Analysis & Applied Math

When does a map defined on a circle admit an extension to the disc with prescribed regularity? This classical question is the starting point of the talk. We begin by discussing some answers for maps into the Euclidean plane, before turning to maps into metric surfaces. In particular, we present a recent result guaranteeing the existence of monotone Sobolev extensions into metric surfaces with locally finite Hausdorff measure. Finally, we outline some applications, including the uniformization of metric surfaces via Sobolev energy minimizers. This is based on joint work with Damaris Meier and Stefan Wenger.