A CLASS OF INVERSE QUOTIENT CURVATURE FLOW IN THE ADS-SCHWARZSCHILD MANIFOLD?

(紀正超)

1. Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China;2. Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

E-mail: jizhengchaode@163.com; jizhengchao@zju.edu.cn

Abstract In this paper,we study the asymptotic behavior of a class of inverse quotient curvature flow in the anti-de Sitter-Schwarzschild manifold.We prove that under suitable convex conditions for the initial hypersurface,one can get the long-time existence for the inverse curvature flow.Moreover,we also get that the principal curvatures of the evolving hypersurface converge to 1 when t →+∞.

Key words star-shaped;quotient curvature;support function

1 Introduction

Geometric curvature flows have been studied extensively during the last decades.It has proven that geometric curvature flows are powerful tools for acquiring differentiable sphere theorems and sharp geometric inequalities.For example,Huisken [13]studied the mean curvature flow (MCF) in the Euclidean space Rn+1and proved that a strictly convex hypersurface in Rn+1converges to a round point during the flow in 1984.subsequently,Chow [7]investigated the powers of the Gauss curvature flow and got a similar result.Andrews generalized these flows by considering the hypersurfaces evolved by some non-linear functions in spaces forms;see [1,2].For hypersurfaces in more general Riemannian manifolds,Huisken [14,15]proved the convergence for the MCF under some suitable curvature pinching conditions.For contracted geometric flows,there has been many interesting results and we only list few of them:[3,5,9,18–20].

At the same time,the expanding geometric flows have also attracted a lot of attention.Expanding curvature flows are scale-invariant,which helps the hypersurfaces in Rn+1not to develop singularities and to become more and more spherical [10,13,29].In 1990,Gerhardt[10]provided the pioneering works for the inverse curvature flow.He proved that if the initial manifoldM0is a star-shapedC2,αhypersurface in Rn+1,then

has a unique solution ofC2,αclass,where the speed functionFis a symmetric,positive function that homogeneously of degree one and is evaluated at the principal curvatures ofX(t).Moreover,he proved that the rescaled hypersurfaces=e-t/nXconverge exponentially quickly to a uniquely determined sphere.We should mention that Urbas[29]also obtained a similar result independently.After these two breakthrough achievements,the inverse curvature flows were investigated widely for the cases of more general ambient manifolds;see[6,8,17,22,25–28,30].Recently,the expanding flows were used to obtain geometric inequalities in several Riemannian warped products.Brendle-Hung-Wang [4]utilized the inverse mean curvature flow (IMCF) in the anti-de Sitter-Schwarzschild (AdS-Schwarzschild) manifold to get a sharp Minkowski inequality for the star-shaped and strictly mean-convex hypersurfaces.Ge-Wang-Wu-Xia [12]studied the IMCF in Kottler spaces and obtained a sharp Penrose inequality.Wang [30]investigated the IMCF in the Reissner-Nordstr?m-anti-de Sitter manifold.Moreover,he proved several Minkowski-type inequalities for the compact mean-convex and star-shaped hypersurfaces.Li-Wei-Xiong [24]used the convergence in [11]for an inverse curvature flow to get a sharp Alexandrov-Fenchel inequalities for closed star-shaped and two-convex hypersurfaces in the Hyperbolic spaces,when

andσkis defined as below.Recently,Lu[25]proved that when the speed function is defined by

the star-shaped,k-convex closed hypersurfaces in the anti-de Sitter-Schwarzschild manifolds satisfy that,along the flow.

In 2018,Chen-Mao [6]successfully obtained the convergence for a wide class of the inverse curvature flow in the AdS-Schwarzschild manifold.Very recently,Li-Xu [23]studied a class of weighted inverse mean curvature flow in the AdS-Schwarzschild manifold for,when the speed function is defined byF=|V|αH,whereα>0 andVis defined by (3.11).They got the longtime existence for the flow and proved that the principal curvatures of the evolving hypersurface converge to 1 whent →+∞.

In this paper,our main aim is to study a class of inverse quotient curvature flow in the AdS-Schwarzschild manifold (see the definition of this in the next section).Before we state our main result,we should introduce some notations.For a smooth hypersurface Σ in Nn+1,we denote by=(λ1,λ2,···,λn) its principal curvatures.It is well-known that the mean curvatureHis defined as

Therefore,by using the normalized elementary symmetric function of orderkfor principal curvaturesλ1,λ2,···,λn,we ca n define thek-th mean curvatureHkas

A hypersurface isk-convex ifHi>0,?i ≤k.Our main theorem is the following:

Theorem 1.1Let Σ0be a star-shaped,k-convex closed hypersurface in the anti-de Sitter-Schwarzschild manifold (Mn+1,).For everyα>0 andVwhich is defined by (3.11),we consider the inverse curvature flow

wherevis the outward unit normal and

Then the solution of (1.2) exists fort ∈[0,∞).Moreover,the second fundamental forms of Σtconverge to 1 as,whereCis a constant depending onα,k,landn.

