Almost maximal volume entropy rigidity for integral Ricci curvature in the non-collapsing case

TL;DR

This paper proves that manifolds with almost maximal volume entropy and lower integral Ricci curvature bounds are close to hyperbolic spaces or manifolds, extending rigidity results to the integral Ricci curvature setting in the non-collapsing case.

Contribution

It establishes almost maximal volume entropy rigidity for manifolds with lower integral Ricci curvature bounds, generalizing classical results to the integral curvature context.

Findings

Universal cover is Gromov-Hausdorff close to hyperbolic space form.

Manifolds are diffeomorphic and Gromov-Hausdorff close to hyperbolic manifolds under volume bounds.

Results depend on integral Ricci curvature bounds, diameter, and volume entropy.

Abstract

In this note we will show the almost maximal volume entropy rigidity for manifolds with lower integral Ricci curvature bound in the non-collapsing case: Given $n, d, p>\frac{n}{2}$, there exist $\delta(n, d, p), \epsilon(n, d, p)>0$, such that for $\delta<\delta(n, d, p)$, $\epsilon<\epsilon(n, d, p)$, if a compact $n$-manifold $M$ satisfies that the integral Ricci curvature has lower bound $\bar k(-1, p)\leq \delta$, the diameter $diam(M)\leq d$ and volume entropy $h(M)\geq n-1-\epsilon$, then the universal cover of $M$ is Gromov-Hausdorff close to a hyperbolic space form $\Bbb H^k$, $k\leq n$; If in addition the volume of $M$, $vol(M)\geq v>0$, then $M$ is diffeomorphic and Gromov-Hausdorff close to a hyperbolic manifold where $\delta, \epsilon$ also depends on $v$.