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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alarcón, A., Drinovec Drnovšek, B., Forstnerič, F., López, F.J.: Minimal surfaces in minimally convex domains. Preprint arXiv:1510.04006
Alarcón, A., Drinovec Drnovšek, B., Forstnerič, F., López, F.J.: Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves. Proc. Lond. Math. Soc. (3) 111(4), 851–886 (2015)
Alarcón, A., Forstnerič, F.: Every conformal minimal surface in \({\mathbb{R}}^3\) is isotopic to the real part of a holomorphic null curve. J. Reine Angew. Math. doi:10.1515/crelle-2015-0069
Alarcón, A., Forstnerič, F.: Null curves and directed immersions of open Riemann surfaces. Invent. Math. 196(3), 733–771 (2014)
Alarcón, A., Forstnerič, F., López, F.J.: Holomorphic Legendrian curves. Preprint arXiv:1607.00634. Compositio Math. (to appear)
Alarcón, A., Forstnerič, F., López, F.J.: Embedded minimal surfaces in \(\mathbb{R}^n\). Math. Z. 283(1), 1–24 (2016)
Bennequin, D.: Entrelacements et équations de Pfaff. In: Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Volume 107 of Astérisque, pp. 87–161. Soc. Math. France, Paris (1983)
Eliashberg, Y.: Classification of contact structures on \({\mathbf{R}}^3\). Internat. Math. Res. Not. 3, 87–91 (1993)
Eliashberg, Y., Mishachev, N.: Introduction to the \(h\)-Principle. Graduate Studies in Mathematics, vol. 48. American Mathematical Society, Providence (2002)
F. Forstnerič. Hyperbolic complex contact structures on \({\mathbb{C}}^{2n+1}\). J. Geom. Anal. doi:10.1007/s12220-017-9800-9
Forstnerič, F.: Stein manifolds and holomorphic mappings, volume 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, Heidelberg (2011)
Forstnerič, F., Lárusson, F.: The parametric h-principle for minimal surfaces in \(\mathbb{R}^n\) and null curves in \({\mathbb{C}}^n\). Preprint arXiv:1602.01529. Commun. Anal. Geom. (to appear)
Fuchs, D., Tabachnikov, S.: Invariants of Legendrian and transverse knots in the standard contact space. Topology 36(5), 1025–1053 (1997)
Geiges, H.: An introduction to contact topology. Cambridge Studies in Advanced Mathematics, vol. 109. Cambridge University Press, Cambridge (2008)
Gromov, M.: Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 9. Springer, Berlin (1986)
Gromov, M.L.: Convex integration of differential relations. I. Izv. Akad. Nauk SSSR Ser. Mat. 37, 329–343 (1973)
Lárusson, F.: Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds. Proc. Am. Math. Soc. 143(3), 1159–1167 (2015)
Mergelyan, S .N.: On the representation of functions by series of polynomials on closed sets. Dokl. Akad. Nauk SSSR (N.S.) 78, 405–408 (1951)
Oka, K.: Sur les fonctions analytiques de plusieurs variables. III. Deuxième problème de Cousin. J. Sci. Hiroshima Univ. Ser. A 9, 7–19 (1939)
Spring, D.: Convex integration theory. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (2010)
Stout, E.L.: Bounded holomorphic functions on finite Riemann surfaces. Trans. Am. Math. Soc. 120, 255–285 (1965)
Strøm, A.: Note on cofibrations. II. Math. Scand. 22, 130–142, (1968, 1969)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Forstnerič, F., Lárusson, F. The Oka principle for holomorphic Legendrian curves in \(\mathbb {C}^{2n+1}\) . Math. Z. 288, 643–663 (2018). https://doi.org/10.1007/s00209-017-1904-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-017-1904-1