L^2 Curvature Bounds on Manifolds with Bounded Ricci Curvature

Consider a Riemannian manifold with bounded Ricci curvature |Ric|\leq n-1 and the noncollapsing lower volume bound Vol(B_1(p))>v>0.  The first main result of this paper is to prove the previously conjectured L^2 curvature bound \fint_{B_1}|\Rm|^2 < C(n,v).  In order to prove this, we will need to first show the following structural result for limits.  Namely, if (M^n_j,d_j,p_j) -> (X,d,p) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates H^{n-4}(S(X)\cap B_1)<C(n,v), which in particular proves the n-4-finiteness conjecture of Cheeger-Colding.  We will see as a consequence of the proof that for n-4 a.e. x\in S(X) that the tangent cone of X at x is unique and isometric to R^{n-4}xC(S^3/G_x) for some G_x\subseteq O(4) which acts freely away from the origin.  The proofs involve several new estimates on spaces with bounded Ricci curvature. This is joint work with Wenshuai Jiang

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