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

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

2007-08-07

arxiv.org/abs/0708.0869v1

Mathematics

Differential Geometry

Scientific paper

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.

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