Orbifold compactness for spaces of Riemannian metrics and applications

TL;DR

This paper establishes orbifold compactness results for spaces of Riemannian metrics, extending previous work for Einstein and Ricci-bounded metrics, and applies these to Bach-flat and other critical point metrics on 4-manifolds.

Contribution

It generalizes orbifold compactness theorems to broader classes of metrics, including Bach-flat and critical point metrics, on 4-manifolds.

Findings

Proved orbifold compactness for spaces of Riemannian metrics.

Extended compactness results to Bach-flat and critical point metrics.

Applied results to specific classes of 4-manifold metrics.

Abstract

This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such compactness for spaces of Bach-flat (for example half-conformally flat) metrics on 4-manifolds, and related results for metrics which are critical points of other natural Riemannian functionals on the space of metrics.