Regularity of Einstein manifolds and the codimension 4 conjecture

• 成果类型: Article
• 署名作者: Cheeger, Jeff; Naber, Aaron
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2015.182.3.5
• 发表日期: 2015
• 页码: 1093-1165
• 关键词: ricci curvature; RIGIDITY; SPACES; bounds; sets

摘要

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (M-n, g) with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces (M-j(n), d(j)) ->(dGH) (X, d), where d(j), denotes the Riemannian distance. Our main result is a solution to the codimension 4 conjecture, namely that X is smooth away from a closed subset of codimension 4. We combine this result with the ideas of quantitative stratification to prove a priori L-q estimates on the full curvature vertical bar Rm vertical bar for all q < 2. In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We apply this to prove a conjecture of Anderson that the collection of 4-manifolds (M-4, g) with vertical bar Ric(M4)vertical bar <= 3, Vol(M) > v > 0, and diam(M) <= D contains at most a finite number of diffeomorphism classes. A local version is used to show that noncollapsed 4-manifolds with bounded Ricci curvature have a priori L-2 Riemannian curvature estimates.