L2 curvature bounds on manifolds with bounded Ricci curvature

• 成果类型: Article
• 署名作者: Jiang, Wenshuai; Naber, Aaron
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.193.1.2
• 发表日期: 2021
• 页码: 107-222
• 关键词: gradient estimate; integral bounds; REGULARITY; RIGIDITY; SURFACES; kernel; SPACES

摘要

Consider a Riemannian manifold with bounded Ricci curvature vertical bar Ric vertical bar <= n - 1 and the noncollapsing lower volume bound Vol vertical bar(B-1 (p)) > v > 0. The first main result of this paper is to prove that we have the L-2 curvature bound f(B1(p)) vertical bar Rm vertical bar(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 (M-j(n),d(j), p(j)) -> (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)boolean AND 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 is an element of S(X), the tangent cone of X at x is unique and isometric to Rn-4 x C(S-3/Gamma(x)) for some Gamma(x )subset of O(4) that acts freely away from the origin.