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

TL;DR

This paper improves the understanding of the regularity and structure of limits of Einstein 5-manifolds, showing uniqueness of tangent cones, the nature of singular sets, and orbifold regularity, with implications for higher-dimensional geometry.

Contribution

It introduces new isolation results for tangent cones and gap theorems for Einstein 4-orbifolds, advancing regularity theory in 5-dimensional Einstein manifolds.

Findings

Tangent cones are unique and of the form R x R^4/Gamma on the top stratum.

Singular sets are contained in Lipschitz curves and points, with curves being smooth geodesics outside a nowhere dense set.

Limits of Einstein manifolds are real-analytic orbifolds with controlled singularities.

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 $\mathbb{R}\times\mathbb{R}^4/\Gamma$ on the top stratum, hence outside a countable set of points; this follows from a new isolation result for cones of the form $\mathbb{R}\times\mathbb{R}^4/\Gamma$ among all tangent cones, (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. The proofs of these gap theorems require a careful analysis of singular metrics and families of metrics that degenerate.