Modulus of elementary domains in the hyperbolic plane

TL;DR

This work defines and analyzes the modulus of curve families in the hyperbolic plane ${\bf H}^1_{\mathbb C}$, focusing on two canonical domains: a normal hyperbolic quadrilateral and a hyperbolic annulus. By employing hyperbolic polar coordinates and extremal density methods, it yields explicit moduli for families connecting boundary components or segments/arcs, highlighting how hyperbolic geometry alters Euclidean intuition. Key results include exact formulas for the quadrilateral: ${\rm Mod}(\Gamma_Q^1)=\frac{4}{A_e(Q_a)}$ and ${\rm Mod}(\Gamma_Q^2)=\frac{(a-1)^2}{A_e(Q_a)+2\left({\bf G}+\frac{\pi}{2}\log a-{ m Ti}_2(a)\right)}$, and for the annulus: ${\rm Mod}(\Gamma_A^1)=\frac{2\pi}{\log\left( \frac{\tanh(R/2)}{\tanh(1/2)}\right)}$, ${\rm Mod}(\Gamma_A^2)=\frac{1}{2\pi}\log\left( \frac{\tanh(R/2)}{\tanh(1/2)}\right)$. These results extend Euclidean modulus theory to hyperbolic geometry and have potential implications for Teichmüller theory and related geometric-function theory settings.

Abstract

We calculate the modulus of curve families inside a hyperbolic quadrilateral and a hyperbolic annulus.

Figures (2)

• Figure 3.1: A normal hyperbolic quadrilateral.
• Figure 4.1: The white region between the circles is the hyperbolic annulus.

Theorems & Definitions (7)

• Definition 2.1
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 4.1
• Theorem 4.2
• Theorem 4.3