Hongbing QIUYunhua YE Anqiang ZHU

1 Introduction

LetMnbe a compactn-dimensional hypersurface without boundary,and letbe a smooth immersion ofinto a Riemannian manifoldConsider the generalized mean curvature flow(abbreviated for GMCF),namely,a smooth one-parameter family of immersions

satisfying the evolution equation

wheref:R→R is a smooth function,depending only on the mean curvature of the immersed surface,andν(·,t)is the outer unit normal onMt:=F(M,t)atF(·,t).If>0 along the GMCF,then the short time existence has been established in[10].It is easy to prove that(1.1)admits a smooth solution on a maximal time intervalwith<∞.

Iffis the identity function,then(1.1)is the classical mean curvature flow.If we choosefto be the power functionxk,then(1.1)is theHkmean curvature flow.In this paper,we mainly pay our attention to theHkmean curvature flow,also we get some results on the GMCF.

The long time existence,convergence,blow up and extension properties are of great interest subjects in curvature flow.Recently,many efforts have been made on the extension theorem for the mean curvature flow under some curvature conditions(see[1,6,11–12]).Le and Sesum[6]showed that if the second fundamental form stays bounded from below all the way toT,then some integral condition of mean curvature is enough to extend the mean curvature flow past timeT.This extension theorem had also been generalized to the setting when the outer space is Riemannian manifold(see[11–12]).In arbitrary codimension,Han and Sun[1]gave an integral condition under which the mean curvature flow can be extended and then they investigated some properties of type I singularity.In[7],Li proved an extension theorem for theHkmean curvature flow in Rn.Motivated by his idea,we prove the following main theorems in our Riemannian setting.

Theorem 1.1Let M be a compact n-dimensional hypersurface without boundary,smoothly immersed intowith bounded geometry by F0.Letbe the maximal time interval of the Hkmean curvature flow withand H(·,0)>0.Then the quantitybecomes unbounded as

Along mean curvature flow,Huisken[3–4]proved that ifT<∞is the first singularity time for a compact MCF,then→∞ast→T.The above theorem is natural for GMCF.

Theorem 1.2Assume k,n∈N,k,n≥2and n+1≥k.Let M be a compact n-dimensional hypersurface without boundary,smoothly immersed intowith bounded geometry by F0.Consider the Hkmean curvature flow on M,

If

(1)along the Hkmean curvature flow for a uniform constant C>0,

(2)for some α≥n+k+1,

then the flow can be extended over the time

2 Preliminaries

In the following,the induced metric and the second fundamental form onMwill be denoted byg=andB=.The mean curvature ofMis the trace of the second fundamental form,i.e.,

The square of the second fundamental form is

The Riemann curvature tensor ofNand its covariant derivative will be denoted byrespectively.We writeRm=for the curvature tensor ofM.Letνbe the unit outer normal to,then for a fixed timet,we can choose a local field of frame,···,inN,such that restricted to,we have

The relation betweenB,Rmandmis then given by the equations of Gauss and Codazzi:

We have the following proposition.

Proposition 2.1(see[4])

3 The Evolution Equations

Theorem 3.1For the GMCF in Riemannian manifold,we have the following evolution equations:

ProofLet us first prove(3.2).

Next we prove(3.3).

Using(2.1)of Proposition 2.1,we have

this proves(3.4).

To prove(3.5),it is easy to get

Hence

this proves(3.5).It is easy to obtain(3.6),we omit the concrete computation.

4 Sobolev Inequalities for the GMCF

Li[7]obtained a Sobolev inequality for the power mean curvature flow by using Michael-Simon inequality(see[8]),which is crucial for the Moser iteration in his situation.In our setting,we also need an inequality which is similar to Michael-Simon inequality.Hence,in this section we first introduce the Hoffman-Spruck Sobolev inequality.

Lemma 4.1(see[2])Let M→N be an isometric immersion of Riemannian manifolds of dimension n and n+p(p≥1),respectively.Assumeand let h be a nonnegative C1function on M vanishing on∂M.Then

provided

and

where(M)is the injectivity radius of N restricted to M and

Here α is a free parameter,0<α<1,and

Following the proof of Theorem 3.4 in[7]and using Lemma 4.1,we obtain the following general result.

Theorem 4.1Suppose that k,n∈N,k,n≥2,or k=1and n=2.Set

Let M be a compact n-dimensional hypersurface without boundary,which is smoothly embedded in.AssumeThen for all nonnegative Lipschitz functions v on M,we have

provided that the function h:=satisfies(4.2)–(4.3),where H is the mean curvatureof M and

Corollary 4.1Under the conditions of Theorem4.1,for any nonnegative Lipschitz function v,we have

where

Similar to the proof of Theorem 3.6 in[7],using Corollary 4.1 and Holder’s inequality,we obtain the following Sobolev type inequality for the GMCF.

