Proof. Let 0 E C°° (R"+I) and suppose that u - 0 has a local maximum at zO = (xo,to). Replacing 0 by the function O(z) = O(z) + Iz - z° I4, if necessary, we may assume that u - 0 has a strict local maximum at zo. By locally uniform convergence Uk - u, for large k also uk - 0 has a local maximum at a point zk, zk -' zo for large k and thus -Ot -

(k

(bij -

oo). If V4(z°) 34 0 we also have Vq(zk) i4 0

IV 012 J

0x,x; < 0 at Zk.

Passing to the limit k - oo, we obtain the same relation also at zo. If V4(zo) =0 we have of - (bij - rlik)rl(k)) 0x,x3 < 0 at Zk

for a sequence of vectors rl(k) E R", I,7(k)l < 1. (Of course, if V¢(zk) ,- 0 we let r7(k) = °m xk ). By passing to a sub-sequence, if necessary, we assume that +)(k) --+,, (k - oo). It follows that Ot - (bij - 7707j) 0.,x, < 0 at 20,

and u is a weak sub-solution, as claimed. The proof that u is a weak super-solution is similar.

Theorem 2.6 (Global existence; [43], Theorem 4.2). Assume uo: R" -- R is continuous and satisfies u°(x) = y° E R for Ixl > R° for some constant RD > 0. Then there exists a weak solution u E CO n LO°(R" x [0, oo[) of (2.4) with initial data 0) = uo and

u(x,t) =yo Moreover, if uo is Lipschitz, so is

for lxl2 +2(n - 1)t> R t) and I[VUIIL-(R^x[o,oo[) : IIVu0IIL-(R^)

.

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

304

Proof. Assume for simplicity that uo is smooth. Extend uo to R"+1 by letting u0(x, xn+1) = u0(x) - Exn+1

Denote points in R"+1 by y = (x,xn+1). The level surface

to = {y E R"+1; uo(y) = 0} is a smooth graph

Hence, by Theorem 2.3 above, to `generates a smooth family of graphs

rt = { 1 X' !((x

t)) ; X E R" 1 ,

evolving by mean curvature. Any other level surface ro = {y;uo(y) = y}

is a parallel translate of to and evolves through parallel translates of I's. Hence uE (x, xn+1, t) = ue (x, t) - Exn+1

is a (classical) solution to (2.4) on R"+1 x [0, oo(. Computing explicitly, thus ue satisfies (2.5)

ut - 6ij -

uex, uex

(Due (2

+ E2)

ux,xj = 0

in R" x [0, oo[ with initial condition 0) = uo. Moreover, for fixed e > 0 by Theorem 2.3 we have local bounds for I Vue I on any slice R" x [0, 71. Thus, by the maximum principle for uniformly parabolic equations, we have uniform a-priori bounds IIuCIIL0(R,,x(O,ooi) = IIUOIIL.O(R°)

Finally, differentiating (2.5) with respect to xI, we have

It - b{j -

uex ue 'ZJ 2 IVueI +E2

ux,xixi

C(E) la'ds' I

IVux, I

and another application of the maximum principle gives the Lipschitz bound IIVudIIL°O(R°xj0,ooi) !5 IIVuoIIL,(R^),

uniformly in e > 0. Thus, passing to a suitable sequence e --+ 0 if necessary, ue - u boundedly and locally uniformly on R" x [0, oo[. A simple variation in the proof of the preceding compactness result shows that u weakly solves (2.4). The characterization of the support of (u - ryo)+ results from comparing the solutions ue above to the solutions yE (x, t) corresponding to the shrinking of spheres. 0

Theorem 2.7 (Comparison principle and uniqueness; (43], Theorem 3.2). If u, v E ConL°°(R" x [0, oo[) are weak solutions of (2.4) with initial data 0) = uo < vo = v(., 0) and if for some R > 0 both u and v are constant on Rn x [0, oo(n{IxI +

t > R}, then u < v. In fact, IIu - VIIL°°(R"x[0.oo() : IIu0 - VOIIL-(Rn)

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

305

Remark 2.3. The construction used above to prove global existence yields weak solutions u < v if the initial data are ordered correspondingly. This follows by applying the maximum principle to the approximate equations (2.5). However, the proof of the corresponding ordering relation among all weak solutions of (2.4) with initial data uo < vo is much more involved. An important concept in the proof is that of sup- and inf-convolution, defined as follows. For w E C° f l LO° (R" x [0, ooD,

e>0let

wE(z) = wc(z)

fER"x(O,oc(

(w()_!Iz_I2) e

fER^

(w()+!Iz_I2). e

sup

0.00(

Then inf w < wt < w < wt < sup w, and wt, wt - w locally uniformly on R" x 10,00f.

Moreover, sup- and inf-convolutions are regularizing in the sense that wt, wt are Lipschitz and almost everywhere twice differentiable. The latter assertion follows from the fact that, for instance, the function

z-+wt(z)+1x12=s{p{wt(t)+E (1x12-Iz-e12)}, being the supremum of a family of affine functions, is convex, hence almost everywhere twice differentiable by a theorem of Alexandroff. Finally, if w is a super-solution of (2.4), wt again is a super-solution. Similarly, if w is a sub-solution of (2.4), so is wt. Now the idea in the proof of the comparison

principle is as follows: Suppose by contradiction that u(z) > v(z) for some z E R"x]0,oo[. By hypothesis, u and v are constants for large 1x1 + t. Thus

max (u-v)=a>0

R" x (0,00(

is attained in R" x]0, oo[. By uniform local convergence uE -+ u, vt -, v then also

max (ut - vt)

R"x[0,o0(

a>0

-2

is attained in R"x]O,oo[ for sufficiently small e > 0. Finally, if we choose a > 0 small enough, also

max (ut - vt - at) > a4 > 0

R"x(o.oo(

will be attained at some interior point zo E R" x10, oo[. Suppose for simplicity that vt is twice differentiable near zo with Ovt (zo) 0. Then a suitable extension of (Vt +at) is admissible as a comparison function 4, for ut. Since ut - 0 by choice of zo has a local maximum at zo and ut is a weak sub-solution of (2.4), we find

vvl E

0 ? mt - 6,j

-

t

(v +a) - (aij - 10u12) v=exs, IV012

contradicting the fact that vt is a super-solution.

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

306

2.6. Monotonicity formula In order to obtain partial regularity results for (2.4) we may try to carry over the estimates obtained for harmonic maps by means of the monotonicity formula. In the following, we derive an analogous monotonicity estimate for the generalized mean curvature flow. In fact, in the parametric setting of problem (2.1) a monotonicity formula analogous to Theorem 1.10 was independently obtained by Huisken [771 in 1990.

For a given point zo = (xo, to) E R" x 10, oo[ let

G"' (x, t) =

I

1

exp

47r(to - t

t))an

(_ [x - xo[ a ` 4(to -

if t < to,

G,, (x, t) = 0 else. Observe that G,, is the fundamental solution to the backward heat equation in any n - 1- dimensional hyperplane through xo. (Any such hypersurface is a stationary solution of (2.1).) Consider a smooth function uo: R" - R which is constant for large [x[ and let u' be the unique global smooth solutions to the approximating problem (2.5) with initial data u° = uo constructed in the proof of Theorem 2.6 above.

Proposition 2.1. For any zo E R" x [0, oo[ the function

01(t) =

[VuE-12+,E2 G0 dx

JR^x{c}

is non-increasing and [2rui + t. Vuf 12

d

JR^x{c} 41rl2(iVudl2+E2)

dL

lVuE [a +E a G,

dx,

where r=t-to, f =x-xo. Proof. Translate zo to (0, 0) and for A > 0 scale u -, U,\ (X, t) = u(Ax, Alt) E-+Ea=AE.

