Statistics

Mathematical analysis

Hausdorff space

Operations management

Hausdorff measure

Database

Physics

Metric (unit)

Markov chain

Euclidean geometry

Riemann–Hurwitz formula

Balance equation

Riemann surface

Computer science

Astronomy

Plane (geometry)

Uniformization theorem

Quantum mechanics

Markov model

Metric space

Pure mathematics

Parametrization (atmospheric modeling)

Mathematics

Bounded function

Radiative transfer

Geometric function theory

Uniformization (probability theory)

Hausdorff dimension

Measure (data warehouse)

Economics

Geometry

SPHERES

Inventiones mathematicae

We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk–Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.

Uniformization of two-dimensional metric surfaces