Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives
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 $j=2$.