Logarithmic-Sobolev inequalities on non-compact Euclidean submanifolds: sharpness and rigidity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Balogh, Zoltán M. Kristály, Alexandru |
| author_facet | Balogh, Zoltán M. Kristály, Alexandru |
| contents | The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $Σ$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for $p=2$, is sharp, it is valid on general submanifolds, and it involves the mean curvature of $Σ$. It implies in particular the main result of S. Brendle [Comm. Pure Appl. Math.}, 2022]. In addition, it turns out that equality can only occur if and only if $Σ$ is isometric to the Euclidean space $\mathbb R^{n}$ and the extremizer is a Gaussian. The second result is a general $L^p$-logarithmic-Sobolev inequality for $p\geq 2$ on Euclidean submanifolds with constants that are codimension-free in case of minimal submanifolds. In order to prove the above results - especially, to deal with the equality cases - we elaborate the theory of optimal mass transport on submanifolds between measures that are not necessarily compactly supported. Applications are provided to sharp hypercontractivity estimates of Hopf-Lax semigroups on submanifolds. The first hypercontractivity estimate is for general submanifolds with bounded mean curvature vector, the second one is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09419 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Logarithmic-Sobolev inequalities on non-compact Euclidean submanifolds: sharpness and rigidity Balogh, Zoltán M. Kristály, Alexandru Differential Geometry Analysis of PDEs The paper is devoted to provide Michael-Simon-type $L^p$-logarithmic-Sobolev inequalities on complete, not necessarily compact $n$-dimensional submanifolds $Σ$ of the Euclidean space $\mathbb R^{n+m}$. Our first result, stated for $p=2$, is sharp, it is valid on general submanifolds, and it involves the mean curvature of $Σ$. It implies in particular the main result of S. Brendle [Comm. Pure Appl. Math.}, 2022]. In addition, it turns out that equality can only occur if and only if $Σ$ is isometric to the Euclidean space $\mathbb R^{n}$ and the extremizer is a Gaussian. The second result is a general $L^p$-logarithmic-Sobolev inequality for $p\geq 2$ on Euclidean submanifolds with constants that are codimension-free in case of minimal submanifolds. In order to prove the above results - especially, to deal with the equality cases - we elaborate the theory of optimal mass transport on submanifolds between measures that are not necessarily compactly supported. Applications are provided to sharp hypercontractivity estimates of Hopf-Lax semigroups on submanifolds. The first hypercontractivity estimate is for general submanifolds with bounded mean curvature vector, the second one is for self-similar shrinkers endowed with the natural Gaussian measure. The equality cases are characterized here as well. |
| title | Logarithmic-Sobolev inequalities on non-compact Euclidean submanifolds: sharpness and rigidity |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2410.09419 |