Hongbing QIU

Abstract In this paper,the author proves that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature by using the Omori-Yau maximum principle.

Keywords Self-Shrinker,Rigidity,Omori-Yau maximum principle,Pseudo-distance

1 Introduction

The mean curvature flow(MCF for short)in Euclidean space is a one-parameter family of immersionsXt=X(·,t):with corresponding imagesMt=Xt(M)such that

is satisfied,whereH(x,t)is the mean curvature vector ofMtatX(x,t)in.

Self-similar shrinkers to the above MCF play an important role in understanding the behavior of the flow since they often occur as singularities.Mmis said to be a self-shrinker if it satisfies a system of quasilinear elliptic PDE of the second order

whereXNis the normal part ofX.

The corresponding MCF could also be studied for the ambient pseudo-Euclidean space(see e.g.[8-12,16]).In this setting,Mmis also called as a self-shrinker if it satisfies(1.2).Ding-Wang[6]investigated self-shrinking graphs with high codimensions in pseudo-Euclidean space and obtained rigidity results under subexponential decay condition.Chau-Chen-Yuan[2]and Huang-Wang[13]showed that any spacelike entire graphic Lagrangian self-shrinkers must be flat under the decay condition on the Hessian of the potential function respectively.Ding-Xin[7]proved that such Lagrangian self-shrinkers must be affine plane which removed the additional condition in[2,13].Later,Liu-Xin[15]derived the rigidity of spacelike self-shrinkers under two different conditions,more specifically,if the spacelike self-shrinker is complete(or a closed subset with respect to the Euclidean topology of the pseudo-Euclidean space,see[5,14]),then it is an affine plane under a growth condition on thew-function(or mean curvature).Some rigidity and classification results were also obtained in[1,3]for spacelike self-shrinkers under various conditions.Recently,Chen-Qiu[4]proved that any completem-dimensional spacelike self-shrinkers inmust be flat by using the Omori-Yau maximum principle,which implies that under the completeness condition,the growth conditions in the previous mentioned results on the spacelike self-shrinkers can be removed.It is natural to ask that how about the corresponding rigidity results when the spacelike self-shrinker is a closed subset(with respect to the Euclidean topology)of.

Along this direction,in the present paper,by establishing a new Omori-Yau maximum principle(see Theorem 2.1),we demonstrate that the spacelike self-shrinker which is closed with respect to the Euclidean topology must be flat under a growth condition on the mean curvature(see Theorem 3.1).

2 An Omori-Yau Maximum Principle for Spacelike Self-shrinkers

Direct computation gives

3 Rigidity Results

Combining with(3.6),it follows that

Remark 3.1In[15],the authors show that ifthenMis a linear subspace.Note that the two curvesy=(x−1)2andshall meet at two distinct points,the one is(0,1)and the other one is far away from the origin in the first quadrant.Over the interval between these two points,the function graph ofy=(x−1)2stays above that ofHence in this interval,the above condition on the mean curvature is weaker than the one in[15].

AcknowledgementThe author would like to express his sincere gratitude to Professor Y.L.Xin for his valuable suggestions.He thanks Dr.Yong Luo for helpful discussion.He also thanks the Shanghai Center for Mathematical Sciences,where part of this work was done during his visit.