Quantitative rigidity of almost maximal volume entropy for both RCD spaces and integral Ricci curvature bound

Abstract: The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian $n$-manifold with a negative lower Ricci curvature bound and a upper diameter bound, it was known that it admits an almost maximal volume entropy if and only if it is diffeomorphic and Gromov-Hausdorff close to a hyperbolic space form. We prove the quantitative rigidity of almost maximal volume entropy for $\operatorname{RCD}$-spaces with a negative lower Ricci curvature bound and Riemannian manifolds with a negative $L^p$-integral Ricci curvature lower bound.