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

No reviews yet.

No comments yet.