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Mathematische Annalen

, Volume 313, Issue 4, pp 585–607 | Cite as

A characterization of products of balls by their isotropy groups

  • Axel Hundemer

Abstract.

In this paper we will characterize products of balls – especially the ball and the polydisc – in \(\mathbb{C}^n\) by properties of the isotropy group of a single point. It will be shown that such a characterization is possible in the class of Siegel domains of the second kind, a class that extends the class of bounded homogeneous domains, and that such a characterization is no longer possible in the class of bounded domains with noncompact automorphism groups. The main result is that a Siegel domain of the second kind \(G\subset\mathbb{C}^n\) is biholomorphically equivalent to a product of balls, iff there is a point \(p\in G\) such that the isotropy group of p contains a torus of dimension n. As an application it will be proved that the only domains biholomorphically equivalent to a Siegel domain of the second kind and to a Reinhardt domain are exactly the domains biholomorphically equivalent to a product of b alls.

Mathematics Subject Classification (1991): 32A07, 32M05, 32M15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Axel Hundemer
    • 1
  1. 1. Department of Mathematics, University of Michigan, East Hall, Ann Arbor, MI 48109–1109, USA (e-mail: hundemer@math.lsa.umich.edu) US

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