Uniqueness of grim hyperplanes for mean curvature flows

Abstract

In this paper we show that an immersed nontrivial translating soliton for a mean curvature flow in $$\mathbb {R}^{n+1}$$ ( $$n=2,3)$$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $$\varSigma $$ with nonnegative scalar curvature, we prove that if the mean curvature of $$\varSigma $$ does not change signs on each end, then $$\varSigma $$ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.