Year: 2017
Type: article
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 $M={M\unicode[STIX]{x0030A}}\,\cup \,bM$ 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 $W$ .
Source: Compositio Mathematica
Authors: Antonio Alarcón, Franc Forstnerič, Francisco J. López
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2467862629