Asymptotic curvature decay and removal of singularities of Bach-flat metrics

Streets, Jeffrey

doi:10.1090/S0002-9947-09-04960-5

Asymptotic curvature decay and removal of singularities of Bach-flat metrics
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by Jeffrey Streets PDF

Trans. Amer. Math. Soc. 362 (2010), 1301-1324 Request permission

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 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 $2$. This result is then used to prove the removal of singularities theorem.

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