The Holomorphic Automorphism Groups of Twisted Fock-Bargmann-Hartogs Domains

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Abstract

We consider a certain class of unbounded nonhyperbolic Reinhardt domains which is called the twisted Fock-Bargmann-Hartogs domains. By showing Cartan’s linearity theorem for our unbounded nonhyperbolic domains, we give a complete description of the automorphism groups of twisted Fock-Bargmann-Hartogs domains.

Keywords

holomorphic automorphism group Bergman kernel Reinhardt domain 

MSC 2010

32M05 32A25 32A07 

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References

  1. [1]
    H. Ahn, J. Byun, J.-D. Park: Automorphisms of the Hartogs type domains over classical symmetric domains. Int. J. Math. 23 (2012), 1250098, 11 pages.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    E. Bi, Z. Feng, Z. Tu: Balanced metrics on the Fock-Bargmann-Hartogs domains. Ann. Global Anal. Geom. 49 (2016), 349–359.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    J. P. D’Angelo: An explicit computation of the Bergman kernel function. J. Geom. Anal. 4 (1994), 23–34.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M. Engliš, G. Zhang: On a generalized Forelli-Rudin construction. Complex Var. Elliptic Equ. 51 (2006), 277-294.Google Scholar
  5. [5]
    Z. Huo: The Bergman kernel on some Hartogs domains. J. Geom. Anal. 27 (2017), 271–299.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    H. Ishi, C. Kai: The representative domain of a homogeneous bounded domain. Kyushu J. Math. 64 (2010), 35–47.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    M. Jarnicki, P. Pflug: First Steps in Several Complex Variables: Reinhardt Domains. EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2008.Google Scholar
  8. [8]
    M. Jarnicki, P. Pflug: Invariant Distances and Metrics in Complex Analysis. De Gruyter Expositions in Mathematics 9, Walter de Gruyter, Berlin, 2013.CrossRefMATHGoogle Scholar
  9. [9]
    H. Kim, V. T. Ninh, A. Yamamori: The automorphism group of a certain unbounded non-hyperbolic domain. J. Math. Anal. Appl. 409 (2014), 637–642.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    H. Kim, A. Yamamori: An application of a Diederich-Ohsawa theorem in characterizing some Hartogs domains. Bull. Sci. Math. 139 (2015), 737–749.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    H. Kim, A. Yamamori, L. Zhang: Invariant metrics on unbounded strongly pseudoconvex domains with non-compact automorphism group. Ann. Global Anal. Geom. 50 (2016), 261–295.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    A. Kodama: On the holomorphic automorphism group of a generalized complex ellipsoid. Complex Var. Elliptic Equ. 59 (2014), 1342–1349.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    E. Ligocka: On the Forelli-Rudin construction and weighted Bergman projections. Stud. Math. 94 (1989), 257–272.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Loi, M. Zedda: Balanced metrics on Cartan and Cartan-Hartogs domains. Math. Z. 270 (2012), 1077–1087.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Q. Lu: On the representative domain. Several Complex Variables. Proc. 1981 Hangzhou Conf., Birkhäuser, Boston, 1984, pp. 199–211.Google Scholar
  16. [16]
    J. Ning, H. Zhang, X. Zhou: Proper holomorphic mappings between invariant domains in ℂn. Trans. Am. Math. Soc. 369 (2017), 517–536.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    F. Rong: On automorphism groups of generalized Hua domains. Math. Proc. Camb. Philos. Soc. 156 (2014), 461–472.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    F. Rong: On automorphisms of quasi-circular domains fixing the origin. Bull. Sci. Math. 140 (2016), 92–98.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    G. Roos: Weighted Bergman kernels and virtual Bergman kernels. Sci. China Ser. A 48 (2005), Suppl., 225–237.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    G. Springer: Pseudo-conformal transformations onto circular domains. Duke Math. J. 18 (1951), 411–424.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    T. Tsuboi: Bergman representative domains and minimal domains. Jap. J. Math. 29 (1959), 141–148.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Z. Tu, L. Wang: Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains. J. Math. Anal. Appl. 419 (2014), 703–714.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Z. Tu, L. Wang: Rigidity of proper holomorphic mappings between equidimensional Hua domains. Math. Ann. 363 (2015), 1–34.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A. Yamamori: A remark on the Bergman kernels of the Cartan-Hartogs domains. C. R., Math. Acad. Sci. Paris 350 (2012), 157–160.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    A. Yamamori: The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function. Complex Var. Elliptic Equ. 58 (2013), 783–793.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    A. Yamamori: Automorphisms of normal quasi-circular domains. Bull. Sci. Math. 138 (2014), 406–415.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    A. Yamamori: A generalization of the Forelli-Rudin construction and deflation identities. Proc. Am. Math. Soc. 143 (2015), 1569–1581.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    A. Yamamori: Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan’s linearity theorem and explicit Bergman kernel. Tohoku Math. J. 69 (2017), 239–260.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    W. Yin: The Bergman kernels on Cartan-Hartogs domains. Chin. Sci. Bull. 44 (1999), 1947–1951.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    M. Zedda: Berezin-Engliš’ quantization of Cartan-Hartogs domains. J. Geom. Phys. 100 (2016), 62–67.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    L. Zhang: Bergman kernel function on Hua construction of the second type. Sci. China Ser. A 48 (2005), Suppl., 400–412.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Center for Mathematical ChallengesKorea Institute for Advanced StudySeoulRepublic of Korea
  2. 2.Academic Support CenterKogakuin UniversityHachioji, TokyoJapan

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