Li CHEN Dan-Dan HU Jing MAO Ni XIANG

Abstract In this paper,for the Lorentz manifold M2 × with M2 a 2-dimensional complete surface with nonnegative Gaussian curvature,the authors investigate its spacelike graphs over compact,strictly convex domains in M2,which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition,and show that solutions converge to ones moving only by translation.

Keywords Translating surfaces,Mean curvature flow,Lorentz manifolds.

1 Introduction

then the submanifoldXt:is called a translating soliton of the MCF(1.1).It is easy to see that the translating soliton gives an eternal solutionXt=X0+tVto(1.1),which is called the translating solution.Translating solitons play an important role in the study of type-II singularities of the MCF.For instance,Angenent and Velázquez[2-3]gave some examples of convergence which implies that type-II singularities of the MCF are modeled by translating surfaces.

From the above brief introduction,we know that translating solutions of the MCF are special solutions to the flow equation and are worthy of being investigated for understanding type-II singularities of the MCF.There exist some interesting results which we prefer to mention here.For instance,Shahriyari[12]proved that for the MCF,there are only three types of complete translating graphs in3,i.e.,entire graphs,graphs between two parallel planes and graphs in one side of a plane.Moreover,in the last two types,graphs are asymptotic to planes next to their boundaries.Xin[14]proved that any complete translating soliton inn+mhas infinite volume and has Euclidean volume growth at least.Moreover,he showed that graphic translating soliton hypersurfaces are weighted area-minimizing.Huisken[9]investigated graphs over bounded domains(withC2,αboundary)inn(n≥2),which are evolving by the MCF with vertical contact angle boundary condition,and proved that the evolution exists for all the time and the evolving graphs converge to a constant function as time tends to infinity(i.e.,t→∞).Altschuler and Wu[1]proved that graphs,defined over compact,strictly convex domains in2,evolved by the non-parametric MCF with prescribed contact angle(not necessary to be vertical),converge to translating surfaces ast→∞.Guan[7]investigated graphs over bounded domains inn,which are evolving by the non-parametric MCF with prescribed contact angle,and proved that the flow exists for all the time.But an extra assumption about the prescribed contact angle should be added in order to get the asymptotical behavior of limiting solutions.Zhou[15]improved Altschuler-Wu’s and Guan’s conclusions to the case of general product spaceswith closed manifoldMnhaving nonnegative Ricci curvature.

The purpose of this paper is to investigate the case of space-like graphs evolved by the non-parametric MCF with the prescribed contact angle boundary condition,and try to get interesting convergence conclusions.

Throughout this paper,M2denotes a 2-dimensional complete Riemannian manifold with a metricσand Ω is a compact,strictly convex domain ofM2with smooth boundary∂Ω.Letκ>0 be the curvature function of∂Ω.Assume that a point on Ω is described by local coordinates{ω1,ω2}.Let∂i,i=1,2,be the corresponding coordinate vector fields andσij=σ(∂i,∂j),i,j=1,2. Similar to the basic introduction of geometry of graphs shown in[4-5],we know that for the space-like graphdefined over Ω ⊂M2,in the Lorentz manifoldwith the metricg:=σijdwi⊗dwj−ds⊗ds,the tangent vectors are given by

and the corresponding upward unit normal vector is given by

For vectorsV,Wor matricesA,B,we will use the shorthand as follows

2 Estimates

2.1 The boundary

2.2 Existence of solutions

The key point of the existence for small time and the uniqueness of solutions to the IBVP(♯)is to show that the evolution equation in(♯)is uniformly parabolic att=0,which can be assured by the assumption that the initial graphic surface over Ω is space-like.In fact,by the linearization theory(see[11])and the inverse function theorem(see[13]),together with the space-like graphic assumption,the short-time existence and the uniqueness of solutions to the IBVP(♯)can be obtained.

Assume that the IBVP(♯)has smooth solutions on the time interval[0,],which means that all derivatives ofuhave bounds on[0,].In the following,we first establish a time independent priori estimate for the gradient of the solution(see Theorem 2.1),which leads to the space-like preserving property for the evolving graphic surfaces inM2×,and then turn the quasilinear evolution equation into a uniformly parabolic equation.Furthermore,by the standard theory of the second-order parabolic PDEs,the higher order regularity follows,which leads to the long-time existence of smooth solutions of the IBVP(♯)-see the end of Subsection 2.4 for a brief explanation.

2.3 The time derivative estimate

by directly applying the weak maximum principle.

2.4 The gradient estimate

Therefore,the evolution equation for the gradient is given as follows

By applying the weak maximum principle to the above evolution inequality,we have

holds.That is,

2.5 Boundary value problems

Together with the fact thatεψ(z)−εuε(z)≥εψ(ξ)−εuε(ξ)for anyz∈Ω,we have

3 Convergence

Now,we can show the following uniqueness conclusion of limit solutions to the IBVP(♯)(up to translation)by applying the strong maximum principle of the second-order linear parabolic PDEs.

which implies

Applying Theorem 2.1 and Corollary 3.1,we know that there exists a positive constantdepending onc1andc8such that

which implies the second assertion of Lemma 3.2.

AcknowledgementThe last part of this paper was carried out when Prof.J.Mao visited the Shanghai Center for Mathematical Sciences(SCMS for short),Fudan University in April 2018,and he is grateful to Prof.Peng Wu for the hospitality during his visit to SCMS.The authors would like to thank Miss Ya Gao for pointing out several typos in the previous version of this paper.The authors also would like to thank the anonymous referee for his or her careful reading and valuable comments such that the article appears as its present version.