Rigidity Of Spacelike Translating Solitons And Self-shrinkers In Pseudo-Euclidean Space

In this paper,we investigate the rigidity theorems of spacelike translating solitons and self-shrinkers under different conditions,and consider the following three problems:the rigidity of spacelike translating solitons in pseudo-Euclidean space,the rigidity of spacelike self-shrinkers in pseudo-Euclidean space,and the rigidity of spacelike self-shrinkers in Euclidean space.In Chapter one,we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space Rnm+n.Firstly,we classify m-dimensional complete spacelike translating solitons in Rnm+n by affine tech-nique and classical gradient estimates.The main result is the following Bernstein-type theorem.Theorem 1.1.7.Let M be an m-dimensional complete spacelike translating soliton in Rnm+n,then it is an affine m-plane.This result provides another proof of a nonexistence theorem for complete spacelike translating solitons in[16],and a simple proof of rigidity theorem in[15].Secondly,we generalize the rigidity theorem of entire spacelike Lagrangian translating solitons in[22]to spacelike translating solitons with general codimensions.Theorem 1.1.8.Let u?(1???n)be smooth functions defined on Rm and their graph M =(x,u1(x),u2(x),...,un(x))be a spacelike translator in Rnm+n.If there exists a number ?>0,the induced metric(gij)satisfies(gij)>?/|x|I,as|x|??,(1)then u1(x),…,un(x)are linear functions on Rm and M is an affine m-plane in Rnm+n.In Chapter two,Li-Xu-Yuan[35]used pointwise approach to provide an elementary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows in Rm+1.We generalize the results to higher codimensions.Theorem 2.1.2.Let X:M ? Rm+n be an m-dimensional complete graphical self-shrinker with flat normal bundle,then M must be an m-plane.In Chapter three,we shall use the idea in[22]to prove the rigidity of spacelike self-shrinker in pseudo-Euclidean space Rnm+n.Theorem 3.1.2.Let X:(M,G)?Rnm+n be an m-dimensional spacelike self-shrinker and z =<X,X>be the pseudo-distance function.If M is complete with respect to a conformal metric G:=exp{z/4mn}G,then it is an affine m-plane.