Resolution of compact Einstein orbifolds in general dimensions

TL;DR

This paper investigates the limits of sequences of compact Einstein manifolds converging to Einstein orbifolds, providing explicit obstructions to such limits and extending previous work in dimension 4.

Contribution

It introduces a new explicit obstruction criterion for negative Einstein orbifolds to appear as limits of smooth Einstein manifolds, generalizing prior results in dimension 4.

Findings

Explicit obstruction for negative Einstein orbifolds as limits

Obstruction does not vanish for hyperbolic orbifolds

Extension of Ozuch's work to higher 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 for a negative Einstein orbifold appearing as a noncollapsing limit of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds. This work extends the work of Ozuch in dimension 4, with significant technical simplifications.