Mean Curvature Flow of Surface in 4-Manifolds

Advances in Mathematics 163, 287–309 (2001) doi:10.1006/aima.2001.2008, available online at http://www.idealibrary.com on

Mean Curvature Flow of Surface in 4-Manifolds 1 Jingyi Chen 2 Department of Mathematics, The University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada E-mail: [email protected]

and Jiayu Li 3 Institute of Mathematics, Fudan University and Academia Sinica, People’s Republic of China E-mail: [email protected] Communicated by Gang Tian Received January 10, 2001; accepted May 22, 2001

1. INTRODUCTION Deforming submanifolds via various geometric parabolic flows has been a powerful method in differential geometry. In [B], Brakke introduced the motion of a submanifold moving by its mean curvature in arbitrary codimension and constructed a generalized varifold solution for all time. For the classical solution of the mean curvature flow, to our knowledge most work has been on hypersurfaces. Huisken showed in [H2, H3] that if the initial hypersurface is compact and uniformly convex in a complete manifold with bounded geometry then it converges to a single point under the mean curvature flow in a finite time and the normalized flow (area is fixed) converges to a sphere of that area in infinite (rescaled) time. As time evolves, the mean curvature flow may develop singularities which can be 1 We thank G. Tian for suggesting to us the problem of studying the mean curvature flow for symplectic surfaces and many helpful discussions. We are grateful to both The University of British Columbia and Academy of Mathematics and System Sciences in Chinese Academy of Sciences where this work was carried out. The authors are grateful for the referee’s useful comments. 2 The research was supported in part by a Sloan Foundation fellowship and a NSERC grant. 3 The research was supported in part by a grant from the National Science Foundation of China.

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287 0001-8708/01 $35.00 © 2001 Elsevier Science All rights reserved.

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classified as Type I and Type II according to the blow-up rate of the second fundamental form with respect to time t. It is shown in [H1] that after appropriate rescaling near the Type I singularities the hypersurfaces St approach a self-similar solution of the mean curvature flow. The existence, uniqueness, and regularity of a weak solution of the mean curvature flow (the so-called viscosity solution) were studied by Chen et al. [CGG], Evans and Spruck [ES], White [W], and others. There is a fairly large amount of work on parabolic equations for plane curves or curves in Riemann surfaces moving by different geometric flows. In this article, we are mainly interested in the mean curvature flow of compact real two dimensional symplectic surfaces in a Kähler–Einstein surface. The aim is to use the mean curvature flow method to produce holomorphic curves from a given initial symplectic surface. This problem is closely related to the well-known symplectic isotopy problem. Recall that two embedded symplectic surfaces are symplectic isotopy if they can be connected by a smooth family of embedded symplectic surfaces. The symplectic isotopy problem asks how many connected embedded symplectic surfaces represent a given 2-dimensional homology class in a simply connected symplectic 4-manifold, up to isotopy. Siebert and Tian conjectured (cf. [ST]) that for an algebraic surface with positive first Chern class the isotopy class of embedded symplectic surfaces in a fixed homology class is unique. The organization of this paper is as follows. We first derive the parabolic differential equations for the second fundamental form and the mean curvature vector for submanifolds moving by the mean curvature flow in arbitrary codimension, which does not seem to be easily found in literature except in the codimension one case. The Laplacian of the second fundamental form was computed by Simons [Sim] a long time ago. Then we concentrate on surfaces in 4-manifolds. In particular, we assume that the initial surface is compact and symplectic in a Kähler surface, which means that the Kähler angle a at every point of the surface lies in [0, p2 ). We recover a result in [CT2] which states that the symplectic property is preserved under the mean curvature flow in a Kähler–Einstein surface as long as the smooth solution exists. The monotonicity formulae for the area functional in the geometric measure theory and for the Dirichlet integral in harmonic mapping theory have been useful tools to provide regularity results. We establish a monotonicity formula for >St cos1 a r(F, t) f dmt , where F: S0 × [0, T) Q M is the mean curvature flow, f is a cut-off function supported in a small geodesic ball in M which is contained in a single coordinate chart, and r is the backward heat kernel on R 4. The idea of using various backward heat kernels (on the domain or the entire target) appeared in the earlier work of other people on the mean curvature flow, the harmonic map heat flow and the Yang–Mills heat flow and so on (cf. [CS, Ha, H1, St], etc.). Using our monotonicity formula and a blowing

MEAN CURVATURE FLOW

289

up argument near singularities, we prove the main result of this article: if the initial surface is compact and symplectic in a Kähler–Einstein surface with nonnegative scalar curvature, then the mean curvature flow has no Type I singularities. We should point out that if the mean curvature flow has a global solution, then F(S, t) will converge to holomorphic curves. We take summation for repeated indices throughout the paper.

