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

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

**Zoom link**: **https://princeton.zoom.us/j/594605776**

It is known by a result of Colding-Naber that for any two points in a Ricci limit space, there exists a minimizing geodesic where the geometry of small balls centred along the interior of the geodesic change in at most a Hölder continuous manner. This was shown using an extrinsic argument and had several key applications for the structure theory of Ricci limits. In this talk, I will discuss how to overcome the use of smooth structure in the Colding-Naber argument in order to generalize this result to the setting of metric measure spaces satisfying the synthetic lower Ricci curvature bound condition RCD(K,N). As an application, I will show that all RCD(K,N) spaces are non-branching, a result which was previously unknown for Ricci limit spaces.