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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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