Hölder continuity of tangent cones in RCD(K,N) spaces and applications to nonbranching

Deng, Qin

doi:10.2140/gt.2025.29.1037

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Hölder continuity of tangent cones in $\mathrm{RCD(K,N)}$ spaces and applications to nonbranching

Qin Deng

Geometry & Topology 29 (2025) 1037–1114

DOI: 10.2140/gt.2025.29.1037

Abstract

We study the structure theory of metric measure spaces $(X, d, m)$ satisfying the synthetic lower Ricci curvature bound condition $RCD (K, N)$ . We prove that such a space is nonbranching and that tangent cones from the same sequence of rescalings are Hölder continuous along the interior of every geodesic in $X$ . More precisely, we show that the geometry of balls of small radius centered in the interior of any geodesic changes in at most a Hölder continuous way along the geodesic in pointed Gromov–Hausdorff distance. This improves a result in the Ricci limit setting by Colding and Naber where the existence of at least one geodesic with such properties between any two points is shown. As in the Ricci limit case, this implies that the regular set of an $RCD (K, N)$ space has $m$ -a.e. constant dimension, a result recently established by Brué and Semola, and is $m$ -a.e convex. It also implies that the top dimension regular set is weakly convex, and therefore connected. In proving the main theorems, we develop in the $RCD (K, N)$ setting the expected second-order interpolation formula for the distance function along the regular Lagrangian flow of some vector field using its covariant derivative.

Keywords

Ricci curvature, RCD spaces, geodesic, nonbranching

Mathematical Subject Classification

Primary: 51K10, 53B20

References

Publication

Received: 11 August 2023

Revised: 14 February 2024

Accepted: 16 March 2024

Published: 21 April 2025

Proposed: Tobias H Colding

Seconded: David Fisher, Gang Tian

Authors

Qin Deng
University of Toronto Toronto, ON Canala	MIT Cambridge, MA United States

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Volume 29, issue 2 (2025)