Uniqueness of grim hyperplanes for mean curvature flows

Abstract

In this paper we show that an immersed nontrivial translating soliton for 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 $\Sigma$ with nonnegative scalar curvature, we prove that if the mean curvature of $\Sigma$ does not change signs on each end, then $\Sigma$ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.