We single out a notion of staticity which applies to any domain in hyperbolic
space whose boundary is a non-compact totally umbilical hypersurface. For
(time-symmetric) initial data sets modeled at infinity on any of these latter
examples, we formulate and prove a positive mass theorem in the spin category
under natural dominant energy conditions (both in the interior and along the
boundary) whose rigidity statement retrieves, among other things, a sharper
version of a recent result by Souam to the effect that no such hypersurface
admits a compactly supported deformation keeping the original lower bound on
the mean curvature. A key ingredient in our approach is the consideration of a
family of elliptic boundary conditions on spinors interpolating between
chirality and MIT bag boundary conditions.
