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  - 
TY  - JOUR
A1  - Eichmair, Michael
A1  - Metzger, Jan
T1  - On large volume preserving stable CMC surfaces in initial data sets
JF  - Journal of differential geometry
N2  - Let (M, g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K subset of M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the results of G. Huisken and S.-T. Yau [17] and of J. Qing and G. Tian [26] on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass m > 0. The analysis in [17] and [26] takes place in the asymptotic regime of M. Here we adapt ideas from the minimal surface proof of the positive mass theorem [32] by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.
Y1  - 2012
SN  - 0022-040X
VL  - 91
IS  - 1
SP  - 81
EP  - 102
PB  - International Press of Boston
CY  - Somerville
ER  - 
TY  - JOUR
A1  - Eichmair, Michael
A1  - Metzger, Jan
T1  - Large isoperimetric surfaces in initial data sets
JF  - Journal of differential geometry
N2  - We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C-0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.
Y1  - 2013
SN  - 0022-040X
VL  - 94
IS  - 1
SP  - 159
EP  - 186
PB  - International Press of Boston
CY  - Somerville
ER  - 
TY  - JOUR
A1  - Eichmair, Michael
A1  - Metzger, Jan
T1  - Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions
JF  - Inventiones mathematicae
Y1  - 2013
U6  - https://doi.org/10.1007/s00222-013-0452-5
SN  - 0020-9910
SN  - 1432-1297
VL  - 194
IS  - 3
SP  - 591
EP  - 630
PB  - Springer
CY  - New York
ER  -