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Uniformization of two-dimensional metric surfaces

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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.

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Acknowledgments

We are grateful to the anonymous referee for the careful reading of the manuscript and the thoughtful comments that lead to several improvements. We thank Mario Bonk, Changyu Guo, Jeff Lindquist, Dimitrios Ntalampekos, Martti Rasimus and Vyron Vellis for their comments and corrections.

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  1. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014, University of Jyväskylä, Finland

    Kai Rajala

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  1. Kai Rajala
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Correspondence to Kai Rajala.

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Research supported by the Academy of Finland, Project Number 257482. Parts of this research were carried out while the author was visiting the University of Michigan. He thanks the Department of Mathematics for hospitality.

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Rajala, K. Uniformization of two-dimensional metric surfaces. Invent. math. 207, 1301–1375 (2017). https://doi.org/10.1007/s00222-016-0686-0

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