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C∞-Convergence of Circle Patterns to Minimal Surfaces

Published online by Cambridge University Press:  11 January 2016

Shi-Yi Lan and
Dao-Qing Dai
Shi-Yi Lan
Affiliation:
Department of Mathematics, Guangxi University for Nationalities, Nanning 530006, P. R. China
Dao-Qing Dai
Affiliation:
Department of Mathematics, Sun Yat-Sen (Zhongshan) University, Guangzhou 510275, P. R. China, stsddq@mail.sysu.edu.cn
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Abstract

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Given a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊ : → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.

Keywords

52C2653A1053C42

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 194 , 2009, pp. 149 - 167
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

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