Theorem 4.2Suppose that k,n∈N,k,n≥2.Let M be a compact n-dimensional hypersurface without boundary,which is smoothly embedded inAssumeConsider the GMCF

where f∈C∞(Ω),Ω⊂R.Suppose(x)>0,and f(x)·x≥0along the GMCF.Then for all nonnegative Lipschitz functions v,we have

provided that the function h:=satisfies(4.2)–(4.3),where

and β=2+>2.

Remark 4.1Ifk=1,then=2.Thus we do not need to use Holder inequality to control theL2-norm,and in this case,is a constant.

5 Reverse Hder and Harnack Inequalities

In this section,we can follow the lines of[7]and[11],and easily derive a soft version of reverse Holder inequality and a Harnack inequality for parabolic inequality along the GMCF in Riemannian manifolds.Suppose thatf∈(Ω)for an open set Ω⊂R,and thatvis a smooth function onM×[0,T]such that its image is contained in Ω.

We start with the following differential inequality:

where the functionG+Chas bounded(M×[0,T])-norm with

Cis a fixed positive constant andLetη(x,t)be a smooth function onM×[0,T]with the property thatη(x,0)=0 for allx∈M.

LetSbe the set of all functionsf∈C∞(Ω)(Ω⊂R)satisfying the following conditions:

(a)fsatisfies the differential inequality(5.1),

(b)(x)>0 for allx∈Ω,

(c)f(x)≥0 wheneverx≥0,

(d)f(H(t))H(t)≥0 along the GMCF,

(e)(v)≥>0 onM×[0,T]for some uniform constant.

Lemma 5.1Let M be a compact n-dimensional hypersurface without boundary,which is smoothly embedded inConsider the differential inequality(5.1).Let β≥2be a fixed number.Then

ProofMultiplying(5.1)by(v),then for anys∈[0,T],we have

Using the integration by parts,the properties ofηand(3.6),we conclude that

Direct calculation gives

And the Cauchy-Schwartz inequality implies

and

Note that

If we choose=,then we can obtain that

Combining the above estimates with

gives

Theorem 5.1Let M be a compact n-dimensional hypersurface without boundary,which is smoothly embedded inAssumeand k,n∈N,k,n≥2.Consider the differential inequality(5.1).Let

and β≥2be a fixed number.Then there exists a positive constantdepending only on n,k,T,β,q,andVol(M),such that for any f∈S,

provided that the functionsatisfies the conditions(4.2)–(4.3)for any t∈[0,T],where

and

In particular,if(G+C)(M×[0,T]),then letting q→∞,we have

where

In this case,we obtain

provided that the functionsatisfies the conditions(4.2)–(4.3)for any t∈[0,T],where

ProofDenote

and

By Lemma 5.1,we have

LetS:=M×[0,T]and let the normbe abbreviated byIf the function(η(v))satisfies the conditions(4.2)–(4.3)for anyt∈[0,T],applying Theorem 4.2 toη(v),we have the following estimate:

Since 1

where

Hence if we choose

then we have

whereis the constant depending only onn,k,T,β,q,Vol(M).From the definition of Λ and noting that 1<≤2,we obtain

where

Next,we shall show that anL∞-norm off(v)over a smaller domain can be bounded by anLβ-norm off(v)over the whole manifoldM×[0,T].

Corollary 5.1Let M be a compact n-dimensional hypersurface without boundary,which is smoothly embedded inAssumeand k,n∈N,k,n≥2.Consider the differential inequality(5.1).Let

and β≥2be a fixed number.Then there exists a uniform constant Cn>0depending only on n,such that for any f∈S,we have

where

ProofSet

Letηi(x,t)be smooth functions satisfying the following properties:

SetNow we claim thatsatisfies the conditions(4.2)–(4.3)for anyt∈[0,T].

In fact,under the GMCF,we observe that

for anyt∈[0,T]by(3.6).Forg(0),there exists a non-positive constantsuch that the sectional curvature ofM0is bounded from below byK.Then by the Bishop-Gromov Volume comparison theorem,we have

where VolK(B(R))denotes the volume of the ball with radiusRin then-dimensional complete simply connected space form with constant curvatureK.Hence

Therefore,we can chooseRsufficiently small such that

whereρ0is defined by(4.4).Here the sufficient smallness ofRcan be achieved by choosing a sufficiently largep.Sosatisfies the conditions(4.2)–(4.3)for anyt∈[0,T].Sinceexists,using Theorem 5.1,we have

Then by the standard Moser iteration process,we have

Corollary 5.2Let M be a compact n-dimensional hypersurface without boundary,which is smoothly embedded inwith bounded geometry.Suppose n,k∈N,k,n≥2,and n+1≥k.Consider the Hkmean curvature flow

If

along the Hkmean curvature flow for some uniform constant C2>0,then there exists a uniform constant Cn,depending only on n,such that

where

ProofLet

From the evolution equation ofH(t),i.e.,(3.5),we have

By Corollary 5.1,there exists a uniform constantCn>0,such that

i.e.,

Chooseβ=then it follows that

Remark 5.1Whenk=1,n+1≥kis obvious,but fork≥2,this assumption is needed in our proof.

