Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 034, 15 pages      arXiv:2307.08537
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

A Weierstrass Representation Formula for Discrete Harmonic Surfaces

Motoko Kotani a and Hisashi Naito b
a) The Advanced Institute for Materials Research (AIMR), Tohoku University, Japan
b) Graduate School of Mathematics, Nagoya University, Japan

Received July 17, 2023, in final form April 12, 2024; Published online April 17, 2024

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.

Key words: discrete harmonic surfaces; minimal surfaces; Weierstrass representation formula.

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