The singular set of mean curvature flow with generic singularities
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 pl...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2016
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| Online Access: | http://hdl.handle.net/1721.1/104371 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
| Summary: | 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. |
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