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

Donald Marshall, Steffen Rohde, Yilin Wang

TL;DR

This work characterizes piecewise geodesic Jordan curves on the Riemann sphere via conformal welding that is piecewise Möbius. It provides explicit constructions of $C^1$ geodesic pairs in the disk and derives the Schwarzian derivatives of the corresponding uniformizing maps, showing a simple pole at the vertices (and, when non-smooth, a double pole with coefficients tied to the spiral geometry). The welding framework ties the geometric edges to Möbius weldings, yielding a classification and uniqueness results for $C^1$ curves, and linking the local edge behavior to global Schwarzian structure. In the four-edge case, it reveals enhanced symmetry and a normalization-based parameterization of $C^1$ four-edge piecewise geodesic curves, connecting edge data to the Teichmüller-like moduli of the punctured sphere.

Abstract

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 $j=2$.

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

TL;DR

This work characterizes piecewise geodesic Jordan curves on the Riemann sphere via conformal welding that is piecewise Möbius. It provides explicit constructions of geodesic pairs in the disk and derives the Schwarzian derivatives of the corresponding uniformizing maps, showing a simple pole at the vertices (and, when non-smooth, a double pole with coefficients tied to the spiral geometry). The welding framework ties the geometric edges to Möbius weldings, yielding a classification and uniqueness results for curves, and linking the local edge behavior to global Schwarzian structure. In the four-edge case, it reveals enhanced symmetry and a normalization-based parameterization of four-edge piecewise geodesic curves, connecting edge data to the Teichmüller-like moduli of the punctured sphere.

Abstract

We consider Jordan curves of the form on the Riemann sphere for which each is a hyperbolic geodesic in . 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 form a rational function with at most second-order poles at the endpoints of and that the poles are simple if the curve has continuous tangents. A key tool is the explicit computation of all geodesic pairs, namely chords in a simply connected domain such that is a hyperbolic geodesic in for both and .
Paper Structure (8 sections, 21 theorems, 49 equations, 2 figures)

This paper contains 8 sections, 21 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geodesic pairs
  3. Welding Maps
  4. Piecewise geodesic curves
  5. General facts
  6. $C^1$ piecewise geodesic Jordan curves
  7. Piecewise geodesic Jordan curves with spirals
  8. $C^1$ piecewise geodesic curves with four edges

Key Result

Lemma 2.2

The function $G_\theta$ is a conformal map of the half disk $D_+=\{z:\textnormal{Re} z >0 ~{\rm and}~ |z| < 1\}$ onto $U_\theta$, where the branch of $\log z$ is chosen so that $-\frac{\pi}{2} \le \arg z \le \frac{\pi}{2}$.

Figures (2)

  • Figure 1: Construction of the geodesic pair $\gamma_1\cup\gamma_2$ in $(\mathbb{D};\mathrm{e}^{\mathfrak{i}\theta},-\mathrm{e}^{-\mathfrak{i}\theta},0)$ using the conformal map $G_\theta \colon D_+\to U_\theta$ sending $0$ to $\infty$. Here, $A = (\pi/2) \sin \theta$, $B = \cos \theta + \theta \sin \theta$, $C = -A-(B+A)$, $D = A - (B-A)$, $\gamma_1 = G_\theta^{-1} ((B,\infty))$, and the analytic extension of $G_\theta^{-1}$ to $\mathbb C \smallsetminus (-\infty, B]$ welds $(-\infty, C)$ to $(-\infty, D)$ to form $\gamma_2$.
  • Figure 2: A piecewise geodesic curve consisting of three spirals with spiral rate 2, reminiscent of the Triskele, one of the oldest Irish Celtic symbols.

Theorems & Definitions (51)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Example 2.6
  • Example 2.7
  • ...and 41 more