Abstract In this work, we present several rigidity results for compact free boundary hypersurfaces in initial data sets with boundary. Specifically, in the first part of the paper, we extend the local splitting theorems from G. J. Galloway and H. C. Jang, Some scalar curvature warped product splitting theorems , Proc. Am. Math. Soc. 148 (2020), no. 6,2617–2629 to the setting of manifolds with boundary. To achieve this, we build on the approach of the original paper, utilizing results on free boundary marginally outer trapped surfaces (MOTS) applied to specific initial data sets. In the second part, we extend the main results from A. Barros and C. Cruz, Free boundary hypersurfaces with non-positive Yamabe invariant in mean convex manifolds , J. Geom. Anal. 30 (2020), no. 4, 3542–3562 to the context of free boundary MOTS in initial data sets with boundary.
Almeida et al. (2026) studied this question.