  Let (M, g, k) be an initial data set for the Einstein equations of general
relativity. We prove that there exist solutions of the Plateau problem for
marginally outer trapped surfaces (MOTSs) that are stable in the sense of
MOTSs. This answers a question of G. Galloway and N. O'Murchadha and is an
ingredient in the proof of the spacetime positive mass theorem given by L.-H.
Huang, D. Lee, R. Schoen and the first author. We show that a canonical
solution of the Jang equation exists in the complement of the union of all
weakly future outer trapped regions in the initial data set with respect to a
given end, provided that this complement contains no weakly past outer trapped
regions. The graph of this solution relates the area of the horizon to the
global geometry of the initial data set in a non-trivial way. We prove the
existence of a Scherk-type solution of the Jang equation outside the union of
all weakly future or past outer trapped regions in the initial data set. This
result is a natural exterior analogue for the Jang equation of the classical
Jenkins--Serrin theory. We extend and complement existence theorems by
Jenkins-Serrin, Spruck, Nelli-Rosenberg, Hauswirth-Rosenberg-Spruck, and
Pinheiro for Scherk-type constant mean curvature graphs over polygonal domains
in (M, g), where (M, g) is a complete Riemannian surface. We can dispense with
the a priori assumptions that a sub solution exists and that (M, g) has
particular symmetries. Also, our method generalizes to higher dimensions.
