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...
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| Format: | Article |
| Language: | English |
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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 |
| _version_ | 1811071294777589760 |
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| author | Colding, Tobias Minicozzi, William |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William |
| author_sort | Colding, Tobias |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T08:48:55Z |
| format | Article |
| id | mit-1721.1/104371 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:48:55Z |
| publishDate | 2016 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1043712022-09-23T14:45:41Z The singular set of mean curvature flow with generic singularities Colding, Tobias Minicozzi, William Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William 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. National Science Foundation (U.S.). (Grants DMS 1404540, DMS 11040934, DMS 1206827) National Science Foundation (U.S.). Focused Research Group (Grants DMS 0854774 and DMS 0853501) 2016-09-22T18:49:50Z 2016-09-22T18:49:50Z 2015-09 2013-02 2016-08-18T15:24:20Z Article http://purl.org/eprint/type/JournalArticle 0020-9910 1432-1297 http://hdl.handle.net/1721.1/104371 Colding, Tobias Holck, and William P. Minicozzi. “The Singular Set of Mean Curvature Flow with Generic Singularities.” Inventiones mathematicae 204.2 (2016): 443–471. https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 en http://dx.doi.org/10.1007/s00222-015-0617-5 Inventiones mathematicae Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Colding, Tobias Minicozzi, William The singular set of mean curvature flow with generic singularities |
| title | The singular set of mean curvature flow with generic singularities |
| title_full | The singular set of mean curvature flow with generic singularities |
| title_fullStr | The singular set of mean curvature flow with generic singularities |
| title_full_unstemmed | The singular set of mean curvature flow with generic singularities |
| title_short | The singular set of mean curvature flow with generic singularities |
| title_sort | singular set of mean curvature flow with generic singularities |
| url | http://hdl.handle.net/1721.1/104371 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
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