# Discrete Minimal Surfaces

Wai Yeung Lam, Ulrich Pinkall

### Description

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 [1].

### References

• 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.
arXiv:1506.08099.
• Wai Yeung Lam.
Infinitesimal deformations of discrete surfaces.
preprint, June 2016.

#### Dr. Wai Yeung Lam   +

Projects: A05
University: TU Berlin

#### Prof. Dr. Ulrich Pinkall   +

Projects: A05, C07
University: TU Berlin
E-Mail: pinkall[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~pinkall/