Uniformization of two-dimensional metric surfaces

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• 作者：Kai Rajala
• 关键词：Mathematics Subject ClassificationPrimary 30L10 ; Secondary 30C65 ; 28A75 ; 51F99 ; 52A38
• 刊名：Inventiones mathematicae
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：207
• 期：3
• 页码：1301-1375
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1297
• 卷排序：207

文摘

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.