Rectifiability of singular sets of noncollapsed limit spaces with Ricci curvature bounded below


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 $\mathrm{Ric}_{M^n_i}\geq -(n-1)$ as well as the noncollapsing assumption $\mathrm{Vol}(B_1(p_i))>\mathrm{v}>0$. 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, $S^k$ is the set of points such that no tangent cone splits off a Euclidean factor $\mathbb{R}^{k+1}$. It is classical from Cheeger-Colding that the Hausdorff dimension of $S^k$ satisfies $\mathrm{dim}\, S^k\leq k$ and $S=S^{n-2}$, i.e., $S^{n-1}\setminus S^{n-2}=\emptyset$. 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 for all $k$. In fact, we will show for $\mathcal H^k$-a.e. $x\in S^k$ that 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 $X_x$. Here $\mathcal{H}^k$ denotes the $k$-dimensional Hausdorff measure. As an application we show for all $0\lt \epsilon\lt \epsilon(n,\mathrm{v})$ there exists an $(n-2)$-rectifiable closed set $S^{n-2}_\epsilon$ with $\mathcal{H}^{n-2}(S_{\epsilon}^{n-2}) < C(n,\mathrm{v},\epsilon)$, such that $X^n\setminus S^{n-2}_\epsilon$ is $\epsilon$-bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, $S=\bigcup_\epsilon S^{n-2}_\epsilon$. As another application, we show that tangent cones are unique $\mathcal H^{n-2}$-a.e.

In the case of limit spaces $X^n$ satisfying a $2$-sided Ricci curvature bound $|\mathrm{Ric}_{M^n_i}|\leq n-1$, we can use these structural results to give a new proof of a conjecture from Cheeger-Colding stating that $S$ is $(n-4)$-rectifiable with uniformly bounded measure. We can also conclude from this structure that tangent cones are unique $\mathcal H^{n-4}$-a.e.

Our analysis builds on the notion of quantitative stratification introduced by Cheeger-Naber, and the neck region analysis developed by Jiang-Naber-Valtorta. Several new ideas and new estimates are required, 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.


Jeff Cheeger

Courant Institute, New York, NY

Wenshuai Jiang

School of Mathematical Sciences, Zhejiang University, Hangzhou, China

Aaron Naber

Northwestern University, Evanston, IL