Mappings of finite distortion on metric surfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910445341769728 |
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| author | Meier, Damaris Rajala, Kai |
| author_facet | Meier, Damaris Rajala, Kai |
| contents | We investigate basic properties of mappings of finite distortion $f:X \to \mathbb{R}^2$, where $X$ is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite $2$-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant $f:X \to \mathbb{R}^2$ with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if $f$ is moreover injective then $f^{-1}$ is a Sobolev map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15615 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mappings of finite distortion on metric surfaces Meier, Damaris Rajala, Kai Metric Geometry Complex Variables 30L10 (Primary) 30C65, 30F10 (Secondary) We investigate basic properties of mappings of finite distortion $f:X \to \mathbb{R}^2$, where $X$ is any metric surface, i.e., metric space homeomorphic to a planar domain with locally finite $2$-dimensional Hausdorff measure. We introduce lower gradients, which complement the upper gradients of Heinonen and Koskela, to study the distortion of non-homeomorphic maps on metric spaces. We extend the Iwaniec-Šverák theorem to metric surfaces: a non-constant $f:X \to \mathbb{R}^2$ with locally square integrable upper gradient and locally integrable distortion is continuous, open and discrete. We also extend the Hencl-Koskela theorem by showing that if $f$ is moreover injective then $f^{-1}$ is a Sobolev map. |
| title | Mappings of finite distortion on metric surfaces |
| topic | Metric Geometry Complex Variables 30L10 (Primary) 30C65, 30F10 (Secondary) |
| url | https://arxiv.org/abs/2309.15615 |