6 Proof of Main Theorem

Proof of Theorem 1.1We shall follow the basic ideas of Schulze[9].If Theorem 1.1 is false,then there exists someC<∞such that

on 0Using the evolution equation and the upper bound forH,it follows that forp∈U⊂M,0<σ<ρwe have that

andF(·,t)converges uniformly to some continuous limit functionF(·,).We want to show thatF(·,)actually represents a smooth limit surfaceThis is then a contradiction to the maximality ofIn order to show thatF(·,)represents a smooth surfacewe only have to establish uniform bounds for all derivatives of the second fundamental form onMt,

In the following,we denote the metric ofNand Graph(u)byandgrespectively.Fork≤1,sinceHkis concave inand it has uniform-bound,then using the estimate of[5](see[5,Theorem 2 in Chapter 5.5]),we can obtain the uniform-bounds.Fork>1,letSbe a fixed reference hypersurface which is tangent to the hypersurfaceF(·,)at some pointp∈N,and assume that we have Gaussian coordinates{···,}in a neighborhood ofponS.Then there exists a local coordinate in the neighborhood ofpinNconstructed from the above coordinate.Suppose thatUis a neighborhood ofpsuch that for every pointq∈Uthere exists a unique minimal geodesic

to the hypersurfaceSsatisfyingL(γ(t))=d(q,S).The coordinate ofqis set to be

By the construction,=0 for anyv∈S.Given1≤i≤n,there exists a curvesuch that(0)=q,and=d(q,S).For any pointγq(s),−δ

such thatF(·,s):[0,d(q,S)]Nis the minimal geodesic fromγq(s)toS.Hence the vector fieldv(t)=dF(∂s)(t,0)is a Jacobi field withv(0)==0.Hence

that is=0,i=1,···,n.Sincewe have

Under this coordinatelocally aroundpwe can writeF(·,t)fort∈(for someε>0)as graphs of functionu(t)onS(see[4,13]).Set

Then|1≤i≤n}gives a basis for the tangent space to Graph(u).It is easy to see that

is the unit inner normal vector onF(·,t)andusatisfies the following evolution equation:

By direct calculation,we have that

since=0 fori=1,···,n,andp=1,···,n+1,where Γ is the Christoffel symbol ofN.Using the expression of,we compute that

Hence

Therefore(6.3)and(6.5)imply that

According to Theorem 2,Chapter 5.3 in[5],with the assumption that|B|is bounded,we can obtain the uniform Hlder-estimates in space and time forSimilarly,by Theorem 4,Chapter 5.2 in[5],we can also have the Holder-estimates for∇u.On the other hand,the mean curvatureHsatisfies the evolution equation

Then letφbe the solution of the ODE

Then we have

Sincek>1,φ(t)>0 for allt>0.

If we considerφas a function onM×[0,we have

Supposet0be the first time that

attaining zero.Then at(p,t0),we have

ByH(p,>0,we have

which is a contradiction.HenceH(x,t)>φ(t)>>0,where

(6.6)implies that

and

are also uniformly Holder-continuous in space and time.Therefore we can write(6.6)as a linear,strictly parabolic PDE

with coefficientsin space and time.The interior Schauder estimates then lead toIn both cases,namely,k≤1 andk>1,using again parabolic Schauder estimates,we get a bound on all the higherCl-norms.

Proof of Theorem 1.2It is sufficient to prove the theorem forα=n+k+1 since by the Hlder inequality,<∞impliesH(t)<∞ifα>n+k+1.Note thatH(t)is invariant under the rescaling of theHkmean curvature flow.

We argue by contradiction.Suppose that the solution to theHkmean curvature flow can not be extended overThenB(t)is unbounded ast→(i=1,···,n)be the principal curvatures.Then

Since(c>0),thusis also unbounded ast→Namely,

Choose an increasing time sequencesuch that=We take a sequence of points∈M,satisfying

then

Therefore there exists a positive integeri0such that≥1 and≥1 fori≥.

Fori≥i0andt∈[0,1],we consider the rescaled flows

Then a simple calculation shows that

whereandare the corresponding induced metric,second fundamental forms,and the mean curvature,respectively.From the definition ofwe must have

As in[12],we can find a subsequence oft∈[0,1],converges to a Riemannian manifoldwhereis an immersion.

Since

it follows thatk(is also bounded onM×[0,1]for anyi≥And since(N,h)has bounded geometry and1 fori≥,(N,also has bounded geometry with the same bounding constants as(N,h)for eachi≥.It follows from Corollary 5.2 that

whereCis a constant independent ofifori≥i0.Hence

since<∞and=∞.

On the other hand,by our construction,we have

This is a contradiction.We complete the proof of Theorem 1.2.

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