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A Weierstrass Representation Formula for Discrete Harmonic Surfaces
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: discrete harmonic surfaces, minimal surfaces, Weierstrass representation formula. 
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     title = {A {Weierstrass} {Representation} {Formula} for {Discrete} {Harmonic} {Surfaces}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a33/}
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Motoko Kotani; Hisashi Naito. A Weierstrass Representation Formula for Discrete Harmonic Surfaces. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a33/

[1] Bobenko A.I., Bücking U., Sechelmann S., “Discrete minimal surfaces of Koebe type”, Modern Approaches to Discrete Curvature, Lecture Notes in Math., 2184, Springer, Cham, 2017, 259–291 | DOI | MR | Zbl

[2] Bobenko A.I., Hoffmann T., Springborn B.A., “Minimal surfaces from circle patterns: geometry from combinatorics”, Ann. of Math., 164 (2006), 231–264, arXiv: math.DG/0305184 | DOI | MR | Zbl

[3] Bobenko A.I., Pinkall U., “Discrete isothermic surfaces”, J. Reine Angew. Math., 475 (1996), 187–208 | DOI | MR | Zbl

[4] Bobenko A.I., Pottmann H., Wallner J., “A curvature theory for discrete surfaces based on mesh parallelity”, Math. Ann., 348 (2010), 1–24, arXiv: 0901.4620 | DOI | MR | Zbl

[5] Kotani M., Naito H., Omori T., “A discrete surface theory”, Comput. Aided Geom. Design, 58 (2017), 24–54, arXiv: 1601.07272 | DOI | MR | Zbl

[6] Kotani M., Naito H., Tao C., “Construction of continuum from a discrete surface by its iterated subdivisions”, Tohoku Math. J., 74 (2022), 229–252, arXiv: 1806.03531 | DOI | MR

[7] Lam W.Y., “Discrete minimal surfaces: critical points of the area functional from integrable systems”, Int. Math. Res. Not., 2018 (2018), 1808–1845, arXiv: 1510.08788 | DOI | MR | Zbl

[8] Lam W.Y., “Minimal surfaces from infinitesimal deformations of circle packings”, Adv. Math., 362 (2020), 106939, 24 pp., arXiv: 2019.10693 | DOI | MR | Zbl

[9] Lam W.Y., Pinkall U., “Holomorphic vector fields and quadratic differentials on planar triangular meshes”, Advances in Discrete Differential Geometry, Springer, Berlin, 2016, 241–265, arXiv: 1506.08099 | DOI | MR | Zbl

[10] Pinkall U., Polthier K., “Computing discrete minimal surfaces and their conjugates”, Experiment. Math., 2 (1993), 15–36 | DOI | MR | Zbl

[11] Rodin B., Sullivan D., “The convergence of circle packings to the Riemann mapping”, J. Differential Geom., 26 (1987), 349–360 | DOI | MR | Zbl

[12] Stephenson K., Introduction to circle packing: The theory of discrete analytic functions, Cambridge University Press, Cambridge, 2005 | MR | Zbl

[13] Tao C., “A construction of converging Goldberg–Coxeter subdivisions of a discrete surface”, Kobe J. Math., 38 (2021), 35–51 | MR | Zbl

[14] Thurston W.P., The geometry and topology of three-manifolds, Princeton University Notes, Priceton, 1979 | MR