Cheeger constant rigidity for cocompact negatively curved manifolds

Abstract: Let $(M<sup>m,g)$, $m\geq2$, be a closed connected Riemannian manifold with $\secg_g\leq-1$, and let $(X,g_X)$ be its universal cover. Yau proved that $\hiso(X)\geq m-1$. We prove that equality is rigid: if $\hiso(X)=m-1$, then $(X,g_X)$ is isometric to $\Hh<sup>m(-1)$.