Rigidity and Curvature Estimates for Graphical Self-shrinkers

TL;DR

This paper establishes new rigidity results for self-shrinkers in mean curvature flow, showing that under certain conditions, such hypersurfaces must be hyperplanes, and provides curvature estimates for these and related solitons.

Contribution

It proves that certain graphical self-shrinkers are necessarily hyperplanes and introduces linear curvature estimates for almost stable shrinkers, extending the understanding of their geometric properties.

Findings

Self-shrinkers in specified conditions are hyperplanes.

Linear curvature estimates for almost stable shrinkers.

Uniform curvature bounds for translating solitons.

Abstract

Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \leq 6$, any smooth, complete self-shrinker $\Sigma^n\subset\mathbf{R}^{n+1}$ that is graphical inside a large, but compact, set must be a hyperplane. In fact, this rigidity holds within a larger class of almost stable self-shrinkers. A key component of this paper is the procurement of linear curvature estimates for almost stable shrinkers, and it is this step that is responsible for the restriction on $n$. Our methods also yield uniform curvature bounds for translating solitons of the mean curvature flow.