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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.14482 |
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| _version_ | 1866916184863014912 |
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| author | Alarcon, Antonio Castro-Infantes, Ildefonso Hidalgo, Jorge |
| author_facet | Alarcon, Antonio Castro-Infantes, Ildefonso Hidalgo, Jorge |
| contents | We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space $\mathbb{H}^3$. We go further and establish a jet interpolation theorem for complete conformal CMC-1 immersions $M\to \mathbb{H}^3$. As a consequence, we show the existence of complete densely immersed CMC-1 surfaces in $\mathbb{H}^3$ with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in $\mathbb{C}^2\times\mathbb{C}^*$ which is also established in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14482 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure Alarcon, Antonio Castro-Infantes, Ildefonso Hidalgo, Jorge Differential Geometry Complex Variables We prove that every open Riemann surface $M$ is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the 3-dimensional hyperbolic space $\mathbb{H}^3$. We go further and establish a jet interpolation theorem for complete conformal CMC-1 immersions $M\to \mathbb{H}^3$. As a consequence, we show the existence of complete densely immersed CMC-1 surfaces in $\mathbb{H}^3$ with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in $\mathbb{C}^2\times\mathbb{C}^*$ which is also established in this paper. |
| title | Complete CMC-1 surfaces in hyperbolic space with arbitrary complex structure |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2306.14482 |