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.
Araujo,Critical points of the total scalar curvature plus total mean curvature functional, Indiana Univ
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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.