Uniformization with Infinitesimally Metric Measures

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 maps $$f:X \rightarrow {{\mathbb {R}}}^2$$ , 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.