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

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 $\bar{k}(-1,p)\le \delta$, the diameter $diam(M)\le d$ and volume entropy $h(M)\ge n-1-\epsilon$, then the universal cover of $M$ is Gromov-Hausdorff close to a hyperbolic space form ${\mathbb{H}}^k$, $k\le n$; If in addition the volume of $M$, $vol(M)\ge v>0$, then $M$ is diffeomorphic and Gromov-Hausdorff close to a hyperbolic manifold where $\delta ,\epsilon$ also depends on $v$.

7 pages