On the rigidity of mean convex self-shrinkers

TL;DR

This paper extends the rigidity results of mean convex self-shrinkers by removing the bounded curvature assumption in certain cases, providing new curvature estimates and rigidity theorems for self-shrinkers and translators.

Contribution

It proves that the bounded curvature condition is unnecessary for cylinder rigidity in dimensions up to 6 or with positive mean curvature, and establishes new curvature estimates for self-shrinkers and translators.

Findings

Rigidity of cylinders without bounded curvature in dimensions ≤6 or with positive mean curvature.

Curvature estimates for strictly mean convex self-shrinkers.

Rigidity theorem for graphical self-shrinkers in all dimensions.

Abstract

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curvature on a large, but compact, set must be a round cylinder. Using this result, Colding and Minicozzi were able to establish uniqueness of blowups at cylindrical singularities, and provide a detailed description of the singular set of generic mean curvature flows. In this paper, we show that the bounded curvature assumption is unnecessary for the rigidity of the cylinder if either n is at most 6, or if the mean curvature is bounded below by a positive constant. These results follow from curvature estimates that we prove for strictly mean convex self-shrinkers. We also obtain a rigidity theorem in all dimensions for graphical self-shrinkers, and curvature estimates for translators of the mean curvature flow.