Rigidity of self-shrinkers and translating solitons of mean curvature flows

In this paper, we prove that any complete m  -dimensional spacelike self-shrinkers in pseudo-Euclidean spaces Rnm+n must be affine planes, and there exists no complete m  -dimensional spacelike translating soliton in Rnm+n. These results are proved by using a new Omori–Yau maximal principle. We also derive a rigidity theorem of self-shrinking hypersurfaces in Euclidean space with Gauss image lies in a regular ball.