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 4 under the assumption of bounded L 2 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 by Tian and Viaclovsby. For the proof we emulate Cheeger and Tian and 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 2 2 . This result is then used to prove the removal of singularities theorem.