Staticity of asymptotically hyperbolic minimal mass extensions

In this paper, we define the Bartnik mass of a domain whose boundary is connected and compact, has scalar curvature bounded below −n(n − 1), and whose extensions are asymptotically hyperbolic manifolds. With this definition, we show that asymptotically hyperbolic admissible extensions of a domain that achieve the Bartnik mass must admit a static potential. Given a non-static admissible extension of a domain, we are able to construct a one-parameter family of metrics that are close to the original metric, have smaller mass, share the same bound on the scalar curvature, and contain the domain isometrically.