Uniformization with Infinitesimally Metric Measures

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to R². Given a measure μ on such a space, we introduce μ-quasiconformal mapsf: X→ R², whose definition involves deforming lengths of curves by μ. We show that if μ 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 μ-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.