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.

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