Mathematische Zeitschrift

, Volume 277, Issue 1–2, pp 325–338 | Cite as

Oka properties of ball complements

Article

Abstract

Let \(n>1\) be an integer. We prove that holomorphic maps from Stein manifolds \(X\) of dimension \({<}n\) to the complement \(\mathbb {C}^n{\setminus } L\) of a compact convex set \(L\subset \mathbb {C}^n\) satisfy the basic Oka property with approximation and interpolation. If \(L\) is polynomially convex then the same holds when \(2\dim X \le n\). We also construct proper holomorphic maps, immersions and embeddings \(X\rightarrow \mathbb {C}^n\) with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.

Keywords

Oka principle Holomorphic flexibility Oka manifold Polynomial convexity 

Mathematics Subject Classification (2010)

Primary 32E10 32E20 32E30 32H02 Secondary 32Q99 

References

  1. 1.
    Acquistapace, F., Broglia, F., Tognoli, A.: A relative embedding theorem for Stein spaces. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 2(4), 507–522 (1975)Google Scholar
  2. 2.
    Andersén, E.: Volume-preserving automorphisms of \({\mathbb{C}}^n\). Complex Var. Theory Appl. 14(1–4), 223–235 (1990)Google Scholar
  3. 3.
    Andersén, E., Lempert, L.: On the group of holomorphic automorphisms of \({\mathbb{C}}^n\). Invent. Math. 110, 371–388 (1992)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Andrist, R., Forstneric, F., Ritter, T., Wold, E.F.: Proper holomorphic embeddings into Stein manifolds with the density property. arxiv.org/abs/1309.6956Google Scholar
  5. 5.
    Andrist, R.B., Wold, E.F.: Riemann surfaces in Stein manifolds with density property. Ann. Inst. Fourier (to appear). arXiv:1106.4416Google Scholar
  6. 6.
    Andrist, R.B., Wold, E.F.: The complement of the closed unit ball in \({\mathbb{C}}^3\) is not Subelliptic. arXiv:1303.1804Google Scholar
  7. 7.
    Bishop, E.: Mappings of partially analytic spaces. Am. J. Math. 83, 209–242 (1961)CrossRefMATHGoogle Scholar
  8. 8.
    Chen, B.-Y., Wang, X.: Holomorphic maps with large images. arxiv:1303.5242Google Scholar
  9. 9.
    Dor, A.: Approximation by proper holomorphic maps into convex domains. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 20(4), 147–162 (1993)Google Scholar
  10. 10.
    Dor, A.: Immersions and embeddings in domains of holomorphy. Trans. Am. Math. Soc. 347, 2813–2849 (1995)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Drinovec Drnovšek, B., Forstnerič, F.: Holomorphic curves in complex spaces. Duke Math. J. 139, 203–254 (2007)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Drinovec Drnovšek, B., Forstnerič, F.: Strongly pseudoconvex Stein domains as subvarieties of complex manifolds. Am. J. Math. 132, 331–360 (2010)CrossRefMATHGoogle Scholar
  13. 13.
    Fornæss, J.E.: Embedding strictly pseudoconvex domains in convex domains. Am. J. Math. 98, 529–569 (1976)CrossRefMATHGoogle Scholar
  14. 14.
    Fornæss, J.E., Stout, E.L.: Spreading polydiscs on complex manifolds. Am. J. Math. 99, 933–960 (1977)Google Scholar
  15. 15.
    Fornæss, J.E., Stout, E.L.: Regular holomorphic images of balls. Ann. Inst. Fourier (Grenoble) 32, 23–36 (1982)Google Scholar
  16. 16.
    Forstnerič, F.: Complements of Runge domains and holomorphic hulls. Mich. Math. J. 41, 297–308 (1994)CrossRefMATHGoogle Scholar
  17. 17.
    Forstnerič, F.: Interpolation by holomorphic automorphisms and embeddings in \({\mathbb{C}}^n\). J. Geom. Anal. 9, 93–118 (1999)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Forstnerič, F.: Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis). Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56. Springer, Berlin (2011)Google Scholar
  19. 19.
    Forstnerič, F., Lárusson, F.: Survey of Oka theory. N.Y. J. Math. 17a, 11–38 (2011)Google Scholar
  20. 20.
    Forstnerič, F., Rosay, J.-P.: Approximation of biholomorphic mappings by automorphisms of \({\mathbb{C}}^n\). Invent. Math. 112, 323–349 (1993). Erratum. Invent. Math. 118, 573–574 (1994)Google Scholar
  21. 21.
    Globevnik, J.: On Fatou-Bieberbach domains. Math. Z. 229, 91–106 (1998)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Gromov, M.: Oka’s principle for holomorphic sections of elliptic bundles. J. Am. Math. Soc. 2, 851–897 (1989)MATHMathSciNetGoogle Scholar
  23. 23.
    Hamm, H.: Zum Homotopietyp Steinscher Räume. J. Reine Ang. Math. 338, 121–135 (1983)MATHMathSciNetGoogle Scholar
  24. 24.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North-Holland Mathematical Library, vol. 7. North-Holland, Amsterdam (1990)Google Scholar
  25. 25.
    Narasimhan, R.: Imbedding of holomorphically complete complex spaces. Am. J. Math. 82, 917–934 (1960)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Remmert, R.: Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes. C. R. Acad. Sci. Paris 243, 118–121 (1956)MATHMathSciNetGoogle Scholar
  27. 27.
    Ritter, T.: A strong Oka principle for embeddings of some planar domains into \({\mathbb{C}}\times {\mathbb{C}}^*\). J. Geom. Anal. 23, 571—597 (2013)Google Scholar
  28. 28.
    Rosay, J.-P., Rudin, W.: Holomorphic maps from \({ C}^n\) to \({ C}^n\). Trans. Am. Math. Soc. 310, 47–86 (1988)MATHMathSciNetGoogle Scholar
  29. 29.
    Stensønes, B.: Fatou-Bieberbach domains with \({\cal C}^\infty \)-smooth boundary. Ann. Math. 145(2), 365–377 (1997)CrossRefGoogle Scholar
  30. 30.
    Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007)MATHGoogle Scholar
  31. 31.
    Varolin, D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11, 135–160 (2001)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Varolin, D.: The density property for complex manifolds and geometric structures II. Int. J. Math. 11, 837–847 (2000)MATHMathSciNetGoogle Scholar
  33. 33.
    Wermer, J.: An example concerning polynomial convexity. Math. Ann. 139, 147–150 (1959)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Winkelmann, J.: Non-degenerate maps and sets. Math. Z. 249, 783–795 (2005)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Institute of Mathematics, Physics, and MechanicsLjubljanaSlovenia
  3. 3.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  4. 4.Matematisk InstituttUniversitetet i OsloOsloNorway

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