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Noncollapsed degeneration of Einstein $4$–manifolds, II

Tristan Ozuch

Geometry & Topology 26 (2022) 1529–1634

In this second article, we prove that any desingularization in the Gromov–Hausdorff sense of an Einstein orbifold by smooth Einstein metrics 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 implies a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.

The description of Einstein metrics as the result of a gluing-perturbation procedure 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 Hölder spaces. This is conjecturally the general case, and 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.

Einstein 4-manifolds, compactness, gluing-perturbation, moduli space
Mathematical Subject Classification 2010
Primary: 53C20, 53C21, 53C25
Secondary: 53C23, 58D27
Received: 29 October 2019
Revised: 3 May 2021
Accepted: 2 June 2021
Published: 28 October 2022
Proposed: Tobias H Colding
Seconded: Bruce Kleiner, Simon Donaldson
Tristan Ozuch
Department of Mathematics
Cambridge, MA
United States