A Remark on Regular Points of Ricci Limit Spaces

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 $1\leq \mathrm{dim}_H(Y)<2$.