L² curvature bounds on manifolds with bounded Ricci curvature

2021/01/01 by Wenshuai Jiang, Aaron Naber · 5 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · doi:10.4007/annals.2021.193.1.2

Abstract

Consider a Riemannian manifold with bounded Ricci curvature |Ric|≤ n-1 and the noncollapsing lower volume bound Vol(B₁(p))>v>0. The first main result of this paper is to prove that we have the L² curvature bound ⨏_B1(p)|Rm|²(x) dx \lt C(n,v),which proves the L² conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if (Mⁿ_j,d_j,p_j) \longrightarrow (X,d,p) is a GH-limit of noncollapsed manifolds with bounded Ricci curvature, then the singular set S(X) is n-4 rectifiable with the uniform Hausdorff measure estimates H^n-4(S(X)∩ B₁) \lt C(n,v) which, in particular, proves the n-4-finiteness conjecture of Cheeger-Colding. We will see as a consequence of the proof that for n-4 a.e. x∈ S(X), the tangent cone of X at x is unique and isometric to ℝ^n-4× C(S³/Γ_x) for some Γ_x⊆ O(4) that acts freely away from the origin.

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• Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications
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• The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow
• Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth
• An ε-Regularity Theorem for Non-collapsed Ricci Flow

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