2. BASIC FORMULAE FOR MEAN CURVATURE FLOW IN ARBITRARY DIMENSION AND CO-DIMENSION In this section, we derive the equations for some geometric quantities. The results in this section hold for the mean curvature flow in arbitrary dimension and codimension. We consider the mean curvature flow from a closed P-dimensional manifold in a Q-dimensional Riemannian manifold M with a Riemannian metric. Given an embedding F0 : S Q M, we consider a one-parameter family of smooth maps Ft =F( · , t): S Q M with corresponding images St =Ft (S) are embedded submanifolds in M and F satisfies the mean curvature flow equation:

˛

d F(x, t)=H(x, t) dt F(x, 0)=F0 (x).

(1)

Here H(x, t) is the mean curvature vector of St at F(x, t) in M. Denote by A the second fundamental form of St in M and the Riemannian metric on M by O · , · P. In a normal coordinates around a point in S, the induced metric on St from O · , · P is given by gij =O“i F, “j FP, where “i (i=1, ..., P) are the partial derivatives with respect to the local coordinates. In the sequel, we denote by D and N the Laplace operator and covariant derivative for the induced metric on St respectively. We choose a local field of orthonormal frames e1 , ..., eP , v1 , ..., vQ − P of M along St such that e1 , ..., eP are tangent vectors of St and v1 , ..., vQ − P are in the normal bundle over St . We can write A=A ava H=−H ava .

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Let A a=(h aij ) where (h aij ) is a matrix, by the Weingarten equation (cf. [Sp]), we have h aij =O“i va , “j FP=O“j va , “i FP=h aji . The trace and the norm of the second fundamental form of St in M are H a=g ijh aij =h aii |A| 2=C |A a| 2=g ijg klh aik h ajl =h aik h aik . a

We first derive the evolution equation of the induced metric. Lemma 2.1. Under the mean curvature flow, the induced metric evolves as “gij =−2H ah aij . “t Proof. Using normal coordinate systems at x in S and at F(x) in M, we have “gij “ = O“i F, “j FP “t “t = − O“i (H ava ), “j FP − O“j (H ava ), “i FP, where “i (i=1, ..., P) are the partial derivatives with respect to the local coordinates. By the Gauss–Weingarten equations (cf. [H3, (5); Sp]), we have “i va =h ail g lk “k F+C bia vb . It is clear that C bia =−C aib . So, we get “gij =−2H ah aij . “t This proves the lemma.

Q.E.D.

We then derive the evolution equation of the frame on the normal bundle. Lemma 2.2. We write O“va /“t, vb P=b ba , then b ba =−b ab and “va =NH a+H cC aic ei +b ba vb . “t

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Proof. Differentiating the identity Ova , vb P=dab yields the skewsymmetry of b ab . Using normal coordinate systems at x in S and at F(x) in M, then translating the identity into normal frames, we have

7

8 g “ F+b v “F = − 7 v , “ 1 28 g “ F+b v “t

“va “va = ,“F “t “t i

b a b

ij

j

ij

a

i

j

b a b

=Ova , “i (H ava )P g ij “j F+b ba vb =NH a+H cOva , C bic vb P ei +b ba vb =NH a+H cC aic ei +b ba vb which concludes the proof of the lemma.

Q.E.D. a b

Note that for hypersurfaces the terms b are identically zero. Lemma 2.3. Assume that the Christoffel symbols C kij of the Levi–Civita connection of the induced metric are zero at a point p ¥ St . Then at p “ a h =Ni Nj H a − H ch cjl h ail +H cC bjc C aib +H c Ni C ajc “t ij +Nj H bC aib +Ni H bC ajb − h bij b ab +H bRajbi . Proof. We will first calculate in local normal coordinates and then translate it into normal frames as before. By the Gauss–Weingarten equations (cf. [H3, (5); Sp]), we have h aij =−O“ 2ij F, va P+C aij . Then we have “ a “ h ij = − O“ 2ij F, va P − H b “b C aij “t “t =O“ 2ij (H ava ), va P − O“ 2ij F, NH a+H cC aic ei +b ba vb P − H b “b C aij . By the Gauss formula (cf. [Sp]) and the Weingarten equations, we have that at the point p “ 2ij F=−h aij va and “i va =h ail g lm “m F+C bia vb ,

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we obtain “ a h =N 2ij H a − H ch cjl h ail +H cC bjc C aib − H b “b C aij +H b “i C ajb “t ij +ONj H cC bic vb , va P+ONi H cC bja vb , vc P+H c Ni C ajc − h bij b ab . This proves the lemma.

Q.E.D.

