Almost Maximal Volume Entropy Rigidity for Integral Ricci Curvature in the Non-collapsing Case

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 \(n, d, p>\frac{n}{2}\) $n,d,p>\frac{n}{2}$ , there exist \(\delta (n, d, p), \epsilon (n, d, p)>0\) $\delta (n,d,p),ϵ(n,d,p)>0$ , such that for \(\delta <\delta (n, d, p)\) $\delta <\delta (n,d,p)$ , \(\epsilon <\epsilon (n, d, p)\) $ϵ<ϵ(n,d,p)$ , if a compact n-manifold M satisfies that the integral Ricci curvature has lower bound \(\bar{k}(-1, p)\le \delta \) $\overline{k}(-1,p)\le \delta$ , the diameter \(\operatorname {diam}(M)\le d\) $diam(M)\le d$ and volume entropy \(h(M)\ge n-1-\epsilon \) $h(M)\ge n-1-ϵ$ , then the universal cover of M is Gromov–Hausdorff close to a hyperbolic space form \(\mathbb H^k\) ${\mathbb{H}}^k$ , \(k\le n\) $k\le n$ ; If in addition the volume of M, \(\operatorname {vol}(M)\ge v>0\) $vol(M)\ge v>0$ , then M is diffeomorphic and Gromov–Hausdorff close to a hyperbolic manifold where \(\delta , \epsilon \) $\delta ,ϵ$ also depend on v.