Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy

Wenshuai Jiang; Guofang Wei

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arXiv · 2405.07390

Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy

Wenshuai Jiang·Guofang Wei

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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 K\"ahler surface, the Riemann curvature $L^2$ bound could be replaced by the scalar curvature $L^2$ bound.

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BibTeX RIS

Wenshuai Jiang, Guofang Wei. 2024-05-12. Finite Diffeomorphism Theorem for manifolds with lower Ricci curvature and bounded energy. https://arxiv.org/abs/2405.07390

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