2017/06/29 by Antonio Alarcón, Franc Forstnerič, Francisco J. López · 4 citations
Mathematics · #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Analytic and geometric function theory
paper · pdf · doi:10.1112/s0010437x1700731x
In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on ℂ2n+1 for any n∈ 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\hookrightarrowℂ2n+1 , and we prove that for every compact bordered Riemann surface M=M\unicode[STIX]x0030A ∪ bM there exists a topological embedding M\hookrightarrowℂ2n+1 whose restriction to the interior is a complete holomorphic Legendrian embedding M\unicode[STIX]x0030A\hookrightarrowℂ2n+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 .