Lemma 2.4. Assume that the Christoffel symbols of the Levi–Civita connection of the induced metric are zero at p ¥ St . Then at the point p we have Ni Nj H a=Dh aij +Nl (h bij C alb ) − Nl (h blj C aib )+h bil h blm h amj − H bh bim h amj +h bij h blm h aml − h bim h blj h aml +Ni (h bjl C alb ) − Ni (H bC ajb ) − Nl Rajil − Ni Raljl +Rillm h amj +Riljm h aml − Rabil h blj . Proof. Let K be the curvature tensor on St . By the Gauss equation and the Codazzi–Mainardi equations, we have the following identities at p, Nk h amn − Nm h akn =h bkn C amb − h bmn C akb − Ranmk and Kijkl =(h bik h bjl − h bil h bjk )+Rijkl , where R is the curvature tensor of M. We have Ni Nj H a=Ni Nj h all =Ni (Nl h ajl +h bjl C alb − h bll C ajb − Raljl ) =Ni Nl h alj +Ni (h bjl C alb ) − Ni (h bll C ajb ) − Ni Raljl =Nl Ni h alj +(h bil h blm − h bim h bll ) h amj +Rillm h amj +(h bij h blm − h bim h blj ) h aml +Riljm h aml − Rabil h blj − Ni Raljl +Ni (h bjl C alb ) − Ni (h bll C ajb ) =Nl (Nl h aij +h bij C alb − h blj C aib − Rajil )+h bil h blm h amj − H bh bim h amj +h bij h blm h aml − h bim h blj h aml +Ni (h bjl C alb ) − Ni (H bC ajb ) +Rillm h amj +Riljm h aml − Rabil h blj − Ni Raljl =Dh aij +Nl (h bij C alb ) − Nl (h blj C aib )+h bil h blm h amj − H bh bim h amj +h bij h blm h aml − h bim h blj h aml +Ni (h bjl C alb ) − Ni (H bC ajb ) − Nl Rajil − Ni Raljl +Rillm h amj +Riljm h aml − Rabil h blj . This proves the lemma.

Q.E.D.

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From the Codazzi–Mainardi equation, it follows that (Nj H b) C aib − Nl (h bjl C aib )=h cjl C blc C aib − H cC bjc C aib − Rbllj C aib − h bjl Nl C aib . Then Lemma 2.3 and Lemma 2.4 immediately imply Lemma 2.5. For the mean curvature flow, the second fundamental form satisfies

1 “t“ − D 2 h =N (h C )+h h ij

l

b ij

a lb

b il

b lm

h amj

− H b(h bim h amj +h bjl h ail )+h bij h blm h aml − h bim h bjl h aml +h bjl (Ni C alb − Nl C aib )+h cjl C blc C aib − Rbllj C aib − Nl Rajil − Ni Raljl +Rillm h amj +Riljm h aml − Rabil h blj − h bij b ab . Now we prove the main result in this section. Proposition 2.6. Suppose that F is the mean curvature flow. Then

1 “t“ − D 2 |A| = − 2 |N4A| +2 |A| − 2h (Nb R 2

2

a ij

4

l

ajil

bi Raljl ) +N

+2h aij h bij Ralbl +4h aij Rillm h amj +4h aij Riljm h aml . and

1 “t“ − D 2 |H| =−2 |N4H| +2 |A| |H| +2H H R 2

2

2

2

a

b

lalb

,

4 is the covariant differentiation on Hom(TSt × TSt , NorSt ) deterwhere N mined by the covariant differentiation on TSt and D on the normal bundle, D b is the normal connection for the embedding St … M (cf. [Sp, p. 55]), and N is the connection on M. Proof. By the skew-symmetry of b ab , it is clear that h aij h bij b ab =0.

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Then using the previous lemmas, we have “ “ |A| 2= (g ikg jlh aij h akl ) “t “t =4H ah aik h bij h bkj +2h aij

“ a h “t ij

=2h aij (Dh aij +H cC bjc C aib +H c Ni C ajc +Nj H bC aib +Ni H bC ajb − h bij b ab +Nl (h bij C alb ) − Nl (h blj C aib )+h bil h blm h amj +h bij h blm h aml − h bim h blj h aml − Nl Rajil − Ni Raljl +Rillm h amj +Riljm h aml − Rabil h blj +H bRajbi +Ni (h bjl C alb ) − Ni (H bC ajb )) =2h aij (Dh aij +H cC bjc C aib +H c Ni C ajc +Nj H bC aib +Ni H bC ajb +Nl h bij C alb +h bij Nl C alb − Nl h blj C aib − h blj Nl C aib +Ni h bjl C alb +h bjl Ni C alb − Ni H bC ajb − H b Ni C ajb +h bil h blm h amj +h bij h blm h aml − h bim h blj h aml − Nl Rajil − Ni Raljl +Rillm h amj +Riljm h aml − Rabil h blj +H bRajbi ). Note that Nl h bjl =Nj h bll +H cC bjc − h cjl C blc − Rbljl Ni h blj =Nl h bij +h cij C blc − h cjl C bic − Rbjli Ni C alb − C cib C alc − Nl C aib +C clb C aic =h alk h bki − h aik h bkl +Rilab . We then obtain “ |A| 2=2(h aij Dh aij − 2h aij Nl h bij C bla − (h aij C bla ) 2)+2h aij h bij h blm h aml “t +2h aij ( − Nl Rajil − Ni Raljl +Rillm h amj +Riljm h aml +H bRajbi ).

