Mean curvature flow from conical singularities

• 成果类型: Article
• 署名作者: Chodosh, Otis; Daniels-Holgate, J. M.; Schulze, Felix
• 署名单位: Stanford University; Hebrew University of Jerusalem; University of Warwick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01296-8
• 发表日期: 2024
• 页码: 1041-1066
• 关键词: uniqueness; EXISTENCE; SURFACES; BEHAVIOR

摘要

We prove Ilmanen's resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface Mn subset of Rn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M<^>{n}\subset \mathbb{R}<^>{n+1}$\end{document}, 2 <= n <= 6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\leq n\leq 6$\end{document}, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.