Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets

TL;DR

This paper establishes new quantitative bounds on the size and structure of singular sets in limits of Ricci curvature bounded manifolds, leading to improved curvature estimates for Einstein and Kähler-Einstein manifolds.

Contribution

It introduces effective singular strata and uses quantitative differentiation to sharpen Hausdorff dimension bounds and volume estimates of singular sets.

Findings

Volume bounds for tubular neighborhoods of singular strata

New curvature estimates for Einstein manifolds

A priori L_p curvature bounds for Kähler-Einstein manifolds

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).