Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 643–663 | Cite as

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

  • Franc Forstnerič
  • Finnur Lárusson


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.


Riemann surface Legendrian curve Oka principle Absolute neighborhood retract 

Mathematics Subject Classification

53D10 32E30 32H02 57R17 



F. Forstnerič is supported in part by research program P1-0291 and Grant J1-7256 from ARRS, Republic of Slovenia. F. Lárusson is supported in part by Australian Research Council Grant DP150103442. The work on this paper was done at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo in the autumn of 2016. The authors would like to warmly thank the Centre for hospitality and financial support. We thank Antonio Alárcon and Francisco J. López for many helpful discussions on this topic, and Jaka Smrekar for his advice on topological issues concerning loop spaces.


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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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