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Holomorphic Legendrian curves

Part of: Holomorphic convexity Holomorphic mappings and correspondences Symplectic geometry, contact geometry

Published online by Cambridge University Press:  29 June 2017

Antonio Alarcón ,
Franc Forstnerič  and
Francisco J. López
Antonio Alarcón
Affiliation:
Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain email alarcon@ugr.es
Franc Forstnerič
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia email franc.forstneric@fmf.uni-lj.si
Francisco J. López
Affiliation:
Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain email fjlopez@ugr.es
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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 $W$.

Keywords

Riemann surfacecomplex contact manifoldLegendrian curve

MSC classification

Primary: 53D10: Contact manifolds, general
Secondary: 32B15: Analytic subsets of affine space 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 9 , September 2017 , pp. 1945 - 1986
Copyright
© The Authors 2017 

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