Uniformization of two-dimensional metric surfaces

By Kai Rajala. 2014

73 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a922079e4510e1bc4ea6915d4a8de7218de7759c

How researchers see this paper

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In Applied Mathematics

1. S. Wenger
2. A. Lytchak
3. U. Lang
4. C. Sormani
5. Shin-ichi Ohta
6. Simeon Reich
7. Eva Kopecká

In Geometry and Topology

1. A. Lytchak
2. S. Wenger
3. U. Lang
4. C. Sormani
5. P. Caprace
6. Simeon Reich
7. Eva Kopecká

Citations in the corpus

• Intrinsic structure of minimal discs in metric spaces (2016, methodology): “This observation is used in Lytchak and Wenger [25] as the starting point of further investigations of all such discs Z , also shedding new light on the well-investigated theory of Ahlfors 2–regular discs; see Bonk and Kleiner [4] and Rajala [36].”