Uniformization of two-dimensional metric surfaces
Abstract
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.
Main Author
Format
Articles
Research article
Published
2017
Series
Subjects
Publication in research information system
Publisher
Springer
The permanent address of the publication
https://urn.fi/URN:NBN:fi:jyu-201702141437Käytä tätä linkitykseen.
Review status
Peer reviewed
ISSN
0020-9910
DOI
https://doi.org/10.1007/s00222-016-0686-0
Language
English
Published in
Inventiones mathematicae
Citation
- Rajala, K. (2017). Uniformization of two-dimensional metric surfaces. Inventiones mathematicae, 207(3), 1301-1375. https://doi.org/10.1007/s00222-016-0686-0
License
Copyright© Springer-Verlag Berlin Heidelberg 2016. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.