Area minimizing discs in metric spaces

In this talk, I will discuss a solution to the classical problem of Plateau in the setting of proper metric spaces. Precisely, I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves then I will show that such a solution is locally Hoelder continuous in the interior and continuous up to the boundary. These results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces. This is joint work with Alexander Lytchak.