Marginally outer trapped surface with capillary boundary and rigidity of initial data sets

Abstract

In this paper, we establish a series of rigidity results for initial data sets with boundary in the presence of marginally outer trapped surfaces (MOTSs) satisfying a capillary boundary condition. Our first result establishes area estimates for stable MOTS with capillary boundary. Building on this, we prove a rigidity theorem for three-dimensional initial data sets with boundary. We further extend the rigidity to higher dimensions by employing the Yamabe constant associated to the MOTS. Finally, under a low-index assumption, we obtain additional topological restrictions on MOTS with capillary boundary.