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arXiv · 2307.08537

A Weierstrass Representation Formula for Discrete Harmonic Surfaces

Motoko Kotani·Hisashi Naito
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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.

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Motoko Kotani, Hisashi Naito. 2023-07-17. A Weierstrass Representation Formula for Discrete Harmonic Surfaces. https://doi.org/10.3842/sigma.2024.034

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