Constancy of the Dimension for RCD(<i>K</i>,<i>N</i>) 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.