Abstract
In this paper, we prove that there are no multiple black holes in an n-dimensional static vacuum space-time having weakly harmonic curvature unless the Ricci curvature is trivial.
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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)
Anderson, M.T.: On the structure of solutions to the static vacuum Einstein equations. Ann. Henri Poincaré 1, 995–1042 (2000)
Besse, A.: Einstein Manifolds. Springer, New York (1987)
Bunting, G.L., Masood-ul-Alam, A.K.M.: Non-existence of multiple black holes in asymptotically Euclidean static vacuum space-times. Gen. Rel. Grav. 19, 147–154 (1987)
Chouikha, A.R.: Existence of metrics with harmonic curvature and non parallel Ricci curvature. Balk. J. Geom. Appl. 8(2), 21–30 (2003)
Derdzinski, A.: On compact Riemannian manifolds with harmonic curvature. Math. Ann. 259, 145–152 (1982)
Derdzinski, A.: Riemannian metrics with harmonic curvature on 2-sphere bundles over compact surfaces. Bull. Soc. Math. Fr. 116, 133–156 (1988)
Elvang, H., Figueras, P.: Black saturn. J. High Energy Phys. 05, 050 (2007)
Emparan, R., Reall, H.S.: Generalized weyl solutions. Phys. Rev. D (3) 65(8), 854025 (2002)
Gibbons, G.W., Ida, D., Shiromizu, T.: Uniqueness and non-uniqueness of static blask holes in higher dimenisons. Phys. Rev. Lett. 89, 041101 (2002)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equation of 2nd Order, 2nd edn. Springer, Berlin (1983)
Gray, A.: Einstein-like manifolds which are not Einstein. Geom. Dedicata 7, 259–280 (1978)
Heusler, M.: Black Hole Uniqueness Theorems. In: Cambridge Lecture Notes in Physics, vol. 6. Cambridge University Press, Cambridge (1966)
Holland, S., Ishibashi A.: Black hole uniqueness theorems in higher dimensional spacetimes. Class. Quantum Grav. 32, 163001 (2012)
Israel, W.: Event horizons in static vacuum space-time. Phys. Rev. 164, 1776–1779 (1967)
Li, P.: Geometric Analysis. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2012)
Milnor, J.: Morse Theory Annals of Mathematical Society. Princeton University Press, New Jersey (1963)
Morrey, C.B.: Multiple Integrals in the Calculus of Variations, Séminaire de Théorie Spectrale et Géométrie. Springer, Berlin (1966)
Reiris, M.: The asymptotic of static isolated systems and a generalised uniqueness for Schwarzschild. Class. Quantum Grav. 32, 195001 (2015)
Robinson, D.C.: Four decades of black hole uniqueness theorems. In: Wiltshire, D.L., Visser, M., Scott, S.M. (eds.) The Kerr Spacetime: Rotating Black Holes in General Relativity, pp. 115–143. Cambridge University Press, Cambridge (2009)
Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975)
Yun, G., Chang, J., Hwang, S.: Total scalar curvature and harmonic curvature. Taiwan. J. Math. 18(5), 1439–1458 (2014)
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Seungsu Hwang was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2010-0010792), Jeongwook Chang by the Ministry of Education, Science and Technology (Grant No. 2010-0011310), and Gabjin Yun by the Ministry of Education, Science and Technology (Grant No. 2011-0007465).
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Hwang, S., Chang, J. & Yun, G. Nonexistence of multiple black holes in static space-times and weakly harmonic curvature. Gen Relativ Gravit 48, 120 (2016). https://doi.org/10.1007/s10714-016-2112-8
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DOI: https://doi.org/10.1007/s10714-016-2112-8