Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below

Abstract

This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M_i^n,g_i,p_i)\stackrel{d_{GH}}{⟶}(X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $R{c}_{M_i^n}\ge -(n-1)$ as well as the noncollapsing assumption $Vol(B_1(p_i))>v>0$. In such cases, there is a filtration of the singular set, $S_0\subset S_1\cdots S_{n-1}:=S$, where $S^k:= \{x\in X:\text{ no tangent cone at $x$ is }(k+1)\text{-symmetric}\}$; equivalently no tangent cone splits off a Euclidean factor ${\mathbb{R}}^{k+1}$ isometrically. Moreover, by \cite{ChCoI}, $\dim S^k\le k$. However, little else has been understood about the structure of the singular set $S$. Our first result for such limit spaces $X^n$ states that $S^k$ is $k$-rectifiable. In fact, we will show that for $k$-a.e. $x\in S^k$, {\it every} tangent cone $X_x$ at $x$ is $k$-symmetric i.e. that $X_x={\mathbb{R}}^k\times C(Y)$ where $C(Y)$ might depend on the particular $X_x$. We use this to show that there exists $\epsilon =\epsilon (n,v)$, and a $(n-2)$-rectifible set $S_{\epsilon }^{n-2}$, with finite $(n-2)$-dimensional Hausdorff measure $H^{n-2}(S_{\epsilon }^{n-2})<C(n,v)$, such that $X^n\setminus S_{\epsilon }^{n-2}$ is bi-Hölder equivalent to a smooth riemannian manifold. This improves the regularity results of \cite{ChCoI}. Additionally, we will see that tangent cones are unique of a subset of Hausdorff $(n-2)$ dimensional measure zero. Our analysis is based on several new ideas, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.