Some splitting and rigidity results for sub-static spaces

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.