Generalized Symplectic Mean Curvature Flows in Almost Einstein Surfaces?

Jiayu LI Liuqing YANG

1 Introduction

Suppose thatis a smooth K?hler manifold of complex dimensionn.Letbe the Ricci tensor of,and then the Ricci formis de fined by

Recently,T.Behrndt[1]proposed a generalized mean curvature flow.Instead of considering the flow in a K?hler-Einstein manifold,he considered the case that the ambient manifold is almost Einstein,that is,ann-dimensional K?hler manifoldwith

for some constantλ∈R and some smooth functionψo(hù)nM(see[1]).

Suppose that the K?hler manifoldis almost Einstein.Given an immersionof ann-dimensional manifold Σ intoM,Behrndt[1]considered a generalized mean curvature flow

where

is a normal vector field along Σ whichis called the generalized mean curvature vector field of Σ.AsKis a differential operator differing fromHjust by lower order terms,it is easy to see that(1.1)has a unique solution on a short time interval(see[1]).

Behrndt[1]proved that if Σ0=F0(Σ)is Lagrangian in the almost Einstein manifoldM,then along the generalized mean curvature flow(1.1),it remains Lagrangian for each time.Therefore,it is reasonable to call such a flow the generalized Lagrangian mean curvature flow.

As a special case,Behrndt[1]also considered the generalized Lagrangian mean curvature flow in an almost Calabi-Yau manifold(see[11]).

In[12],we studied the generalized Lagrangian mean curvature flow in an almost Einstein manifold.We proved that the singularity of this flow is characterized by the second fundamental form.We also proved that the type-I singularity of the generalized Lagrangian mean curvature flow in an almost Calabi-Yau manifold is a stationary cone.In particular,the generalized Lagrangian mean curvature flow has no type-I singularity.

Letbe a K?hler surface.For a compact oriented real surface Σ whichis smoothly immersed inM,αis the K?hler angle of Σ inM(see[5]).We say that Σ is a symplectic surface if cosα>0.

In this paper,we mainly study the generalized mean curvature flow in an almost Einstein surface with the initial surface symplectic.We show that if the initial surface Σ0is symplectic,then along the generalized mean curvature flow(1.1),it remains symplectic for each time.Therefore,we can call this flow the generalized symplectic mean curvature flow.

In general,the mean curvature flow may develop singularities as time evolves.According to the blow-up rate of the second fundamental form,Huisken[8]classified the singularities of the mean curvature flow into two types:type I and type II.Chen and Li[2]and Wang[13]independently proved that ifMis a K?hler-Einstein surface,then the symplectic mean curvature flow has no type-I singularity.Following the idea in[8],we can also define type-I and type-II singularity for our flow.And we can also prove that ifMis an almost Einstein surface,then the generalized symplectic mean curvature flow has no type-I singularity(see Theorem 5.1).Note that ifψ=const.,our flow is just the symplectic mean curvature flow in a K?hler-Einstein surface,so our result is a generalization of theirs.

In this paper,we also consider the graph case.Suppose thatM=M1×M2,whereandare Riemann surfaces with the same average scalar curvaturer.ThenMis an almost Einstein surface withSuppose that the initial surface is a graph withwhere{e1,e2}is an orthonormal frame of the initial surface.We show that the generalized mean curvature flow(1.1)exists globally and the global solutionF(·,t)sub-converges toF∞inC2ast→∞,possibly outside a finite set of points,andis a minimal surface inChen,Li and Tian[4]and Wang[13]proved the global existence and convergence of the mean curvature flow in the graph case thatM1andM2are of the same constant curvature.Han and Li[7]proved a similar result for the K?hler-Ricci mean curvature flow.In[7]and this paper,we only assume thatM1andM2have the same average scalar curvature.

2 Evolution Equations

In[12],we computed the evolution equations of the induced metric and the second fundamental form of Σtalong the generalized mean curvature flow(1.1).We will omit the proof and state them here in this section.

Lemma 2.1Along the generalized mean curvature flow(1.1),the induced metric evolves by

Consequently,we have the following corollary.

