Skip to main content

Proper holomorphic mappings of balanced domains in \(\mathbb {C}^n\)

Mathematische Zeitschrift volume 280, pages 257–268 (2015)Cite this article

Abstract

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in \(\mathbb {C}^n, \ n > 1\). Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in \(\mathbb {C}^n, \ n > 1\), is an automorphism. The main novelty of our proof is the use of a recent result of Opshtein on the behaviour of the iterates of holomorphic self-maps of a certain class of domains. We use Opshtein’s theorem, together with the tools made available by finiteness of type, to deduce that the aforementioned map is unbranched. The monodromy theorem then delivers the result.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Abate, M.: Iteration Theory of Holomorphic Maps on Taut Manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry. Mediterranean Press, Rende (1989)

    Google Scholar 

  2. Alexander, H.: Proper holomorphic mappings in \(C^{n}\). Indiana Univ. Math. J. 26(1), 137–146 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bedford, E., Bell, S.: Proper self-maps of weakly pseudoconvex domains. Math. Ann. 261(1), 47–49 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bell, S., Catlin, D.: Boundary regularity of proper holomorphic mappings. Duke Math. J. 49(2), 385–396 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bedford, E.: Proper holomorphic mappings. Bull. Am. Math. Soc. (N.S.) 10(2), 157–175 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Steven, R.: Bell, Proper holomorphic mappings between circular domains. Comment. Math. Helv. 57(4), 532–538 (1982)

    MATH  MathSciNet  Google Scholar 

  7. Bell, S.: Local boundary behavior of proper holomorphic mappings, Complex Analysis of Several Variables (Madison, Wis., 1982). In: Proceedings, Symposia in Pure Mathematics, vol. 41, American Mathematical Society, Providence, pp. 1–7 (1984)

  8. Berteloot, F.: Holomorphic vector fields and proper holomorphic self-maps of Reinhardt domains. Ark. Mat. 36(2), 241–254 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bharali, G., Janardhanan, J.: Proper holomorphic maps between bounded symmetric domains revisited. Pac. J. Math. 271(1), 1–24 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  10. Catlin, D.: Boundary invariants of pseudoconvex domains. Ann. Math. (2) 120, 529–586 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  11. Catlin, D.: Subelliptic estimates for the \(\overline{\partial }\)-Neumann problem on pseudoconvex domains. Ann. Math. (2) 126(1), 131–191 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chirka, E.M.: Complex Analytic Sets, Mathematics and its Applications (Soviet Series), vol. 46. Kluwer Academic Publishers Group, Dordrecht (1989)

    Book  Google Scholar 

  13. Coffman, A., Pan, Y.: Proper holomorphic maps from domains in \({\mathbb{C}}^2\) with transverse circle action. Chin. Ann. Math. Ser. B 28(5), 533–542 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Coupet, B., Pan, Y., Sukhov, A.: On proper holomorphic mappings from domains with \( T\)-action. Nagoya Math. J. 154, 57–72 (1999)

    MATH  MathSciNet  Google Scholar 

  15. Coupet, B., Pan, Y., Sukhov, A.: Proper holomorphic self-maps of quasi-circular domains in \(\mathbf{C}^2\). Nagoya Math. J. 164, 1–16 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Diederich, K., Fornæss, J.E.: Boundary regularity of proper holomorphic mappings. Invent. Math. 67(3), 363–384 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. D’Angelo, J.P., Kohn, J.J.: Subelliptic estimates and finite type. In: Schneider, M., Siu, Y.-T. (eds.) Several Complex Variables (Berkeley, CA, 1995–1996). Mathematical Sciences Research Institute Publications, vol. 37, pp. 199–232. Cambridge University Press, Cambridge (1999)

  18. Forstnerič, F.: Proper holomorphic mappings: a survey, Several Complex Variables (Stockholm, 1987/1988), Mathematical Notes, vol. 38, Princeton University Press, Princeton, NJ, pp. 297–363 (1993)

  19. Hurewicz, W., Wallman, H., Theory, D.: Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton (1941)

    Google Scholar 

  20. Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, de Gruyter Expositions in Mathematics, vol. 9. Walter de Gruyter & Co., Berlin (1993)

    Book  Google Scholar 

  21. Nicoara, C.: Effective vanishing order of the Levi determinant. Math. Ann. 354(4), 1223–1245 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Opshtein, E.: Dynamique des applications holomorphes propres de domaines réguliers et problème de l’injectivité. Math. Ann. 335(1), 1–30 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pan, Y.: Proper holomorphic self-mappings of Reinhardt domains. Math. Z. 208(2), 289–295 (1991)

    Article  MathSciNet  Google Scholar 

  24. Sibony, N.: Une classe de domaines pseudoconvexes. Duke Math. J. 55, 299–319 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  25. Vesentini, E.: Complex geodesics. Compos. Math. 44(1–3), 375–394 (1981)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

I would like to thank my advisor Gautam Bharali for his support during the course of this work, and for suggesting several useful ideas. The idea behind Lemma 5.1 was given by him. I would also like to thank my colleague and friend G. P. Balakumar for some informative discussions related to this work. I am truly grateful to the anonymous referee of this article for the many suggestions for improving the exposition.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    Jaikrishnan Janardhanan

Authors
  1. Jaikrishnan Janardhanan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaikrishnan Janardhanan.

Additional information

This work is supported by a UGC Centre for Advanced Study Grant and by a scholarship from the IISc.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Janardhanan, J. Proper holomorphic mappings of balanced domains in \(\mathbb {C}^n\) . Math. Z. 280, 257–268 (2015). https://doi.org/10.1007/s00209-015-1421-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-015-1421-z

Keywords

  • Balanced domains
  • Proper holomorphic maps
  • Alexander’s theorem

Mathematics Subject Classification

  • Primary 32H35