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.

我们考虑拟形映射的扩展和统一定理，以度量空间X同胚为\（{{\ mathbb {R}}} ^ 2 \）的设置。给定这样一个空间上的度量\（\ mu \），我们引入\（\ mu \） -拟形映射 \（f：X \ rightarrow {{\ mathbb {R}}} ^ 2 \），其定义涉及变形曲线的长度以\（\ mu \）表示。我们证明如果\（\ mu \）是一个无限量度度量，即它满足David和Semmes的度量倍增条件的无穷小形式，则该\（\ mu \）-准保形映射存在。我们应用该结果来给出度量空间的特征，以允许无限模拟准对称的参数化。