Abstract
In this note, we show that compact static near-horizon geometries with negative cosmological constant are either Einstein or the product of a circle and an Einstein metric. Chruściel, Reall, and Todd proved rigidity when the cosmological constant vanishes, in which case one get the stronger result that the space is Ricci flat (Chruściel et al. in Class Quantum Gravity 23:549–554, 2006). It has been previously asserted that a stronger rigidity statement also holds for negative cosmological constant, but Bahuaud, Gunasekaran, Kunduri, and Woolgar recently pointed out that this was not the case (Bahuaud et al. in Lett Math Phys 112(6):116, 2022). They showed, moreover, that for a compact static near-horizon geometry with negative cosmological constant, the potential vector field X is constant length and divergence-free. We give an argument using the Bochner formula to improve their conclusion to X being a parallel field, which implies the optimal rigidity result. The result also holds more generally for m-Quasi Einstein metrics with \(m>0\).
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References
Bahuaud, E., Gunasekaran, S., Kunduri, H., Woolgar, E.: Static near-horizon geometries and rigidity of quasi-Einstein manifolds. Lett. Math. Phys. 112(6), 116 (2022)
Besse, A.L.: Einstein manifolds. Ergeb. Math. Grenzgeb. (3) 10. Springer, Berlin (1987)
Böhm, C.: Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math. 134(1), 145–176 (1998)
Case, J., Shu, Y.-J., Wei, G.: Rigidity of quasi-Einstein metrics Differ. Geom. Appl. 29, 93–100 (2011)
Chen, Z., Liang, K., Zhu, F.: Non-trivial m-quasi-Einstein metrics on simple Lie groups. Ann. Mat. Pura Appl. (4) 195, 1093–1109 (2016)
Chruściel, P., Reall, H., Tod, P.: On non-existence of static vacuum black holes with degenerate components of the event horizon. Class. Quantum Gravity 23, 549–554 (2006)
He, C., Petersen, P., Wylie, W.: On the classification of warped product Einstein metrics. Commun. Anal. Geom. 20(2), 271–312 (2012)
Hollands, S., Ishibashi, A.: All vacuum near horizon geometries in D-dimensions with (D-3) commuting rotational symmetries. Ann. Henri Poincaré 10(8), 1537–1557 (2010)
Ivey, T.: Ricci solitons on compact three-manifolds. Differ. Geom. Appl. 3(4), 301–307 (1993)
Kim, D.-S., Kim, Y.: Compact Einstein warped product spaces with nonpositive scalar curvature. Proc. Am. Math. Soc. 131(8), 2573–2576 (2003)
Kunduri, H., Lucietti, J.: Classification of near-horizon geometries of extremal black holes. Living Rev. Relativ. 16(8), 1–71 (2013)
Lim, A.: Locally homogeneous non-gradient quasi-Einstein 3-manifolds. Adv. Geom. 22(1), 79–93 (2022)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
Acknowledgements
The author is supported by NSF grant #1654034 and would like to thank Eric Woolgar for making him aware of this problem and for helpful feedback in preparing this note. We also thank the referee for very helpful feedback.
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Wylie, W. Rigidity of compact static near-horizon geometries with negative cosmological constant. Lett Math Phys 113, 29 (2023). https://doi.org/10.1007/s11005-023-01654-2
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DOI: https://doi.org/10.1007/s11005-023-01654-2