Noncollapsed degeneration of Einstein 4-manifolds II

TL;DR

This paper develops a gluing-perturbation method to analyze the desingularization of Einstein 4-manifolds, extending previous obstructions and providing new insights into the local structure of the Einstein moduli space near its boundary.

Contribution

It introduces a new gluing-perturbation approach for Einstein orbifolds, extending known obstructions and analyzing desingularizations under weaker convergence assumptions.

Findings

Desingularization can be achieved via a gluing-perturbation procedure.

Obstructions to desingularization are extended to broader cases.

Identifies obstructions to desingularizing certain orbifolds with Ricci-flat ALE spaces.

Abstract

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted H\"older spaces, in which the analysis of the present paper takes place. The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we extend the obstruction to the desingularization of Einstein orbifolds found by Biquard, and prove that it holds for any desingularization by trees of quotients of gravitational instantons only assuming a mere Gromov-Hausdorff convergence instead of specific weighted H\"older spaces. This is conjecturally the general case, but it can at least be ensured by topological assumptions such as a spin structure on the degenerating manifolds. We also identify an obstruction to desingularizing spherical and hyperbolic orbifolds by general Ricci-flat ALE spaces.