Uniformization with Infinitesimally Metric Measures

Abstract

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to {{mathbb {R}}}^2. Given a measure mu on such a space, we introduce mu -quasiconformal mapsf:X rightarrow {{mathbb {R}}}^2, whose definition involves deforming lengths of curves by mu . We show that if mu is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a mu -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.