Calculus of Variations and Geometric Measure Theory

E. Bruè - D. Semola

Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows

created by bruè on 19 Apr 2018
modified by semola on 30 May 2020


Published Paper

Inserted: 19 apr 2018
Last Updated: 30 may 2020

Journal: Comm. Pure and Applied Math.
Volume: 73
Pages: 1141-1204
Year: 2020


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 abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.