Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings

Abstract

In 1991, Beardon and Stephenson [2] generalized the classical Schwarz-Pick lemma in hyperbolic geometry to the discrete Schwarz-Pick lemma for Andreev circle packings. This paper continues to investigate the discrete Schwarz-Pick lemma for generalized circle packings (including circle, horocycle or hypercycle) in hyperbolic background geometry. Since the discrete Schwarz-Pick lemma is to compare some geometric quantities of two generalized circle packings with different boundary values, we first show the existence and rigidity of generalized circle packings with boundary values, and then we introduce the method of combinatorial Calabi flows to find the generalized circle packings with boundary values. Moreover, motivated by the method of He [21], we propose the maximum principle for generalized circle packings. Finally, we use the maximum principle to prove the discrete Schwarz-Pick lemma for generalized circle packings.

Figures (12)

• Figure 1: The generalized circles on $\mathbb{H}^2$.
• Figure 2: The inner angle $\theta$ of the arc $C$ for hypercycle.
• Figure 3: The generalized circle packing of the triangle $f$.
• Figure 4: The triangle $f$ and hyperbolic triangle $\hat{f}$.
• Figure 5: The hyperbolic triangles of $f$.

Theorems & Definitions (31)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Remark 2.2
• Lemma 3.1
• proof