Invariant hyperbolic Stein domains. (English) Zbl 0791.32014

This paper deals with invariant hyperbolic Stein domains in complex Lie groups. Conditions which ensure hyperbolicity (in the sense of Kobayashi) are the existence of sufficiently many bounded holomorphic functions or the existence of a hermitian metric whose holomorphic sectional curvatures are bounded from above by a negative number. The author’s main results are that if \(G_ C\) is a complex Lie group and \(G_ R\) a real form then the coset \(\overline e\) in \(G_ C/G_ R\) of the neutral element \(e\) of \(G_ C\) admits a neighbourhood basis whose pre-images in \(G_ C\) are complete hyperbolic and Stein; moreover, if \(G_ R\) has a compact maximal connected semisimple subgroup then the preimage in \(G_ C\) of any relatively compact open subset in \(G_ C/G_ R\) is hyperbolic. He applies these results to obtain the following interesting theorem: Given a real connected Lie group \(G\) there exists a hyperbolic Stein complete manifold on which \(G\) acts freely and effectively. This result generalizes earlier results of E. Bedford and J. Dadok [Comment. Math. Helv. 62, 561-572 (1987; Zbl 0647.32027)], R. Saerens and W. R. Zame [Trans. Am. Math. Soc. 301, No. 1, 413-429 (1987; Zbl 0621.32025)] and A. E. Tumanov and G. B. Shabat [Funct. Anal. Appl. 24, No. 3, 255-257 (1991); translation from Funkts. Anal. Prilozh. 24, No. 3, 94-95 (1990; Zbl 0717.32019)].
Reviewer: H.Azad (Islamabad)


32M05 Complex Lie groups, group actions on complex spaces
32M10 Homogeneous complex manifolds
22E10 General properties and structure of complex Lie groups
Full Text: DOI EuDML


[1] Bedford, E.; Dadok, J., Bounded domains with prescribed group of automorphisms, Comm. Math. Helv., 62, 561-572, (1987) · Zbl 0647.32027
[2] Grauert, H.; Reckziegel, H.: Hermitesche Metriken und normale Familien holomorpher Abbildungen. Math. Z.89, 108-125 (1965) · Zbl 0135.12503
[3] Hochschild, G.: The Structure of Lie groups. Holden Day Inc. San Francisco. (1965) · Zbl 0131.02702
[4] Kiernan, P. J., Hyperbolically imbedded Spaces and the Big Picard Theorem, Math. Ann., 204, 203-209, (1973) · Zbl 0244.32010
[5] Kobayashi, S.: Hyperbolic manifolds and holomorphic mappings. Dekker Inc., New York 1970 · Zbl 0207.37902
[6] Lang, S.: Introduction to Complex Hyperbolic Spaces. Springer 1987 · Zbl 0628.32001
[7] Matsushima, Y.; Morimoto, A.: Sur certaines espaces fibrés holomorphes sur une variété de Stein. Bull. Soc. Math. France88, 137-155 (1960) · Zbl 0094.28104
[8] Oeljeklaus, K.: Une remarcque sur le groupe des automorphismes holomorphes de domaines tube dans ℂ \(n\). C.R. Acad. Sci. Paris312, 967-968 (1991) · Zbl 0744.32010
[9] Stein, K.: Überlagerungen holomorph-vollständiger komplexer Räume. Arch. Math.VIII, 354-361 (1956) · Zbl 0072.08002
[10] Saerens, R.; Zame, W. R., The Isometry Groups of Manifolds and the Automorphism Groups of Domains, Trans. A.M.S., 301, 413-429, (1987) · Zbl 0621.32025
[11] Tumanov, A. E.; Shabat, G. B., Realization of linear Lie Groups by Biholomorphic Automorphisms Funct, Anal. Appl., 24, 255-257, (1991) · Zbl 0717.32019
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