TY  - JOUR
A1  - Eichmair, Michael
A1  - Metzger, Jan
T1  - JENKINS-SERRIN-TYPE RESULTS FOR THE JANG EQUATION
JF  - Journal of differential geometry
N2  - Let (M, g, k) be an initial data set for the Einstein equations of general relativity. 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 [19, 20, 40, 29, 18, 31, 11] 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.
Y1  - 2016
U6  - https://doi.org/10.4310/jdg/1453910454
SN  - 0022-040X
SN  - 1945-743X
VL  - 102
SP  - 207
EP  - 242
PB  - International Press of Boston
CY  - Somerville
ER  -