On the entropy of closed hypersurfaces and singular self-shrinkers

By Jonathan J. Zhu. Journal of differential geometry, 2016

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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 $\mathbf{S}^k(\sqrt{2k})\times \mathbf{R}^{n-k}$.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a0f12b93b3d352694af38720047eb08c5306b05

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In Applied Mathematics

  1. Otis Chodosh
  2. F. Schulze
  3. Guofang Wang
  4. Chao Xia
  5. Antoine Song
  6. Junfang Li
  7. Pengfei Guan

In Geometry and Topology

  1. Otis Chodosh
  2. F. Schulze
  3. Guofang Wang
  4. Chao Xia
  5. Junfang Li
  6. Pengfei Guan

In Mathematical Physics

  1. Otis Chodosh
  2. F. Schulze

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