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

TL;DR

This paper establishes a finiteness result for the diffeomorphism types of manifolds with lower Ricci bounds, bounded volume, diameter, and energy, extending previous theorems by removing the upper Ricci bound.

Contribution

It proves a finite diffeomorphism theorem for manifolds with only lower Ricci bounds and bounded energy, generalizing Anderson-Cheeger's result by removing the upper Ricci curvature constraint.

Findings

Finiteness of diffeomorphism types under specified geometric bounds.

Extension of previous results to manifolds with only lower Ricci bounds.

In Kähler surfaces, scalar curvature bounds suffice for similar finiteness results.

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.