2 Preliminary

In this section,we will give some basic formulas on (Mn+1,).First,we recall some definitions of the AdS-Schwarzschild manifold (one can find these notations in [4]).For a given positive numberm,we guarantee that the following equation has positive solutions:

For convenience,lets0be the largest one.The AdS-Schwarzschild manifold is a manifoldM=Sn×[s0,∞) with the Riemannian metric

wheregSnis the standard metric onSn(1).Since the sectional curvature ofMapproach-1,is asymptotically hyperbolic.In addition,one can infer that the scalar curvature ofMis-n(n+1) and its the boundary is?M=Sn×{s0}.If we definethenf(s) satisfies the following equation

With the above notations in hand,we state the following lemmas form [4]:

Lemma 2.1By a change of variable,the metric can be written as=dr ?dr+λ(r)gSn,whereλ(r) satisfies ODEand the asymptotic expansion

Let{eα}α=1,···,nbe an orthonormal frame,θ={θj}j=1,2,···,nbe the coordinate system onSnand letbe the corresponding coordinate vector.Then we have the asymptotic expansion of the Riemannian curvature tensors.

Lemma 2.2Lettingαβγμbe the Riemannian curvature tensor of the AdS-Schwarzschild metric,then

Moreover,the Ricci tensor satisfies that

whereσij=gsn(?θi,?θj).

The following result can be found in [23]that:Lemma 2.3λ′(r) andλ′′(r) satisfy

Given someε>0,bothcan be bounded by two positive constants forr ∈[ε,∞).Furthermore,they tend to 1 asr →∞.

For convenience,we introduce the following equations:

Lemma 2.4We have the Gauss equation,Codazzi equation and the interchanging formula as

3 Long Time Existence

In this section,we will prove the long time existence for the flow by giving theC0,C1andC2estimates.

It is well-known that a star-shaped hypersurface Σ?Mcan be considered as a graph onSn,i.e.,Σ=(r(θ),θ) for smooth functionronSn,whereθ ∈Sn.Define

where Φ(r) is a positive function which satisfiesLet?i=?i?and?ij=?i?j?denote the first order and the second order derivative of?under the metricgSn.Letting

then one gets thatgij=λ2(σij+?i?j),wheregijis the induced metric on the star-shaped hypersurface Σ.The second fundamental formhijcan be parameterized by

Moreover,we also have:

Lemma 3.1The induced metricgon Σ is given bygij:=λ2(σij+?i?j)=λ2σij+rirj,and its inverse is

ProofThese equations can be found in [23].

The following is important to get theC1estimate:

Lemma 3.2We have that

ProofBy using the inverse ofg,one has that

Therefore,we obtain the first equation.The proof of the second equation can be found in [23].

whereνis the outward normal vector,andk>l ≥0.Due to the works of Huisken and Polden [16],we have the following evolution equations:

Lemma 3.3During the flow (3.1),we have that

By utilizing the interchanging formula,one can obtain a more detailed version of the evolution for

Lemma 3.4During the flow (3.1),we have that

ProofSee [25].

As the flow exists,if the Σtis still star-shaped,it can be parameterized as

Moreover,the equation of the flow (3.1) is equivalent to

For convenience,we userto denoter(θ,t) for the rest of this paper.Let?(θ,t) :=Ψ(r(θ,t)),where Ψ(r)>0 and satisfies thatThen,we have that

The support function is defined as

The first and the second covariant derivatives foruwith respect to the metricgSncan be found in [25].

Lemma 3.5The support functionusatisfies that

Now we will give theC0estimate.

Thus,we get that?i=0,Hess(?)≤0 at (x0,t0).Since

one concludes that

at (x0,t0).Therefore,we obtain that

which implies that

From the above inequality,we obtain the desired inequality.Similarly,we can also prove the other inequalities.

We have the following estimate for:

Lemma 3.7Along the flow (3.1) we have||≤C,whereCis a constant depending on Σ0,n,k.

ProofIn this proof,we will use the fact thatvis uniformly bounded during the flow(see Lemma 3.12).Letg=g(t) be a smooth function which is determined later,and letG=g.Therefore,we have that

According to the maximal principle,we get that|˙?| has an upper bound.Putting (3.22) into(3.21),we have that

Assume thatGattains its maximum at(x0,t0)andG(x0,t0)is big enough.If there exists somet ∈[0,t0]such thatG(x0,t0)≥C,from (3.23) and the gradient estimate in Lemma 3.12,we get that

Using theC0estimate forλ,we have that

wherec′=c′(Σ0,α,n).Due to Lemma 3.11,we also have that

Next,we will give theC1estimate.Before we give our proof,we should provide several basic properties of elementary symmetric functions in [21].

Lettingζ=(ζ1,···,ζn)∈Rn,we denote byσk(ζ|i) the symmetric function withζi=0 and byσk(ζ|ij) the symmetric function withζi=ζj=0.

Proposition 3.8Letζ=(ζ1,···,ζn)∈Rnandk=0,1,···,n.Then for any 1≤i ≤n,the following equations hold:

Proposition 3.9Supposing thatW=(Wij)is diagonal and thatmis a positive integer,we have that

Moreover,we also have the generalized Newton-MacLaurin inequality.

