Calculus of Variations and Geometric Measure Theory
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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 on 11 Sep 2019


Accepted Paper

Inserted: 19 apr 2018
Last Updated: 11 sep 2019

Journal: Comm. Pure and Applied Math.
Pages: 44
Year: 2018


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.


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