Minimal surfaces in Euclidean space are classical in differential geometry. They arise in the calculus of variations, in complex analysis and are related to integrable systems. A surface is minimal if it is a critical point of the area functional, or equivalently, its mean curvature vanishes identically. The Weierstrass representation for minimal surfaces asserts that locally each minimal surface is given by a pair of holomorphic functions. New minimal surfaces can be obtained from a given minimal surface via Bonnet, Goursat and Darboux transforms.
Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector $Y_i \in C$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. Each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field a certain holomorphic quadratic differential can be associated in a Möbius invariant fashion. A quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. A Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gauß map and prescribed Hopf differential is derived in .
W. Y. Lam and U. Pinkall.
Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes.
In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
Wai Yeung Lam.
Infinitesimal deformations of discrete surfaces.
preprint, June 2016.