Uniqueness and energy bounds for static AdS metrics

Chruściel, Piotr T.; Galloway, Gregory J.; Potaux, Yohan

doi:10.1103/physrevd.101.064034

Cited by 8 publications

(11 citation statements)

References 38 publications

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“…Among the contributions that motivated this work, we primarily mention the classical isoperimetric inequality and a result due to Shen [35] and Boucher, Gibbons and Horowitz [11] that asserts that the boundary ∂M of a compact three-dimensional oriented triple static space (i.e., 1-quasi-Einstein manifold) with connected boundary and scalar curvature 6 must be a 2-sphere whose area satisfies the inequality |∂M | ≤ 4π, with equality if and only if M 3 is equivalent to the standard hemisphere. In the same spirit, boundary estimates for V -static metrics and static spaces were established in, e.g., [1,3,6,7,20,21,24,29,32]. In the recent work [22, Theorem 1], Diógenes and Gadelha proved an analogous boundary estimate for compact m-quasi-Einstein manifolds M n with connected boundary ∂M by assuming the following conditions:…”

Section: Example 2 ([22]mentioning

confidence: 96%

Geometry of compact quasi-Einstein manifolds with boundary

Diógenes¹,

Gadelha²,

Ribeiro³

2020

Preprint

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In this article, we study the geometry of compact quasi-Einstein manifolds with boundary. We establish sharp boundary estimates for compact quasi-Einstein manifolds with boundary that improve some previous results. Moreover, we obtain a characterization theorem for such manifolds in terms of the surface gravity of the boundary components, which leads to a new sharp geometric inequality. In addition, we prove an integral curvature estimate involving the Yamabe constant, which allows us to get an obstruction result to the existence of new examples.

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Section: Example 2 ([22]mentioning

confidence: 96%

Geometry of compact quasi-Einstein manifolds with boundary

Diógenes¹,

Gadelha²,

Ribeiro³

2020

Preprint

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“…Inspired by [15], we consider the Schwarzschild-de Sitter spacetime. The Schwarzschild-de Sitter metric in d + 1 dimensions is given by [8]…”

Section: Schwarzschild-de Sittermentioning

confidence: 99%

The Penrose property with a cosmological constant

Cameron

2022

Class. Quantum Grav.

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A spacetime satisfies the non-timelike boundary version of the Penrose property if the timelike future of any point on I<sup>−< contains the whole of I<sup>+<. This property was first discussed for asymptotically flat spacetimes by Penrose, along with an equivalent definition (the finite version). In this paper we consider the Penrose property in greater generality. In particular we consider spacetimes with a non-zero cosmological constant and we note that the two versions of the property are no longer equivalent. In asymptotically AdS spacetimes it is necessary to re-state the property in a way which is more suited to spacetimes with a timelike boundary. We arrive at a property previously considered by Gao and Wald. Curiously, this property was shown to fail in spacetimes which focus null geodesics. This is in contrast to our findings in asymptotically flat and asymptotically de Sitter spacetimes. We then move on to consider some further example spacetimes (with zero cosmological constant) which highlight features of the Penrose property not previously considered. We discuss spacetimes which are the product of a Lorentzian and a compact Riemannian manifold. Perhaps surprisingly, we find that both versions of the Penrose property are satisfied in this product spacetime if and only if they are satisfied in the Lorentzian spacetime only. We also discuss the Ellis-Bronnikov wormhole (an example of a spacetime with more than one asymptotically flat end) and the Hayward metric (an example of a non-singular black hole spacetime).

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“…The consequences obtained in Theorem 4 enable us to establish static uniqueness. There are several existing static uniqueness for ALH manifolds assuming various boundary conditions by, for example, X. Wang, G. Galloway and E. Woolgar, P. Chruściel, G. Galloway, and Y. Potaux [13,19,34]. It appears that the boundary conditions which naturally arise in our mass minimizing problem are different.…”

Section: ]): (mentioning

confidence: 99%

“…T. Paetz [12,Theorem 1.1] (see also the remark from [13,Theorem VI.4] on the mean curvature assumption).…”

Section: ]): (mentioning

confidence: 99%

“…By Theorem 4, it suffices to establish static uniqueness, Corollary 6.3 below. There have been various static uniqueness results by, for example, [13,19,34]. The boundary conditions that naturally arise in our setting seem to be different from theirs, but the proof is also based on the following fundamental identity of Y. Shen [31] and X. Wang [34] (see also Lemma 4.4 and Proof of Theorem 2 in [22]): Let (M, g) be a static, ALH manifold with boundary Σ.…”

Section: Static Uniqueness and Rigidity Of The Positive Mass Theoremsmentioning

confidence: 99%

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Scalar curvature deformation and mass rigidity for ALH manifolds with boundary

Huang¹,

Jang²

2021

Preprint

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We study scalar curvature deformation for asymptotically locally hyperbolic (ALH) manifolds with nonempty compact boundary. We show that the scalar curvature map is locally surjective among either (1) the space of metrics that coincide exponentially toward the boundary, or (2) the space of metrics with arbitrarily prescribed nearby Bartnik boundary data. Using those results, we characterize the ALH manifolds that minimize the Wang-Chruściel-Herzlich mass integrals in great generality and establish the rigidity of the positive mass theorems.

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Uniqueness and energy bounds for static AdS metrics

Cited by 8 publications

References 38 publications

Geometry of compact quasi-Einstein manifolds with boundary

Geometry of compact quasi-Einstein manifolds with boundary

The Penrose property with a cosmological constant

Scalar curvature deformation and mass rigidity for ALH manifolds with boundary

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