Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911940873289728 |
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| author | Bernstein, Jacob |
| author_facet | Bernstein, Jacob |
| contents | We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from below. This bound is independent of codimension and sharp on planes. As an application we verify a conjecture of Colding-Minicozzi about the entropy of closed self-shrinkers of arbitrary codimension for self-shrinkers that are topologically two-dimensional real projective planes. As part of the proof we introduce two auxiliary functionals which we call stable conformal volume and virtual entropy which should be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_02332 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers Bernstein, Jacob Differential Geometry Analysis of PDEs 53E10, 53C18, 53C42 We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from below. This bound is independent of codimension and sharp on planes. As an application we verify a conjecture of Colding-Minicozzi about the entropy of closed self-shrinkers of arbitrary codimension for self-shrinkers that are topologically two-dimensional real projective planes. As part of the proof we introduce two auxiliary functionals which we call stable conformal volume and virtual entropy which should be of independent interest. |
| title | Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers |
| topic | Differential Geometry Analysis of PDEs 53E10, 53C18, 53C42 |
| url | https://arxiv.org/abs/2407.02332 |