On the entropy of closed hypersurfaces and singular self-shrinkers

TL;DR

This paper proves that all closed hypersurfaces in Euclidean space have entropy at least that of the sphere, extending previous results and classifying entropy-stable singular self-shrinkers under certain conditions.

Contribution

It confirms the entropy conjecture for all dimensions and extends classification of entropy-stable self-shrinkers to singular cases with structural hypotheses.

Findings

Any closed hypersurface has entropy at least that of the sphere.

Entropy-stable self-shrinkers with certain singularities are round cylinders.

The conjecture holds in all dimensions.

Abstract

Self-shrinkers are the special solutions of mean curvature flow in $\mathbf{R}^{n+1}$ that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow. In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White, namely that any closed hypersurface in $\mathbf{R}^{n+1}$ has entropy at least that of the round sphere, holds in any dimension $n$. This result had previously been established for the cases $n\leq 6$ by Bernstein-Wang using a carefully constructed weak flow. The main technical result of this paper is an extension of Colding-Minicozzi's classification of entropy-stable self-shrinkers to the singular setting. In particular, we show that any entropy-stable self-shrinker whose singular set satisfies Wickramasekera's $\alpha$-structural hypothesis must be a round cylinder $\mathbf{S}^k(\sqrt{2k})\times \mathbf{R}^{n-k}$.