(2)

Covariant differentiation of the curvature tensor leads to the following formula (cf. [SSY]), bq Ramnp − Rambp h bnq − Ramnb h bpq +Rsmnp h asq , Nq Ramnp =N and in turn we have bl Rajil − Rajbl h bil − Rajib h bll +Rmjil h aml , Nl Rajil =N

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295

and bi Raljl − Ralbl h bij − Raljb h bil +Rmljl h ami . Ni Raljl =N It is clear that D |A| 2=2h aij Dh aij +2Nl h aij Nl h aij . Substituting the last three identities into (2), we get the first identity in the proposition. The second one can be proved by a similar argument. Q.E.D.

3. MOVING SYMPLECTIC CURVES IN KÄHLER–EINSTEIN SURFACES In this section, we assume that M is a Kähler–Einstein surface. Let w be the symplectic form on M and let J be a complex structure compatible with w in the sense that for any tangent vectors U, V in TM, we have w(JU, JV)=w(U, V). The Riemannian metric O , P on M is defined by OU, VP=w(U, JV). Let JSt be an almost complex structure in a tubular neighborhood of St on M with

˛

JSt e1 = e2 JSt e2 = − e1 JSt v1 = v2 JSt v2 = − v1 .

It is not difficult to check that bi JS ) e1 =(h 2i1 +h 1i2 ) v1 +(h 2i2 − h 1i1 ) v2 (N t and bi JS ) e2 =(h 2i1 +h 1i2 ) v2 +(h 2i2 − h 1i1 ) v1 . (N t So we have b JS | 2=|h 211 +h 112 | 2+|h 221 +h 122 | 2+|h 212 − h 111 | 2+|h 222 − h 121 | 2. |N t

(3)

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b JS | 2 and |H| 2 which A simple computation shows the relation between |N t we state as follows (cf. [CT]). Lemma 3.1. We have b JS | 2=1 |H| 2+1 (((h 111 +h 122 )+2(h 212 − h 122 )) 2+(h 211 +h 222 +2(h 121 − h 211 )) 2). |N 2 2 t The contraction of the Kähler 2-form w with the area element of St determines the Kähler angle a of St in M. Using the orthonormal frame e1 , e2 , we have cos a=Oe1 N e2 , wP. The function a is continuous everywhere on St and smooth possibly except at the complex or anti-complex points, i.e., where a=0, p. Whenever a is smooth, we have Proposition 3.2. Assume that M is a Kähler–Einstein surface with scalar curvature R. Let a be the Kähler angle of the surface St which evolves by the mean curvature flow. Then

1 “t“ − D 2 cos a=|NbJ | cos a+R sin a cos a. 2

2

St

Remark. The above equation is a simplified version of the evolution equation on cos a in [CT2]. This simplified version was pointed out to us by G. Tian. Proof. We may choose the local orthonormal frame {e1 , e2 , e3 , e4 } on M along St so that the symplectic form w takes the following form (cf. [CT, CW]), w=cos au1 N u2 +cos au3 N u4 +sin au1 N u3 − sin au2 N u4 ,

(4)

where {u1 , u2 , u3 , u4 } is the dual frame of {e1 , e2 , e3 , e4 }. Then along the surface St the complex structure on M takes the form

R

0 cos a − cos a 0 J= − sin a 0 0 sin a

sin a 0 0 − cos a

0 − sin a cos a 0

S

(5)

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MEAN CURVATURE FLOW

and we use J ij to denote the coordinate components of the type (1,1) tensor J. We set v=cos a=Oe1 N e2 , wP. From Lemma 2.2, we have

7 8 e +7 “t“ e , v 8 v “ “ =7 e , e 8 e − 7 e , v 8 v “t “t “ =7 e , e 8 e − N H v − H C v . “t

“ “ ei = e,e “t “t i j

j

i

j

j

i

j

j

i

i

a

i

a

a

a

a

c

a

a ic a

Since w is independent of t, it follows

7

8 7

“ “ “ v= e N e , w + e1 N e2 , w “t “t 1 2 “t

8

= − (N1 H a+H cC a1c )Ova N e2 , wP − (N2 H a+H cC a2c )Oe1 N va , wP. For convenience of summing over indices, we denote v1 by e3 and v2 by e4 . For the covariant derivatives of the orthonormal frame e1 , e2 , e3 , e4 in M, we set be ej =C kij ek . N i b is torsion-free and its connection matrix is skew-symmetric in the That N orthonormal frame {e1 , ..., e4 } asserts C kij =C kji

and

C kij =−C jik .

