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

Rigidity of non-compact static domains in hyperbolic space via positive mass theorems