2-regular points in Ricci limit spaces

TL;DR

The paper addresses the structure of Ricci limit spaces in the collapsing regime by focusing on $2$-regular points. It shows that if a $2$-regular point lies in the interior of a geodesic, then the entire interior is $2$-regular, leading to $X$ being $2$-rectifiable and $\mathcal{R}=\,\mathcal{R}_2$. The proof leverages a geometric stability argument: intersecting minimal geodesics across a regular geodesic produce a $(2,\epsilon)$-strainer, and sharp Hölder continuity of tangent cones propagates regularity to the full interior. The results clarify the local-to-global regularity structure for collapsing Ricci limit spaces and contrast the behavior of $2$-regular points with higher regularities, contributing to the broader understanding of rectifiability and tangential structure in $\text{RCD}$ and related spaces.

Abstract

In this note, we will show that if a measured Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with lower Ricci curvature bound contains a 2-regular point which lies in the interior of a geodesic, then it is 2-rectifiable. And we will also give some properties about 2-regular points.

Theorems & Definitions (21)

• Theorem 1.2
• Proposition 1.3
• Theorem 1.5
• Remark 1.6
• Theorem 2.1: Splitting theorem CC
• Theorem 2.2: Deng
• Theorem 2.3: Sharp Hölder continuity CN
• Theorem 2.4: CN
• Theorem 2.5: KL
• Lemma 2.6