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

2019/06/30 by Elia Brué, Daniele Semola · 94 citations
Mathematics · #Computer science #Constant (computer programming) #Dimension (graph theory) #Function (biology) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Lagrangian #Laplace operator #Limit (mathematics) #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Metric (unit) #Metric space #Nonlinear Partial Differential Equations #Pure mathematics #Sobolev space

paper · open access · doi:10.1002/cpa.21849

published in Communications on Pure and Applied Mathematics 73(6), 1141-1204 (Wiley)

openalex publication_date 2019/06/30 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/27

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.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

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