Minimal surfaces from infinitesimal deformations of circle packings

Abstract

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of Koebe type. Furthermore, every minimal surface of Koebe type can be extended naturally to a discrete minimal surface of general type. In this way, discrete minimal surfaces via Steiner's formula are unified.