Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy
Abstract
In this paper we prove that the space $\cM(n,\rv,D,\Lambda):=\{(M^n,g) \text{ closed }: ~~\Ric\ge -(n-1),~\Vol(M)\ge \rv>0, \diam(M)\le D \text{ and } \int_{M}|\Rm|^{n/2}\le \Lambda\}$ has at most $C(n,\rv,D,\Lambda)$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.07390
- arXiv:
- arXiv:2405.07390
- Bibcode:
- 2024arXiv240507390J
- Keywords:
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- Mathematics - Differential Geometry
- E-Print:
- 18 pages