Minimal surfaces from infinitesimal deformations of circle packings (Q2287941)

In complex analysis, there exist Weierstrass representation that every simply connected surface in space with vanishing mean curvature corresponds to a pair of holomorphic functions. The problem is to generalize this representation to discrete case. Thurston shows that circle packing and circle patterns can be considered as a discrete analogue of holomorphic functions. But there are three different definitions of the discrete minimal surfaces depending on the choice of so called vertex normal vectors. For two of them a one-to-one correspondence between infinitesimal deformations of circle patterns and discrete minimal surfaces of general type via a Weierstrass representation formula was previously obtained by author. Third case (so called discrete minimal surfaces of Koebe type) remained unsolved. Now it is shown 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.