Open AccessUniformization of metric surfaces using isothermal coordinates
Author(s) -
Toni Ikonen
Publication year - 2021
Publication title -
annales fennici mathematici
Language(s) - English
Resource type - Journals
eISSN - 2737-114X
pISSN - 2737-0690
DOI - 10.54330/afm.112781
Subject(s) - uniformization (probability theory) , metric (unit) , mathematics , uniformization theorem , intrinsic metric , hausdorff distance , equivalence of metrics , injective metric space , euclidean geometry , pure mathematics , topology (electrical circuits) , surface (topology) , hausdorff space , metric space , convex metric space , fisher information metric , mathematical analysis , geometry , geometric function theory , combinatorics , riemann surface , riemann–hurwitz formula , statistics , balance equation , operations management , markov model , markov chain , economics
We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.