Definitions of quasiconformality on metric surfaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929341267443712 |
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| author | Meier, Damaris Rajala, Kai |
| author_facet | Meier, Damaris Rajala, Kai |
| contents | We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Definitions of quasiconformality on metric surfaces Meier, Damaris Rajala, Kai Metric Geometry 30L10 (Primary) 30C65, 30F10 (Secondary) We explore the interplay between different definitions of distortion for mappings $f\colon X\to \mathbb{R}^2$, where $X$ is any metric surface, meaning that $X$ is homeomorphic to a domain in $\mathbb{R}^2$ and has locally finite 2-dimensional Hausdorff measure. We establish that finite distortion in terms of the familiar analytic definition always implies finite distortion in terms of maximal and minimal stretchings along paths. The converse holds for maps with locally integrable distortion. In particular, we prove the equivalence of various notions of quasiconformality, implying a novel uniformization result for metric surfaces. |
| title | Definitions of quasiconformality on metric surfaces |
| topic | Metric Geometry 30L10 (Primary) 30C65, 30F10 (Secondary) |
| url | https://arxiv.org/abs/2405.07476 |