Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929425499553792 |
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| author | Ren, Tianyin Rong, Xiaochun |
| author_facet | Ren, Tianyin Rong, Xiaochun |
| contents | In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete $n$-manifold $M$ of Ricci curvature, $\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $π$, then $M$ is isometric to the unit sphere $S^n_1$. The main result in this paper is a quantitative maximal diameter rigidity: if $M$ satisfies that $\operatorname{Ric}_M\ge n-1$, $\operatorname{diam}(M)\approx π$, and the Riemannian universal cover of every metric ball in $M$ of a definite radius satisfies a Riefenberg condition, then $M$ is diffeomorphic and bi-Hölder close to $S^n_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01592 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature Ren, Tianyin Rong, Xiaochun Differential Geometry In Riemannian geometry, the Cheng's maximal diameter rigidity theorem says that if a complete $n$-manifold $M$ of Ricci curvature, $\operatorname{Ric}_M\ge (n-1)$, has the maximal diameter $π$, then $M$ is isometric to the unit sphere $S^n_1$. The main result in this paper is a quantitative maximal diameter rigidity: if $M$ satisfies that $\operatorname{Ric}_M\ge n-1$, $\operatorname{diam}(M)\approx π$, and the Riemannian universal cover of every metric ball in $M$ of a definite radius satisfies a Riefenberg condition, then $M$ is diffeomorphic and bi-Hölder close to $S^n_1$. |
| title | Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2308.01592 |