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

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.