On the Uniqueness of Schwarzschild–de Sitter Spacetime

Abstract

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 Łojasiewicz inequality, which holds in a neighborhood of the extremal points of any smooth function. We further prove the 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 vacuum Einstein equations with a positive cosmological constant, based on the geometry of the maximum-set of the lapse.

Geometry of static perfect fluid space-time

“…Lemma 1 can be seen as a Robinson-Shen type identity [47] for manifolds satisfying equation (2.1), which is a large class of spaces; see also [1,9,10]. In the case of constant scalar curvature, for example, static space or V-static space, we deduce that the expression in the left hand side of (2.6) is necessarily non-negative.…”

A Green’s Function Proof of the Positive Mass Theorem

“…This latter approach, inspired by [40], turned out to be flexible enough to also allow for the treatment of the space-time case [13,24], and, together with some of the computations carried out in [28,29], is probably the method that displays the largest number of analogy with ours. On this regard, it should be mentioned that level set methods, combined with Bochner technique and integral identities, have recently found some applications to the study of static metrics in general relativity [3,6], even when the function under consideration is not necessarily harmonic [10][11][12].…”