Differentiable Rigidity under Ricci curvature lower bound

Abstract

In this article we prove a differentiable rigidity result. Let $(Y,g)$ and $(X,g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n⩾3$) 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 $\epsilon :=\epsilon (n,d)>0$ 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.

33 pages, 1 dessin

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