Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces
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| Autores principales: | , , , , |
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| Formato: | Preprint |
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2024
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| _version_ | 1866913547521359872 |
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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. This paper is the first in a two paper series, in which we answer that question in the positive. In this paper we introduce a correspondence between s-embeddings (incircular nets) and congruences of touching Lorentz spheres. This geometric interpretation of s-embeddings enables us to apply the tools of discrete differential geometry. We identify a subclass of s-embeddings -- isothermic s-embeddings -- that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall. S-isothermic surfaces are the key component that will allow us to obtain discrete maximal surfaces in the follow-up paper. Moreover, we show here that the Ising weights of an isothermic s-embedding are in a subvariety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces Affolter, Niklas Christoph Dellinger, Felix Müller, Christian Polly, Denis Smeenk, Nina Differential Geometry Mathematical Physics 53A70 (Primary), 82B20 (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. This paper is the first in a two paper series, in which we answer that question in the positive. In this paper we introduce a correspondence between s-embeddings (incircular nets) and congruences of touching Lorentz spheres. This geometric interpretation of s-embeddings enables us to apply the tools of discrete differential geometry. We identify a subclass of s-embeddings -- isothermic s-embeddings -- that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall. S-isothermic surfaces are the key component that will allow us to obtain discrete maximal surfaces in the follow-up paper. Moreover, we show here that the Ising weights of an isothermic s-embedding are in a subvariety. |
| title | Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces |
| topic | Differential Geometry Mathematical Physics 53A70 (Primary), 82B20 (Secondary) |
| url | https://arxiv.org/abs/2410.11575 |