Then ua solves (2.5) for Ea with u'(.,0) = uo.\ and 4 (-A2;u`) = 0af(-l;u1). Hence (2.6)

Wt--01 (t) _

I`(t;10) =

2

[t,

T- (0A'(- 1; ul))

Since u` is smooth, moreover d

tat(-1; ua) _

Vu, V( ua)+AE2Gdx [Vu I r+ - JR^x{-1} Vua V(a,,ua) Gdx

fR-X(-Il Vu' +A E where G = G(o.o) and we used the fact that A > 0. Integrating by parts and using the scaled equation (2.5), that is, div

Vu A

7u7\

uac E

)

VuT + A 7 '

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

307

and the relation VG(x, t) _ -&G(x, t), we thus obtain 4

)

(- 1; u`

JR"X(-1) -2t Mu

z Dua }

E

1 2 W \ ' + x qua 12

IVuj +,\2f2

JR^x{-l} 2AItl

_

12tui + x Vu` 12

IVuc 2 +E2

- JR^x{-A2} 2It13/2

d

(dAuG a)

dx

G dx

Gdx.

Thus, finally, we deduce from (2.6) that (2.7)

dv(t)<- JR^x{t} 4ItI2(IVU,12+E2) Vrj VW2+EGdx,

dt

as claimed.

Passing to the limit c -i 0 in Proposition 2.1, from Fatou's lemma we thus obtain: Theorem 2.8. Let U E CO ft L°O(R" x [0, oo(), u =_ coast for large I xl + t, weakly solve (2.4), and let zo E R" x [0, oo[. Then for any T > 0 we have

f

IVuIGdx < f IVu°IG0 dx. " x {T}

^

A similar result was obtained by IImanen [78], approximating the level-set motion u by solutions to the Allen-Cahn equation.

2.7. Consequences of the monotonicity estimate As a first consequence we obtain bounds on the volume of hypersurfaces evolving under the mean curvature flow. Note that by the co-area formula ([57]; Theorem 1.32) for any lZ C R", any function f E BV(11), and any a E R, letting 0. = {x E 11; f(x) < s} and denoting Xn, the characteristic function, we have

fIDfI =

f

da

jIDxnI.

The monotonicity estimate thus gives a bound on the average of the volume of the level sets of u, weighted with G,z0. To narrow in on a particular level surface, say I't = {u(x, t) = 0}, choose smooth functions *k converging locally uniformly away from a = 0 to the signum function

'1'(s)=

+1,s>0 0,s=0

-1,s<0

Since the monotonicity estimate holds for Wk(U) as well, from lower semi-continuity of the BV-norm ([57]; Theorem 1.9) we deduce the estimate

I

IDvIGso < f ^

x {T}

R

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

308

for v = 4'(u), vo ='P(uo), provided Vuo 4 0 on ro = {uo(x) = 0}. Multiply by t(0 n" 1)12 and let to - oo to conclude

fit. x(T}

IDvI < IRn IDvoI =

t he (n - 1)-dimensional Hausdorff-measure of the initial surface. In particular, rT = (x; u(x, T) = 0) has finite perimeter; that is, rt is a Caccioppoli set (see [57]; p.6). By De Giorgi's structure theorem for Caccioppoli sets (see [57]; Theorem 4.4), therefore 8rT possesses a unit normal

v(x,T) = lim

fBo(Z) Dv(.,T)

P-0 fBo(=)

and hence a tangent hyperplane, everywhere. Thus we can also define a generalized mean curvature H = div v and generalized mean curvature vector everywhere on BrT, for any T > 0.

Of course, v and H coincide with the classical notions as long as rT is a smooth hypersurface. In the latter case, from the monotonicity estimate (2.7) for vk = 'Yk(u), upon passing to the limit k -+ oo, we recover the sharper estimate T

f

H+

-vl 2r

2

IDvIG,. dt < fR. IDvoIG.0, 0 ^ fan x(T) corresponding to Huisken's formula [77], Theorem 3.1. On the other hand, we can recover some of the estimates of [45]; see for instance Theorem 4.2. If rT is the closure of an open set with smooth boundary 8rT then IDvIG,to + I

(2.8)

I

IDvI = n' '(SrT) < 2nn-1(r0) Examples show that an initial (non-smooth) hypersurface may indeed "double up", as suggested by this formula. (Take a pair of crossed diagonals in the plane, for example.) As a further consequence we can obtain improved estimates on extinction times for compact hypersurfaces of finite (n -1)-dimensional measure, analogous to [45], Theorem 4.1.

Theorem 2.9. Suppose u E C° n L°O(Rn x [0,oo[),u = coast for large Ixi + t, solves (2.4) with initial data uo. Let rt = {x; u(x, t) = 01 fort > 0 and suppose C(wn"1(r0))2/(n-1) with a constant C = C(n). rT 76 0. Then we have T!5 Proof. Let xo be a point of density for the (n -1)-dimensional Hausdorff measure on 8rT. Then with ith constants C, = C,(n) > 0 we have lim

to\T

^ x (T}

IDvIG, > C1 lim P1 -n r P-0 Bo(xo)

C2 > 0;

see [57], Lemma 3.5. On the other hand, the left hand side of this inequality is bounded by

4r°)

lim J IDvoiG0 <- "r4'

to\T

^

v;r

_r_('01

11

Similarly, a result analogous to Brakke's "clearing out" lemma can be deduced.

PART 2. EVOLUTION OF HYPERSURFACES BY MEAN CURVATURE

309

Proposition 2.2 ([11], §6.3; [45],Theorem 3.1). Suppose Co = ?fn-1(ro) < oo. Given p > 0 there exist constants a, q > 0, possibly depending on Co, such that, if xn-1(r0 n B2p({)) 5 r1p"-1, then we have arc n Bp({) = 0 fort E [ape, 2ap2].

Proof. After scaling we may assume that p = 1. Let t E [ape, 2ap2], a > 0 to be determined, and let xo E 8rcnBp(t) be a point of density for the (n-1)-dimensional Hausdorff measure. Then

0 < C2 < lim

r

to\t JR^ x (t)

< lim

r

to\t JR^

<

jDvoIGzo < 2

j

fro a-Hn-

2 47ra

ffr.nB.(c)

«

<

1DvIGzo

_1

(r}+Coexp

4

x-xo et

p

2

1

47rltI

+?{n- I(ro)exp

(-)l

- )

.

Now choose a = a(Co) > 0 and then n =, (a) > 0 to achieve that the right of the above inequality is smaller than C2. In view of the above results, it seems reasonable to extend the classsical motion of surfaces Mt by mean curvature beyond the first singular time T by letting Mt =

ert for t > T.

2.8. Singularities A point zo = (xo, to) with xo E rc, is a singular point of a C2-solution u of (2.4) if Vu(zp) = 0. (Conversely, if so is regular, that is, Vu(zo) 0 0, then rt, is a C2-surface near xo.) Suppose, for instance, that A = V2u(zo) is positive definite. Then by (2.4) we

have a := ut = (bid -

0 at zo and hence for t < to close to to there

holds

rt

{x; (x - xo)T A(x - x0) = alto - t)1 . 2

Hence we expect the 0-level surfaces of the rescaled functions UR(X,

t) = u \XO + R'to + R2 /

at t = -1 to approach a smooth limit. More generally, consider the signum function ' above and for a weak solution u of (2.4) let v = 111(u). Moreover, for zo = (x0, to) E R" x [0, oo[, R > 0, let

x t VR(x, t) = v x0 + R, to +

R2

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

310

Suppose the level surfaces 1'R = {VR(-,t) = 0} are smooth. Then by the monotonicity estimate (2.8) for any T > 0 we have

f

0

T

2

JHR+ 2tvRl IDVRIGdt R°

IH

to

= fo_TR-2 JR. t

2

+

vI IDvIGzo dt 2r

0

(R - oo),

I

where v denotes the normal v = Tsf and H the mean curvature. Generalizing [77], we call a singularity of Type 1 if the surfaces TR smoothly converge to a smooth

surface Fr for every t < 0, as R - oo. Let H,,, v. denote the mean curvature and normal on Fr. Then by the above we have

2ItIHH = (x,v.). Represent rt ° = Ft (I'' 1). Then by (2.1) it follows that d (F, v_)

Ft=-H.v. _ -21t1

voo

whence up to tangential diffeomorphism 1'i° is contracting by self-similar motion. All smooth star-shaped hypersurfaces in R^ satisfying

H=(x,v)>0, and hence giving rise to self-similar solutions of (2.1) or (2.4), have been classified by Huisken [77], Theorem 5.1. In particular, we have:

Theorem 2.10 ([77], Theorem 4.1). If M"-1 C Ilt° is compact with non-negative mean curvature H = (x, v), then M is a sphere of radius n - 1. There seems to be a large variety of self-similar solutions of (2.1) that do not satisfy H > 0; in particular, the torus-shaped hypersurface of Example 2.2. Moreover, there are singularities that evolve at a different (faster) rate, which we call of Type 2. The study of singularities for the mean curvature flow is a very active field of current research.

