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March 2020 On the entropy of closed hypersurfaces and singular self-shrinkers
Jonathan J. Zhu
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J. Differential Geom. 114(3): 551-593 (March 2020). DOI: 10.4310/jdg/1583377215


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


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Jonathan J. Zhu. "On the entropy of closed hypersurfaces and singular self-shrinkers." J. Differential Geom. 114 (3) 551 - 593, March 2020.


Received: 8 September 2016; Published: March 2020
First available in Project Euclid: 5 March 2020

zbMATH: 07179186
MathSciNet: MR4072205
Digital Object Identifier: 10.4310/jdg/1583377215

Rights: Copyright © 2020 Lehigh University


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Vol.114 • No. 3 • March 2020
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