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.
Keywords
Einstein 4-manifolds, compactness, gluing-perturbation,
moduli space
