Uniformization of metric surfaces: A survey
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915273772105728 |
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| author | Ntalampekos, Dimitrios |
| author_facet | Ntalampekos, Dimitrios |
| contents | In this survey we present the most recent developments in the uniformization of metric surfaces, i.e., metric spaces homeomorphic to two-dimensional topological manifolds. We start from the classical conformal uniformization theorem of Koebe and Poincaré. Then we discuss the Bonk-Kleiner theorem on the quasisymmetric uniformization of metric spheres, which marks the beginning of the study of the uniformization problem on fractal surfaces. The next result presented is Rajala's theorem on the quasiconformal uniformization of metric spheres. We conclude with the final result in this series of works, due to Romney and the author, on the weakly quasiconformal uniformization of arbitrary metric surfaces of locally finite area under no further assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02697 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniformization of metric surfaces: A survey Ntalampekos, Dimitrios Complex Variables Geometric Topology Metric Geometry Primary 30C62, 30C65, 30F10, 30L10, 53A05, Secondary 28A75, 51F99, 53C23, 53C45 In this survey we present the most recent developments in the uniformization of metric surfaces, i.e., metric spaces homeomorphic to two-dimensional topological manifolds. We start from the classical conformal uniformization theorem of Koebe and Poincaré. Then we discuss the Bonk-Kleiner theorem on the quasisymmetric uniformization of metric spheres, which marks the beginning of the study of the uniformization problem on fractal surfaces. The next result presented is Rajala's theorem on the quasiconformal uniformization of metric spheres. We conclude with the final result in this series of works, due to Romney and the author, on the weakly quasiconformal uniformization of arbitrary metric surfaces of locally finite area under no further assumption. |
| title | Uniformization of metric surfaces: A survey |
| topic | Complex Variables Geometric Topology Metric Geometry Primary 30C62, 30C65, 30F10, 30L10, 53A05, Secondary 28A75, 51F99, 53C23, 53C45 |
| url | https://arxiv.org/abs/2505.02697 |