The Round Sphere Minimizes Entropy among Closed Self-Shrinkers
The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Th...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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International Press of Boston, Inc.
2015
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| Online Access: | http://hdl.handle.net/1721.1/93156 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
| _version_ | 1826202143172853760 |
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| author | Colding, Tobias Minicozzi, William White, Brian Ilmanen, Tom |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William White, Brian Ilmanen, Tom |
| author_sort | Colding, Tobias |
| collection | MIT |
| description | The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest. |
| first_indexed | 2024-09-23T12:02:33Z |
| format | Article |
| id | mit-1721.1/93156 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:02:33Z |
| publishDate | 2015 |
| publisher | International Press of Boston, Inc. |
| record_format | dspace |
| spelling | mit-1721.1/931562022-10-01T07:46:38Z The Round Sphere Minimizes Entropy among Closed Self-Shrinkers Colding, Tobias Minicozzi, William White, Brian Ilmanen, Tom Massachusetts Institute of Technology. Department of Mathematics Colding, Tobias Minicozzi, William The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest. National Science Foundation (U.S.) (Grant DMS 11040934) National Science Foundation (U.S.) (Grant DMS 0906233) National Science Foundation (U.S.). Focused Research Group (Grant DMS 0854774) National Science Foundation (U.S.). Focused Research Group (Grant DMS 0853501) 2015-01-22T20:41:53Z 2015-01-22T20:41:53Z 2013-07 2012-05 Article http://purl.org/eprint/type/JournalArticle 0022-040X 1945-743X http://hdl.handle.net/1721.1/93156 Colding, Tobias Holck, Tom Ilmanen, William P. Minicozzi, and Brian White. "The Round Sphere Minimizes Entropy Among Closed Self-Shrinkers." J. Differential Geom. 95.1 (2013): 53-69. https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 en_US http://projecteuclid.org/euclid.jdg/1375124609 Journal of Differential Geometry Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf International Press of Boston, Inc. arXiv |
| spellingShingle | Colding, Tobias Minicozzi, William White, Brian Ilmanen, Tom The Round Sphere Minimizes Entropy among Closed Self-Shrinkers |
| title | The Round Sphere Minimizes Entropy among Closed Self-Shrinkers |
| title_full | The Round Sphere Minimizes Entropy among Closed Self-Shrinkers |
| title_fullStr | The Round Sphere Minimizes Entropy among Closed Self-Shrinkers |
| title_full_unstemmed | The Round Sphere Minimizes Entropy among Closed Self-Shrinkers |
| title_short | The Round Sphere Minimizes Entropy among Closed Self-Shrinkers |
| title_sort | round sphere minimizes entropy among closed self shrinkers |
| url | http://hdl.handle.net/1721.1/93156 https://orcid.org/0000-0001-6208-384X https://orcid.org/0000-0003-4211-6354 |
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