A Remark on Regular Points of Ricci Limit Spaces

TL;DR

This paper investigates the structure of Ricci limit spaces, showing that if such a space contains a regular point of dimension one, then it must be a one-dimensional topological manifold, refining previous results on the dimensionality of these spaces.

Contribution

The paper proves that the existence of a 1-regular point in a Ricci limit space implies the space is a one-dimensional topological manifold, improving prior results on the space's dimensionality.

Findings

If $\\mathcal{R}_1 \neq \emptyset$, then $Y$ is a 1-dimensional topological manifold.

The result extends Handa's theorem to broader conditions.

Provides insight into the structure of Ricci limit spaces with regular points.

Abstract

Let $Y$ be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in $Y$ is called $k$-regular, if its tangent is unique and is isometric to an $k$-dimensional Euclidean space. By \cite{B5}, there is $k>0$ such that the set of all $k$-regular point $\mathcal{R}_k$ has a full renormalized measure. An open problem is if $\mathcal{R}_l=\emptyset$ for all $l<k$? The main result in this paper asserts that if $\mathcal{R}_1\ne \emptyset$, then $Y$ is a one dimensional topological manifold. Our result improves the Handa's result \cite{Honda} that under the assumption that $1\leq \mathrm{dim}_H(Y)<2$.