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

Qin Deng

TL;DR

This work proves that $RCD(K,N)$ spaces are non-branching and that tangent cones along the same geodesic depend Hölder continuously on the geodesic parameter in the pointed Gromov-Hausdorff sense. The authors adapt the CN12 strategy to the metric-measure setting by developing a second-order interpolation formula for the distance along Regular Lagrangian flows and by using heat-flow approximations to distance/excess functions, establishing robust control without relying on smooth Hessians. As a consequence, the regular set has $m$-a.e. constant dimension, is $m$-a.e. convex, and the top-dimensional regular set is weakly convex and connected; non-branching follows as a corollary. Overall, the paper extends Ricci-limit techniques to the $RCD(K,N)$ setting, providing a powerful framework for understanding geometric and measure-theoretic rigidity in non-smooth spaces and informing the structure theory of Ricci-type spaces. The methods combine optimal transport, heat-flow analysis, and Regular Lagrangian flow theory to yield quantitative Hölder control of geometry along geodesics and global non-branching consequences in metric measure geometry.

Abstract

In this paper we prove that a metric measure space $(X,d,m)$ satisfying the finite Riemannian curvature-dimension condition ${\sf RCD}(K,N)$ is non-branching 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 centred 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-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 ${\sf RCD}(K,N)$ space has $m$-a.e. constant dimension, a result already established by Bruè-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 ${\sf RCD}(K,N)$ setting the expected second order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative.

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

TL;DR

This work proves that spaces are non-branching and that tangent cones along the same geodesic depend Hölder continuously on the geodesic parameter in the pointed Gromov-Hausdorff sense. The authors adapt the CN12 strategy to the metric-measure setting by developing a second-order interpolation formula for the distance along Regular Lagrangian flows and by using heat-flow approximations to distance/excess functions, establishing robust control without relying on smooth Hessians. As a consequence, the regular set has -a.e. constant dimension, is -a.e. convex, and the top-dimensional regular set is weakly convex and connected; non-branching follows as a corollary. Overall, the paper extends Ricci-limit techniques to the setting, providing a powerful framework for understanding geometric and measure-theoretic rigidity in non-smooth spaces and informing the structure theory of Ricci-type spaces. The methods combine optimal transport, heat-flow analysis, and Regular Lagrangian flow theory to yield quantitative Hölder control of geometry along geodesics and global non-branching consequences in metric measure geometry.

Abstract

In this paper we prove that a metric measure space satisfying the finite Riemannian curvature-dimension condition is non-branching and that tangent cones from the same sequence of rescalings are Hölder continuous along the interior of every geodesic in . More precisely, we show that the geometry of balls of small radius centred 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-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 space has -a.e. constant dimension, a result already established by Bruè-Semola, and is -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 setting the expected second order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative.

Paper Structure

This paper contains 26 sections, 64 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. Outline of Paper and Proof
  3. Acknowledgements
  4. Preliminaries
  5. Curvature-dimension condition preliminaries
  6. First order calculus on metric measure spaces
  7. Tangent, cotangent, and tensor modules
  8. RCD(K,N) and Bakry-Émery conditions
  9. Heat flow and Bakry-Ledoux estimates
  10. Second order calculus and improved Bochner inequality
  11. Non-branching and essentially non-branching spaces
  12. RCD(K,N) structure theory
  13. Additional RCD(K,N) theory
  14. Mean value and integral excess inequalities
  15. Differentiation formulas for Regular Lagrangian flows
  16. ...and 11 more sections

Key Result

Theorem 1.1

(Hölder continuity of geometry of small balls with same radius) Let $(X,d,m)$ be an $\mathop{\mathrm{\sf RCD}}\nolimits(K,N)$ space for some $K \in \mathbb{R}$ and $N \in (1, \infty)$. Let $p, q \in X$ and $d(p,q)=\ell$. Define $K' = (\frac{K}{-(N-1)} \vee 1)^{1 \slash 2}$ and $\ell' = \ell \wedge 1

Theorems & Definitions (135)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 125 more