Some splitting and rigidity results for sub‐static spaces

Abstract In this paper, we study the rigidity problem for sub‐static systems with possibly nonempty 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 Chruściel and Boucher–Gibbons–Horowitz to nonvacuum spaces. Even in the vacuum static case, the inequalities improve on known ones. Finally, we consider the system arising from static solutions to the Einstein field equations coupled with a ‐model. The Liouville theorem we obtain allows for positively curved target manifolds, generalizing a result by Reiris.