TY - JOUR

T1 - Rigidity of compact static near-horizon geometries with negative cosmological constant

AU - Wylie, William

N1 - Funding Information:
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.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2023/4

Y1 - 2023/4

N2 - 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.

AB - 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.

KW - 53C24

KW - 53C25

KW - 83C05

KW - 83C57

KW - Near horizon geometry

KW - Quasi-Einstein manifold

KW - Rigidity

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U2 - 10.1007/s11005-023-01654-2

DO - 10.1007/s11005-023-01654-2

M3 - Article

AN - SCOPUS:85149965799

SN - 0377-9017

VL - 113

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 2

M1 - 29

ER -