Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Yⁿ denote the Gromov-Hausdorff limit of v-noncollapsed Riemannian manifolds with Ric_Mni ≥ - (n - 1). The singular set S ⊂ Y has a stratification S ⊂ S ⊂. ⊂S, where y ∈S 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 X_η,r^k satisfying ∪_η ∩_r S_n,r^k = S^k. Sharpening the known Hausdorff dimension bound dim S^k ≤k, we prove that for all y, the volume of the r-tubular neighborhood of S_η,r^k satisfies Vol(T_r(S_η,r^k) ∩ B1/2(y)) ≤C(n,v,η)r^n-k-η. The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let B_r denote the set of points at which the C²-harmonic radius is ≤r. If also the M_iⁿ are Kähler-Einstein with L₂ curvature bound, {double pipe}Rm{double pipe}L₂ ≤ C, then Vol(B_r ∩B 1/2(y)) ≤c(n,v,C)r for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the M_iⁿ, we obtain a slightly weaker volume bound on B_r which yields an a priori L_p curvature bound for all p <2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.