A theorem of Anderson and Bando, Kasue and Nakajima from 1989 states that to
compactify the set of normalized Einstein metrics with a lower bound on the volume
and an upper bound on the diameter in the Gromov–Hausdorff sense, one has to add
singular spaces, called Einstein orbifolds, and the singularities form as blow-downs of
Ricci-flat ALE spaces.
This raises some natural issues, in particular: Can all Einstein orbifolds be
Gromov–Hausdorff limits of smooth Einstein manifolds? Can we describe more
precisely the smooth Einstein metrics close to a given singular one?
In this first paper, we prove that Einstein manifolds sufficiently close, in the
Gromov–Hausdorff sense, to an orbifold are actually close to a gluing of
model spaces in suitable weighted Hölder spaces. The proof consists in
controlling the metric in the neck regions thanks to the construction of optimal
coordinates.
This refined convergence is the cornerstone of our subsequent work on the
degeneration of Einstein metrics or, equivalently, on the desingularization of Einstein
orbifolds, in which we show that all Einstein metrics Gromov–Hausdorff close to an
Einstein orbifold are the result of a gluing-perturbation procedure. This procedure
turns out to be generally obstructed, and this provides the first obstructions to a
Gromov–Hausdorff desingularization of Einstein orbifolds.