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

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关于封闭超曲面和奇异自收缩的熵

自收缩是 $\mathbf{R}^{n+1}$ 中平均曲率流的特解，通过同位收缩演化；它们用作流动的奇点模型。Colding-Minicozzi 引入的超曲面的熵是平均曲率流的李雅普诺夫函数，是他们的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 $\mathbf{R}^{n+1}$ 中的任何封闭超曲面的熵至少是圆球的熵，在任何维度上都成立$n$。这个结果之前由 Bernstein-Wang 使用精心构造的弱流在 $n\leq 6$ 的情况下建立。本文的主要技术成果是将 Colding-Minicozzi 的熵稳定自收缩分类扩展到奇异设置。