Remark on a geometric inequality for closed hypersurfaces in weighted manifolds
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917050137444352 |
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| author | Rudnik, Adam |
| author_facet | Rudnik, Adam |
| contents | In this paper we consider noncompact smooth metric measure spaces $(M, g,e^{-f}dvol_{g})$ of nonnegative Bakry-Émery Ricci curvature, i.e. $Ric + D^{2}f - \frac{1}{N}df \otimes df \geq 0$, for $0< N \leq \infty$, in order to obtain geometric inequalities for the boundary of a given open and bounded set $Ω\subset M$, with regular boundary $\partial Ω$. Our inequalities are sharp for both the cases $N< \infty$ and $N= \infty$, provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the $N=\infty$ case holds true precisely when $M \setminus Ω$ is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26062 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remark on a geometric inequality for closed hypersurfaces in weighted manifolds Rudnik, Adam Differential Geometry 53C24 (Primary) 53C21 (Secondary) In this paper we consider noncompact smooth metric measure spaces $(M, g,e^{-f}dvol_{g})$ of nonnegative Bakry-Émery Ricci curvature, i.e. $Ric + D^{2}f - \frac{1}{N}df \otimes df \geq 0$, for $0< N \leq \infty$, in order to obtain geometric inequalities for the boundary of a given open and bounded set $Ω\subset M$, with regular boundary $\partial Ω$. Our inequalities are sharp for both the cases $N< \infty$ and $N= \infty$, provided that the underlying ambient space has large weighted volume growth. The rigidity obtained for the $N=\infty$ case holds true precisely when $M \setminus Ω$ is isometric to a twisted product metric and, as such, is a generalization of the Willmore-type inequality for nonnegative Ricci curvature from Agostiniani, Fagagnolo and Mazzieri to the context of weighted manifolds. |
| title | Remark on a geometric inequality for closed hypersurfaces in weighted manifolds |
| topic | Differential Geometry 53C24 (Primary) 53C21 (Secondary) |
| url | https://arxiv.org/abs/2510.26062 |