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

摘要

Given a smooth minimal surface $F : Omega ightarrow mathbb{R}^{3}$ 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 $F^{epsilon} : V_{Omega}^{epsilon} ightarrow mathbb{R}^{3}$ 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 $Gamma^{epsilon} : V_{Omega}^{epsilon} ightarrow mathbb{R}^{3}$, which converges in $C^{infty}(Omega)$ to the minimal surface $F : Omega ightarrow mathbb{R}^{3}$ as $epsilon ightarrow 0$.