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

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

DOI: https://doi.org/10.1090/S0002-9947-09-04960-5

Published electronically: October 20, 2009

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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.

References

Jeff Cheeger and Gang Tian, On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994), no. 3, 493–571. MR 1296356, DOI 10.1007/BF01231543
S. Chen, Optimal curvature decays on asymptotically locally euclidean manifolds, preprint.
Xiuxiong Chen, Claude Lebrun, and Brian Weber, On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc. 21 (2008), no. 4, 1137–1168. MR 2425183, DOI 10.1090/S0894-0347-08-00594-8
Andrzej Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405–433. MR 707181
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
G. Tian and J. Viaclovsky, Moduli spaces of critical Riemannian metrics in dimension four, arXiv:math.DG/0312318.
G. Tian and J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, arXiv:math.DG/0310302.