Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces
Abstract Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient...
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| Format: | Article |
| Language: | English |
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Springer US
2021
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| Online Access: | https://hdl.handle.net/1721.1/136747 |
| _version_ | 1811088998597132288 |
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| author | Sun, Ao |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Sun, Ao |
| author_sort | Sun, Ao |
| collection | MIT |
| description | Abstract
Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. |
| first_indexed | 2024-09-23T14:12:13Z |
| format | Article |
| id | mit-1721.1/136747 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:12:13Z |
| publishDate | 2021 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1367472023-03-15T19:53:09Z Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces Sun, Ao Massachusetts Institute of Technology. Department of Mathematics Abstract Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. 2021-10-29T18:31:29Z 2021-10-29T18:31:29Z 2020-08-11 2021-06-03T03:28:07Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/136747 Sun, Ao. 2020. "Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces." en https://doi.org/10.1007/s12220-020-00494-z Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Mathematica Josephina, Inc. application/pdf Springer US Springer US |
| spellingShingle | Sun, Ao Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces |
| title | Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces |
| title_full | Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces |
| title_fullStr | Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces |
| title_full_unstemmed | Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces |
| title_short | Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces |
| title_sort | entropy in a closed manifold and partial regularity of mean curvature flow limit of surfaces |
| url | https://hdl.handle.net/1721.1/136747 |
| work_keys_str_mv | AT sunao entropyinaclosedmanifoldandpartialregularityofmeancurvatureflowlimitofsurfaces |