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 (Mⁿᵢ, gᵢ, pᵢ) d₆₇ (Xⁿ, d, p) of Riemannian manifolds satisfying a uniform lower Ricci curvature bound Ric₌䂞㶁 - (n-1) as well as the noncollapsing assumption Vol (B₁ (pᵢ) ) >v>0. In such cases, there is a filtration of the singular set, S⁰ S¹ S^n-1: = S, where Sᵏ: =x X: no tangent cone at x is (k+1) -symmetric. Equivalently, Sᵏ is the set of points such that no tangent cone splits off a Euclidean factor R^k+1. It is classical from Cheeger-Colding that the Hausdorff dimension of Sᵏ satisfies dim\, Sᵏ k and S=S^n-2, i. e. , S^n-1 S^n-2=. However, little else has been understood about the structure of the singular set S. Our first result for such limit spaces Xⁿ states that Sᵏ is k-rectifiable for all k. In fact, we will show for Hᵏ-a. e. \ x Sᵏ that every tangent cone Xₓ at x is k-symmetric, i. e. , that Xₓ= Rᵏ C (Y) where C (Y) might depend on the particular Xₓ. Here Hᵏ denotes the k-dimensional Hausdorff measure. As an application we show for all 0 (n, v) there exists an (n-2) -rectifiable closed set S^n-2_ with H^n-2 (S_^n-2) C (n, v, ), such that Xⁿ S^n-2_ is -bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, S=_ S^n-2_. As another application, we show that tangent cones are unique H^n-2-a. e. In the case of limit spaces Xⁿ satisfying a 2-sided Ricci curvature bound |Ric₌䂞㶁| 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 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.