Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Abstract

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N) metric measure spaces; regularity is understood with respect to a newly defined quasi‐metric built from the Green function of the Laplacian. Its main application is that RCD(K, N) spaces have constant dimension. In this way we generalize to such an abstract framework a result proved by Colding‐Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting. © 2019 Wiley Periodicals, Inc.