Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets

Abstract

Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S of Y has a stratification S^0\subset S^1\subset\...\subset S, where y\in S^k if no tangent cone at y splits off a factor R^{k+1} isometrically. There is a known Hausdorff dimension bound dimS^k\leq k. Here, we define for all \eta>0, 0<r\leq 1, 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\leq k, 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 further explanation, see Section 2. The applications give new curvature estimates for Einstein 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 are K\"ahler-Einstein with ||Rm||_{L_2}\leq C, we get the volume bound Vol(\cB_r\cap B_{1/2}(y))\leq c(n,v,C)r^4$ for all y. In the K\"ahler-Einstein case, without assuming any integral curvature bound on the M^n_i, we obtain a slightly weaker volume bound on \cB_r, which yields an a priori L_p curvature bound for all p<2; see Section 1 the for the precise statement (which is sharper).