2024/11/09 by Guangming Hu, Hu, Guangming, Ziping Lei +5 · 1 citation
Engineering · #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Manufacturing Process and Optimization #Optimization and Packing Problems
paper · pdf · doi:10.48550/arxiv.2411.06274
openalex publication_date 2024/11/09 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28
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.