The Plateau Problem in Alexandrov spaces

We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in a certain class of Alexandrov spaces. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison along with an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel'man. The key is to 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 Holder continuous in the interior and continuous up to the boundary.