$C^{\infty}$-convergence of circle patterns to minimal surfaces

Abstract

Given a smooth minimal surface $F : \Omega \rightarrow \mathbb{R}^{3}$ defined on a simply connected region $\Omega$ in the complex plane $\mathbb{C}$, there is a regular SG circle pattern $Q_{\Omega}^{\epsilon}$. By the Weierstrass representation of $F$ and the existence theorem of SG circle patterns, there exists an associated SG circle pattern $P_{\Omega}^{\epsilon}$ in $\mathbb{C}$ with the combinatoric of $Q_{\Omega}^{\epsilon}$. Based on the relationship between the circle pattern $P_{\Omega}^{\epsilon}$ and the corresponding discrete minimal surface $F^{\epsilon} : V_{\Omega}^{\epsilon} \rightarrow \mathbb{R}^{3}$ defined on the vertex set $V_{\Omega}^{\epsilon}$ of the graph of $Q_{\Omega}^{\epsilon}$, we show that there exists a family of discrete minimal surface $\Gamma^{\epsilon} : V_{\Omega}^{\epsilon} \rightarrow \mathbb{R}^{3}$, which converges in $C^{\infty}(\Omega)$ to the minimal surface $F : \Omega \rightarrow \mathbb{R}^{3}$ as $\epsilon \rightarrow 0$.