From the Gauss equation Ni ej =−(h aij − C aij ) va +C kij ek , we see that the function v satisfies a a+2 N1 v=−((h a11 − C a+2 11 )Ova N e2 , wP+(h 12 − C 12 )Oe1 N va , wP), a a+2 N2 v=−((h a21 − C a+2 21 )Ova N e2 , wP+(h 22 − C 22 )Oe1 N va , wP).

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For the sake of simplicity, we assume that C kij =0 at the point under consideration, for i, j, k=1, ..., 4. Then a a b N 21 v= − N1 (h a11 − C a+2 11 )Ova N e2 , wP − h 11 ON1 va N e2 , wP+h 11 h 12 Ova N vb , wP

− N1 (h a12 − C a12 )Oe1 N va , wP − h a12 Oe1 N N1 va , wP+h a11 h b12 Ova N vb , wP = − N1 h a11 Ova N e2 , wP − (h a11 ) 2 Oe1 N e2 , wP − h c11 C a1c Ova N e2 , wP − N1 h a12 Oe1 N va , wP − (h a12 ) 2 Oe1 N e2 , wP − h c12 C a1c Oe1 N va , wP a+2 +2h a11 h b12 Ova N vb , wP+N1 C a+2 11 Ova N e2 , wP+N1 C 12 Oe1 N va , wP

and similarly N 22 v= − N2 h a21 Ova N e2 , wP − (h a21 ) 2 Oe1 N e2 , wP − h c21 C a2c Ova N e2 , wP − N2 h a22 Oe1 N va , wP − (h a22 ) 2 Oe1 N e2 , wP − h c22 C a2c Oe1 N va , wP a+2 +2h a21 h b22 Ova N vb , wP+N2 C a+2 21 Ova N e2 , wP+N2 C 22 Oe1 N va , wP.

By the Coddazi–Mainardi equation N1 h a21 =N2 h a11 +h c11 C a2c − h c21 C a1c − R(a+2) 121 , N2 h a12 =N1 h a22 +h c22 C a1c − h c12 C a2c − R(a+2) 212 and Ov1 N v2 , wP=Oe1 N e2 , wP Oe1 N v1 , wP=−Oe2 N v2 , wP Oe1 N v2 , wP=Oe2 N v1 , wP, we obtain

1 “t“ − D 2 v=((h

) +2(h a12 ) 2+(h a22 ) 2 − 2h 111 h 212 +2h 211 h 112

a 2 11

− 2h 121 h 222 +2h 221 h 122 )Oe1 N e2 , wP − R(a+2) 121 Oe1 N va , wP − R(a+2) 212 Ova N e2 , wP − (N1 C 311 − N1 C 412 +N2 C 321 − N2 C 422 )Oe1 N v2 , wP − (N1 C 411 +N1 C 312 +N2 C 421 +N2 C 322 )Oe1 N v1 , wP =|DJSt | 2v − (R1213 +R2124 )Oe1 N v1 , wP − (R1214 − R2123 )Oe1 N v2 , wP − (N1 C 311 − N1 C 412 +N2 C 321 − N2 C 422 )Oe1 N v2 , wP − (N1 C 411 +N1 C 312 +N2 C 421 +N2 C 322 )Oe1 N v1 , wP.

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By the formula (1.16) in [Ya, Chap. IV], we get H12 =cos a(R1212 +R1234 )+sin a(R1213 − R1224 ), where Hij is defined in [Ya]. By the formula (1.17) in [Ya, Chapter IV], we get H12 =R1s J s2 . Since M is a Kähler–Einstein manifold, we have H12 =−

R cos a, 2

where R is the scalar curvature. By the formula (1.10) in Chapter IV in [Ya], we get R1212 =R1212 cos 2 a − R1234 sin 2 a+sin a cos a(R1213 − R1224 ), so we obtain cos a R1212 +R1234 = (R − R1224 ). sin a 1213 It follows that 1 1 − R cos a= (R − R1224 ). 2 sin a 1213 Thus R1213 − R1224 =− 12 R sin a cos a. One checks directly that N1 C 411 = − N1 C 141 =0 N1 C 312 = − N1 C 123 =R1213 N2 C 421 = − N2 C 214 =−R1224 N2 C 322 = − N2 C 232 =0.

(6)

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We therefore have

1 “t“ − D 2 v=|NbJ | v − 2(R 2

1213

St

− R1224 )Oe1 N v1 , wP.

