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

References in corpus (8)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Semiconcave functions in Alexandrov's geometry
• Well posedness of Lagrangian flows and continuity equations in metric measure spaces
• Independence on $p$ of weak upper gradients on RCD spaces
• Bakry-Émery conditions on almost smooth metric measure spaces

Cited by in corpus (2)

• On the structure of RCD spaces with upper curvature bounds
• Bakry-Émery conditions on almost smooth metric measure spaces