In arXiv:1310.3002, Lee and Neves proved a Penrose inequality for spacetimes with negative cosmological constant and nonpositive mass aspect. As an application, they were able to obtain a static uniqueness theorem for Kottler spacetimes with nonpositive mass. In this paper, we propose an alternative more elementary proof of this static uniqueness result and we discuss some possible generalizations, most notably to degenerate horizons.
We show that Wang's proof of uniqueness of Anti-de Sitter spacetime can be adapted to provide uniqueness results for strictly static asymptotically locally hyperbolic vacuum metrics with toroidal infinity, and to prove negativity of the free energy $E-TS$ of asymptotically AdS black holes with higher-genus horizons.
We show a uniqueness result for the n-dimensional spatial Reissner-Nordström manifold: a static, electrovacuum, asymptotically flat system which is asymptotically Reissner-Nordström is a subextremal Reissner-Nordström manifold with positive mass, provided that its inner boundary is a (possibly disconnected) photon sphere that fulfils a suitably defined quasilocal subextremality condition. Our result implies a number of earlier uniqueness results for the Schwarzschild and the Reissner-Nordström manifolds in the static, (electro-)vacuum, asymptotically flat context, both for photon sphere and bl…
Uniqueness of equipotential photon surfaces in 4-dimensional static vacuum asymptotically flat spacetimes for positive, negative, and zero mass -- and a new partial proof of the Willmore inequality
We establish a new uniqueness theorem for the three dimensional Schwarzschild-de Sitter metrics. For this some new or improved tools are developed. These include a reverse Lojasiewicz inequality, which holds in a neighborhood of the extremal points of any smooth function. We further prove smoothness of the set of maxima of the lapse, whenever this set contains a topological hypersurface. This leads to a new strategy for the classification of well behaved static solutions of Einstein equations with a positive cosmological constant, based on the geometry of the maximum-set of the lapse.
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.…
This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumption on the topology or the asymptotic. Specifically, it is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black h…
The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we extend this result by proving a classification theorem for all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without making any further assumption on the topology or the asymptotic. It is shown that any …
