Rigidity of outermost MOTS: the initial data version

In the paper Commun Anal Geom 16(1):217–229, 2008, a rigidity result was obtained for outermost marginally outer trapped surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This allowed one to treat the “borderline case” in the author’s work with R. Schoen concerning the topology of higher dimensional black holes (Commun Math Phys 266(2):571–576, 2006). The proof of this rigidity result involved bending the initial data manifold in the vicinity of the MOTS within the ambient spacetime. In this note we show how to circumvent this step, and thereby obtain a pure initial data version of this rigidity result and its consequence concerning the topology of black holes.