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 > n/2, there exist delta(n, d, p),is an element of(n, d, p)>0, such that for delta < delta(n, d, p), is an element of < is an element of(n, d, p), if a compact n-manifold M satisfies that the integral Ricci curvature has lower bound k(-1,p) <= delta, the diameter diam(M) <= d and volume entropy h(M) >= n-1-is an element of, then the universal cover of M is Gromov-Hausdorff close to a hyperbolic space form H-k, k <= n; If in addition the volume of M, vol(M) >= v > 0, then M is diffeomorphic and Gromov-Hausdorff close to a hyperbolic manifold where delta,is an element of also depend on nu.