@article{AnderssonMetzger2010,
  author    = {Andersson, Lars and Metzger, Jan},
  title     = {Curvature estimates for stable marginally trapped surfaces},
  issn      = {0022-040X},
  year      = {2010},
  abstract  = {We derive local integral and sup-estimates for the curvature of stable marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situations of physical interest, such as dynamical horizons.},
  language  = {en}
}
@article{EichmairMetzger2013,
  author    = {Eichmair, Michael and Metzger, Jan},
  title     = {Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions},
  series = {Inventiones mathematicae},
  volume    = {194},
  journal   = {Inventiones mathematicae},
  number    = {3},
  publisher = {Springer},
  address   = {New York},
  issn      = {0020-9910},
  doi       = {10.1007/s00222-013-0452-5},
  pages     = {591 -- 630},
  year      = {2013},
  language  = {en}
}
@article{EichmairMetzger2013,
  author    = {Eichmair, Michael and Metzger, Jan},
  title     = {Large isoperimetric surfaces in initial data sets},
  series = {Journal of differential geometry},
  volume    = {94},
  journal   = {Journal of differential geometry},
  number    = {1},
  publisher = {International Press of Boston},
  address   = {Somerville},
  issn      = {0022-040X},
  pages     = {159 -- 186},
  year      = {2013},
  abstract  = {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.},
  language  = {en}
}
@article{EichmairMetzger2012,
  author    = {Eichmair, Michael and Metzger, Jan},
  title     = {On large volume preserving stable CMC surfaces in initial data sets},
  series = {Journal of differential geometry},
  volume    = {91},
  journal   = {Journal of differential geometry},
  number    = {1},
  publisher = {International Press of Boston},
  address   = {Somerville},
  issn      = {0022-040X},
  pages     = {81 -- 102},
  year      = {2012},
  abstract  = {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.},
  language  = {en}
}
@article{EichmairMetzger2016,
  author    = {Eichmair, Michael and Metzger, Jan},
  title     = {JENKINS-SERRIN-TYPE RESULTS FOR THE JANG EQUATION},
  series = {Journal of differential geometry},
  volume    = {102},
  journal   = {Journal of differential geometry},
  publisher = {International Press of Boston},
  address   = {Somerville},
  issn      = {0022-040X},
  doi       = {10.4310/jdg/1453910454},
  pages     = {207 -- 242},
  year      = {2016},
  abstract  = {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.},
  language  = {en}
}
@article{LammMetzger2013,
  author    = {Lamm, Tobias and Metzger, Jan},
  title     = {Minimizers of the willmore functional with a small area constraint},
  series = {ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE},
  volume    = {30},
  journal   = {ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE},
  number    = {3},
  publisher = {Elsevier},
  address   = {Paris},
  issn      = {0294-1449},
  doi       = {10.1016/j.anihpc.2012.10.003},
  pages     = {497 -- 518},
  year      = {2013},
  abstract  = {We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we partially classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.},
  language  = {en}
}
@article{LammMetzgerSchulze2011,
  author    = {Lamm, Tobias and Metzger, Jan and Schulze, Felix},
  title     = {Foliations of asymptotically flat manifolds by surfaces of Willmore type},
  series = {Mathematische Annalen},
  volume    = {350},
  journal   = {Mathematische Annalen},
  number    = {1},
  publisher = {Springer},
  address   = {New York},
  issn      = {0025-5831},
  doi       = {10.1007/s00208-010-0550-2},
  pages     = {1 -- 78},
  year      = {2011},
  abstract  = {The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Thus our result has applications in the theory of general relativity.},
  language  = {en}
}
@article{MoestaAnderssonMetzgeretal.2015,
  author    = {Moesta, Philip and Andersson, Lars and Metzger, Jan and Szilagyi, Bela and Winicour, Jeffrey},
  title     = {The merger of small and large black holes},
  series = {Classical and quantum gravit},
  volume    = {32},
  journal   = {Classical and quantum gravit},
  number    = {23},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0264-9381},
  doi       = {10.1088/0264-9381/32/23/235003},
  pages     = {20},
  year      = {2015},
  abstract  = {We present simulations of binary black-hole mergers in which, after the common outer horizon has formed, the marginally outer trapped surfaces (MOTSs) corresponding to the individual black holes continue to approach and eventually penetrate each other. This has very interesting consequences according to recent results in the theory of MOTSs. Uniqueness and stability theorems imply that two MOTSs which touch with a common outer normal must be identical. This suggests a possible dramatic consequence of the collision between a small and large black hole. If the penetration were to continue to completion, then the two MOTSs would have to coalesce, by some combination of the small one growing and the big one shrinking. Here we explore the relationship between theory and numerical simulations, in which a small black hole has halfway penetrated a large one.},
  language  = {en}
}