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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.01574 |
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| _version_ | 1866908357737054208 |
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| author | Carberry, Emma Klein, Sebastian Schmidt, Martin Ulrich |
| author_facet | Carberry, Emma Klein, Sebastian Schmidt, Martin Ulrich |
| contents | This paper is motivated by the question of whether a sequence of solutions of a given integrable system can be blown up to obtain a solution of a different integrable system in the limit. We study a specific example of this phenomenon. Namely, we describe a blow-up for immersed constant mean curvature (cmc) planes of finite type with unbounded principal curvatures and derive sufficient conditions under which this blow-up converges to a minimal surface immersion. Passing to the respective Gauss-Codazzi equations, we are blowing up a sequence of solutions to the sinh-Gordon integrable system to obtain a solution to Liouville's equation, whose integrable system will turn out to be closely related to the Korteweg-de Vries integrable system. Our most important tool for this investigation is the algebraic-geometric correspondence that was established by Pinkall/Sterling and by Hitchin for cmc planes of finite type, which include all cmc tori. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_01574 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Blowing up sequences of constant mean curvature tori in $\mathbb{R}^3$ to minimal surfaces Carberry, Emma Klein, Sebastian Schmidt, Martin Ulrich Differential Geometry 53A10, 58E12 (Primary), 14H55 (Secondary) This paper is motivated by the question of whether a sequence of solutions of a given integrable system can be blown up to obtain a solution of a different integrable system in the limit. We study a specific example of this phenomenon. Namely, we describe a blow-up for immersed constant mean curvature (cmc) planes of finite type with unbounded principal curvatures and derive sufficient conditions under which this blow-up converges to a minimal surface immersion. Passing to the respective Gauss-Codazzi equations, we are blowing up a sequence of solutions to the sinh-Gordon integrable system to obtain a solution to Liouville's equation, whose integrable system will turn out to be closely related to the Korteweg-de Vries integrable system. Our most important tool for this investigation is the algebraic-geometric correspondence that was established by Pinkall/Sterling and by Hitchin for cmc planes of finite type, which include all cmc tori. |
| title | Blowing up sequences of constant mean curvature tori in $\mathbb{R}^3$ to minimal surfaces |
| topic | Differential Geometry 53A10, 58E12 (Primary), 14H55 (Secondary) |
| url | https://arxiv.org/abs/2110.01574 |