A Weierstrass Representation Formula for Discrete Harmonic Surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929316038705152 |
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| author | Kotani, Motoko Naito, Hisashi |
| author_facet | Kotani, Motoko Naito, Hisashi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08537 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Weierstrass Representation Formula for Discrete Harmonic Surfaces Kotani, Motoko Naito, Hisashi Differential Geometry 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. |
| title | A Weierstrass Representation Formula for Discrete Harmonic Surfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2307.08537 |