Regularity of Einstein manifolds and the codimension 4 conjecture
Regularity of Einstein manifolds and the codimension 4 conjecture
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DOI：
10.4007/annals.2015.182.3.5
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发表时间：
2014-06
影响因子：
4.9
通讯作者：
J. Cheeger;A. Naber
中科院分区：
文献类型：
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作者：
J. Cheeger;A. Naber
在本文中，我们关注具有有界里奇曲率的非塌陷黎曼流形 (M n ;g) 的正则性，以及它们的 Gromov-Hausdor 极限空间 (M n ;dj) dGH ！ (X;d)，其中 dj 表示黎曼距离。我们的主要结果是余维 4 猜想的解，即 X 远离余维 4 的闭子集是平滑的。我们将此结果与定量分层的思想结合起来，证明对所有 q v > 0 的全曲率 jRmj 的先验 L q 估计，并且 diam(M) D 最多包含有限数量的异态类。局部版本用于表明具有有界 Ricci 曲率的非塌陷 4 流形具有先验 L 2 黎曼曲率估计。
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-Hausdor limit spaces ( M n ;dj) dGH ! (X;d), where dj denotes the Riemannian distance. Our main result is a solution to the codimension 4 conjecture, namely thatX is smooth away from a closed subset of codimension 4. We combine this result with the ideas of quantitative stratication to prove a priori L q estimates on the full curvaturejRmj for all q v > 0, and diam(M) D contains at most a nite number of dieomorphism classes. A local version is used to show that noncollapsed 4-manifolds with bounded Ricci curvature have a priori L 2 Riemannian curvature estimates.