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 $n, d, p>\frac{n}{2}$, there exist $δ(n, d, p), ε(n, d, p)>0$, such that for $δ<δ(n, d, p)$, $ε<ε(n, d, p)$, if a compact $n$-manifold $M$ satisfies that the integral Ricci curvature has lower bound $\bar k(-1, p)\leq δ$, the diameter $diam(M)\leq d$ and volume entropy $h(M)\geq n-1-ε$, 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 $δ, ε$ also depends on $v$.

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