Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Y n denote the Gromov-Hausdorff limit of v-noncollapsed Riemannian manifolds with . The singular set has a stratification , where if no tangent cone at y splits off a factor ℝ k +1 isometrically. Here, we define for all η >0, 0< r ≤1, the k-th effective singular stratum satisfying . Sharpening the known Hausdorff dimension bound , we prove that for all y , the volume of the r -tubular neighborhood of satisfies . The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let denote the set of points at which the C 2 -harmonic radius is ≤ r . If also the are Kähler-Einstein with L 2 curvature bound, , then for all y . In the Kähler-Einstein case, without assuming any integral curvature bound on the , we obtain a slightly weaker volume bound on which yields an a priori L p curvature bound for all p