L 2 curvature bounds on manifolds with bounded Ricci curvature

• Published in: Annals of mathematics 2021-01, Vol.193 (1), p.107-222
• Main Authors: Jiang, Wenshuai, Naber, Aaron
• Format: Article
• Language: English
• ISSN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2021.193.1.2

Consider a Riemannian manifold with bounded Ricci curvature |Ric| ≤ n — 1 and the noncollapsing lower volume bound Vol(B 1 (p)) > v > 0. The first main result of this paper is to prove that we have the L 2 curvature bound ⨍ B 1(p)|Rm|2 (x) dx < C(n, v), which proves the L 2 conjecture. In order to prove this, we will need to first show the following structural result for limits. Namely, if (Mj n,dj,pj ) → (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 Hn -4(S(X)∩B 1) < 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 Rn-4 × C (S 3/Γ x ) for some Γ x ⊆ O(4) that acts freely away from the origin.