PART 3 Harmonic maps of Minkowsky space

3.1. The Cauchy problem for harmonic maps Consider Mm+l = R x R', equipped with the Minkowsky metric 1

g = (gap) =

0 and let N be a compact Riemannian manifold, isometrically embedded in RI. By

analogy with harmonic maps of Riemannian manifolds, we define a map u: M'"+1 -+ N C R to be harmonic if it is stationary for the Lagrangian L(u; R) =

Jx l (u) dz,

where

1(u) =

29Q08Qu.80u`

= 2 (IVuI2 - lug12)

and where z = (t, x) = (x°, xl, ... , x"). Here, V = ( DiT , ... , 88 ) denotes the spatial derivatives and, for brevity, ea = g , a = 0, ... , m. Moreover, it will be convenient to denote D = (Oo...... m) _ (at, 0) and 8° = g°1980. For 0 E Co (M'"+l; u- TN) supported on a space-time domain R C M'+1 we have de

L ("N (u + CO); R) 11=0 = J O u O dz = 0, R

where _02

0

2-0

is the wave operator, acting component-wise on u; that is, u is harmonic if (3.1)

Ou 1 TuN. 311

M. 8TRUWE, GEOMETRIC EVOLUTION PROBLEMS

312

Suppose v1+1,. .. , v is a smooth (local) orthonormal frame field for the normal bundle T1N C TR". Then in the spirit of 1.1 we may write

u = E Akvk

u,

k

where Ak : M"'+I -. R is given by Ak = (E] u, vk a u) = -Sa (8°u, vk . U) + (8°u, So (vk ° u))

= A"(u)(BQu,8°u) = Ak(u)(Vu, Vu) - Ak(u)(ut,ut), with (3.2)

denoting the scalar product in R". Hence

Du = E(vk ° u)Ak(u)

(Sau,&,u)

= A(u)(Du,Du).

k

Introducing the "null form"

Q* 0) = VOW L' - ot0t, note that (3.2) also may be written in the form (3.3)

uk = a (u)Q(u`, uJ ),

k = 1, ... , n.

Finally, in local coordinates (u',. .. , uI) on N, (3.2) may be written (3.4)

uk = f'j (u)QW, uj ),

where I''if denote the Christoffel symbols on N. Note that Q(u, u) = 21(u), and Q is associated with the wave operator in the same way as the Dirichlet energy density is associated with the Laplacian. For hyperbolic problems it is natural to consider the Cauchy problem: Given

initial data uo,ul at t = 0 we seek u: M'+I -+ N C R" satisfying (3.2) and the initial condition (3.5)

Unt .o = uo,

ut 1t-o = ul

on R,".

The questions we consider are local and global existence of classical or weak solutions, and development of singularities. We will not consider the problem of determining the asymptotic behavior of solutions or scattering theory. Equation (3.2) shares the property of the homogeneous wave equation 0 = 0 that initial disturbances propagate with speed < 1. Thus, as far as the above topics are concerned, we may restrict our attention to initial data with "compact support"

in the sense that (3.6)

uo - const,

Ui = 0

on R' \ Sao

for some compact set Sao CC R"'. In what follows we discuss various mechanisms for proving existence for (3.2), (3.5).

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

313

3.2. Local existence Existence via the fundamental solution Consider the Cauchy problem for u: M"+1 -, R satisfying

u=f

in M'11

with initial data uo and u1. If m = 1, the solution is given by u(t, x) =21 (uo(x + t) + uo(x - t)) + 1 2

-t

x+a

:

l I f -a f(t - s, y) dy ds.

+2

Ifm=3,wehave u(t' x)

f'+t u1(v) dy

0

)

dt

(4ir , u0(x + t{) do

+f'

8

47r

S2

+

_L

J u1(x + tc) do

f (t - s, x + s£) do ds,

whereas, if m > 3, the representation formula involves also derivatives of f transverse to the backward light cone M(z) from z = (t, x), given by

M(z) _ {(s,y) ; t -s = Ix - vI} . Note that, even in dimension m = 1, there is no "gain" in derivatives. Thus, already

in dimension m = 1, for nonlinearities f = f (u, Du) as in (3.2) a simple-minded iteration procedure will fail to yield a local existence result due to loss of derivatives.

Energy method First consider a smooth solution u of the homogeneous wave equation

u=0

in M'"+I

having compact support on any slice it = cont.} Multiplying by ut we obtain the conservation law (3.7)

0 = attar - Auut =

d f utI1+IVu12\) dt

2

I

- div(Vuut).

Denote e(u) = IutI2 + IDuI2 = 1 IDuj2 2

2

the energy density of u and let

E(u(t)) = f

e(u) dx. t }xA^

Then, since u(t) has compact support, upon integrating the above conservation identity we obtain dtE(u(t)) = 0; in particular

E(u(t)) E(u(0)). Similarly, since derivatives v = D°u = 00 '0 ... &,W-u for any multi-index a = (ao,... , a,) again solve v = 0 with supp(v(t)) cc It'", we have E(D°u(t)) < E(D°u(0)).

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

314

Note that D°u(0) can be computed entirely in terms of uo and ul, in view of the equation utt = Au. 0. Observe that (3.7) remains true if instead of Du = 0 we only require On account of (3.1), the latter condition is satisfied for harmonic maps, and we obtain Lemma 3.1 (Energy inequality). Let u be a smooth solution of the Cauchy problem (3.2), (3.5) having compact support in the sense of (3.6). Then for all t we have

E(u(t)) < E(u(O)). (In fact, for smooth u equality holds.)

In order to obtain corresponding bounds for higher derivatives, however, we have to consider also inhomogeneous equations. It is useful to introduce the spacetime norms 00

1 1/4 I

/

OL+

,

1 < p, q < oo, and the corresponding spaces L9.P(Mm+1) = L9 (R; LP(Rm)). On a finite space-time cylinder SIT = [0, T] x Cl corresponding norms may be defined.

Consider now a smooth solution u: M+1 --, R of

Ou = f with initial data uo, ul, where u(t) has compact support for any t. Multiplying by ut we obtain

je(u) - div(Vuut) = fut <- IfIIutI Integrating over {t} x R'", by the Cauchy-Schwartz inequality this gives

dE(u(t)) t

< I{t}XRm IfIIutldx <_ IIf(t)IIL3IIut(t)IIL3 <_ IIf(t)IIL2(2E(u(t)))1J2

That is, E(u(t)) <- IIf(t)IILv,

a

neglecting a factor V2_. Hence we obtain E(u(t)) < 2E(u(O)) + 2111)tL1.2

for all t, and similarly E(D°u(t)) < 2E(D°u(0)) + 2IID°f112 2, (3.8) for all a = (ao,...,a,,,).

