Area, Volume and Capacity in non--compact $3$--manifolds with non--negative scalar curvature
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| Format: | Preprint |
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2025
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| _version_ | 1866909567128961024 |
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| author | Oronzio, Francesca |
| author_facet | Oronzio, Francesca |
| contents | Let $(M,g)$ be a $3$--dimensional, complete, one--ended Riemannian manifold, with a minimal, compact and connected boundary. We assume that $M$ has a simple topology and that the scalar curvature of $(M,g)$ is non--negative. Moreover, we suppose that $(M,g)$ admits a $2$--capacitary potential $v$ with $v,\,\vert \nabla v\vert\to 0$ at infinity. In this note, we provide a gradient integral estimate for the level sets of the function $u=1-v$. This estimate leads to a sharp volume comparison for the sub--level sets of $u$, and a sharp area comparison of the level sets of $u$. From this last comparison it follows a sharp area--capacity inequality, originally derived by Bray and Miao, thereby extending its cases of validity. This work is based on a recent paper by Colding and Minicozzi. Finally, for completeness, we also show the same type of area and volume comparison, in the case where $(M,g)$ has no boundary, replacing the function $u$ with one related to the minimal positive Green's function. This volume comparison leads to a more geometric proof of the positive mass inequality than the one given in \cite{Ago_Maz_Oro}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04093 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Area, Volume and Capacity in non--compact $3$--manifolds with non--negative scalar curvature Oronzio, Francesca Differential Geometry Let $(M,g)$ be a $3$--dimensional, complete, one--ended Riemannian manifold, with a minimal, compact and connected boundary. We assume that $M$ has a simple topology and that the scalar curvature of $(M,g)$ is non--negative. Moreover, we suppose that $(M,g)$ admits a $2$--capacitary potential $v$ with $v,\,\vert \nabla v\vert\to 0$ at infinity. In this note, we provide a gradient integral estimate for the level sets of the function $u=1-v$. This estimate leads to a sharp volume comparison for the sub--level sets of $u$, and a sharp area comparison of the level sets of $u$. From this last comparison it follows a sharp area--capacity inequality, originally derived by Bray and Miao, thereby extending its cases of validity. This work is based on a recent paper by Colding and Minicozzi. Finally, for completeness, we also show the same type of area and volume comparison, in the case where $(M,g)$ has no boundary, replacing the function $u$ with one related to the minimal positive Green's function. This volume comparison leads to a more geometric proof of the positive mass inequality than the one given in \cite{Ago_Maz_Oro}. |
| title | Area, Volume and Capacity in non--compact $3$--manifolds with non--negative scalar curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.04093 |