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Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n ofv-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S ofY has a stratification S^0\subset S^1\subset\...\subset S, where y\in S^k if notangent cone at y splits off a factor R^{k+1} isometrically. There is a knownHausdorff dimension bound dimS^k\leq k. Here, we define for all \eta>0, 0<r\leq1, the {\it k-th effective singular stratum} S^k_{\eta,r} such that\bigcup_\eta\bigcap_r \,\cS^k_{\eta,r}= \cS^k. Sharpening the bound dim S^k\leqk, we prove that the r-tubular neighborhood satisfies:Vol(T_r(S^k_{\eta,r})\cap B_{1/2}(y))\leq c(n,v,\eta)r^{n-k-\eta}, for all y.The proof depends on a {\it quantitative differentiation} argument; for furtherexplanation, see Section 2. The applications give new curvature estimates forEinstein manifolds. Let Rm denote the curvature tensor and regard |Rm(y)|=\infty unless Y^n is smooth in some neighborhood of y. Put \cB_r=\{y:|sup_{B_r(y)}|Rm|\geq r^{-2}\}. Assuming in addition that the M^n_i areK\"ahler-Einstein with ||Rm||_{L_2}\leq C, we get the volume boundVol(\cB_r\cap B_{1/2}(y))\leq c(n,v,C)r^4$ for all y. In the K\"ahler-Einsteincase, without assuming any integral curvature bound on the M^n_i, we obtain aslightly weaker volume bound on \cB_r, which yields an a priori L_p curvaturebound for all p<2; see Section 1 the for the precise statement (which issharper).

Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets