For stable MOTS with capillary boundary, equality in the area estimate forces the contact angle to 90 degrees and the ambient initial data set to split as a product, under stated energy assumptions.
Rigidity results for free boundary hypersurfaces in initial data sets with boundary
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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.
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Marginally outer trap surface with capillary boundary and rigidity of initial data sets
For stable MOTS with capillary boundary, equality in the area estimate forces the contact angle to 90 degrees and the ambient initial data set to split as a product, under stated energy assumptions.