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

Abstract

In this paper we prove that a metric measure space $(X,d,m)$ satisfying the finite Riemannian curvature-dimension condition $\mathsf{RCD}(K,N)$ 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 $\mathsf{RCD}(K,N)$ 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 $\mathsf{RCD}(K,N)$ setting the expected second order interpolation formula for the distance function along the Regular Lagrangian flow of some vector field using its covariant derivative.

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