Euclidean Domains with Nearly Maximal Yamabe Quotient
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910792441397248 |
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| author | Mazurowski, Liam Yao, Xuan |
| author_facet | Mazurowski, Liam Yao, Xuan |
| contents | Let $Ω$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $Ω$ is at most the Yamabe quotient of a ball, and equality holds if and only if $Ω$ is a ball. We show that if equality almost holds then the following things are true:
(i)$Ω$ is diffeomorphic to a ball;
(ii) There is a small number $ε> 0$ such that $B(x,r) \subset Ω\subset B(x,r(1+ε))$; (iii) After suitable scaling, $Ω$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its induced length metric.
We also give a qualitative comparison between $Q$ and the coefficient of quasi-conformality studied in the theory of quasi-conformal maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Euclidean Domains with Nearly Maximal Yamabe Quotient Mazurowski, Liam Yao, Xuan Differential Geometry Analysis of PDEs Let $Ω$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $Ω$ is at most the Yamabe quotient of a ball, and equality holds if and only if $Ω$ is a ball. We show that if equality almost holds then the following things are true: (i)$Ω$ is diffeomorphic to a ball; (ii) There is a small number $ε> 0$ such that $B(x,r) \subset Ω\subset B(x,r(1+ε))$; (iii) After suitable scaling, $Ω$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its induced length metric. We also give a qualitative comparison between $Q$ and the coefficient of quasi-conformality studied in the theory of quasi-conformal maps. |
| title | Euclidean Domains with Nearly Maximal Yamabe Quotient |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2501.12347 |