On marginally outer trapped surfaces in stationary and static spacetimes

In this paper, we prove that for any spacelike hypersurface containing an untrapped barrier in a stationary spacetime satisfying the null energy condition, no bounding marginally outer trapped surface lying in the exterior region where the stationary Killing vector is causal and penetrating into the timelike region can exist. In the static case, we prove that any bounding marginally outer trapped surface cannot penetrate into the exterior region where the static Killing vector is timelike. In fact, we prove this result at an initial data level, without even assuming the existence of a spacetime. The proof relies on a powerful theorem by Andersson and Metzger on the existence of an outermost marginally outer trapped surface.