Jenkins-Serrin type results for the Jang equation

TL;DR

This paper establishes existence results for solutions to the Jang equation and related geometric problems in general relativity, extending classical theories and providing tools for the positive mass theorem.

Contribution

It proves the existence of solutions to the Jang equation and Plateau problem for MOTSs, extending Jenkins-Serrin theory to higher dimensions and removing previous assumptions.

Findings

Existence of stable MOTSs solutions in initial data sets.

Construction of Scherk-type solutions outside trapped regions.

Generalization of Jenkins-Serrin theory to higher dimensions.

Abstract

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.