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Tautness and complete hyperbolicity of domains in \(\mathbb{C}^n\). (English) Zbl 0912.32025
A domain \(\Omega\) in \({\mathbb{C}}^n\) is hyperbolic if the Kobayashi pseudo-metric on \(\Omega\) is a metric. A hyperbolic domain \(\Omega\) is said to be complete hyperbolic if the Kobayashi distance is complete.
In this paper, the author gives sufficient conditions on an unbounded domain in \({\mathbb{C}}^n\), in terms of hyperbolicity of the domain, for the normality of holomorphic maps taking their values in this domain. More precisely, he proves the following:
Theorem. Let \(\Omega\) be an unbounded domain in \({\mathbb{C}}^n\). Assume that there is a local peak holomorphic function at each point in \(\partial\Omega\cup\{\infty\}\). Then, \(\Omega\) is a complete hyperbolic domain.
This result generalizes a known result in the case of a bounded domain.
The author gives also a relation between local tautness and global tautness of a domain. We recall that a domain \(\Omega\) in \({\mathbb{C}}^n\) is taut if the family of holomorphic maps from the unit disk in \(\mathbb{C}\) to \(\Omega\) is a normal family. Finally, the author gives examples of taut and complete hyperbolic domains which illustrate the results. The examples include domains with noncompact automorphism groups.

MSC:
32M05 Complex Lie groups, group actions on complex spaces
32F45 Invariant metrics and pseudodistances in several complex variables
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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