Discrete Lorentz surfaces and s-embeddings II: maximal surfaces
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| Format: | Preprint |
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2024
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| author | Affolter, Niklas Christoph Dellinger, Felix Müller, Christian Polly, Denis Smeenk, Nina |
| author_facet | Affolter, Niklas Christoph Dellinger, Felix Müller, Christian Polly, Denis Smeenk, Nina |
| contents | S-embeddings were introduced by Chelkak as a tool to study the conformal invariance of the thermodynamic limit of the Ising model. Moreover, Chelkak, Laslier and Russkikh introduced a lift of s-embeddings to Lorentz space, and showed that in the limit the lift converges to a maximal surface. They posed the question whether there are s-embeddings that lift to maximal surfaces already at the discrete level, before taking the limit. We answer this question in the positive. In a previous paper we identified a subclass of s-embeddings--isothermic s-embeddings--that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall as a discretization of isothermic surfaces. In this paper we identify a special class of isothermic s-embeddings that correspond to discrete S-maximal surfaces, translating an approach of Bobenko, Hoffmann and Springborn introduced for discrete S-minimal surfaces in Euclidean space. Additionally, each S-maximal surface comes with a 1-parameter family of associated surfaces that are isometric. This enables us to obtain an associated family of s-embeddings for each maximal s-embedding. We show that the Ising weights are constant in the associated family. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Lorentz surfaces and s-embeddings II: maximal surfaces Affolter, Niklas Christoph Dellinger, Felix Müller, Christian Polly, Denis Smeenk, Nina Differential Geometry Mathematical Physics 53A70 (primary), 82B20, 53A10, 52C26 (secondary) S-embeddings were introduced by Chelkak as a tool to study the conformal invariance of the thermodynamic limit of the Ising model. Moreover, Chelkak, Laslier and Russkikh introduced a lift of s-embeddings to Lorentz space, and showed that in the limit the lift converges to a maximal surface. They posed the question whether there are s-embeddings that lift to maximal surfaces already at the discrete level, before taking the limit. We answer this question in the positive. In a previous paper we identified a subclass of s-embeddings--isothermic s-embeddings--that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall as a discretization of isothermic surfaces. In this paper we identify a special class of isothermic s-embeddings that correspond to discrete S-maximal surfaces, translating an approach of Bobenko, Hoffmann and Springborn introduced for discrete S-minimal surfaces in Euclidean space. Additionally, each S-maximal surface comes with a 1-parameter family of associated surfaces that are isometric. This enables us to obtain an associated family of s-embeddings for each maximal s-embedding. We show that the Ising weights are constant in the associated family. |
| title | Discrete Lorentz surfaces and s-embeddings II: maximal surfaces |
| topic | Differential Geometry Mathematical Physics 53A70 (primary), 82B20, 53A10, 52C26 (secondary) |
| url | https://arxiv.org/abs/2411.19055 |