6533b872fe1ef96bd12d422a

RESEARCH PRODUCT

Uniformization of two-dimensional metric surfaces

Kai Rajala

subject

metric surfacesPure mathematicsMathematics - Complex VariablesGeneral Mathematics010102 general mathematicsPrimary 30L10 Secondary 30C65 28A75 51F99 52A38Hausdorff spaceMetric Geometry (math.MG)01 natural sciencesUpper and lower boundsMetric spaceMathematics - Metric GeometryBounded function0103 physical sciencesMetric (mathematics)Euclidean geometryFOS: MathematicsMathematics::Metric Geometry010307 mathematical physicsComplex Variables (math.CV)0101 mathematicsUniformization (set theory)ParametrizationMathematics

description

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. peerReviewed

yearjournalcountryeditionlanguage
2014-12-10Inventiones mathematicae
https://doi.org/10.1007/s00222-016-0686-0