2019/07/16 by Kai Rajala, Rajala, Kai, Martti Rasimus +3
Mathematics · #28A75 #30C65 #30L10 (Primary) #51F99 (Secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1907.07124
openalex publication_date 2019/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to \mathbb R2. Given a measure μ on such a space, we introduce μ-quasiconformal maps f:X → \mathbb 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.