A note on the topological stability theorem from RCD spaces to Riemannian manifolds

Inspired by a recent work of Wang-Zhao in [ 72 ], in this note we prove that for a fixed n -dimensional closed Riemannian manifold ( M n , g ) , if an RCD ( K , n ) space ( X , d , m ) is Gromov-Hausdorff close to M n , then there exists a regular homeomorphism F from X to M n such that F is Lipschitz continuous and that F - 1 is Hölder continuous, where the Lipschitz constant of F , the Hölder exponent and the Hölder constant of F - 1 can be chosen arbitrary close to 1. This is sharp in the sense that in general such a map cannot be improved to being bi-Lipschitz. Moreover if X is smooth, then such a homeomorphism can be chosen as a diffeomorphism. It is worth mentioning that the Lipschitz-Hölder continuity of F improves the intrinsic Reifenberg theorem for closed manifolds with Ricci curvature bounded below established by Cheeger-Colding. The Nash embedding theorem plays a key role in the proof.