Canonical parameterizations of metric disks

Abstract

We use the recently established existence and regularity of area and energy minimizing disks in metric spaces to obtain canonical parameterizations of metric surfaces. Our approach yields a new and conceptually simple proof of a well-known theorem of Bonk and Kleiner on the existence of quasisymmetric parameterizations of linearly locally connected, Ahlfors $2$-regular metric $2$-spheres. Generalizations and applications to the geometry of such surfaces are described.