Mean curvature flow with generic initial data

• 成果类型: Article
• 署名作者: Chodosh, Otis; Choi, Kyeongsu; Mantoulidis, Christos; Schulze, Felix
• 署名单位: Stanford University; Korea Institute for Advanced Study (KIAS); Rice University; University of Warwick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01258-0
• 发表日期: 2024
• 页码: 121-220
• 关键词: convex ancient solutions; strong maximum principle; closed hypersurfaces; ricci flow; compact solutions; self-shrinkers; SINGULAR SET; level sets; uniqueness; entropy

摘要

We show that the mean curvature flow of generic closed surfaces in R 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}<^>{3}$\end{document} avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}<^>{4}$\end{document} is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.