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 $\bar k(-1, p)\leq \delta$, 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 $v$.