$\mathrm{CMC\text{-}1}$ surfaces in hyperbolic and de Sitter spaces with Cantor ends
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| Format: | Preprint |
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2024
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| _version_ | 1866911883309613056 |
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| author | Castro-Infantes, Ildefonso Hidalgo, Jorge |
| author_facet | Castro-Infantes, Ildefonso Hidalgo, Jorge |
| contents | We prove that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space $\mathbb{H}^3$. Moreover, we obtain that every bordered Riemann surface admits an almost proper $\mathrm{CMC\text{-}1}$ face into de Sitter $3$-space $\mathbb{S}_1^3$, and we show that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits an almost proper $\mathrm{CMC\text{-}1}$ face into $\mathbb{S}_1^3$. These results follow from different uniform approximation theorems for holomorphic null curves in $\mathbb{C}^2 \times \mathbb{C}^*$ that we also establish in this paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12723 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathrm{CMC\text{-}1}$ surfaces in hyperbolic and de Sitter spaces with Cantor ends Castro-Infantes, Ildefonso Hidalgo, Jorge Differential Geometry Complex Variables 53A10, 53C42, 32H02, 32E30 We prove that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space $\mathbb{H}^3$. Moreover, we obtain that every bordered Riemann surface admits an almost proper $\mathrm{CMC\text{-}1}$ face into de Sitter $3$-space $\mathbb{S}_1^3$, and we show that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits an almost proper $\mathrm{CMC\text{-}1}$ face into $\mathbb{S}_1^3$. These results follow from different uniform approximation theorems for holomorphic null curves in $\mathbb{C}^2 \times \mathbb{C}^*$ that we also establish in this paper. |
| title | $\mathrm{CMC\text{-}1}$ surfaces in hyperbolic and de Sitter spaces with Cantor ends |
| topic | Differential Geometry Complex Variables 53A10, 53C42, 32H02, 32E30 |
| url | https://arxiv.org/abs/2405.12723 |