Noncollapsed degeneration of Einstein 4-manifolds II

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 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 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 Hölder 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.

Integrability of Einstein deformations and desingularizations

“…We finally recover the obstructions of [Biq13] to the desingularization of Einstein metrics and many of the additional obstructions of [Ozu19b] (but not the higher order obstructions of [Ozu20]). We conclude this section by proving that some Einstein orbifolds cannot be desingularized by smooth Einstein manifolds.…”

“…Let us now show that we recover the obstructions of [Biq13,Ozu19b] and find another expression for them. This new expression will further highlight the link between these obstructions and the lack of integrability of Einstein desingularizations of [Ozu20t, Chapter 4].…”

“…Any smooth Einstein 4-manifold close to a compact Einstein orbifold in a mere Gromov-Hausdorff sense has been recently been produced by a gluing-perturbation procedure [Ozu19a,Ozu19b]. Here, this lets us understand the obstructions to the d GH -desingularization of an Einstein orbifold through obstructions similar to (5) and (7).…”

“…We call Einstein modulo obstructions metrics such deformations which have been constructed in [Ozu19b] (see also [Koi83] in the smooth compact case). We study the leading order of the obstruction along curves s → g sv for s ∈ (−1, 1) at s = 0.…”

“…• for the conformal Killing vector field r∂ r , we have Proof. Let us come back to the origin of the obstructions of [Ozu19b,Ozu20t]. They are the obstructions to solving:…”