The singular set of mean curvature flow with generic singularities

A mean curvature flow starting from a closed embedded hypersurface in R 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...

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Bibliographic Details
Published in:Inventiones mathematicae 2016-05, Vol.204 (2), p.443-471
Main Authors: Colding, Tobias Holck, Minicozzi, William P.
Format: Article
Language:English
Subjects:
Curvature
Evolution
Geometry
Hyperspaces
Manifolds (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Optimization
Singularities
Singularity (mathematics)
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Summary:A mean curvature flow starting from a closed embedded hypersurface in R 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 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.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-015-0617-5