Holomorphic Legendrian curves

Abstract

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 $M=\mathring{M}\cup bM$ there exists a topological embedding $M\hookrightarrow {\mathbb{C}}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding $\mathring{M}\hookrightarrow {\mathbb{C}}^{2n+1}$. 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 $W$.

Compos. Math., in press

References in corpus (4)

• Bordered Riemann surfaces in C^2
• Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves
• Every meromorphic function is the Gauss map of a conformal minimal surface
• Hyperbolic complex contact structures on ${\mathbb{C}}^{2n+1}$

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• Mergelyan approximation theorem for holomorphic Legendrian curves
• Hyperbolic complex contact structures on ${\mathbb{C}}^{2n+1}$
• Holomorphic Legendrian curves in ${\mathbb{CP}}^3$ and superminimal surfaces in ${\mathbb{S}}^4$
• Proper superminimal surfaces of given conformal types in the hyperbolic four-space
• A properly embedded holomorphic disc in the ball with finite area and dense boundary curve
• Existence of Horizontal Immersions in Fat Distributions
• On Horizontal Immersions of Discs in Fat Distributions of Type $(4,6)$
• Oka-1 manifolds
• Direct Products for the Hamiltonian Density Property