Shape of Alexandrov spaces with positive Ricci curvature

TL;DR

This paper extends a classical geometric stability result from Riemannian manifolds to Alexandrov spaces with positive Ricci curvature, under a small cut points volume assumption, broadening the understanding of geometric convergence.

Contribution

It generalizes Colding's sphere stability theorem to Alexandrov spaces with Ricci curvature bounds, incorporating a new assumption on cut points volume.

Findings

Alexandrov spaces with Ricci ≥ n-1 are close to spheres if volume is near maximal.

The result holds under a small cut points volume condition.

The approach adapts Riemannian techniques to Alexandrov spaces with limited differentiability.

Abstract

Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we generalize a result by Colding that an n dimentional manifold with Ricci curvature greater or equal to n minus 1 and volume close to that of the unit n sphere is close (in the Gromov-Hausdorff distance) to the sphere, from the case of Riemannian manifolds to the case of Alexandrov spaces, with an additional assumption, roughly speaking, that the rough volume of the set of short cut points is small, following the basic idea in the Riemannian case with necessary modifications because of the only almost everywhere second differentiability of distance functions.