STEFAN HILDEBRANDT and HEIKO VON DER MOSEL
Conformal representation of surfaces, and Plateau's problem for Cartan functionals
Received: 12 July 2005 Revised: 8 August 2005
Mathematics Subject Classification (2000): 49J45 - 49N60 - 49Q05 - 49Q10 - 53A07 - 53A10 - 53C42
Abstract: This survey article presents the existence and regularity theory for Cartan functionals, i.e., for general parameter invariant double integrals defined on parametric surfaces with arbitrary codimension. We also discuss the closely related problem of finding globally conformal parametrizations for surfaces or two-dimensional Riemannian metrics by direct minimization of the area functional as a particular Cartan functional. With this new approach we also establish conformal representations of Fréchet surfaces and provide an alternative proof of Lichtenstein's theorem on globally conformal mappings.