2024/05/13 by Damaris Meier, Meier, Damaris, Kai Rajala +1 · 1 citation
Mathematics · #30F10 (Secondary) #30L10 (Primary) 30C65 #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2405.07476
openalex publication_date 2024/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explore the interplay between different definitions of distortion for mappings f\colon X→ ℝ2, where X is any metric surface, meaning that X is homeomorphic to a domain in ℝ2 and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces.