The singular set of mean curvature flow with generic singularities

• 成果类型: Article
• 署名作者: Colding, Tobias Holck; Minicozzi, William P., II
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0617-5
• 发表日期: 2016
• 页码: 443-471
• 关键词: minimal-surfaces; level sets; uniqueness; rectifiability; motion

摘要

A mean curvature flow starting from a closed embedded hyper-surface in Rn+1 must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded (n - 1)-dimensional Lipschitz submanifolds plus a set of dimension at most n-2. If the initial hypersurface is mean convex, then all singularities are generic and the results apply. In R-3 and R-4, we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time-slice. For 2 or 3-convex hypersurfaces in all dimensions, the same arguments lead to the same conclusion: the flow is completely smooth at almost all times and connected components of the singular set are contained in time-slices. A key technical point is a strong parabolic Reifenberg property that we show in all dimensions and for all flows with only generic singularities. We also show that the entire flow clears out very rapidly after a generic singularity. These results are essentially optimal.