On the uniqueness of Schwarzschild-de Sitter spacetime

TL;DR

This paper proves a new uniqueness theorem for three-dimensional Schwarzschild-de Sitter spacetimes, introducing novel mathematical tools like a reverse Lojasiewicz inequality and analyzing the geometry of the lapse function's maxima.

Contribution

It develops new mathematical tools and a classification strategy for static Einstein solutions with positive cosmological constant, enhancing understanding of Schwarzschild-de Sitter spacetime uniqueness.

Findings

Established a new uniqueness theorem for 3D Schwarzschild-de Sitter metrics.

Developed a reverse Lojasiewicz inequality near extremal points.

Proved smoothness of the maximum set of the lapse function.

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 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.