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

TL;DR

This paper proves that the singular sets in noncollapsed Ricci limit spaces are rectifiable and have a well-understood structure, improving regularity results and analyzing tangent cone uniqueness.

Contribution

It establishes the rectifiability of singular sets in Ricci limit spaces and shows that tangent cones are mostly unique, with new techniques like cone-splitting and geometric transformation theorems.

Findings

Singular sets $S^k$ are $k$-rectifiable.

Almost all tangent cones at points in $S^k$ are $k$-symmetric.

Existence of a large $(n-2)$-rectifiable set outside which the space is bi-H"older to a smooth manifold.

Abstract

This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $Rc_{M^n_i}\geq -(n-1)$ 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}, $\dim S^k\leq k$. 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 $S^{n-2}_\epsilon$, with finite $(n-2)$-dimensional Hausdorff measure $H^{n-2}(S_\epsilon^{n-2})<C(n,v)$, such that $X^n\setminus S^{n-2}_\epsilon$ is bi-H\"older 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.