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Holomorphic Legendrian curves
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 9 / September 2017
- Published online by Cambridge University Press:
- 29 June 2017, pp. 1945-1986
- Print publication:
- September 2017
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- Article
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In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on
$\mathbb{C}^{2n+1}$ for any 
$n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface 
$M$ admits a proper holomorphic Legendrian embedding 
$M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface 
$M={M\unicode[STIX]{x0030A}}\,\cup \,bM$ there exists a topological embedding 
$M{\hookrightarrow}\mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding 
${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold 
$W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on 
$W$.
