Uniqueness of grim hyperplanes for mean curvature flows

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 $\mathrm{\Sigma }$ with nonnegative scalar curvature, we prove that if the mean curvature of $\mathrm{\Sigma }$ does not change signs on each end, then $\mathrm{\Sigma }$ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.