  We establish the positive energy theorem and a Penrose-type inequality for
3-dimensional asymptotically hyperboloidal initial data sets with toroidal
infinity, weakly trapped boundary, and satisfying the dominant energy
condition. In the umbilic case, a rigidity statement is proven showing that the
total energy vanishes precisely when the initial data manifold is isometric to
a portion of the canonical slice of the associated Kottler spacetime.
Furthermore, we provide a new proof of the recent rigidity theorems of
Eichmair-Galloway-Mendes [10] in dimension 3, with weakened hypotheses in
certain cases. These results are obtained through an analysis of the level sets
of spacetime harmonic functions.
