Regularity of Einstein 5-manifolds via 4-dimensional gap theorems

TL;DR

This work analyzes noncollapsed limits of Einstein manifolds, establishing a sharp regularity picture in dimension five by leveraging a new 4D gap/isolation theory. The authors prove uniqueness of tangent cones that split off a line, refine the singular set to lie along Lipschitz curves, and show that interiors of these curves admit a real-analytic orbifold extension with bounded curvature away from endpoints. A central innovation is an isolation theorem for spherical and hyperbolic 4-orbifolds among Einstein 4-orbifolds, enabling precise gap theorems and controlled desingularizations which undergird the 5D regularity theory. The resulting orbifold-regularity off a codimension-5 set advances the program toward a codimension-5 regularity conjecture in arbitrary dimension and opens avenues for understanding higher-dimensional degenerations of Einstein metrics. The methods combine refined gluing, obstruction analysis, and cone-splitting techniques to produce quantitative curvature and coordinate regularity near singular curves, culminating in a robust, analytic orbifold structure on the regular set.

Abstract

We refine the regularity of noncollapsed limits of 5-dimensional manifolds with bounded Ricci curvature. In particular, for noncollapsed limits of Einstein 5-manifolds, we prove that (1) tangent cones are unique of the form $\mathbb{R}\times\mathbb{R}^4/Γ$ on the top stratum, hence outside a countable set of points, (2) the singular set is entirely contained in a countable union of Lipschitz curves and points, (3) away from a nowhere dense subset, these Lipschitz curves consist of smooth geodesics, (4) the interior of any geodesic is removable: limits of Einstein manifolds are real-analytic orbifolds with singularities along geodesic and bounded curvature away from their extreme points, and (5) if an asymptotically Ricci-flat 5-manifold with Euclidean volume growth has one tangent cone at infinity that splits off a line, then it is the unique tangent cone at infinity. These results prompt the question of the orbifold regularity of noncollapsed limits of Einstein manifolds off a codimension 5 set in arbitrary dimension. The proofs rely on a new result of independent interest: all spherical and hyperbolic 4-orbifolds are isolated among Einstein 4-orbifolds in the Gromov-Hausdorff sense. This yields various gap theorems for Einstein 4-orbifolds, which do not extend to higher dimensions.

Figures (1)

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Theorems & Definitions (134)

• Theorem 1.1: Isolation theorem
• Remark 1.2
• Corollary 1.3: Gap theorems
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9: Isolation of $1$-symmetric Ricci-flat cones
• Remark 1.10