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

25 pages, 1 figure