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

Abstract

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.

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

“…Further evidences in favour of the virtual mass will be presented in the forthcoming paper [11], where sharp area bounds will be obtained for horizons of black hole and cosmological type, the equality case being characterised by the Schwarzschild-de Sitter solution (1.8).…”

Monotonicity formulas in potential theory

“…The reason for the name is that when u is rotationally symmetric, then g is the cylindrical metric. Before proceeding, we recall that the same strategy described here is at the basis of the results of [5] and of [8,7], where static metrics and the associated static potentials are considered in place of the Euclidean metric and the corresponding electrostatic potential. The cylindrical ansatz leads to a reformulation of problem (1.1) where the new metric g and the g-harmonic function ϕ = − log u fulfill the quasi-Einstein type equation…”

Symmetry results for Serrin-type problems in doubly connected domains

“…In fact, the most part of our arguments (with the notable exception of the Pohozaev identity) do not rely on the structure of the Euclidean space at all. As a couple of examples of further applications, this method has proven to be quite successful to characterize static spacetimes in General Relativity [10,11,12,13] and somewhat similar techniques have been employed for other problems in General Relativity [4,21] and for p-harmonic functions in manifolds with nonnegative Ricci curvature in [3,20].…”

“…One of the most notable ones is the paper [9] and the subsequent developments in [15,26], where a gradient comparison argument very much resembling Theorem 3.1 has been obtained and exploited for static spacetimes in General Relativity. The introduction of the pseudo-radial function is instead more recent (it was exploited in the series of papers [11,12,13] for static spacetimes). This function is really helpful as it allows to have an explicit formula for W 0 .…”