## Journal of Differential Geometry

### The plateau problem in Alexandrov spaces

#### Abstract

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).

#### Article information

Source
J. Differential Geom., Volume 85, Number 2 (2010), 315-356.

Dates
First available in Project Euclid: 20 October 2010

https://projecteuclid.org/euclid.jdg/1287580967

Digital Object Identifier
doi:10.4310/jdg/1287580967

Mathematical Reviews number (MathSciNet)
MR2732979

Zentralblatt MATH identifier
1250.53066

#### Citation

Mese, Chikako; Zulkowski, Patrick R. The plateau problem in Alexandrov spaces. J. Differential Geom. 85 (2010), no. 2, 315--356. doi:10.4310/jdg/1287580967. https://projecteuclid.org/euclid.jdg/1287580967