分段测地若尔当曲线I：焊接、显式计算与施瓦茨导数
Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives
AI总结：
本文研究黎曼球面上的分段测地若尔当曲线，证明其共形焊接为分段莫比乌斯映射，且两区域均匀化映射的施瓦茨导数为有理函数，通过显式计算$C^1$测地对作为关键工具推进研究。
摘要：
我们研究黎曼球面上形如$\gamma=\cup_{j=1}^n \gamma_j$的若尔当曲线，其中每个$\gamma_j$都是$(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$中的双曲测地线。这类若尔当曲线的特征是其共形焊接为分段莫比乌斯映射。我们证明，$\widehat{\mathbb C} \smallsetminus \gamma$中两个区域的共形均匀化映射的施瓦茨导数构成一个有理函数，其极点最多在$\gamma_j$的端点处为二阶，且若曲线具有连续切线，则极点为单极点。关键工具是对所有$C^1$测地对的显式计算，即单连通域$D$中的$C^1$弦$\gamma=\gamma_1\cup\gamma_2$，满足对$j=1$和$j=2$，$\gamma_j$都是$D\smallsetminus \gamma_{3-j}$中的双曲测地线。
英文摘要：
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 $j=2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
