Uniformization of two-dimensional metric surfaces

• 成果类型: Article
• 署名作者: Rajala, Kai
• 署名单位: University of Jyvaskyla
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0686-0
• 发表日期: 2017
• 页码: 1301-1375
• 关键词: quasi-symmetric parametrizations; good parameterizations; Absolute continuity; conformal mappings; measure-spaces; MAPS; quasiconformality; inequalities; EMBEDDINGS; CURVATURE

摘要

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.