## Journal of Differential Geometry

- J. Differential Geom.
- Volume 102, Number 2 (2016), 207-242.

### Jenkins–Serrin-type results for the Jang equation

Michael Eichmair and Jan Metzger

#### 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 raised in “Some remarks on the size of bodies and black holes,” [Classical Quantum Gravity 25 (2008), no. 10, 105009, 9. MR 2416045] 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 named 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 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.

#### Article information

**Source**

J. Differential Geom., Volume 102, Number 2 (2016), 207-242.

**Dates**

First available in Project Euclid: 27 January 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1453910454

**Digital Object Identifier**

doi:10.4310/jdg/1453910454

**Mathematical Reviews number (MathSciNet)**

MR3454546

**Zentralblatt MATH identifier**

1338.53089

#### Citation

Eichmair, Michael; Metzger, Jan. Jenkins–Serrin-type results for the Jang equation. J. Differential Geom. 102 (2016), no. 2, 207--242. doi:10.4310/jdg/1453910454. https://projecteuclid.org/euclid.jdg/1453910454