Corollary 2.1The area element ofΣtsatisfies the following equation:

and consequently,

We also have the following lemma.

Lemma 2.2Along the generalized mean curvature flow(1.1),the norm of the second fundamental form satisfies

where C depends on the ambient space M and‖ψ‖C2(M).

Theorem 2.1If the second fundamental form ofΣtis uniformly bounded under the generalized mean curvature flow(1.1)for all time t∈[0,T),then the solution can be extended beyond T.

3 The Evolution of the K?hler Angle Along the Flow

In this section,we consider the casen=2.That is to say,Mis an almost Einstein surface,and Σ0is a symplectic surface inM.

Letbe the almost complex structure in a tubular neighborhood of ΣtonMwith

It is proved in[2]that

Choose an orthonormal basis{e1,e2,e3,e4}onalong Σtsuch that{e1,e2}is the basis of Σtand the symplectic formωttakes the form

where{u1,u2,u3,u4}is the dual basis of{e1,e2,e3,e4}.Then along the surface Σtthe complex structure onMtakes the form(see[2])

Theorem 3.1The evolution equation forcosαalongΣtis

As a corollary,if the initial surfaceΣ0is symplectic,then along the flow,at each time t,Σtis symplectic.

ProofUsing Lemma 2.1,(3.3),and the fact thatω=0,we have

Recall the equation in Proposition 3.1 and Lemma 3.2 in[6]for cosαto have

Thus we have

DenoteThen

It is computed in[12]that

Recalling that(see[6])

and

we get

SinceMis an almost Einstein surface,we have

Moreover,

Hence

Thus we have

Putting(3.8)–(3.9)into(3.6),we get

This proves the theorem.

The above theorem motivates the following definition.

Definition 3.1A family of symplectic surfaces satisfying(1.1)is said to evolve by the generalized symplectic mean curvature flow.

4 Monotonicity Formula

LetH(X,X0,t,t0)be the backward heat kernel on4.Let Σtbe a smooth family of surfaces in4defined byFt:Σ→4.Define

fort

We also have

Combining(4.1)with(4.2)gives us

Applying the evolution equation for cosα,we have

whereCdepends on‖ψ‖C2(M)andλ.

On Σt,we set

whereCis the constant in(4.4).Denote the injectivity radius ofbyiM.ForX0∈M,take a normal coordinate neighborhoodUand letbe a cut-offfunction withUsing the local coordinates inUwe may regardF(x,t)as a point in RkwheneverF(x,t)lies inU.We define

The following monotonicity formula generalizes Proposition 4.2 of[2]to the almost Einstein case.In[12],we got the similar monotonicity formula for the generalized Lagrangian mean curvature flow.Some of the estimates in the proof of the following proposition have appeared in[12].For completeness,we sketch the proof below.

Proposition 4.1Let Ft:Σ→M be a generalized symplectic mean curvature flow in a compact almost Einstein surface M.Then there exist positive constants c1,c2,c3and c4depending only on M,F0,t0and r whichis the constant in the definition ofΨ,such that

ProofBy(3.4),we have

Note that

Using(2.1),(3.7),(4.3)and(4.6),we have

Again,by(2.1)and(3.7),we have

which implies that

Therefore,we have

The same estimate as in[2]implies

Aswe have(see[9,Lemma 6.6])

By Young’s inequality,

Using the fact that|and H¨older’s inequality,we have

Since

we have

In a way similar to the proof of(13)in[3],we have

Finally,we need to estimate the termWe claim that(see[12]for the proof)

Especially,if we chooseα=andβ=,then we have

Putting(4.9),(4.12)–(4.14)and(4.16)into(4.7),we obtain

Rearranging(4.17)yields the desired inequality.

5 No Type-I Singularity

Using(2.3),we can argue in the same way as that of the mean curvature flow(for example,Lemma 4.6 of[2])to obtain the lower bound of the blow-up rate of the maximal norm of the second fundamental form at a finite singular timeT.