By (4) and (6), we have

1 “t“ − D 2 cos a=|NbJ | cos a+R sin a cos a. 2

2

St

This proves the proposition.

Q.E.D.

Then by the parabolic minimum principle, we obtain (cf. [CT2]): Theorem 3.3. Let M be a Kähler–Einstein surface with nonnegative scalar curvature. Suppose that the smooth solution of the mean curvature flow exists on [0, T), T [ .. Set v=cos a where a is the Kähler angle of St in M. If v(x, 0) \ v0 > 0, then v(x, t) \ v0 ,

1 “t“ − D 2 v \ |NbJ | v 2

St

for all t ¥ [0, T).

4. MONOTONICITY FORMULAS AND TYPE I SINGULARITIES Let H(X, X0 , t) be the backward heat kernel on R 4. Define

1

1 |X − X0 | 2 r(X, t)=4p(t0 − t) H(X, X0 , t)= exp − 4p(t0 − t) 4(t0 − t)

2

for t < t0 . We prove two monotonicity inequalities in this section, the first one was essentially proved by Huisken [H1] and Hamilton [Ha]. Using the second monotonicity derived in this section in our blow-up analysis at singularities, we prove that the mean curvature flow has no Type I singularity provided v(x, 0) \ v0 > 0. We first prove the monotonicity inequality on R 4. The monotonicity inequality for heat flow of harmonic maps was proved by Struwe [St] in the Euclidean case and was proved by Chen and Struwe [CS] in the general case.

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MEAN CURVATURE FLOW

Proposition 4.1. Let M 4=R 4. We have

:

“ (F − X0 ) + F r(F, t) dmt =−F r(F, t) H+ “t St 2(t0 − t) St

: dm . 2

t

Proof. It is clear that “ “ F r(F, t) dmt =F r(F, t) dmt − F r(F, t) |H| 2 dmt “t St St “t St =F St

1 “t“ +D 2 r(F, t) dm − F t

r(F, t) |H| 2 dmt .

St

Straight computation leads to

1

2

|X − X0 | 2 1 “ 1 r(X, t) r(X, t)= − OH, X − X0 P − “t t0 − t 2(t0 − t) 4(t0 − t) 2 and

1

N exp −

2

1

2

|X − X0 | 2 |X − X0 | 2 OX − X0 , NXP =−exp= − 4(t0 − t) 4(t0 − t) 2(t0 − t)

and

1

D exp −

|X − X0 | 2 4(t0 − t)

1

=exp −

2

|X − X0 | 2 4(t0 − t)

2 1 |OX4(t− X−, t)NXP| − OX2(t− X−, t)DXP − 2(t|NX|− t) 2 . 2

2

0

0

2

0

0

0

Note that in the induced metric on St |NF| 2=2

and

DF=H,

so we have

1 “t“ +D 2 r(F, t)=−1 OF(t− X− t), HP+|(F4(t− X− t)) | 2 r(F, t) + 2

0

0

2

0

Then the proposition follows.

(7)

0

Q.E.D.

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CHEN AND LI

We then prove the monotonicity inequality on a curved manifold M 4. Let iM be the injective radius of M 4. We choose a cut off function f ¥ C. 0 (B2r (X0 )) with f — 1 in Br (X0 ), where X0 ¥ M, 0 < 2r < iM . Choose a normal coordinates in B2r (X0 ) and express F using the coordinates (F 1, F 2, F 3, F 4) as a surface in R 4. We define F(X0 , t0 , t)=F f(F) r(F, t) dmt . St

Hamilton used the backward heat kernel on M 4 in his monotonicity in [Ha]. We use the backward heat kernel on R 4 to localize near a point. For harmonic map heat flow, similar localization was taken in [CS] on the domain rather then on the target manifold. Proposition 4.2. Let M 4 be a compact Riemannian 4-manifold. There are positive constants c1 and c2 depending only on M 4, F0 and r which is the constant in the definition of f, such that

:

(F − X0 ) + “ c1 `t0 − t e F(X0 , t0 , t) [ − e c1 `t0 − t F fr(F, t) H+ “t 2(t0 − t) St

: dm +c (t − t). 2

t

2

0

Proof. In this case, we have “ F(X0 , t0 , t)=F “t St =F St

1 “t“ +D 2 fr(F, t) dm − F fr(F, t) |H| dm “ f 1 +D 2 r(F, t) dm − F fr(F, t) |H| dm “t 2

t

t

St

2

t

t

St

+F Dfr(F, t) dmt +2 F Nf · Nr(F, t) dmt . St

St

Notice that DF=H+g ijC aij va and we then have

1 “t“ +D 2 r(F, t) OF − X , HP |(F − X ) | OF − X , g C v P 2 =− 1 + + r(F, t). (t − t) 4(t − t) (t − t) + 2

