Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives

Abstract

We consider Jordan curves of the form ${\hat{I}}^3={\cup }_{j=1}^n{\hat{I}}_j^3$ on the Riemann sphere for which each ${\hat{I}}_j^3$ is a hyperbolic geodesic in $(\hat{\mathbb{C}}\setminus {\hat{I}}^3)\cup {\hat{I}}_j^3$. 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 $\hat{\mathbb{C}}\setminus {\hat{I}}^3$ form a rational function with at most second-order poles at the endpoints of ${\hat{I}}_j^3$ 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 ${\hat{I}}^3={\hat{I}}_1^3\cup {\hat{I}}_2^3$ in a simply connected domain $D$ such that ${\hat{I}}_j^3$ is a hyperbolic geodesic in $D\setminus {\hat{I}}_{3-j}^3$ for both $j=1$ and $j=2$.

26 pages, 2 figures, minor revision according to referee's comments. To appear in Ark. Mat. Part II can be found: arXiv:2410.22275