Regularity of Einstein manifolds and the codimension 4 conjecture

2015/10/13 by Jeff Cheeger, Aaron Naber · 18 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research

paper · doi:10.4007/annals.2015.182.3.5

Abstract

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.

Citations

• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products

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• Splitting and Slow Volume Growth for Open Manifolds with Nonnegative Ricci Curvature
• On the structure of Ricci shrinkers
• Complete Kähler manifolds with nonnegative Ricci curvature II
• Uniqueness of the asymptotic limits for Ricci-flat manifolds with linear volume growth
• Heat kernel on Ricci shrinkers
• On the Poincaré-Einstein manifolds with cylindrical conformal infinity
• Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds
• Compactness of conformally compact Einstein 4-manifolds II
• L² Curvature Bounds on Manifolds with Bounded Ricci Curvature
• Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below, II
• On manifolds with almost non-negative Ricci curvature and integrally-positive k th -scalar curvature. [europepmc]
• On the Dimension of the Singular Set of Perimeter Minimizers in Spaces with a Two-Sided Bound on the Ricci Curvature [europepmc]

Related

• Regularity of Einstein Manifolds and the Codimension 4 Conjecture
• Codimension four regularity of generalized Einstein structures
• Regularity of Einstein 5-manifolds via 4-dimensional gap theorems
• Curvature Estimates for Critical 4-Manifolds with a Lower Ricci Curvature Bound
• L² curvature bounds on manifolds with bounded Ricci curvature