Canonical parametrizations of metric discs

Abstract

We use the recently established existence and regularity of area and energy minimizing discs in metric spaces to obtain canonical parametrizations 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 parametrizations of linearly locally connected, Ahlfors $2$-regular metric $2$-spheres. Generalizations and applications to the geometry of such surfaces are described.

References in corpus (2)

• Gromov hyperbolic spaces and the sharp isoperimetric constant
• Area minimizing discs in metric spaces

Cited by in corpus (6)

• Polyhedral approximation of metric surfaces and applications to uniformization
• Non-removability of the Sierpinski Gasket
• Potential theory on Sierpinski carpets with applications to uniformization
• Area minimizing discs in locally non-compact metric spaces
• Isoperimetric inequalities vs. upper curvature bounds
• Quasispheres and metric doubling measures