On the structure of locally conformally flat orbifolds and ALE manifolds

TL;DR

This paper establishes structure theorems for locally conformally flat orbifolds and ALE manifolds, relating their properties through conformal transformations, and applies these results to classification, moduli space analysis, and the orbifold Yamabe problem.

Contribution

It provides new classification results and structural insights for locally conformally flat orbifolds and ALE manifolds, including their fundamental groups and conformal relations.

Findings

Orbifolds admit a manifold cover.

Fundamental group homomorphism for ALE spaces is injective.

The positive mass theorem applies to these ALE ends.

Abstract

In this paper, we prove several structure theorems for locally conformally flat, positive Yamabe orbifolds and nonnegative scalar curvature, ALE manifolds. These two kinds of spaces can be related by conformal blow-up and conformal compactification. For the orbifolds, we prove that such orbifolds admit a manifold cover. For the ALE manifolds, the homomorphism of the fundamental group for the ALE space induced by the embedding of the ALE end is always injective. Using these properties, several classifications of such ALE manifolds and orbifolds are given in low dimensions. As an application to the moduli space, we prove that the football orbifold $\mathbb{S}^4/\Gamma$ cannot be realized as the Gromov-Hausdorff limit. In addition, we prove the positive mass theorem of these ALE ends and give a simple proof for the optimal decay rate. Using the positive mass theorem, we can solve the orbifold Yamabe problem in the locally conformally flat case.