Resolution of compact Einstein orbifolds in general dimensions

Abstract

Given a noncollapsing sequence of $m$-dimensional compact Einstein manifolds with a uniform energy bound, the Gromov-Hausdorff limit is a compact Einstein orbifold with at most finitely many singularities. Conversely, starting with a compact Einstein orbifold, we are interested in whether there exists a sequence of smooth Einstein metrics converging to it. In this paper, we provide a negative answer. We give an explicit obstruction, such that if an Einstein orbifold with negative scalar curvature appears as a noncollapsing limit of compact Einstein manifolds, then the obstruction must vanish. Such an obstruction links the curvature at the orbifold singularity and the geometry of the blow-up limit. As an example, the obstruction never vanishes for hyperbolic orbifolds, so they can not be approximated by smooth Einstein manifolds. This work partially extends the work of Ozuch in dimension 4. We are assuming the Einstein orbifold has negative scalar curvature, to facilitate the exposition. Also, this work includes an obstruction found by Morteza-Viaclovski as a special case, which holds when the blow-up limit comes from the Calabi ansatz.