A Weierstrass Representation Formula for Discrete Harmonic Surfaces

TL;DR

This paper introduces a Weierstrass representation formula for discrete harmonic surfaces, enabling the approximation of classical minimal surfaces like the Enneper surface through discrete models.

Contribution

It proposes a novel discrete Weierstrass formula linking holomorphic data to discrete harmonic surfaces, bridging discrete and smooth minimal surface theory.

Findings

Constructed a sequence of discrete harmonic surfaces converging to classical minimal surfaces.

Provided a discrete approximation method for the Enneper surface.

Established a theoretical foundation connecting discrete and smooth minimal surface representations.

Abstract

A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3-dimensional Euclidean space and achieves energy minimizing under local deformations. Given a topological trivalent graph, a holomorphic function, and an associated discrete holomorphic quadratic form, a version of the Weierstrass representation formula for discrete harmonic surfaces in the 3-dimensional Euclidean space is proposed. By using the formula, a smooth converging sequence of discrete harmonic surfaces is constructed, and its limit is a classical minimal surface defined with the same holomorphic data. As an application, we have a discrete approximation of the Enneper surface.