Lemma 5.1LetIf the generalized mean curvature flow(1.1)blows up at afinite time T>0,there exists a positive c depending only on M,such that ifthen the function Utsatisfies

According to the lower bound of the blow-up rate,we can classify the singularities of the generalized symplectic mean curvature flow(1.1)into two types,whichis similar to that of the mean curvature flow defined by Huisken[8].This definition was given in[12].

Definition 5.1We say that the generalized mean curvature flow(1.1)develops type-I singularity at T>0,if

for some positive constant C.Otherwise,we say that the generalized mean curvature flow(1.1)develops type-II singularity.

Arguing as in[2],we have

Theorem 5.1The generalized symplectic mean curvature flow has no type-I singularity at any T>0.

ProofSuppose that the generalized mean curvature flow develops a type-I singularity at a finite timet0>0.Assume that

As Σ is closed,we may assume thatxk→p∈Σ andtk→t0ask→∞.We choose a local coordinate system onaroundsuch thatThen we rescale the generalized mean curvature flow to have

Denote bythe scaled surfaceFk(·,t).Then the induced metric satisfies

The scaled surface satisfies

By Lemma 5.1,we have

for some uniform constantscandCindependent ofk.We then have

and

so there exists a subsequence ofFkwhich we also denote byFk,such thatFk→F∞in any ballBR(0)?4,andF∞satisfies

with

SetandIt is easy to see that

where?is the function defined in the definition of Ψ.Notice thatfor any fixedt.By Proposition 4.1,

and it then follows thatexists.This implies that,for any fixeds1ands2with?∞

Integrating(4.5)froms1tos2,we obtain

Since the singularity is of type I and the vector fieldis bounded,we know that there exists a constantC>0 such that fortclosed tot0,

Therefore,

where the last inequality follows from the type-I singularity assumption.Without loss of generality,we can assume thatλkF(p,tk)→Qask→∞.Lettingk→ ∞in(5.4)and using(5.3),we get that

That is,

It follows that forα=3,4,

SinceH∞=0,we also have forα=3,4,

Thus,for alli,j=1,2,α=3,4,which yields that|A∞|≡0.This contradicts(5.2).

This finishes the proof of the theorem.

6 The Graph Case

In this section we study the generalized symplectic mean curvature flow(1.1)in a special case.Suppose thatMis a product of compact Riemann surfacesM1andM2,i.e.We denote byr1andr2the average scalar curvature ofM1andM2,respectively.We assume thatr1=r2.Suppose that Σ is a graph inM=M1×M2.Recall the definition of the graph in[4].A surface Σ is a graph inM1×M2ifv=?e1∧e2,ω1?≥c0>0,whereω1is a unit K?hler form onM1,and{e1,e2}is an orthonormal frame on Σ.In this section,we use some ideas in[4,7,13].We first prove a proposition.

Proposition 6.1Each Riemann surfaceis an almost Einstein curve with=for some smooth function φon N,where r is the average scalar curvature of N.

ProofSinceBy the Hodge theorem,there exists a smooth functionφsuch thatR=r+ ?φ.Since the complex dimension ofNis 1,we haveThis finishes the proof of the proposition.

Then we can get the following theorem.

Theorem 6.1Letandbe Riemann surfaces which have the sameaverage scalar curvature.Suppose thatΣ0evolves along the generalized mean curvature flow inM1×M2.then the generalized mean curvature flow exists for all time.

ProofSetr≡r1=r2.By the above proposition,there exist smooth functionsψ1onM1andψ2onM2such thatandFor each pointonIt follows thatψis a smooth function onM1×M2,andwhich means thatM1×M2is an almost Einstein surface.

