Skip to main content
SpringerLink
Log in

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ℂ×ℂ∗, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ℂ×ℂ∗ of recent results of Wold and Forstnerič on the long-standing problem of properly embedding open Riemann surfaces into ℂ2, with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd ed. Advanced Book Program. Benjamin/Cummings, Reading (1978)

    MATH  Google Scholar 

  2. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications, 2nd ed. Applied Mathematical Sciences, vol. 75. Springer, New York (1988)

    Book  MATH  Google Scholar 

  3. Alexander, H.: Explicit imbedding of the (punctured) disc into C 2. Comment. Math. Helv. 52(4), 539–544 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Andersén, E., Lempert, L.: On the group of holomorphic automorphisms of C n. Invent. Math. 110(2), 371–388 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bishop, E.: Mappings of partially analytic spaces. Am. J. Math. 83, 209–242 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dixon, P.G., Esterle, J.: Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon. Bull., New Ser., Am. Math. Soc. 15(2), 127–187 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drinovec-Drnovšek, B., Forstnerič, F.: Strongly pseudoconvex domains as subvarieties of complex manifolds. Am. J. Math. 132(2), 331–360 (2010)

    Article  MATH  Google Scholar 

  8. Eliashberg, Y., Gromov, M.: Embeddings of Stein manifolds of dimension n into the affine space of dimension 3n/2+1. Ann. Math. 136(1), 123–135 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Forstnerič, F.: Interpolation by holomorphic automorphisms and embeddings in C n. J. Geom. Anal. 9(1), 93–117 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Forstnerič, F., Lárusson, F.: Survey of Oka theory. N.Y. J. Math. 17a, 11–38 (2011)

    MATH  Google Scholar 

  11. Forstnerič, F., Løw, E.: Global holomorphic equivalence of smooth submanifolds in C n. Indiana Univ. Math. J. 46(1), 133–153 (1997)

    MathSciNet  MATH  Google Scholar 

  12. Forstnerič, F., Løw, E., Øvrelid, N.: Solving the d- and \(\overline{\partial}\)-equations in thin tubes and applications to mappings. Mich. Math. J. 49(2), 369–416 (2001)

    Article  MATH  Google Scholar 

  13. Forstnerič, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of C n. Invent. Math. 112(2), 323–349 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Forstnerič, F., Rosay, J.-P.: Erratum: “Approximation of biholomorphic mappings by automorphisms of \(\bold C^{n}\)”. Invent. Math. 118(3), 573–574 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Forstnerič, F., Wold, E.F.: Bordered Riemann surfaces in ℂ2. J. Math. Pures Appl. 91(1), 100–114 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Globevnik, J., Stensønes, B.: Holomorphic embeddings of planar domains into C 2. Math. Ann. 303(4), 579–597 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. Translations of Mathematical Monographs, vol. 26. Am. Math. Soc., Providence (1969)

    MATH  Google Scholar 

  18. Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2(4), 851–897 (1989)

    MathSciNet  MATH  Google Scholar 

  19. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd ed. North-Holland Mathematical Library, vol. 7. North-Holland, Amsterdam (1990)

    MATH  Google Scholar 

  20. Kaliman, S., Kutzschebauch, F.: Criteria for the density property of complex manifolds. Invent. Math. 172(1), 71–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kaliman, S., Kutzschebauch, F.: Density property for hypersurfaces \(UV=P(\overline{X})\). Math. Z. 258(1), 115–131 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kutzschebauch, F., Løw, E., Wold, E.F.: Embedding some Riemann surfaces into ℂ2 with interpolation. Math. Z. 262(3), 603–611 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lárusson, F.: Model structures and the Oka principle. J. Pure Appl. Algebra 192(1–3), 203–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lárusson, F.: Mapping cylinders and the Oka principle. Indiana Univ. Math. J. 54(4), 1145–1159 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Laufer, H.B.: Imbedding annuli in C 2. J. Anal. Math. 26, 187–215 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  26. Narasimhan, R.: Imbedding of holomorphically complete complex spaces. Am. J. Math. 82, 917–934 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  27. Remmert, R.: Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes. C. R. Acad. Sci. Paris 243, 118–121 (1956)

    MathSciNet  MATH  Google Scholar 

  28. Rosay, J.-P.: Straightening of arcs. Astérisque 217, 217–225 (1993). Colloque d’Analyse Complexe et Géométrie (Marseille, 1992)

    MathSciNet  Google Scholar 

  29. Schürmann, J.: Embeddings of Stein spaces into affine spaces of minimal dimension. Math. Ann. 307(3), 381–399 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stehlé, J.-L.: Plongements du disque dans C 2. In: Séminaire Pierre Lelong (Analyse), Année 1970–1971. Lecture Notes in Math., vol. 275, pp. 119–130. Springer, Berlin (1972)

    Google Scholar 

  31. Stolzenberg, G.: Uniform approximation on smooth curves. Acta Math. 115, 185–198 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stout, E.L.: Polynomial Convexity. Progress in Mathematics, vol. 261. Birkhäuser Boston, Boston (2007)

    MATH  Google Scholar 

  33. Tóth, Á., Varolin, D.: Holomorphic diffeomorphisms of complex semisimple Lie groups. Invent. Math. 139(2), 351–369 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tóth, Á., Varolin, D.: Holomorphic diffeomorphisms of semisimple homogeneous spaces. Compos. Math. 142(5), 1308–1326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Varolin, D.: The density property for complex manifolds and geometric structures. II. Int. J. Math. 11(6), 837–847 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11(1), 135–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wold, E.F.: Embedding Riemann surfaces properly into ℂ2. Int. J. Math. 17(8), 963–974 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wold, E.F.: Proper holomorphic embeddings of finitely and some infinitely connected subsets of ℂ into ℂ2. Math. Z. 252(1), 1–9 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia

    Tyson Ritter

Authors
  1. Tyson Ritter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tyson Ritter.

Additional information

Communicated by Marco Abate.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ritter, T. A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗ . J Geom Anal 23, 571–597 (2013). https://doi.org/10.1007/s12220-011-9254-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-011-9254-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation