Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910444149538816 |
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| author | Jiang, Wenshuai Wei, Guofang |
| author_facet | Jiang, Wenshuai Wei, Guofang |
| contents | In this paper we prove that the space $\cM(n,\rv,D,Λ):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv>0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le Λ\}$ has at most $C(n,\rv,D,Λ)$ many diffeomorphism types. This removes the upper Ricci curvature bound of Anderson-Cheeger's finite diffeomorphism theorem in \cite{AnCh}. Furthermore, if $M$ is Kähler surface, the Riemann curvature $L^2$ bound could be replaced by the scalar curvature $L^2$ bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy Jiang, Wenshuai Wei, Guofang Differential Geometry In this paper we prove that the space $\cM(n,\rv,D,Λ):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv>0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le Λ\}$ has at most $C(n,\rv,D,Λ)$ many diffeomorphism types. This removes the upper Ricci curvature bound of Anderson-Cheeger's finite diffeomorphism theorem in \cite{AnCh}. Furthermore, if $M$ is Kähler surface, the Riemann curvature $L^2$ bound could be replaced by the scalar curvature $L^2$ bound. |
| title | Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2405.07390 |