Local existence for the Cauchy problem for harmonic maps We now apply (3.8) to solve (3.2) and (3.5) near t = 0. This result is quite standard. A survey of local existence results is given in Kato (91). Denote

H I.2(Rm;N), uI E Hja(Rm;u01TN),(uo,ul)satisfy (3.6)}.

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

315

where Hm,2(Rm; N), etc., is defined as in Part 1, sec 1, and let HH+1 = H:+I(Rm;TRn). Then we have:

Theorem 3.1. Suppose (u°iul) E H:+1(Rm;TN), where s > 2 . Then for some T > 0 problem (3.2), (3.5) admits a unique solution u: [-T,T] x Rt --' N such that sup IIDu(t)IIH. < oo.

ltI
Proof. Let u(0) E L°O(R;HH+1) solve Ou(°) = 0 with Cauchy date (u°iul). We claim there exists T > 0 and a convex neighborhood U of u(0) in LO° ([-T, TJ;

such that the map S: v -+ u = S(v) given by solving Ou = A(v)(Dv, Dv) with initial data u°, ul is a strictly contracting map of U to itself. Indeed, by (3.8) we have the estimate

suPII(Du - Du)(t)II,. < Csup E E(D°(u - u)(t)) t

e

(3.9)

< C E II D° (A(v)(Dv, Dv) - A(v)(Dv, D8))II2

,.3

101<8

for the difference of any two solutions u = S(v), u = S(v). Moreover, for any IaI < s, any Itt < T, by Leibniz' rule we can estimate

F(t) := ID° (A(v)(Dv, Dv) - A(i)(Di), D0))I


IDO(v - v)I

IQISI°I+1 `

r D"(,)vI

\H `I

+ ID"c,>vI))

where in the product ry = E, -Y(i) ranges over all multi-indices such that I?'I < 2 + Ial - 1,01 and Iry(')I < IaI + 1. We use Sobolev's embedding theorem

Hl.a(R-), where

if 2(k -1) < m,

1_1k-1 q

2

m'

and

if 2(k -1) > m. Choosing 2s 0 N we can achieve that 2((s + 1) - 1) # m for all 1; otherwise we Hk,2(Rm) _.> HI,°°(R n),

may use the extended Holder-Sobolev inequalities (see for example F iedman [481 or Mazja [1071) to justify the following estimates. Note that by Holder's inequality

IIF(t)IIL2 < C ; 1ID'3(v - v)IlLrco

(pD"" vIIL.c+ + IID""'VIIV(o)

ICI<_I°I+1

where -y = E, y('), IyI < 2+IaI-I/I, Iryi`iI < IaI+1, and where for each 101 :5 IaI+1 and each ry we assume

>r()+rW

316

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

Let ,Q = -t(o) and for i = 0, 1, ... set

(s+1)-Iry(')I

1

if 2((s+1)-I7(0I) <m

m and set r(') = oo in the remaining cases. Note that r(') > 2 for all i. Hence we may assume that at least two distinct numbers r(') are finite. Since 2

EIryi'li < 2 + IoI < 2 + s

ryW < 1 + Iop < 1 + s,

i>O

then, if s > m/2, indeed we have

i< (21

i>O

-Y(i) - (s + 1) m

yi'l - 2(s + 1) ) <1 + Ei>o <1-m 1-m <2. m 1

Hence we obtain

IIFIIL2 <- CII(Dv - Di)IIf. (1 + IIDvII. + IIDvII .

.

Define

- Dut°i)(t)IIN. < 1 } . u = {v; supll(Dv t Integrating the above inequality and combining with (3.9) we obtain supll(Du - Du)(t)IIH. < CIIFI1L..2 <- CT2supIIF(t)II t

t

,2

(I+ sup (IIDv(t)IIH. + IIDi(t)IIff. ))

< CT2 supll(Dv - Dv)(t)IIff.

2

< 6supll(Dv - Dv)(t)IIH. t

with 6 < 1 if v, ii E U, and if T > 0 is sufficiently small. Similarly, we have

supll(Du-DuiOi)(t)IIH
and this is < 1, if v E U and T > 0 is sufficiently small.

It follows that S has a unique fixed point u = S(u) E U which is the local solution we seek.

In the derivation of Theorem 3.1 neither the geometric interpretation (3.1) of the harmonic map equation nor its special form (3.3) or (3.4) were used. An example of how this structure may be exploited is given by the following result of Klainerman-Machedon 197].

Theorem 3.2. Let u, v: M3+1 - R be solutions to

Ou=F,

Dv=G

in m3+1

with initial conditions alt=o = uo,

ut1t_o = u1,

vlt=o = Vol

vt It_o = v1,

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

317

respectively, and let Q(u,v) = Du Vv - utvt be a null-form as above. Then for any T > 0 we have IIDQ(u, v)IIL-

< C (IlDu(0)1111, + f

T IIG(t)IIHI

dt)

with an absolute constant C.

Q may be replaced by other "null-forms"; however, the result is false for general

bilinear forms in Du and Dv. As a consequence they derive the following slight improvement of Theorem 3.1.

Theorem 3.3. Let m < 3 and suppose (uo, u1) E H.. Then there exists T > 0 and a unique local solution u: [-T, 71 x R3 - N of (3.2), (3.5), satisfying supllDu(t)IIH, < CIIDu(O)IIHi

A simple proof of this fact, based, however, on the geometric condition (3.1) and energy estimates, can be given als follows.

Proof of Theorem 3.3. For any a E 0, ... , m we have dtE(&u) < j

Cf But be rewritten as

1

m

IDul3ID2uI dx + 2 fit m A(u)(D&u, Du) . 8aut dx.

thus the integrand in the second term on the right may

-(8QA(u))(D80u, Du) ut < CIDuI3ID2uI and we obtain

dtE(Du) < Cf IDuI3ID2uldx < CIIDuIIielID2uIIL2. R-^

Finally, since Du(t) has compact support for any t, by Sobolev's embedding theorem and since m:5 3 this implies dtE(Du) <_ CIID2UIII2 < C(E(Du) )2

for 0 < t < T, where C may depend on T and the size of the support of the initial data. Local existence thus follows from Gronwall's inequality. To obtain uniqueness, note that by (3.1) the difference of equations (3.2) for two H2-solutions u and v after multiplication by us - vg gives

0 (u - v) (u - v)t = (A(u)(Du, Du) - A(v)(Dv, Dv)) (ut - vt) = -A(u)(Du, Du)vt - A(v)(Dv, Dv)ut = (A(u) - A(v)) (Dv, Dv)ut - (A(u) - A(v)) (Du, Du)vt < Clu - vIID(u - v)I(IDuI2 + IDvI2).

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

318

Integrating over R"', for m < 3 by Sobolev's embedding theorem therefore we obtain dt

E(u - v) < Cllu - vIILOIID(u - v)IIL= - (IID(u)IIi6 +11D(v)11' e) CE(u - v) - sup(IID(u)IIH, + IID(v)IIHi )

and uniqueness in the class of H2-solutions follows.

Klainerman-Machedon conjecture that (uo, ul) E H,,, s > 2 , suffices. In fact, many experts believe that local existence and uniqueness should hold true in

the class HH+1, s > 2 In particular, for m = 2 such a result would imply the global existence of a unique solution to the Cauchy problem (3.2) (3.5) in the energy class. However, apart from some special cases, this problem still is completely open. Observe that the following simple model problem from Klainerman-Machedon

[97] demonstrates that such an improvement is not possible without using the geometric structure of the harmonic map equation.

