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Jenkins-Serrin-type results for the Jang equation. (English) Zbl 1338.53089
The authors prove the existence of solutions for a Plateau problem for marginally trapped outer stable surfaces (MOTs). This is an ingredient in the proof of the spacetime positive mass theorem given by the first author et al. [J. Eur. Math. Soc. (JEMS) 18, No. 1, 83–121 (2016; Zbl 1341.53067)]. The authors show that a canonical solution of the Jang equation exists in the complement of the union of all weakly future outer trapped surfaces. The graph of this solution relates the area of horizon to the global geomety of the initial data set in a non-trivial way.
The authors prove the existence of Scherk-type solutions of the Jang equations outside the union of all weakly future or past outer trapped regions in the initial data set. This result is a natural exterior analog of the Jang equation from the classical Jenkins-Serrin theory.
The authors extend and complement existence theorems for Scherk-type constant mean curvature graphs. The article has a connection with the papers by G. J. Galloway and R. Schoen [Commun. Math. Phys. 266, No. 2, 571–576 (2006; Zbl 1190.53070)] and G. J. Galloway and N. Ó. Murchadha [Classical Quantum Gravity 25, No. 10, Article ID 105009, 9 p. (2008; Zbl 1140.83373)], treating problems of black holes physics and geometry.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
83C10 Equations of motion in general relativity and gravitational theory
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