0

0

ij

0

2

0

0

0

a ij a

(8)

303

MEAN CURVATURE FLOW

Note that Df=0, Nf=0 in Br (X0 ), we can see that and

|Dfr(F, t)| [ C

|NfNr(F, t)| [ C,

where the constant C depends on r and M, hence F Dfr(F, t) dmt [ C F dmt [ C St

St

F Nf · Nr(F, t) dmt [ C F dmt [ C. St

St

Since C aij (X0 )=0, we have |g ijC aij (F)| [ C |F − X0 |. thus OF − X0 , g ijC aij va P |F − X0 | 2 [C . (t0 − t) t0 − t Hence OF − X0 , g ijC aij va P r(F, t) r(F, t) [ c1 +C. (t0 − t) `t0 − t It concludes that

:

“ (F − X0 ) + F(X0 , t0 , t) [ − F fr(F, t) H+ “t 2(t0 − t) St

: dm +`tc − t F(X , t , t)+c . 2

1

t

0

0

2

0

The proposition follows.

Q.E.D.

By an argument similar to the one used in the proof of Proposition 4.2, we can prove the following monotonicity inequality. Theorem 4.3. Let M be a Kähler–Einstein surface with nonnegative scalar curvature. Let a be the Kähler angle of the surface St in M. Suppose that cos a( · , 0) has a positive lower bound. Let v=cos a. Then we have “ c1 `t0 − t 1 e F fr(F, t) dmt “t St v

1

[ − e c1 `t0 − t F St

+F St

:

1 (F − X0 ) + fr(F, t) H+ v 2(t0 − t)

: dm 2

t

1 b JS | 2 dmt +F 2 |Nv| 2 fr(F, t) dmt fr(F, t) |N 3 t v St v

+c2 (t0 − t).

2

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CHEN AND LI

Here the positive constants c1 and c2 depend on M, F0 and r which is the constant in the definition of f. Proof. By Theorem 3.3, we have

1 “t“ − D 2 1v [ − |NbJ | 1v − v2 |Nv| . 2

St

2

3

So, 1 “ 1 “ 1 “ F fr(F, t) dmt =F fr(F, t) dmt +F r(F, t) f dmt “t St v v St “t v St “t − F r(F, t) f |H| 2 dmt St

[F f St

1 “t“ +D 2 r(F, t) 1v dm − F t

St

r(F, t) f |H| 2 dmt

b JS | 2 1 fr(F, t) dmt − F f 2 |Nv| 2 r(F, t) dmt − F |N t v v3 St St 1 1 +F Df r(F, t) dmt +2 F Nf · Nr(F, t) dmt . v v St St Using (8) and the estimates in the proof of Proposition 4.2, we obtain the monotonicity inequality. Q.E.D. The standard parabolic theory implies that (1) has a smooth solution for short time. We state it in the following theorem. Theorem 4.4. There exists T > 0 such that (1) has a smooth solution in the time interval [0, T). If maxSt |A| 2 is bounded near T, the solution can be extended to [0, T+E) for some E > 0. However, in general maxSt |A| 2 becomes unbounded as t Q T. Definition 4.5. We say that the mean curvature flow F has Type I singularity at T > 0, if lim (T − t) max |A| 2 [ C, tQT

for some positive constant C.

St

MEAN CURVATURE FLOW

305

Lemma 4.6. Let U(t)=maxSt |A| 2. If the mean curvature flow blows up at T > 0, there is a positive constant c depending only on M 4, such that, if 0 < T − t < p/16 `c, the function U(t) satisfies U(t) \

1 4 `2 (T − t)

.

Proof. By Proposition 2.6 and the parabolic maximum principle, we have “ U(t) [ 2(U(t)) 2+c1 U(t)+c2 `U(t) “t [ 4(U(t)) 2+4c, where c1 and c2 are constants which only depend on max |R|, max |NR| for the Riemannian curvature R on M and the constant c depends on c1 , c2 . so,

1

1 22 \ − 4 `c .

“ `c arctan “t U(t)

(9)

Since the mean curvature flow blows up at T > 0, we have U(t) Q . as t Q T. Integrating the last inequality from t to T, we get, if 0 < T − t < 16 p`c , `c U(t)

[ tan(4 `c (T − t)) [ `2 · 4 `c (T − t).

This implies the desired inequality.

Q.E.D.

Theorem 4.7. Let M 4 be a Kähler–Einstein surface with nonnegative scalar curvature. If the Kähler angle of the initial surface v(x, 0) \ v0 > 0, then the mean curvature flow (1) has no Type I singularity at any T > 0. Proof. Suppose that the mean curvature flow has a Type I singularity at t0 > 0. Assume that l 2k =|A| 2 (xk , tk )=max |A| 2 t [ tk

and xk Q p ¥ S, tk Q T as k Q .. We choose a local coordinate system on M 4 such that F(p, t0 )=0. We rescale the mean curvature flow at 0, more precisely, we set Fk (x, t)=lk (F(x, l k−2 t+tk ) − F(p, tk )).

