论文标题
分段地球Jordan曲线I：焊接，显式计算和Schwarzian衍生物
Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives
论文作者
论文摘要
我们考虑Jordan曲线$γ= \ cup_ {j = 1}^nγ_j$在Riemann Sphere中，每个$γ_j$均为剩余弧$γ\ SmallSetSetMinusγ_j$的补充，每个$γ_j$是一个双曲线测量。这些曲线的特征在于它们的保形焊接是分段的莫比乌斯。除其他事项外，我们计算了两个区域的Riemann地图的Schwarzian衍生物，$ \ hat {\ Mathbb c} \SmallSetMinusγ$，表明它们在$γ_j，$的端点上具有二阶杆形成合理功能，如果波雷斯的端点很简单，并且表明curve converve conters curve contents contents持续使用。 Our key tool is the explicit computation of all geodesic pairs, namely pairs of 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 $j=2$.
We consider Jordan curves $γ=\cup_{j=1}^n γ_j$ in the Riemann sphere for which each $γ_j$ is a hyperbolic geodesic in the complement of the remaining arcs $γ\smallsetminus γ_j$. These curves are characterized by the property that their conformal welding is piecewise Möbius. Among other things, we compute the Schwarzian derivatives of the Riemann maps of the two regions in $\hat {\mathbb C}\smallsetminus γ$, show that they form a rational function with second order poles at the endpoints of the $γ_j,$ and show that the poles are simple if the curve has continuous tangents. Our key tool is the explicit computation of all geodesic pairs, namely pairs of 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 $j=2$.
