Open Access
June 2010 The plateau problem in Alexandrov spaces
Chikako Mese, Patrick R. Zulkowski
J. Differential Geom. 85(2): 315-356 (June 2010). DOI: 10.4310/jdg/1287580967


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 Hölder 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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Chikako Mese. Patrick R. Zulkowski. "The plateau problem in Alexandrov spaces." J. Differential Geom. 85 (2) 315 - 356, June 2010.


Published: June 2010
First available in Project Euclid: 20 October 2010

MathSciNet: MR2732979
zbMATH: 1250.53066
Digital Object Identifier: 10.4310/jdg/1287580967

Rights: Copyright © 2010 Lehigh University

Vol.85 • No. 2 • June 2010
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