Uniqueness of grim hyperplanes for mean curvature flows

TL;DR

This paper characterizes grim hyperplanes as unique translating solitons in mean curvature flow under specific convexity and curvature growth conditions, providing new geometric classification results.

Contribution

It establishes a characterization of grim hyperplanes among translating solitons based on mean convexity and curvature growth, extending understanding of their uniqueness.

Findings

Translating solitons with certain curvature conditions are grim hyperplanes.

Embedded solitons with nonnegative scalar curvature and sign-preserving mean curvature are either hyperplanes or grim hyperplanes.

Provides conditions under which mean curvature flow solitons are uniquely classified as grim hyperplanes.

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.