On the mass of static metrics with positive cosmological constant: I

Borghini, Stefano; Mazzieri, Lorenzo

doi:10.1088/1361-6382/aac081

Cited by 31 publications

(43 citation statements)

References 58 publications

(97 reference statements)

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“…The virtual mass. As already discussed in [13], in the case of a positive cosmological constant there does not seem to be a general consensus about what the right notion of mass should be. For some possible approaches, as well as for more insights on the problems posed by the case Λ > 0, we refer the reader to the following references [1,6,7,8,22,34,43,50,51,54].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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On the Mass of Static Metrics with Positive Cosmological Constant: II

Borghini

Mazzieri

2020

Commun. Math. Phys.

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This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds à la Boucher–Gibbons–Horowitz, respectively. Building on the related rigidity statements, we also deduce a uniqueness result for the Schwarzschild–de Sitter spacetime.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…For some possible approaches, as well as for more insights on the problems posed by the case Λ > 0, we refer the reader to the following references [1,6,7,8,22,34,43,50,51,54]. In our previous work [13], we have introduced a different point of view, leading to a new notion of mass, that we now recall.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Mass of Static Metrics with Positive Cosmological Constant: II

Borghini

Mazzieri

2020

Commun. Math. Phys.

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“…Equation (VIII.5) below.When Λ is positive we review the argument of [2] (compare [4]), that the identity mentioned above provides a simple proof of an upper bound for entropy of horizons, Equation (IX.10) below. This upper bound has been rediscovered in [5,6], with different proofs.This work can be thought-of as a continuation of [7]. It reviews, in a slightly different manner, some results from [7] and considers further applications.…”

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confidence: 99%

“…When Λ is positive we review the argument of [2] (compare [4]), that the identity mentioned above provides a simple proof of an upper bound for entropy of horizons, Equation (IX.10) below. This upper bound has been rediscovered in [5,6], with different proofs.…”

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Uniqueness and energy bounds for static AdS metrics

2020

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3. We prove an upper bound, suggested in [3], on the free energy of a connected static black hole in terms of the genus of the horizon, cf. Equation (VIII.5) below.When Λ is positive we review the argument of [2] (compare [4]), that the identity mentioned above provides a simple proof of an upper bound for entropy of horizons, Equation (IX.10) below. This upper bound has been rediscovered in [5,6], with different proofs.This work can be thought-of as a continuation of [7]. It reviews, in a slightly different manner, some results from [7] and considers further applications. Similar ideas have been independently pursued in [8]. II. THE EQUATIONSWe consider the vacuum Einstein equations with a cosmological constant Λ for a static spacetime metric which we denote byḡ:ḡ(II.1)We always assume that V ≡ 0 and that (M, g) is complete, possibly with boundary. We shall say that (M, g, V ) is strictly static if V > 0 on M .

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Monotonicity Formulas for Static Metrics with Non-zero Cosmological Constant

Borghini

Mazzieri

2019

Contemporary Research in Elliptic PDEs and Related Topics

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In this paper we adopt the approach presented in [1,2] to study non-singular vacuum static space-times with non-zero cosmological constant. We introduce new integral quantities, and under suitable assumptions we prove their monotonicity along the level set flow of the static potential. We then show how to use these properties to derive a number of sharp geometric and analytic inequalities, whose equality case can be used to characterize the rotational symmetry of the underlying static solutions. As a consequence, we are able to prove some new uniqueness statements for the de Sitter and the anti-de Sitter metrics. In particular, we show that the de Sitter solution has the least possible surface gravity among three-dimensional static metrics with connected boundary and positive cosmological constant. MSC (2010): 35B06, 53C21, 83C57, 35N25.

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On the mass of static metrics with positive cosmological constant: I

Cited by 31 publications

References 58 publications

On the Mass of Static Metrics with Positive Cosmological Constant: II

On the Mass of Static Metrics with Positive Cosmological Constant: II

Uniqueness and energy bounds for static AdS metrics

Monotonicity Formulas for Static Metrics with Non-zero Cosmological Constant

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