2017/02/28 by Antonio Alarcón, Antonio Alarcon, Franc Forstnerič +1 · 1 citation
Mathematics · #Complex manifold #Contractible space #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Holomorphic function #Identity theorem #Immersion (mathematics) #Mathematical analysis #Mathematics #Pure mathematics #Riemann surface #math.CV #math.DG #msc:32E30 #msc:32H02 #msc:37J55 #msc:53D10
paper · pdf · doi:10.1093/imrn/rnx158
published as Internat. Math. Res. Not. (IMRN) (2019), no. 3, 893-922 · Internat. Math. Res. Not. (IMRN), to appear
arxiv created 2017/06/16 · openalex publication_date 2017/06/26 · arxiv updated 2019/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this article, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold |(X,ξ)|. By using such a chart, we show that every holomorphic Legendrian immersion |R→ X| from an open Riemann surface can be approximated on relatively compact subsets of |R| by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion |M→ X| from a compact bordered Riemann surface is a uniform limit of topological embeddings |M\hookrightarrow X| such that |\mathring M\hookrightarrow X| is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds in complex contact manifolds.