Integrability of Einstein deformations and desingularizations

TL;DR

This paper investigates the integrability of Einstein deformations and their role in desingularizing Einstein 4-orbifolds, providing new obstructions and showing that certain orbifolds cannot be approximated by smooth Einstein metrics.

Contribution

It introduces preserved integral quantities as obstructions to Einstein deformation integrability and demonstrates their equivalence to known desingularization obstructions.

Findings

Certain spherical and hyperbolic 4-orbifolds cannot be limits of smooth Einstein 4-manifolds.

New integral obstructions are introduced based on symmetries and variations of Ricci-flat metrics.

Obstructions to desingularization are interpreted as failures of deformation integrability.

Abstract

We study the question of the integrability of Einstein deformations and relate it to the question of the desingularization of Einstein metrics. Our main application is a negative answer to the long-standing question of whether or not every Einstein $4$-orbifold (which is an Einstein metric space in a synthetic sense) is limit of smooth Einstein $4$-manifolds. We more precisely show that spherical and hyperbolic $4$-orbifolds with the simplest singularities cannot be Gromov-Hausdorff limits of smooth Einstein $4$-metrics without relying on previous integrability assumptions. For this, we analyze the integrability of deformations of Ricci-flat ALE metrics through variations of Schoen's Pohozaev identity. Inspired by Taub's preserved quantity in General Relativity, we also introduce preserved integral quantities based on the symmetries of Einstein metrics. These quantities are obstructions to the integrability of infinitesimal Einstein deformations "closing up" inside a hypersurface - even with change of topology. We show that many previously identified obstructions to the desingularization of Einstein $4$-metrics are equivalent to these quantities on Ricci-flat cones. In particular, all of the obstructions to desingularizations bubbling out Eguchi-Hanson metrics are recovered. This lets us further interpret the obstructions to the desingularization of Einstein metrics as a defect of integrability.