On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

TL;DR

This paper demonstrates that in limit spaces with lower Ricci curvature bounds, tangent cones along a geodesic vary Hölder continuously and maintain consistent dimension, revealing stability properties of the geometric structure.

Contribution

It establishes Hölder continuity of tangent cones along geodesics in Ricci limit spaces and confirms their dimension stability in the sense of Colding-Naber.

Findings

Tangent cones are Hölder continuous along geodesics.

Tangent cones have consistent dimension along the geodesic.

The results apply to limit spaces of Ricci curvature bounded below.

Abstract

We show that if $X$ is a limit of $n$-dimensional Riemannian manifolds with Ricci curvature bounded below and $\gamma$ is a limit geodesic in $X$ then along the interior of $\gamma$ same scale measure metric tangent cones $T_{\gamma(t)}X$ are H\"older continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.