Uniqueness of Kottler spacetime and Besse conjecture

TL;DR

This paper proves a uniqueness theorem for Schwarzschild-de Sitter (Kottler) spacetime, characterizing its interior domain and confirming the Besse conjecture using PDE and differential geometry techniques.

Contribution

It establishes a new uniqueness theorem for Kottler spacetime and verifies the Besse conjecture through novel geometric analysis methods.

Findings

Characterization of the interior domain of Kottler spacetime

Proof of the black hole uniqueness theorem for Schwarzschild-de Sitter

Validation of the Besse conjecture in the Riemannian setting

Abstract

We establish a black hole uniqueness theorem for Schwarzschild-de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular maximal level set of the lapse function. We provide a characterization of the interior domain of communication of the Kottler spacetime, which surrounds an inner horizon and is surrounded by a cosmological horizon. The proposed proof combines arguments from the theory of partial differential equations and differential geometry, and is centered on a detailed study of a possibly singular foliation. We also apply our technique in the Riemannian setting, and establish the validity of the so-called Besse conjecture.