Holomorphic Legendrian curves in convex domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909307463794688 |
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| author | Svetina, Andrej |
| author_facet | Svetina, Andrej |
| contents | We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in $\mathbb{C}^{2n+1}$, $n \geq 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface $M$, whose image lies in the interior of a convex domain $\mathscr{D} \subset \mathbb{C}^{2n+1}$, may be approximated uniformly on compacts in the interior $\mathrm{Int} \, M$ by holomorphic Legendrian curves $\mathrm{Int} \, M \to \mathscr{D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in $\mathrm{Int} \, M$ to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any bordered Riemann surface properly embeds into a convex domain as a complete holomorphic Legendrian curve under a suitable geometric condition on the boundary of the codomain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic Legendrian curves in convex domains Svetina, Andrej Complex Variables 53D10 (Primary), 32E30 (Secondary) We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in $\mathbb{C}^{2n+1}$, $n \geq 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface $M$, whose image lies in the interior of a convex domain $\mathscr{D} \subset \mathbb{C}^{2n+1}$, may be approximated uniformly on compacts in the interior $\mathrm{Int} \, M$ by holomorphic Legendrian curves $\mathrm{Int} \, M \to \mathscr{D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in $\mathrm{Int} \, M$ to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any bordered Riemann surface properly embeds into a convex domain as a complete holomorphic Legendrian curve under a suitable geometric condition on the boundary of the codomain. |
| title | Holomorphic Legendrian curves in convex domains |
| topic | Complex Variables 53D10 (Primary), 32E30 (Secondary) |
| url | https://arxiv.org/abs/2409.04197 |