Abstract
In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.
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Anderson, M.T.: Scalar curvature, metric degenerations and the static vacuum Einstein equations on \(3\)-manifolds I. Geom. Funct. Anal. 9(5), 855–967 (1999)
Bach, R.: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. 9, 110–1359 (1921)
Besse, A.L.: Einstein Manifolds. Springer, New York (1987)
Bourguignon, J.P.: Une stratification de l’espace des structures riemanniennes. Compos. Math. 30(1), 1–41 (1975)
Cao, H.D., Chen, Q.: On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162(6), 1149–1169 (2013)
Catino, G., Mastrolia, P., Monticelli, D.D.: Gradient Ricci solitons with vanishing conditions on Weyl. J. Math. Pures Appl. 108(1), 1–13 (2017)
Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000)
Ejiri, N.: A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J. Math. Soc. Jpn. 33, 261–266 (1981)
Fischer, A.E., Marsden, J.E.: Manifolds of Riemannian metrics with prescribed scalar curvature. Bull. Am. Math. Soc. 80, 479–484 (1974)
Hwang, S., Chang, J., Yun, G.: Nonexistence of multiple black holes in static space-times and weakly harmonic curvature. Gen. Relat. Gravit. 48, 120 (2016)
Kobayashi, O.: A differential equation arising from scalar curvature function. J. Math. Soc. Jpn. 34(4), 665–675 (1982)
Kobayashi, O., Obata, M.: Conformally-Flatness and Static Space-Times, Manifolds and Lie Groups. Progress in Mathematics, vol. 14, pp. 197–206. Birkhäuser, Basel (1981)
Kim, J., Shin, J.: Four-dimensional static and related critical spaces with harmonic curvature. Pac. J. Math. 295(2), 429–462 (2018)
Lafontaine, J.: Sur la géométrie d’une généralisation de l’équation différentielle d’Obata. J. Math. Pures Appl. 62(1), 63–72 (1983)
Listing, M.: Conformally invariant Cotton and Bach tensor in \(N\)-dimensions. arXiv:math/0408224v1 [math.DG] (2004)
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14(3), 333–340 (1962)
Petersen, P., Wiley, W.: Rigidity of gradient Ricci solitons. Pac. J. Math. 241(2), 329–345 (2009)
Shen, Y.: A note on Fischer-Marsden’s conjecture. Proc. Am. Math. Soc. 125, 901–905 (1997)
Qing, J., Yuan, W.: A note on static spaces and related problems. J. Geom. Phys. 74, 13–27 (2013)
Yun, G., Chang, J., Hwang, S.: Total scalar curvature and harmonic curvature. Taiwan. J. Math. 18(5), 1439–1458 (2014)
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The first author was supported by the National Research Foundation of Korea (NRF-2018R1D1A1B05042186) and the second author was supported by the National Research Foundation of Korea (NRF-2019R1A2C1004948).
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Hwang, S., Yun, G. Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor. J Geom Anal 31, 3060–3084 (2021). https://doi.org/10.1007/s12220-020-00384-4
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DOI: https://doi.org/10.1007/s12220-020-00384-4