Proposition 3.10Letζ ∈Γkandk>l ≥0,r>s ≥0,k ≥r,l ≥s.Then we have that

The following inequalities offwill be important (these results are standard,but some of the computations will be used through out the rest of this paper,so convenience,we show the proof here):

Lemma 3.11Let,whereλ=(λ1,···,λn) and (λi)∈Γk.Then we have that

whereC3andC4are constants depending on Σ0,k,l,nand

ProofFirst,we have that

Combining Proposition 3.8,Proposition 3.9 and Proposition 3.10,we obtain that

whereC3is a constant depending onn,k,l.We can also get the lower and the upper bounds forOn the one hand,by using Lemma 3.14,we have that

On the other hand,we can get the lower bound based on the direct computation:

This completes our proof.

For convenience,we define that

TheC1estimate will given in the next lemma.

Lemma 3.12Along the flow (3.1),we have that|??|≤C,whereCis a constant depending only on Σ0,nandα.Moreover,it holds that

whereC2is a constant depending on Σ0,nandα.ProofBy (3.18),we get that

Differentiating (3.35),we obtain that

To complete the evolution ofω,we also need to compute the Hessian ofω.By directly calculating,we get that

Hence,we arrive at

whereC2=C2(Σ0,α,n).

Due to the maximal principle,we obtain

This completes the proof.

To give the long time existence of the flow (3.1),it remains to prove theC2estimate.The evolution equation for the support functionuwill be critical.

Lemma 3.13The support function of Σtsatisfies that

ProofAccording to Lemma 3.5,we have that

Hence,we obtain that

At the end of this section,we are going to get the upper bounds of the principal curvatures.

Lemma 3.14The principal curvaturesλiof Σtare bounded by a constantC1which depends only on Σ0,αandn.

ProofSet thatω=logη-logu+logg,whereη=sup{hijξiξj:gijξiξj=1},g=

Assume thatωattains its maximum at(x0,t0)and thatη=,then we have that?ω=0 and Hessω ≤0 at (x0,t0);i.e.,

By the Codazzi equation,the Ricci identity and the Gauss equation,we get that

Putting (3.38) into (3.37),one gets that

Notice that

Moreover,by utilizing the Cauchy-Schwarz inequality and the fact thatf>c′,we also have that

Therefore,we get that

In addition,we also have that

By using the evolution equations ofanduand Lemma 2.2,we get that

Corollary 3.15The solution of the flow (3.1) is defined on [0,+∞).

ProofFrom theC0,C1andC2estimates and the Evans-Krylov theorem,we obtain theC2,αestimate.Together with Schauder estimate,we have all of the high order estimates,so the result follows immediately.

4 Convergence of the Flow

In this section,we will investigate the limit behavior of the principal curvatureλi=.

Lemma 4.1The principal curvatureλisatisfies that

ProofLet us consider the test functionω:=(logη-logu+r-log(2))log(t+1).By utilizing theC0estimate and Lemma 2.1,we have that

Combining this with (4.1),we get that

for some positive constantC6.Similarly,we can get that

Without lose of generality,we suppose thatωattains its maximum at (x0,t0) and thatη=.

Therefore,we have that

Notice that

Hence,we have that

Taking the above formula into (4.2),we obtain that

Next,we are going to consider the gradient terms.From the critical equations,we get that

and|?logu|2≤C|?r|2,we get

Thus,combining this with (4.3),we arrive at

Next,we will give the limit behavior of the functionF.

Lemma 4.2The functionsatisfies that

According to Lemma 3.7,we get that

we have that

By using the maximum principle,we get that

On the other hand,we have that

which yields that

Hence,we get that

whereβ>0 is a constant.From the above inequality,we get that

where we use (3.33).Therefore,we have that

Due to Lemma 4.1,we also have that≤c0ast →∞.

Therefore,we obtain

Combining this with the asymptotic behaviors ofλiandF,we have the following result:

Corollary 4.3The asymptotic behavior ofsatisfies that|-|→0 ast →∞.

By using the above estimate of the second fundamental forms,we can improve theC0estimates andC1estimates.

Lemma 4.4For any constant 0<τ<1,whent →∞,we have that

ProofAccording theC0estimate and Corollary 4.3,we have that

For any fixed constant 0<τ<1,since

the constantC2in (3.36) can be replaced byτwhent →∞.Hence,we get

From the above improved estimate,we get that

This completes the proof.

At the end of this section,we will give the exponential convergence for the second fundamental forms.

Lemma 4.5For,we have the following exponential convergence:

ProofConsidering the test function

By computation,we have that

Assume thatGattain its maximum at some point;at this critical point,we get that

By using the critical equation,we have that

From Corollary 4.3,the gradient terms can be controlled byand thus we get that

where we chooseτ →1 as in Lemma 4.4.Then,we obtain that

Without loss of generality,we can assume thatgij=δij,hij=λiδij,λ1≤λ2···≤λnat the critical point.Hence,we get that

This completes our proof.

Conflict of InterestThe author declares no conflict of interest.