Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets (1103.1819v4)
Abstract: Let Yn denote the Gromov-Hausdorff limit of a sequence Mn_i-> Yn of v-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S of Y has a stratification S0\subset S1\subset...\subset S, where y\in Sk if no tangent cone at y splits off a factor R{k+1} isometrically. There is a known Hausdorff dimension bound dimSk\leq k. Here, we define for all \eta>0, 0<r\leq 1, the {\it k-th effective singular stratum} Sk_{\eta,r} such that \bigcup_\eta\bigcap_r \,\cSk_{\eta,r}= \cSk. Sharpening the bound dim Sk\leq k, we prove that the r-tubular neighborhood satisfies: Vol(T_r(Sk_{\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 Yn is smooth in some neighborhood of y. Put \cB_r={y: |sup_{B_r(y)}|Rm|\geq r{-2}}. Assuming in addition that the Mn_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)r4$ for all y. In the K\"ahler-Einstein case, without assuming any integral curvature bound on the Mn_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).