## 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)].

### MSC:

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

### Citations:

Zbl 0647.32027; Zbl 0621.32025; Zbl 0717.32019
Full Text:

### References:

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