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)dGH (Xⁿ,d,p) of Riemannian manifolds satisfying a uniform lower Ricci curvature bound RicMⁿᵢ≥ -(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⋯ Sⁿ⁻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 Rᵏ⁺¹. It is classical from Cheeger-Colding that the Hausdorff dimension of Sᵏ satisfies dim\, Sᵏ≤ k and S=Sⁿ⁻², i.e., Sⁿ⁻¹ Sⁿ⁻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< ε< ε(n,v) there exists an $(n-2)$-rectifiable closed set Sⁿ⁻²_ε with Hⁿ⁻²(Sεⁿ⁻²) < C(n,v,ε), such that Xⁿ Sⁿ⁻²_ε is ε-bi-Hölder equivalent to a smooth Riemannian manifold. Moreover, S=_ε Sⁿ⁻²_ε. As another application, we show that tangent cones are unique Hⁿ⁻²-a.e. In the case of limit spaces Xⁿ satisfying a $2$-sided Ricci curvature bound |RicMⁿᵢ|≤ 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.