Differentiable Rigidity under Ricci curvature lower bound

TL;DR

This paper establishes a differentiable rigidity result for closed Riemannian manifolds with Ricci curvature bounded below, showing that near-volume hyperbolic manifolds are diffeomorphic under certain conditions.

Contribution

It proves a new differentiable rigidity theorem linking Ricci curvature bounds, volume closeness, and diffeomorphism of manifolds with hyperbolic metrics.

Findings

Manifolds with Ricci curvature ≥ -n(n-1) and nearly hyperbolic volume are diffeomorphic.

The proof uses Cheeger-Colding theory on limits of Riemannian manifolds.

A quantitative epsilon depends on dimension and diameter.

Abstract

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of $(X,g_0)$. We show that there exists a number $\varepsilon:=\varepsilon (n, d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by $-n(n-1)$ and its volume satisfies $\vol_g (Y)\leqslant (1+\varepsilon) \vol_{g_0} (X)$ then the manifolds are diffeomorphic. The proof relies on Cheeger-Colding's theory of limits of Riemannian manifolds under lower Ricci curvature bound.