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The plateau problem in alexandrov spaces

Chikako Mese Johns Hopkins University Patrick R. Zulkowski University of California

Differential Geometry mathscidoc:1609.10185

Journal of Differential Geometry, 85, (2), 315-356, 2010
We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in Alexandrov spaces with curvature ≥ κ. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison. Imposing an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel’man, we develop a harmonic map theory from two dimensional domains into these spaces. In particular, we show that the solution to the Dirichlet problem from a disk is H¨older continuous in the interior and continuous up to the boundary. Using this theory, we solve the Plateau Problem in this setting generalizing classical results in Euclidean space (due to J. Douglas and T. Rado) and in Riemannian manifolds (due to C.B. Morrey).
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@inproceedings{chikako2010the,
  title={THE PLATEAU PROBLEM IN ALEXANDROV SPACES},
  author={Chikako Mese, and Patrick R. Zulkowski},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911220222005762842},
  booktitle={Journal of Differential Geometry},
  volume={85},
  number={2},
  pages={315-356},
  year={2010},
}
Chikako Mese, and Patrick R. Zulkowski. THE PLATEAU PROBLEM IN ALEXANDROV SPACES. 2010. Vol. 85. In Journal of Differential Geometry. pp.315-356. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911220222005762842.
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