The Oka principle for holomorphic Legendrian curves in \(\mathbb {C}^{2n+1}\)

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Abstract

Let M be a connected open Riemann surface. We prove that the space \(\mathscr {L}(M,\mathbb {C}^{2n+1})\) of all holomorphic Legendrian immersions of M to \(\mathbb {C}^{2n+1}\), \(n\ge 1\), endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space \(\mathscr {C}(M,\mathbb {S}^{4n-1})\) of continuous maps from M to the sphere \(\mathbb {S}^{4n-1}\). If M has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of \(\mathscr {L}(M,\mathbb {C}^{2n+1})\) in terms of the homotopy groups of \(\mathbb {S}^{4n-1}\). It follows that \(\mathscr {L}(M,\mathbb {C}^{2n+1})\) is \((4n-3)\)-connected.

Keywords

Riemann surface Legendrian curve Oka principle Absolute neighborhood retract 

Mathematics Subject Classification

53D10 32E30 32H02 57R17 

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Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Department of Mathematics, Institute of MathematicsPhysics and MechanicsLjubljanaSlovenia
  3. 3.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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