Quasiconformal geometry of monotone mappings

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 Holder 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.