Some splitting and rigidity results for sub-static spaces

TL;DR

This paper investigates rigidity and splitting theorems for sub-static spaces, extending boundary inequalities and analyzing static Einstein solutions with a generalized Liouville theorem.

Contribution

It provides new local and global splitting results, boundary inequalities for non-vacuum spaces, and a generalized Liouville theorem for static Einstein systems.

Findings

Established local and global splitting theorems under minimal hypersurface conditions.

Extended boundary integral inequalities to non-vacuum sub-static spaces.

Proved a Liouville theorem allowing positively curved target manifolds in Einstein-sigma models.

Abstract

In this paper we study the rigidity problem for sub-static systems with possibly non-empty boundary. First, we get local and global splitting theorems by assuming the existence of suitable compact minimal hypersurfaces, complementing recent results in the literature. Next, we prove some boundary integral inequalities that extend works by Chr\'usciel and Boucher-Gibbons-Horowitz to non-vacuum spaces. Even in the vacuum static case, the inequalities improve on known ones. Lastly, we consider the system arising from static solutions to the Einstein field equations coupled with a $\sigma$-model. The Liouville theorem we obtain allows for positively curved target manifolds, generalizing a result by Reiris.