Asymptotic Curvature Decay and Removal of Singularities of Bach-Flat Metrics

TL;DR

This paper proves a removal of singularities theorem for 4-dimensional Bach-flat metrics under certain boundedness conditions, extending previous results and analyzing decay rates of solutions to the Bach-flat equation.

Contribution

It extends singularity removal results for Bach-flat metrics by analyzing decay rates and classifying Bach-flat cones as ALE of any order less than 2.

Findings

Bach-flat cones are ALE of any order less than 2

Singularities can be removed under bounded curvature and volume growth conditions

Decay rates of solutions to the Bach-flat equation are crucial for the analysis

Abstract

We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the proof we analyze the decay rates of solutions to the Bach-flat equation linearized around a flat metric. This classification is used to prove that Bach-flat cones are in fact ALE of order $\tau$ for any $\tau < 2$. This result is then used to prove the removal of singularities theorem.