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Uniqueness theorems for static spacetimes containing marginally outer trapped surfaces

Marginally outer trapped surfaces are widely considered as the best quasi-local replacements for event horizons of black holes in General Relativity. However, this equivalence is far from being proved, even in stationary and static situations. In this paper we study an important aspect of this equivalence, namely whether classic uniqueness theorems of static black holes can be extended to static spacetimes containing weakly outer trapped surfaces or not. Our main theorem states that, under reasonable hypotheses, a static spacetime satisfying the null energy condition and containing an asymptotically flat initial data set, possibly with boundary, which possesses a bounding weakly outer trapped surface is a unique spacetime. A related result to this theorem was given in arXiv:0711.1299, where we proved that no bounding weakly outer trapped surface can penetrate into the exterior region of the initial data where the static Killing vector is timelike. In this paper, we also fill some gaps in arXiv:0711.1299 and extend this confinement result to initial data sets with boundary.