2015/02/28 by Alexander Lytchak, Stefan Wenger
Mathematics · #math.DG #math.AP #math.MG #msc:49Q05 #msc:53C23
paper · pdf · doi:10.1007/s00205-016-1054-3
typos corrected, some comments added, minor changes to the exposition at some places
arxiv created 2015/07/16 · arxiv updated 2016/11/23
We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove 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 we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.