Skip to main content
Log in

Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains

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

Abstract

The Fock–Bargmann–Hartogs domain \(D_{n,m}(\mu )\) (\(\mu >0\)) in \(\mathbb {C}^{n+m}\) is defined by the inequality \(\Vert w\Vert ^2<e^{-\mu \Vert z\Vert ^2},\) where \((z,w)\in \mathbb {C}^n\times \mathbb {C}^m\), which is an unbounded non-hyperbolic domain in \(\mathbb {C}^{n+m}\). Recently, Tu–Wang obtained the rigidity result that proper holomorphic self-mappings of \(D_{n,m}(\mu )\) are automorphisms for \(m\ge 2\), and found a counter-example to show that the rigidity result is not true for \(D_{n,1}(\mu )\). In this article, we obtain a classification of proper holomorphic mappings between \(D_{n,1}(\mu )\) and \(D_{N,1}(\mu )\) with \(N<2n\).

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

Access this article

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. Alexander, H.: Holomorphic mappings from the ball and polydisc. Math. Ann. 209, 249–256 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Cima, J., Suffridge, T.J.: A reflection principle with applications to proper holomorphic mappins. Math. Ann. 265, 489–500 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Diederich, K., Fornæss, J.E.: Proper holomorphic images of strictly pseudoconvex domains. Math. Ann. 259, 279–286 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dini, G., Primicerio, A.S.: Proper holomorphic mappings between generalized pseudoellipsoids. Ann. Mat. Pura Appl. 158, 219–229 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ebenfelt, P., Son, D.N.: Holomorphic mappings between pseudoellipsoids in different dimensions. Methods Appl. Anal. 21(3), 365–378 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Faran, J.: Maps from the two ball to the three ball. Invent. Math. 68, 441–475 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, X.J.: On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimansions. J. Differ. Geom. 51, 13–33 (1999)

    Article  MATH  Google Scholar 

  10. Kim, H., Ninh, V.T., Yamamori, A.: The automorphism group of a certain unbounded non-hyperbolic domain. J. Math. Anal. Appl. 409, 637–642 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Landucci, M., Pinchuk, S.: Proper mappings between Reinhardt domains with an analytic variety on the boundary. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(3), 363–373 (1995)

    MathSciNet  MATH  Google Scholar 

  12. Pelles, D.: Proper holomorphic self-maps of the unit ball, Math. Ann. 190, 298–305 (1971). Correction, Math. Ann. 202, 135–136 (1973)

  13. Pinčuk, S.I.: On the analytic continuation of biholomorphic mappings, Math. Sb. 98(140), no. 3, 416–435 (1975); English transl., Math. USSR-Sb. 27, 375–392 (1975)

  14. Pinčuk, S.I.: On holomorphic mappings of real-analytic hypersurfaces, Math. Sb. 105(147), no. 4, 574–593 (1978); English transl., Math. USSR-Sb. 34, 503–519 (1978)

  15. Poincaré, H.: Les Fonctions analytiques de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo 23, 185–220 (1907)

    Article  MATH  Google Scholar 

  16. Spiro, A.: Classification of proper holomorphic maps between Reinhardt domains in \(\mathbb{C}^2\). Math. Z. 227, 27–44 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Su, G.C., Tu, Z.H., Wang, L.: Rigidity of proper holomorphic self-mappings of the pentablock. J. Math. Anal. Appl. 424, 460–469 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tanaka, N.: On the pseudo-conformal geometry of hypersurfaces of the space of \(n\) complex variables. J. Math. Soc. Jpn. 14, 397–429 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tu, Z.H.: Rigidity of proper holomorphic maps between equidimensional bounded symmetric domains. Proc. Am. Math. Soc. 130, 1035–1042 (2002)

    Article  MATH  Google Scholar 

  20. Tu, Z.H., Wang, L.: Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains. J. Math. Anal. Appl. 419, 703–714 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tu, Z.H., Wang, L.: Rigidity of proper holomorphic mappings between equidimensional Hua domains. Math. Ann. 363, 1–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Webster, S.: On mappings an \(n\)-ball into an \((n+1)\)-ball in complex spaces. Pacific J. Math. 81, 267–272 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zapalowski, P.: Proper holomorphic mappings between generalized Hartogs triangles. Ann. Mat. Pura Appl. 196, 1055–1071 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Xianyu Zhou for his helpful discussions, and thank the referees for useful comments. The first author was supported by the National Natural Science Foundation of China (No. 11671306), and the second author was partially supported by China Postdoctoral Science Foundation (No. 2016M601150).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lei Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tu, Z., Wang, L. Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains. J Geom Anal 29, 378–391 (2019). https://doi.org/10.1007/s12220-018-9995-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-018-9995-4

Keywords

Mathematics Subject Classification

Navigation