The singular set of mean curvature flow with generic singularities
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222_2015_Article_617.pdf
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Author(s)
Colding, Tobias
Minicozzi, William
Date Issued
September 2015
Journal
Inventiones mathematicae
Publisher
Springer Berlin Heidelberg
Citation
Colding, Tobias Holck, and William P. Minicozzi. “The Singular Set of Mean Curvature Flow with Generic Singularities.” Inventiones mathematicae 204.2 (2016): 443–471.
Version
Author's final manuscript
Abstract
A mean curvature flow starting from a closed embedded hypersurface in R[superscript n+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 R³ and R[superscript 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00222-015-0617-5