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CHEN AND LI

Denote by S kt the scaled surface Fk ( · , t) and take g kij =l 2k gij , (g k) ij=l k−2 g ij. We therefore have “Fk “F =l k−1 “t “t

(10)

Dg k Fk =l k−1 DF

(11)

and “Fk =Hk “t

(12)

|Ak | 2=l k−2 |A| 2.

(13)

By Lemma 4.6, we have c C [ |A| 2 (x, tk ) [ (t0 − tk ) (t0 − tk ) for some uniform constants c and C independent of k, so |Ak | 2 [ 1

and

|Ak | 2 (p, 0) \ c > 0.

There is a subsequence of Fk which we also denote by Fk , such that Fk Q F. as k Q . in any ball BR (0) … R 4, and F. satisfies “F. =H. “t

with |A. (0, 0)| \ c > 0.

(14)

Recall that St is the curve defined by F( · , t). For any R > 0, we choose a cut-off function fR ¥ C . 0 (B2R (0)) with fR — 1 in BR (0) where Br (0) is the metric ball centered at 0 with radius r. It is clear that, for k sufficiently large, F

1

k

St

2 |F(x, t +l t)| 2 exp 1 − dm , t) 4(t − (t +l t))

1 1 |F +lk F(p, tk )| 2 fR (Fk ) exp − k dm kt v 0−t 4(0 − t) =F St

k

−2 t +l k

1 1 fR (F) v tk − (tk +l k−2

k

k

k

−2 k −2 k

2

t

307

MEAN CURVATURE FLOW

where f is the function defined in Theorem 4.3. Note that tk +l k−2 t Q t0 for any fixed t. Using Theorem 4.3 by taking the initial time to be tk , we see that for any − . < s1 < s2 < 0, −2

e c1 `tk − (tk +l k

s2 )

1

F

k 2

Ss

1 1 |F +lk F(p, tk )| 2 fR exp − k v 0 − s2 4(0 − s2 ) −2

− e c1 `tk − (tk +l k

s1 )

2 dm

k s2

1

F

k St 1

2

1 1 |F +lk F(p, tk )| 2 fR exp − k dm ks1 v 0 − s1 4(0 − s1 )

as k Q ..

Q0

By Theorem 4.3, if we take the initial time to be 0, then for any − . < s1 < s2 < 0, −2

− e c1 ` − l k

s2

F

1

k 2

Ss

−2

+e c1 ` − l k

s1

1

F

k 1

:

1 (F +lk F(p, tk )) + fR r(Fk , t) Hk + k v 2(t0 − t)

−2

\ F e c1 ` − l k t F

k St

s1 s2

−2

+F e c1 ` − l k t F

k St

s1 s2

−2

+F e c1 ` − l k t F

k St

s1

2

1 1 |F +lk F(p, tk )| 2 fR exp − k dm ks1 v 0 − s1 4(0 − s1 )

Ss s2

2

1 1 |F +lk F(p, tk )| 2 fR exp − k dm ks2 v 0 − s2 4(0 − s2 )

: dm 2

t

1 b JS | 2 dm kt f r(Fk , t) |N tk v R 2 |Nv| 2 fR r(Fk , t) dm kt v3

− c2 l k−1 (s2 − s1 ).

(15)

Since |F(p, tk )| [ F

t0

tk

: “F“t : dt [ F k

t0

tk

|H| dt [ C `(t0 − tk ) [

C , lk

we assume that lk F(p, tk ) Q q as k Q .. Letting k Q ., by (15), we get DJ. — 0, where D is the derivatives in R 4. By Lemma 3.1, we can see that H. — 0

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CHEN AND LI

and hence “F. — 0. “t Applying (15) again, we get (F. +q) + — 0. That is, OF. +q, va P=0. Note that the above inner product is taken in R 4, and differentiating in R 4 then yields 0=O“i F. , va P+OF. +q, “i va P=OF. +q, “i va P because “i F. is tangential to S. and then by the Weingarten equation we have (h. ) aij OF. +q, ej P=0

for all a, i=1, 2.

So for a=1, 2, we have det((h. ) aij )=0. Since H=0, we also have for a=1, 2 we also have tr((h. ) aij )=0. These result in that the symmetric matrix ((h. ) aij ) is the zero matrix. In other words, we have (h. ) aij =0 for all i, j, a=1, 2, which yields that |A. | — 0. However, (14) asserts |A. | – 0, and hence we get a contradiction. Q.E.D.

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