On the entropy of closed hypersurfaces and singular self-shrinkers
On the entropy of closed hypersurfaces and singular self-shrinkers
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关于闭超曲面和奇异自收缩器的熵
DOI：
10.4310/jdg/1583377215
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发表时间：
2016
影响因子：
2.5
通讯作者：
Jonathan J. Zhu
中科院分区：
文献类型：
--
作者：
Jonathan J. Zhu
自收缩器是 $\mathbf{R}^{n+1}$ 中平均曲率流的特殊解，通过相似收缩演化；它们充当流动的奇点模型。 Colding-Minicozzi 引入的超曲面的熵是平均曲率流的李亚普诺夫函数，并且是其一般平均曲率流理论的基础。
在本文中，我们证明了 Colding-Ilmanen-Minicozzi-White 的猜想，即 $\mathbf{R}^{n+1}$ 中的任何闭超曲面都至少具有圆球的熵，在任何维度 $n$ 中都成立。 Bernstein-Wang 之前已使用精心构造的弱流针对 $n\leq 6$ 的情况建立了此结果。
本文的主要技术成果是将Colding-Minicozzi的熵稳定自收缩器分类扩展到奇异设置。特别是，我们证明任何奇异集满足 Wickramasekera 的 $\alpha$ 结构假设的熵稳定自收缩器必须是圆柱体 $\mathbf{S}^k(\sqrt{2k})\times \mathbf{R}^{n-k}$。
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 $\mathbf{S}^k(\sqrt{2k})\times \mathbf{R}^{n-k}$.