Holomorphic vector fields and quadratic differentials on planar triangular meshes

TL;DR

This paper explores the theory of holomorphic vector fields and quadratic differentials on planar triangular meshes, linking discrete harmonic functions to infinitesimal deformations and minimal surface construction.

Contribution

It introduces a novel framework connecting discrete holomorphic vector fields, harmonic functions, and quadratic differentials on triangulated meshes, with applications to minimal surface modeling.

Findings

Holomorphic vector fields can be constructed from discrete harmonic functions.

A M"obius invariant quadratic differential is associated with each holomorphic vector field.

A Weierstrass representation formula for discrete minimal surfaces is derived.

Abstract

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that 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 we associate in a M\"obius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gau{\ss} map and prescribed Hopf differential.