Quasiconformal geometry of monotone mappings

Abstract

This paper concerns a class of monotone mappings, in a Hilbert space, that can be viewed as a nonlinear version of the class of positive invertible operators. Such mappings are proved to be open, locally Hölder continuous, and quasisymmetric. They arise naturally from the Beurling–Ahlfors extension and from Brenier's polar factorization and find applications in the geometry of metric spaces and the theory of elliptic partial differential equations.