Lower bounds on Ricci curvature and quantitative behavior of singular sets

Abstract

Let Yn denote the Gromov-Hausdorff limit M[superscript n][subscript i][d[subscript GH] over ⟶]Y[superscript n] of v-noncollapsed Riemannian manifolds with Ric[subscript M[superscript n][subscript i]] ≥ −(n−1). The singular set S ⊂ Y has a stratification S[superscript 0] ⊂ S[superscript 1] ⊂ ⋯ ⊂ S, where y ∈ S[superscript k] if no tangent cone at y splits off a factor ℝ[superscript k+1] isometrically. Here, we define for all η > 0, 0 < r ≤1, the k-th effective singular stratum S[superscript k][subscript η,r] satisfying ⋃[subscript η]⋂[subscript r]S[superscript k][subscript η,r] = S[superscript k]. Sharpening the known Hausdorff dimension bound dim S[superscript k] ≤ k, we prove that for all y, the volume of the r-tubular neighborhood of S[superscript k][subscript η,r] satisfies Vol(T[subscript r](S[superscript k][subscript η,r])∩B[subscript 1/2](y)) ≤ c(n,v,η)r[superscript n−k−η]. The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let B[subscript r] denote the set of points at which the C[superscript 2]-harmonic radius is ≤ r. If also the M[superscript n][subscript i] are Kähler-Einstein with L[subscript 2] curvature bound, ∥Rm∥[subscript L2] ≤ C, then Vol(B[subscript r]∩B[subscript 1/2](y)) ≤ c(n,v,C)r[superscript 4] for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the M[superscript n][subscript i], we obtain a slightly weaker volume bound on B[subscript r] which yields an a priori L[subscript 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.