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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.

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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