Example 3.1. Let u satisfy the nonlinear wave equation Ou = Q(u, u) = IDul2 - IutI2 in M'"+1

with initial data NO - 0, u1 E Ho'2(Rm). Then v = e' satisfies the linear wave equation

v=eu(D.-Q(u,u))=0

in M'"+1

with initial data vo = eu0 - 1, v1 = u1 and t - v(t) is continuous in HH+1. To ensure that v = e" > 0 for small t > 0 we need continuity of v(t), and hence, by Sobolev's embedding theorem, we need s > m2 2

3.3. Global existence Example 3.2. Geodesics (Sideris 1131]). If y: R -' N is a geodesic on N and if v: M"t+1 -, R satisfies

v=0,

then u = y

v solves

Ou = y'(v) Dv + y"(v)Q(v, v) 1 TuN, because y" 1 T.N. That is, u is harmonic. Note that u preserves the regularity of the initial data. Example 3.3. Let m = 1. In this case, global existence and regularity was established by Gu [68] and Ginibre-Velo [54]. A surprisingly simple proof was given by Shatah [127] based on the following observation: Multiply (3.1) by ut, respectively by ux, to obtain the system of conservation laws

0=Duut=Bte-8ym, 0=Duu,, =Ojm - Ore, where e = e(u) = z (IutI2 + Iu=12) is the energy density and m = m(u) = u=ut is the density of momentum; compare (3.7). Thus, e satisfies the linear wave equation

De=0.

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

319

and from the representation formula we obtain pointwise bounds on e, and hence on Du, in terms of the initial data. Higher regularity for smooth data then follows by using the energy inequality (3.8) to iteratively derive Gronwall-type estimates

d E(D°u(t)) < C + CE D- (u(t))) for any a.

Remark 3.4. For m = 3 and rather smooth initial data uo, ul whose norm is small in H', s > 10, Sideris [131] has constructed global solutions to (3.2), (3.5) in H' by combining the local existence result above with certain decay estimates that may be obtained by using the invariant norms of Klainerman [95]. The latter are defined by means of the generators of the Poincare group and scale changes. In addition to the standard differentials 8°, a = 0, ... , 3, denoted I',. .. , r4 in the following, these include the generators r5,. . . , r 10 of Lorentz and proper rotations

f3°6 =x°Y -xp8°,

0
and the generator of dilations

r0 = xo8°. Note that Lorentz invariance of the wave operator implies the commutation relation H°a] = 0; moreover,

The family r spanned by r0,. .. , r,o forms a Lie-algebra. The invariant norms II

II r,9 of functions u : M3+1 - R now are defined as follows: 1/2

IIu(t)IIr,9 -

where r° =

Ilr°u(t)IIL2(R9)

rj;°o r°' for any multi-index a = (ao,... , alo). Also let E0 = E and

E(r°u(t)),

E9(u(t)) _

S E N.

1a1<9

Note that E.(u(t)), s > 1 involves weighted norms of derivatives of u. Hence, finiteness of E.(u(t)) for sufficiently large s will imply some decay of u and its derivatives. For instance, we have the generalized Sobolev inequality, due to Klainerman [95], which gives the bound

suplDu(t,x)I <

1 + it + IxI E4(u(t)),

for any smooth function u on M3+1. More generally, Sideris has constructed global solutions for smooth initial data H10-close to a "geodesic". For m = 2, Kovalyov [99] has obtained similar global existence results for small data The above results should be compared with long-time existence and blow-up results for general hyperbolic equations

u = f (u, Du, D2u)

in M'"+1+

in particular, in case m = 3 again a "null condition" for the quadratic part of f is required; see Klainerman [96], John [85], Christodoulou [24].

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

320

3.4. Finite-time blow-up Following Shatah [127] and Shatah-Tahvildar-Zadeh [129], [130], in order to produce examples of solutions to the Cauchy problem for harmonic maps that blow up in finite time, as in Lecture 1.2 we consider equivariant maps into spheres or, more generally, into hypersurfaces of revolution N, locally diffeomorphic to If"` with metric dal = dh2 + g2(h)dw2

in spherical coordinates h > 0, w E S-'. Moreover, on M'"+I we consider the standard metric -dt2 + dr2 + r2d62,

written in terms of spherical coordinates (r, 0) on Rm. Consider equivariant maps

u: M+' - N of the form u(t, r, 0) = (h(t, r), w(6)) where w is a homogeneous harmonic polynomial of degree d > 0. Then

L(u) = 2 f {Ihri2 - Ihtl2 +

r

92(h)Ioew121

r'"-' dtdrd6,

and u is harmonic if and only if h : M'"+1 - R satisfies kf(h) = 0, Clh +

(3.10)

r2

where k = d(d + m - 2), and f (h) = g(h)g'(h). Observe that for N = S'" we have g(h) = sin h. More generally we require g(0) = 0, g'(0) = 1, g(h) > 0,

(3.11)

for h > 0, and g(-h) = -g(h).

Note that the ball of radius ho around 0 in N is convex iff g'(h) > 0 for

0
Self-similar equivariant solutions Using invariance of (3.10) under dilation of space-time and time reversal t - -t, in order to achieve a further simplification we try to construct solutions h to (3.10) of the form

h(t,r)=0(t) with smooth Cauchy data 0 at t = 1. For self-similar solutions of this kind, we compute

- h rr - m lhr r (r , (r 2r r2 =

h=

h tt t3

= where p =

f

\t )+

\t )

N

1

t2

(r

t (2W( ) + ( 2 - 1)4/'( ) - M p

P

p

_ m-1

t) P

rt

,

r) (t )

l e( P)

and 0' = A 0, etc. That is, m-1

(P) + (P(1-p2)

_

2p

-t2 O h

1-p2) '(P) - 1-P2

-

kf (0)

p2(1-p2)

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

321

Simplifying, we finally obtain the equation (3.12)

+ <002

m - 1+(m - 3)P p 1-p2 <2now let 2(0)_02-

4)

,_

kf (4') p2(1-p2) = 0 .

if.0 <00,

24,

whence

f (0) = 9(4))9 (4)) = 92(0), = 0 - 03,

if 0 < ¢o, and extend g smoothly elsewhere. Then for d = 1, k = m - 1, the function 4'(P) = cp,

where c = m? solves (3.12) for p < vi;ii- By finiteness of propagation speed, the behavior of h at 0 is determined by the Cauchy data on B1(0) at time 11

t = 1 only. Hence the solution h to (3.10) with initial data 0 in a neighborhood of B1 (0) at time t = I will blow up in finite time. Note that for g as above, the radius of convexity of N around 0 is

4).=1 which is larger than c form > 4 and equals c if m = 3. By changing the metric g(¢) on N suitably for 0 > c, and by changing the initial data for h off B1(0), we may thus construct solutions to (3.2), (3.5) which blow up in finite time, with initial data having compact support and such that the target manifold is convex, if m > 4, and only slightly fails to be convex, if m = 3. A more detailed analysis shows that in 3 space dimensions blow-up may occur also for more general metrics on the target surface:

Theorem 3.4 (Shatah-Tahvildar-Zadeh [1301). Suppose g E C°° satisfies 9(0) = 0, g'(0) = 1 and suppose g' has a smallest positive zero ¢.. Also suppose that g"(4).) 0 0. Then there is a class of regular initial data such that the corresponding Cauchy problem for equivariant harmonic maps from M3+1 into N has a solution that blows up in finite time.

Non-uniqueness of weak solutions In particular, Theorem 3.4 applies to the sphere, where g(¢) = sin 0, 0. = z Shatah-Tahvildar-Zadeh construct a solution 0 to (3.12) on [0, oo[, satisfying

0(1) _ 0. and having the asymptotic expansion for p - oo: gy(p)

CA

=

b a+P+;+O( )

+o(

_

)

.

They consider the corresponding function h(t, r) = ¢ (f) as a weak solution of the equivariant harmonic map problem (3.10), that is, (3.13)

2

h« - hrr - - hr +

sin.2h = 0, r2

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

322

with singular initial data at t = 0, given by

h(0, r) = ho(r) = a = limb (1t)

(3.14)

ht(O,r)

hi (r)

lim Wt-0

r

(r jA 0) , (r 96 0) .