Choose an orthonormal basis{e1,e2,e3,e4}onMalong Σtsuch that{e1,e2}is the basis of Σt.Setu1=?e1∧e2,ω1+ω2?andu2=?e1∧e2,ω1?ω2?,whereω2is a unit K?hler form onM2.Since bothω1+ω2andω1?ω2are parallel K?hler forms onM1×M2,we see that Theorem 3.1 is applicable.Therefore,

where

By switchinge3ande4,we get that

where

It is clear that

The initial conditionv(x,0)>implies thatui(x,0)≥v(x,0)?≥c0>0,i=1,2.By(6.1)–(6.2),

Applying the maximum principle for parabolic equations,we obtain thatui(x,t)have positive lower bounds at any finite time.Suppose thatui≥δfor 0≤t

Setu=u1+u2.Adding(6.1)to(6.2),we get

Sinceu≥2δ+|u1?u2|,using the Cauchy-Schwarz inequality,we get

Assume that(X0,t0)is a singularity point.As in the proof of Proposition 4.1,we can derive a weighted monotonicity formula forwhere?is the cut-offfunction in Proposition 4.1.

It follows that

From this we see thatexists.

Let 0<λi→∞and letFibe the blow-up sequence:

Letdenote the induced volume form onbyFi.It is obvious that

where

Therefore we get that

Note thatfor any fixedsasi→∞and thatexists.By the above monotonicity formula,we have,for any fixeds1ands2,

Sinceuiis bounded below,we have

Therefore,for any ballBR(0)?4,

Becauseuhas a positive lower bound,we see that Σtcan locally be written as the graph of a mapft:??M1→M2with uniformly bounded|dft|.Consider the blow up ofas

It is clear that|dft|is also uniformly bounded andBy the Arzela’s theorem,on any compact set.By the inequality(29)in[10],we have

where?dfiis measured with respect to the induced metric onFrom(6.6)it follows that,for any ballBR(0)?4,

which implies thatfi→f∞inand the second derivative off∞is 0.It is then clear that→Σ∞and Σ∞is the graph of a linear function.Therefore,

We therefore have

By[14,Theorem 4.1](note thatβ(M)in this theorem for our flow istraceII(x)|V,whereX=(x,t)andII(x)is the second fundamental form ofMinNatx),we know that(X0,t0)is a regular point.This proves the theorem.

Now we consider the convergence of the generalized mean curvature flow.We follow the idea in[4].We do not require the ambient spaceMto have a product structure in the following Theorem 6.2.

Theorem 6.2Let M be a K?hler surface.Suppose that the smooth solution of the generalized mean curvature flow(1.1)exists on[0,∞).Then there exists a finite set of points Sand a sequence of ti→∞such thatΣticonverges to a surface satisfyingandthe convergence is in C2outside S.In particular,ifis an almost Einstein surface withthen the limit surface is a minimal surface in

ProofBy the Gauss equation,we have

wheregis the genus of the initial surface Σ0.Because Σtis a continuous deformation of Σ0,so its genus is alsog.De fine two conformally rescaled Riemannian metricsandonMby

Proposition 2 in[1]gives

from which we get

So,

and there exists a sequenceti→∞such that

Hence,

It follows that

and then

Suppose that Σtiblows up around a pointp∈M.We have

Assume thatλi=|A|(xi)and thatF(xi,ti)→pasi→∞.Considering the blow-up sequence

we can see thatFi→F∞asi→∞andF∞is a minimal surface in R4with|A|≤|A(0)|=1.

By[4,Lemma 5.3],we have

By(6.8),one can see that the blow-up set is at most a finite set of points which we denote byS.We can see from(6.7)that Σ∞is a surface withK=0,i.e.,As mentioned in[1],given a surface Σ inwhereeHis the mean curvature vector field on Σ with respect to the metric on Σ whichis induced by Consequently,K=0 is equivalent to=0.This proves the theorem.

Combining Theorem 6.1 and Theorem 6.2,we have the following corollary.

Corollary 6.1Assume that M=M1×M2,M1and M2are Riemann surfaces with thesame average scalar curvature r.Then M is an almost Einstein surface withLetΣ0be a graph in M.Ifthen the glabal solution F(·,t)of(1.1)exists andsub-converges to F∞in C2as t→∞,possibly outside a finite set of points,andΣ∞=F∞(Σ)is a minimal surface in

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