Improved regularity estimates for Lagrangian flows on $\text{RCD}(K,N)$ spaces

Abstract

This paper gives a contribution to the study of regularity of Lagrangian flows on non-smooth spaces with lower Ricci curvature bounds. The main novelties with respect to the existing literature are the better behaviour with respect to time and the local nature of the regularity estimates. These are obtained by sharpening previous results of the first and third authors, in combination with some tools recently developed by the second author (adapting to the synthetic framework ideas introduced in [Colding-Naber 12]. The estimates are suitable for applications to the fine study of $\text{RCD}$ spaces and play a central role in the construction of a parallel transport in this setting.

References in corpus (6)

• 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
• Structure of fundamental groups of manifolds with Ricci curvature bounded below
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below

Cited by in corpus (1)

• Parallel transport on non-collapsed $\mathsf{RCD}(K,N)$ spaces