\t/

Thereby, h is a weak solution of (3.13) if there holds

/1/00 (3.15)

{ -htit + hrtbr + 11

°°

1

p sin 2h y r2 dr dt = fo

r

6

,(0, r) r2 dr

r

11

for any 1' E CO0 ([0,1] x R3) such that O(t, x) = t1(t, r),,0(1, )

0, and supp li(t) C

BR(0) for some R > 0. Moreover, h assumes the initial data (3.14) in the sense

that IIh(t, r) - aII6l. (R3) -+ 0

(t -+ 0),

b

(t - 0).

IIht(t, r) -

JIL2(R3) -> 0

Note that ho E Him, hi E L«. On the other hand, also the function

r>t

h(t,r)-f O(i),

r
0.,

weakly satisfies (3.13), (3.14) on [0,1] x R3, with Dh E LO° ([0,1]; L2(BR(0))) for any R > 0, showing that weak solutions are in general not unique. To verify that h solves (3.15), for any 0 we split 1

11100 {_hP + hrlr + r

_

sin 2h y r2 dr dt - Joo y(0, r)r2 1

b

i,(0,r) r2dr}+

...}r2drdt-j00

r

fo, f 0" e

111

f1f 0

{...}r2drdt=I+11.

0

Clearly, since Dh(t, r) = 0 for r < t, the second integral II = 0. Moreover, since h h for r > t, and since h satisfies (3.15) the first integral reduces to the boundary term

I=- 75 f (ht(t, t) + hr(t, t)) (t, t)t2 dt I

which also vanishes on account of ht+hr=_

t

ci/(t/+tq'\t/ (l (l (1

t) '( ) =0

for r = t.

Observe that h induces a solution u of (3.2) with E(u(t); BI(0)) < E(u(t); BI(0)) for any t E10, 11, where u is the solution corresponding to h. Hence there may be a chance of restoring uniqueness by some entropy principle.

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

323

Self-similar solutions, general theory More generally, self-similar solutions u(t, x) = v (A) to the harmonic map equation satisfy -vPP -

(3.16)

(m_p 1

+ (1 - pep) VP - p2(11

p2)

D,,,v J

where p, w denote spherical coordinates on Rm. This may either be verified by direct computations or by introducing similarity coordinates

T=

t2-r2,

r

p= !,

w=8

on {r < t} and writing the Minkowski metric on Mm+I as

ds2 = -dr2 + r2 {

dw2} -I,2)2 dp2 + 1 - p2 For u(t, r, 0) = v(-r, p, w) the Lagrangian then becomes (1

(3.17)

L(v) = 1

[ f_IVrl2 +

(1 - p2)2Iv I2 + (1 - p2)

Tmpm-I

dpdwdT. 2J T2 r2p2 l } (1 _ p2)(.+1)/2 In particular, if v = v(p,w) is stationary for L, we obtain (3.16). Note that (3.16) is an elliptic harmonic map problem on the m-dimensional hypersurface jr = 1} with the (hyperbolic) metric a

d802 = (1

vWI2

ip2)2 dp2 + 1

p2 dw2,

as was pointed out by Shatah-Tahvildar-Zadeh [130). Now observe that, if m = 3, for v to be regular (C2) at p = 1 we need

atp=1; that is, v(1, ) : 52 --+ N has to be harmonic. By the maximum principle (Jiiger-Kaul )81]) for harmonic maps into convex manifolds, therefore either v(1, ) - const. or

v(1, ) cannot be contained in a strictly convex part of N. Moreover, if v(1, ) cont., then for p < 1 sufficiently close to 1 the image of v(p, ) is contained in an arbitrarily small strictly convex part of N and again we may apply the Jiiger-Kaul maximum principle to conclude:

Theorem 3.5. If m = 3 and if u(t, x) = v (i) is a self-similar solution to the harmonic map equation (3.2), where v: R3 - N is smooth in a neighborhood of Bl (0) and such that the image v (BI (0)) is contained in a strictly convex part of

N, then v - cont. on B, (0). By Theorem 3.4 the above result is best possible in dimension m = 3. In case m = 2, due to the following result we can rule out self-similar solutions altogether.

Theorem 3.6. If m = 2 and if u(t, x) = v (1) solves (3.16), where v : D C R2 N is smooth in a neighborhood of BI (0), then v =- const. on B1 (0).

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

324

Proof. If m = 2 we may write (3.16) in the form (p

A"v

1 - p2vp)p + p

Multiplying by pvp E Wp

1 TN. P2

and integrating over w E S1, we obtain

(J' p(1-p2)Ivpl2dw- f IV"vI2d)

0.

Integrating in p, we find

JS,p(1-p2)Ivp12dw-f

v12dw=Co. JV

Inspection at p = 0 shows that Co = 0. Hence for p = 1 we obtain V ,,v = 0 that is, vIsa,(o) = cont. Finally, note that in dimension m = 2 Dirichlet's integral and Thence the harmonic map equation is conformally invariant. Theorem 3.6 thus is a consequence of Lemaire's result, Example 1.3

3.5. Global existence and regularity for equivariant harmonic maps for m = 2 The preceding examples of finite-time blow-up hardly leave any hope to achieve a satisfactory existence and regularity theory for the Cauchy problem for harmonic maps, except in dimension m = 2. In fact, we may state the following

Conjecture 3.1. If m = 2, then for any compactly supported initial data uo, ul with finite energy there exists a unique global weak solution u to the Cauchy problem (3.2) (3.5) satisfying the energy inequality. If E(u(0)) < co = CO(N) is sufficiently small, or if the range u(M2+1) lies in a strictly convex part of N or, more generally, does not contain the image of a harmonic sphere u: S2 - N, then u is globally smooth, provided uo and u1 are. At present the theory is still a long way from affording a proof of this conjecture

in general. Partial results, however, are known; in particular, Conjecture 3.1 has been rigorously established for equivariant harmonic maps into convex surfaces of revolution. In the following, we review these latter results, due to Shatah-Tahvildar-Zadeh [129]. A simplified proof was given by Shatah-Struwe [128]; moreover, Grillakis

[64] has recntly been able to relax the convexity assumption. Similar results for radially symmetric harmonic maps (m = 2) have been obtained by ChristodoulouTahvildar-Zadeh [25]. Thus we consider the Cauchy problem (3.10), that is, (3.18)

f

utt - Du +

r2 )

=0

in M2+1

for smooth, radially symmetric data (3.19)

ult--.o

= u0, ut 1t=o = u1

having compact support and such that uo(0) = 0 By uniqueness, also u will be radially symmetric u(t, x) = u(t, r), and u(t, 0) = 0 if u is smooth.

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

325

Regularity for small energy In a first step we shall show that this problem admits a unique smooth solution for all time provided the initial energy E(u(O))

= JR2

IuII2 +2IVuo 2

+ F(uo)1 dx r2

J

is sufficiently small. Here F(u) _ g2(u) = fo f (v) dv. Note that the energy inequality

E(u(t)) < E(u(0)) holds. Moreover, by radial symmetry

E(u(t))

= 2a ( Iut12 + I'url2 r+ F(u) r

2

0

} dr 111

_: 21rErad(u(t))Finally, it will be useful to denote

G(u) =

l

ix(Ia)

dx = 2,r f = F(u) dr = 27rGrad(u)

the potential energy of u.

Lemma 3.2. Given Co > 0, there are constants C1, el > 0 such that any solution u to (3.18), (3.19) with E(u(0)) < Co satisfies IIu(t)IIL.e <_ C1, provided G(u(t)) < el, for any t.

Proof. Denote

r

H(u)

rv

g(v) dv = f

0

2F(v) dv.

0

By radial symmetry, for any (t, ro) we have

H(u(t, r))r dr) 2

IH(u(t,ro))12 = ( I

U 00

<_ ( I fOO

lurIIg(u)I dr)2 00

dr u2rdr fo g2u r

< 4Erad(t(t)) - Grad(U(t)) 5 Ir-2E(u(0))G(u(t)) Thus, if we fix 2

el < a2Co 1 sup H2(u) _ 1r2Co 1 U

the claim follows.

