On stationary vacuum solutions to the Einstein equations

TL;DR

This paper proves that the only complete stationary vacuum solutions to Einstein's equations are Minkowski spaces or their quotients, and provides curvature bounds away from boundaries or horizons.

Contribution

It generalizes Lichnerowicz's classical result by classifying all geodesically complete stationary vacuum solutions and establishing curvature bounds.

Findings

Only Minkowski space or its quotients are geodesically complete stationary vacuum solutions.

Established an a priori curvature bound away from boundaries or horizons.

Extended classical results in Einstein vacuum solutions.

Abstract

It is proved that the only geodesically complete stationary vacuum solution of the Einstein equations is the empty Minkowski space, or a quotient of it by a discrete group of isometries, generalizing a classical result of Lichnerowicz. In addition, we obtain an apriori bound on the curvature of stationary vacuum solutions away from the boundary or horizon.