Further Steps Towards Classifying Homogeneous Kobayashi-Hyperbolic Manifolds with High-Dimensional Automorphism Group

  • Alexander IsaevEmail author
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We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension \(n\ge 4\) whose group of holomorphic automorphisms has dimension either \(n^2-4\), or \(n^2-5\), or \(n^2-6\). This paper continues a series of articles that achieve classifications for automorphism group dimension \(n^2-3\) and greater.


Kobayashi-hyperbolic manifolds Homogeneous complex manifolds The group of holomorphic automorphisms 

Mathematics Subject Classification

53C30 53C35 32Q45 32M05 32M10 



Most of the work in this paper was done during the author’s visit to the Steklov Mathematical Institute in Moscow, which the author thanks for its hospitality. We are also grateful to M. Jarnicki and P. Pflug for their help with the editorial procedures required to process this paper for publication.


  1. 1.
    Dorfmeister, J.: Homogeneous Siegel domains. Nagoya Math. J. 86, 39–83 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Isaev, A.V.: Hyperbolic manifolds of dimension \(n\) with automorphism group of dimension \(n^2-1\). J. Geom. Anal. 15, 239–259 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Isaev, A.V.: Hyperbolic \(n\)-dimensional manifolds with automorphism group of dimension \(n^2\). Geom. Funct. Anal. 17, 192–219 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Isaev, A.V.: Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds. Lecture Notes in Mathematics, vol. 1902. Springer, Berlin (2007)zbMATHGoogle Scholar
  5. 5.
    Isaev, A.V.: Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group. Geom. Topol. 12, 643–711 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Isaev, A.V.: Proper actions of high-dimensional groups on complex manifolds. Bull. Math. Sci. 5, 251–285 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Isaev, A.V.: Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms, to appear in Asian J. Math. Google Scholar
  8. 8.
    Isaev, A.V.: Homogeneous Kobayashi-hyperbolic manifolds with automorphism group of subcritical dimension. Complex Var. Elliptic Equ. 63, 1752–1766 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Isaev, A.V., Krantz, S.G.: On the automorphism groups of hyperbolic manifolds. J. reine angew. Math. 534, 187–194 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ishihara, S.: Homogeneous Riemannian spaces of four dimensions. J. Math. Soc. Japan 7, 345–370 (1955)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaneyuki, S.: On the automorphism groups of homogeneous bounded domains. J. Fac. Sci. Univ. Tokyo 14, 89–130 (1967)zbMATHGoogle Scholar
  12. 12.
    Kaneyuki, S., Tsuji, T.: Classification of homogeneous bounded domains of lower dimension. Nagoya Math. J. 53, 1–46 (1974)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kaup, W., Matsushima, Y., Ochiai, T.: On the automorphisms and equivalences of generalized Siegel domains. Am. J. Math. 92, 475–497 (1970)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. Marcel Dekker, New York (1970)zbMATHGoogle Scholar
  15. 15.
    Kobayashi, S.: Transformation Groups in Differential Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 70. Springer, New York (1972)CrossRefGoogle Scholar
  16. 16.
    Kobayashi, S.: Hyperbolic Complex Spaces, Grundlehren der Mathematischen Wissenschaften 318. Springer, Berlin (1998)Google Scholar
  17. 17.
    Nakajima, K.: Some studies on Siegel domains. J. Math. Soc. Japan 27, 54–75 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nakajima, K.: Homogeneous hyperbolic manifolds and homogeneous Siegel domains. J. Math. Kyoto Univ. 25, 269–291 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pyatetskii-Shapiro, I.I.: Automorphic Functions and the Geometry of Classical Domains. Gordon and Breach, New York (1969)Google Scholar
  20. 20.
    Rudin, W.: Function Theory in the Unit Ball of \({\mathbb{C}}^n\), Grundlehren der Mathematischen Wissenschaften 241. Springer, New York (1980)CrossRefGoogle Scholar
  21. 21.
    Satake, I.: Algebraic Structures of Symmetric Domains. \(\text{Kan}\hat{\text{ o }}\) Memorial Lectures 4. Princeton University Press, Tokyo (1980)Google Scholar
  22. 22.
    Verma, K.: A characterization of domains in \(\mathbb{C}^2\) with noncompact automorphism group. Math. Ann. 344, 645–701 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vinberg, E.B., Gindikin, S.G., Pjateckiĭ-Šapiro, I.I.: Classification and canonical realization of complex bounded homogeneous domains. Trans. Mosc. Math. Soc. 12, 404–437 (1963)Google Scholar

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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

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