9(v) dv) < 00,

O

Remark 3.5. In particular, there are constants el > 0, C1 such that any solution u to (3.18), (3.19) satisfies IIuIIL' 5 C1 provided E(u(0)) < el. More generally, if 00

L that is, if N does not include "sphere at infinity" in the words of Shatah-TahvildarZadeh [1291, then any solution u to (3.18), (3.19) with finite initial energy is uniformly bounded.

326

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

By hypothesis (3.11) on g, given C1 > 0 there exists a constant C2 > 0 such

that CZ lug < F(u) < C2u2

(3.20)

for Jul < C1. Introduce w, defined by w(t, r)

_ u(tr,r) .

Note that w is smooth if u is and satisfies /

wtt - Wrr - 3 wr = u - J (u) = w3 [a + p(rw)],

r

where a = - 3

r3

(0) and p is a smooth function; that is, w solves a semi-linear

wave equation

wtt - Aw = w3[a + p(rw)]

in M4+1,

with nonlinearity growing at a "critical" rate. More generally, a semi-linear wave equation

in M-+1

wtt - Ow + n(u) = 0 has critical growth if n(u) exponent in R"'.

for large u, where 2' _ mm2 is the Sobolev

In 1988 the model case

utt - Au + u5 = 0

in M3+1

was solved by Struwe [142] for large radially symmetric data. Grillakis [62] then was able to remove the symmetry condition; see also Struwe [146] for a survey of these developments. More recently, also the higher dimensional cases have become accessible, in particular, the case m = 4 needed here. Note that u = rw is bounded. Moreover,

IOwIz = wrz =

uT

r2

U2

- 2 uru + 74' r3

whence

jlVw(t)!2 dx u2

cwTr3 dr = cu }{u2r+ dr < CE(u(t)) < CE(u(0)). r

fo"o

fooo

JJJ

Hence by Sobolev's embedding theorem w(t) E L4(R4) and IIw(t)IIi4 <_ CE(u(0)). However, (3.20) gives a more convenient bound 00

IIW(t)114

L4 = C

J

00

r4 r3 dr < C

J

T2

dr < CG(u(t)) < C (E(u(0))) .

Finally, wt = r implies

r

jIwtI2dx < cJ Iut12rdr < CE(u(t)) < CE(u(0)). We proceed to show:

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

327

Theorem 3.7 (Shatah-Struwe [128]). Given Co > 0 there exists a constant > 0 such that for any smooth, compactly supported data no, ul having energy E(u(0)) < Co the Cauchy problem (3.18), (3.19) admits a unique smooth local solution u. The solution extends globally, provided the estimate suplH 0. Note that it is easy to obtain global weak solutions of (3.18), (3.19) by energy methods; see Strauss [134] or Struwe [146]. Moreover, by adapting the arguments of Kapitanskii [90] one can show uniqueness of weak solutions in a suitable class. For the proof of Theorem 3.7 the following basic regularity result for the linear wave equation

Dv = f

(3.21) (3.22)

in M'"+1

vlt=o = Vol vtlt=o = Vi

is needed.

Proposition 3.1 (Strichartz [135] [1361). Suppose v is a (smooth) solution of (3.21), (3.22), m > 2. Then for any T > 0 we have the estimate IIVIIL-(Rmx(-T,T]) < C (IIfIIL9'(Rmx(-T,T]) + IIVOIIHI(R,) + IIt1IIH-I(Rm)) where

_ 1-

1

1

1

q

2

m+1

1

q

Hi = H'2 is the interpolation space between L2 and H1.2, H-4 its dual. Remark 3.6. Kapitanskii [90] was the first to note the importance of Strichartz' estimate for semi-linear wave equations with critical nonlinearity. Combining this result with the dilation estimates of Morawetz [109], it was possible to extend the regularity results of Struwe [142] and Grillakis [62] from m = 3 to m:5 5 (Grillakis [83]) and, more recently, even to m < 7 (Shatah-Struwe [128]).

Proof of Theorem 3.7. We apply Proposition 3.1 in dimension m = 4 with q = 3 , q' = to obtain bounds for to and its spatial gradient Vw; that is to,.. Interpolating between the two, we obtain bounds for "half a derivative" of to in Lq. More precisely, denote VTq = {u E L2(R'" x [-T,T]);u(t) E BQ'q(Rm) T

1/q

(LIIut)II, j dt

)

<

for a.e. t,

},

where the Besov space Bq (R'") denotes the interpolation space between Lq(Rm) and W I -e (R"' ). Note that by Sobolev's embedding Bq

L

,

1

1

1

s

q

2m'

that is, s = 7 , if m = 4. Then by Proposition 3.1 we have (3.23)

IIwIIvT 5 CIIw3(a+p(rw))II%.;' +CI)Vw(0)IIL2 +CIIwt(0)IIL2.

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

328

The last two terms are bounded by E(u(O)). To bound the first term on the right for fixed t we estimate the Lq - and W I,q'-norms, using boundedness of rw = u and Holder's inequality IIw3(a+p(rw))IILQ'
(3.24)

IIV[w3(a+p(rw))]IILQ' S CIIIVuIIWI2 + IwI'IILQ' (3.25)

< C(IIVwIILQ + IIIw121ILQ)IlwllL

<-

where 1 2 1-q=4'=q+4i; 1

that is,

1

111_ _ _ 1 ql

2

_

m+1

q

1

5

Note that we also used Sobolev's embedding to bound IIw21ILQ < CIIVWIILqIIWIIL4 <- CsuPIIw(t)IIL4OVWIILQ <- C (E(u(0))) IIVwIILQ

Note that ao = e > 5 = ql > 4. Similarly, by interpolation we obtain

with

2

ql

_

11W11 2..

< Ilwll$

IIw3(a+p(rw))II 2

2-a

_ a

8+

5

4

3IIwII .IIwII ;a

<_ CIIwII

7a

2-a

40+

4

that is, a = 3. By Sobolev's embedding theorem, therefore, for fixed t we may estimate IIw3(a+P(rw))II .,y <_ C Sup IIw(t)IIL 3IIwII7..13 < CE(u(o))I16IIwII713 Ali

ltl:ST

Note that, miraculously, 7

_

3

10

3

_= q.

Thus, when integrating in t we obtain T

(3.26)

IIw3(a+p(rw))IIVT =

jiiwa p(rw))IIB

VT

CSUPIIW(t)112,311WI1713

dt } < CE(u(0))1/8IIw1I7Q

yv Combining this estimate with (3.23), there results the inequality

IIwIIvv 5 C3(E(u(0)))1/2 +C3SUPIIw(t)II L43IIwIIya t

for any T > 0, with a constant C3 = C3(E(u(0))) > 0. Suppose E(u(0)) < Co. By our local existence result Theorem 3.1, IIwIIv; < 2Co/2C3 for small T > 0. If we then have

C38upIIw(t)IIL 3 (2Co/2C3)a/3 < 27

by continuity in T the estimate (3.27)

IIwlIv. < 2Co12C3

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

329

will persist for all T > 0. Using (3.27) and applying Proposition 3.1 again to the differentiated equation (3.18), we can now conclude in a similar way IIVwIILQ(Rm+1) < C(r o,u1) + C4 SUpIIW(t)IIL 3IIVWIILQ(Rm+1),

and thus Vw is bounded in LQ(IC"+I) in terms of the initial data, if sup IIw(t)112. < C sup G(u(t)) < Co t

is sufficiently small.

We can now proceed by energy methods. Indeed, since Vw E LQ we have = 20 is the Sobolev exponent, and thus _

w E LQ Q', where q'

IV [w3(a+p(rw))]I CIVwIw2+wq E It follows that D2w E L1'2, whence w E L°D-p for all p < oo, and Dw E L°°'4. Differentiating one more time and using that for any t we can estimate LI,2.

