On the entropy of closed hypersurfaces and singular self-shrinkers

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 entropystable self-shrinker whose singular set satisfies Wickramasekera’s α-structural hypothesis must be a round cylinder $\mathbf{S}^k (\sqrt{2k}) \times \mathbf{R}^{n-k}$.