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

TL;DR

This paper establishes a quantitative rigidity result for spaces with almost maximal volume entropy, extending known geometric stability results to RCD spaces and manifolds with integral Ricci bounds.

Contribution

It proves the quantitative rigidity of almost maximal volume entropy for RCD spaces and manifolds with integral Ricci curvature bounds, generalizing previous results.

Findings

Rigidity results for RCD spaces with negative Ricci bounds

Extension of volume entropy rigidity to integral Ricci curvature bounds

Quantitative stability near hyperbolic space forms

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.