Mass, Conformal Capacity, and the Volumetric Penrose Inequality
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| Format: | Preprint |
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2024
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| _version_ | 1866916435542933504 |
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| author | Mazurowski, Liam Yao, Xuan |
| author_facet | Mazurowski, Liam Yao, Xuan |
| contents | Let $Ω$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus Ω$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $Σ= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $Ω$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $Ω$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09626 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mass, Conformal Capacity, and the Volumetric Penrose Inequality Mazurowski, Liam Yao, Xuan Differential Geometry Mathematical Physics Classical Analysis and ODEs Let $Ω$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus Ω$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $Σ= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $Ω$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $Ω$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function. |
| title | Mass, Conformal Capacity, and the Volumetric Penrose Inequality |
| topic | Differential Geometry Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2410.09626 |