Rigidity of free boundary MOTS

The aim of this work is to present an initial data version of Hawking's theorem on the topology of back hole spacetimes in the context of manifolds with boundary. More precisely, we generalize the results of G. J. Galloway and R. Schoen [13] and G. J. Galloway [11, 12] by proving that a compact free boundary stable marginally outer trapped surface (MOTS) $\Sigma$ in an initial data set with boundary satisfying natural dominant energy conditions (DEC) is of positive Yamabe type, i.e. $\Sigma$ admits a metric of positive scalar curvature with minimal boundary, provided $\Sigma$ is outermost. To do so, we prove that if $\Sigma$ is a compact free boundary stable MOTS which does not admit a metric of positive scalar curvature with minimal boundary in an initial data set satisfying the interior and the boundary DEC, then an outer neighborhood of $\Sigma$ can be foliated by free boundary MOTS $\Sigma_t$, assuming that $\Sigma$ is weakly outermost. Moreover, each $\Sigma_t$ has vanishing outward null second fundamental form, is Ricci flat with totally geodesic boundary, and the dominant energy conditions saturate on $\Sigma_t$.