
SIGMA 20 (2024), 034, 15 pages arXiv:2307.08537
https://doi.org/10.3842/SIGMA.2024.034
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of JeanPierre 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
Abstract
A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3dimensional 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 3dimensional 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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References
 Bobenko A.I., Bücking U., Sechelmann S., Discrete minimal surfaces of Koebe type, in Modern Approaches to Discrete Curvature, Lecture Notes in Math., Vol. 2184, Springer, Cham, 2017, 259291.
 Bobenko A.I., Hoffmann T., Springborn B.A., Minimal surfaces from circle patterns: geometry from combinatorics, Ann. of Math. 164 (2006), 231264, arXiv:math.DG/0305184.
 Bobenko A.I., Pinkall U., Discrete isothermic surfaces, J. Reine Angew. Math. 475 (1996), 187208.
 Bobenko A.I., Pottmann H., Wallner J., A curvature theory for discrete surfaces based on mesh parallelity, Math. Ann. 348 (2010), 124, arXiv:0901.4620.
 Kotani M., Naito H., Omori T., A discrete surface theory, Comput. Aided Geom. Design 58 (2017), 2454, arXiv:1601.07272.
 Kotani M., Naito H., Tao C., Construction of continuum from a discrete surface by its iterated subdivisions, Tohoku Math. J. 74 (2022), 229252, arXiv:1806.03531.
 Lam W.Y., Discrete minimal surfaces: critical points of the area functional from integrable systems, Int. Math. Res. Not. 2018 (2018), 18081845, arXiv:1510.08788.
 Lam W.Y., Minimal surfaces from infinitesimal deformations of circle packings, Adv. Math. 362 (2020), 106939, 24 pages, arXiv:2019.10693.
 Lam W.Y., Pinkall U., Holomorphic vector fields and quadratic differentials on planar triangular meshes, in Advances in Discrete Differential Geometry, Springer, Berlin, 2016, 241265, arXiv:1506.08099.
 Pinkall U., Polthier K., Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993), 1536.
 Rodin B., Sullivan D., The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), 349360.
 Stephenson K., Introduction to circle packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005.
 Tao C., A construction of converging GoldbergCoxeter subdivisions of a discrete surface, Kobe J. Math. 38 (2021), 3551.
 Thurston W.P., The geometry and topology of threemanifolds, Princeton University Notes, Priceton, 1979.

