The Logarithmic Sobolev inequality on non-compact self-shrinkers
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916442937491456 |
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| author | Wang, Guofang Xia, Chao Zhang, Xiqiang |
| author_facet | Wang, Guofang Xia, Chao Zhang, Xiqiang |
| contents | In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers. We first provide a proof for the logarithmic Sobolev inequality in the Euclidean space by using the Alexandrov-Bakelman-Pucci (ABP) method. Then we use this approach to show an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which is a sharp version of the result of Ecker in \cite{Ecker}. The proof is a noncompact modification of Brendle's proof for closed submanifolds and has a big potential to provide new inequalities in noncompact manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13601 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Logarithmic Sobolev inequality on non-compact self-shrinkers Wang, Guofang Xia, Chao Zhang, Xiqiang Analysis of PDEs Differential Geometry In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers. We first provide a proof for the logarithmic Sobolev inequality in the Euclidean space by using the Alexandrov-Bakelman-Pucci (ABP) method. Then we use this approach to show an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which is a sharp version of the result of Ecker in \cite{Ecker}. The proof is a noncompact modification of Brendle's proof for closed submanifolds and has a big potential to provide new inequalities in noncompact manifolds. |
| title | The Logarithmic Sobolev inequality on non-compact self-shrinkers |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2410.13601 |