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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2202.01967 |
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| _version_ | 1866916983308550144 |
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| author | Marshall, Donald Rohde, Steffen Wang, Yilin |
| author_facet | Marshall, Donald Rohde, Steffen Wang, Yilin |
| contents | We consider Jordan curves of the form $γ=\cup_{j=1}^n γ_j$ on the Riemann sphere for which each $γ_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus γ)\cup γ_j$. These Jordan curves are characterized by their conformal welding being piecewise Möbius. We show that the Schwarzian derivatives of the uniformizing mappings of the two regions in $\widehat{\mathbb C} \smallsetminus γ$ form a rational function with at most second-order poles at the endpoints of $γ_j$ and that the poles are simple if the curve has continuous tangents. A key tool is the explicit computation of all $C^1$ geodesic pairs, namely $C^1$ chords $γ=γ_1\cupγ_2$ in a simply connected domain $D$ such that $γ_j$ is a hyperbolic geodesic in $D\smallsetminus γ_{3-j}$ for both $j=1$ and $j=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_01967 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives Marshall, Donald Rohde, Steffen Wang, Yilin Complex Variables We consider Jordan curves of the form $γ=\cup_{j=1}^n γ_j$ on the Riemann sphere for which each $γ_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus γ)\cup γ_j$. These Jordan curves are characterized by their conformal welding being piecewise Möbius. We show that the Schwarzian derivatives of the uniformizing mappings of the two regions in $\widehat{\mathbb C} \smallsetminus γ$ form a rational function with at most second-order poles at the endpoints of $γ_j$ and that the poles are simple if the curve has continuous tangents. A key tool is the explicit computation of all $C^1$ geodesic pairs, namely $C^1$ chords $γ=γ_1\cupγ_2$ in a simply connected domain $D$ such that $γ_j$ is a hyperbolic geodesic in $D\smallsetminus γ_{3-j}$ for both $j=1$ and $j=2$. |
| title | Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2202.01967 |