IID2 [w3(a +p(rw))a] IIL2 5 CIIID2wllw12 + IDW12IwI + IDwHIwI3 + IWI5IIL2

< CIID3wIIL2 +C,

upon multiplying the equation for Dew by D2wc we obtain a Gronwall-type inequality d E(D2w) < C(1 + E(D2w)) and hence D3w E L `2. But then by Sobolev's embedding theorem, w E L a°° _ LOO L. Continuing in this manner, also the derivatives of w - and hence of u - can be bounded.

By finite speed of propagation the above argument can be localized. To state this precisely, introduce the following notation. Given a point zo = (to, xo) E M'"+I, to > 0, denote K(zo) = {(t, x) E Mm ; Ix - xo1 c to - t, t > 0} the backward light cone with vertex at zo, and for t fixed denote by D(t;zo) = {(t,x) E K(zo)}, its spatial sections. (zo = (0, 0) will be omitted.) The proof of Theorem 3.7 implies

Lemma 3.3. Let zo = (to, xo), to > 0, be given. Suppose u E C- (R2 x [0, to[) is a smooth solution of (3.18), (3.19) with E(u(0)) < Co and suppose (3.28)

lim inf G(u(t); D(t; zo)) < co.

Then u extends smoothly to a neighborhood of zo.

Proof. Indeed, by the proof of Theorem 3.7 we obtain uniform bounds for u and its derivatives on K(zo). In particular, then also the condition E(u(t); D(t; zo)) < co (3.29) is satisfied for t close to to. Since u is smooth on R2 x [0, to[, condition (3.29) holds

for z in a full neighborhood of zo for any (fixed) tI close to to. Finally, by the energy inequality, (3.29) will persist for t > tI on any cone K(z) for such points z and the argument of Theorem 3.7 implies smoothness of u at all such points. Lemma 3.3 will be important for our next section.

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

330

Regularity for equivarlant harmonic maps with convex range We establish the following result of Shatah-Tahvildar-Zadeh [129).

Theorem 3.8. Suppose U: M2+I -. R is a weak solution of (3.18),(3.19) for smooth, compactly supported data uo, uI and of finite energy E(u(t)) < E(u(O)) < oo. Moreover, suppose that f (u) > 0 on the range of u. Then u E CO°. Proof. For the proof we need a local version of the energy inequality.

0

Lemma 3.4. Let u E C°D(K(zo)) be a smooth solution of (3.18). Then for any 0 < S < T < to there holds E(u;

D(T;

zo)) +

f

JMTZO) et

(u) do< E(u; D(S; zo)).

Here,

Ks (zo) = (z = (t, x) E K(zo); S < t < T} denotes a truncated cone and Ms (zo) = {z = (t, x) E Ks (zo); Is - xol = t - to} its mantle. Moreover, et8n(u) =

I(x - xo)ut + (t - to)Du12

F(u)

2It - t012

Ix12

denotes the tangential energy density. Finally, do is the element of surface area on MsT.

Proof. Multiply (3.18) by ut to obtain

jje(u) - div(utVu) = 0 where e (U) -

IutI2 + IVu12 2

F(2)

+ Ixl

Integrating over KS (zo), and noting that the outward normal on Ms (zo) is given by v = 1 (1, _ ), on account of

e(u) - x - xoutVu = eten(u) to - t

0

we obtain the claim.

Remark 3.7. In particular, letting T -, to we obtain

f

et.. (u) do < o0

Ms(ro)

and hence

M.(w) asS -'to.

et. (u) do - 0

PART 3. HARMONIC MAPS OF MINKOWSKY SPACE

331

As a consequence of Lemma 3.4, if u is a smooth local solution of (3.18), (3.19) on R2 x [0, to[ which becomes singular at zo = (to, xo), then xo = 0. Else, by radial

symmetry any point is = (to, x), Ixl = Ixol, would also be singular. By Lemma 3.3 and 3.4 for any such 2 and any T < to we then should have E(u; D(T; z)) > co.

But if Ixol # 0, given K E N, for T sufficiently close to to disjoint discs D(T; zk), zk = (to, --k), IxkI = 1X0 1, 1 < k < K, can be found, whence K

E(u(T)) > E E(u; D(T, zk)) > Keo k=1

for any K. This, however, contradicts our assumption that E(u(t)) < E(u(0)) < 00 for all t. Hence a first singularity must appear on the line x = 0. Suppose zo = (to, 0) is singular and shift time by to to achieve zo = 0. Denote K = K(0), KS = KS(0), etc. We need the following Morawetz-type dilation estimate:

Lemma 3.5. For a smooth solution u of (3.18) on a cone KS there holds z 1 uf(Z) F(a) F(u) dxdt + 4(s) j 1 1 - IXs IVUI2 + dx -' 0 Ixl2 S

ISI JK2IxI Ixll2 \

as

S-+0.

Proof. Multiply (3.18) by tut + xVu + u and use Remark 3.7 to control the z details. boundary terms; see Shatah-Struwe [128] for

Lemma 3.5 implies Theorem 3.8. Indgiven T < 0 consider the set AT

t E [T, O[ ; G(u; D(t)) =

f

F(u) dx > sup G(u, D(t)) - 62 T
(t) IxI2

JJJ

where 6 > 0 will be determined later. Since u f (u) > 0 by assumption, for large T < 0 Lemma 3.5 implies z

(t2 1- l

r

J

for any t E AT. In particular,

) IDu12 dx < 262

t(1-6)

u2(t)rdr < C6, and hence by the argument of Lemma 3.2 we obtain sup

Iu(t,x)I <<

IxISISI(1-6)

2

for any p > 0 and any such t, if 0 < 6 < 6(p). Moreover, estimating z

Iu(t,x)-u(t,y)Iz =

1

JIvi

urdr <

J

I=I dr

Ivl

)

(fIXI U2 urrdr
for IyI = t(1 - 6) < lxi < t, we obtain the inequality

sup lu(t,x)I <- p I=ISItl

vl

332

M. STRUWE, GEOMETRIC EVOLUTION PROBLEMS

foranyp,anytEAT,provided 0T>T(b). Choose p > 0 such that for 0 < Jul < p there holds F(u) < Cu f (u) for some C > 0 and fix b E]0, b(p)]. Consider T E [T(b), 0[. Replacing T by a suitable number S E AT, if necessary, we may assume that 52 TsupoG(u(t),

D(t)) -

2

< G(u(T), D(T)),

while by Lemma 3.5 the right hand side is

!F(u)

1

[TI fK(T)

l

uf(u)1 2[x[2 I

X12

2C)

([T[I l 1

+11-

LITAT, [

dxdt +o(l) T
l follJows

po G(u(t), D(t)) - b2] +0(i),

[T!5t

where o(1) -. 0 as T -. 0. It JATj

01-T1

that

T<poG(u(t),D(t))+ (!_ SIT-, i b2 < o(1),

where o(1) -. 0 (T - 0). Since b < b(p) is arbitrary, th/erefore we can achieve that sup G(u, D(t)) < to T
for T < 0 sufficiently large. Theorem 3.8 now is a consequence of Lemma 3.3.

Note that it suffices to assume u f (u) > -b2 for some suitable number b > 0; that is, the convexity condition may be slightly relaxed. In fact, by a recent result of Grillakis [64] the condition 2F(u) + f (u)u > 0 or, equivalently, the condition g(u) + g'(u)u > 0 suffices.

Global existence of weak solutions Apart from the equivariant or rotationally symmetric case, the only case where global existence of weak solutions (in absence of classical solutions) so far has been established is the case of N = SL, due to Shatah [127]. Shatah independently devised the same penalty approach for (3.1) as Chen [17], Keller-Rubinstein-Sternberg [93] did for the parabolic evolution problem, see section 1.3.

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