On the non-degenerate and degenerate generic singularities formed by mean curvature flow

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-09-09 DOI:10.1016/j.aim.2024.109937

Zhou Gang

引用次数: 0

Abstract

We study a neighborhood of generic singularities formed by mean curvature flow (MCF). For various possibilities when the singularities are modeled on $S^{3} \times R$ , we provide a detailed description for a small, but fixed, neighborhood of singularity, including proving that a small neighborhood is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidence to support it. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead of a standard straight one.

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论平均曲率流形成的非退化和退化泛奇点

我们研究了由平均曲率流（MCF）形成的一般奇点邻域。针对奇点在 S3×R 上建模的各种可能性，我们对奇点的一个小但固定的邻域进行了详细描述，包括证明一个小邻域是均值凸的，奇点是孤立的。对于其余的可能性，我们猜想奇点的整个邻域在爆炸时会变成奇点，并提出了支持这一猜想的证据。一个关键技术是，在为重标定 MCF 寻找主导方向时，我们需要一个正态形式变换，因此，重标定 MCF 的参数是在某个选定的曲线圆柱体上，而不是在标准的直线圆柱体上。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.