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

This paper is concerned with the structure of Gromov-Hausdorff limit d spaces (M-i(n), gi, pi) ->(dGH )(X-n ,d,p) of Riemannian manifolds satisfying a uniform lower Ricci curvature bound RicM(i)(n) >= -(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 of S-1 center dot center dot center dot Sn-1 := S, where S-k := {x is an element of X : no tangent cone at x is (k +1)-symmetric}. Equivalently, S-k is the set of points such that no tangent cone splits off a Euclidean factor Rk+1. It is classical from Cheeger-Colding that the Hausdorff dimension of S-k satisfies dim S-k <= k and S = Sn-2, i.e., Sn-1 \ Sn-2 = empty set. 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 H-k-a.e. x is an element of S-k that every tangent cone X-x at x is k-symmetric, i.e., that X-x = R-k x C(Y) where C(Y) might depend on the particular X-x. Here H-k denotes the k-dimensional Hausdorff measure. As an application we show for all 0 < epsilon < epsilon(n, v) there exists an (n - 2)-rectifiable closed set S-epsilon(n)-2 with Hn-2 (S-epsilon(n-2)) < C(n, v, epsilon), such that X-n \S-epsilon(n-2) is epsilon-bi-Holder equivalent to a smooth Riemannian manifold. Moreover, S = boolean OR(epsilon) S-epsilon(n-2). As another application, we show that tangent cones are unique Hn-2-a.e. In the case of limit spaces X-n satisfying a 2-sided Ricci curvature bound vertical bar Ric (Min)vertical bar <= 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 Hn-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.