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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Complete domains with respect to the Carathéodory distance

Author: Dong S. Kim
Journal: Proc. Amer. Math. Soc. 49 (1975), 169-174
MSC: Primary 32H15
MathSciNet review: 0367297
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Abstract: Concerning completeness with respect to the Carathéodory distance ($ c$-completeness), the following theorems are shown. A bounded convex (in geometric sense) domain $ D$ in $ {{\mathbf{C}}^n}({{\mathbf{R}}^{2n}})$ is $ c$-complete, so that it is boundedly holomorphic convex. To preserve $ c$-completeness in complex spaces, it is sufficient to have a proper local biholomorphic mapping as follows: Let $ \alpha $ be a proper spread map of a $ c$-hyperbolic complex space $ (X,A)$ onto a $ c$-hyperbolic complex space $ (\tilde X,\tilde A)$; then $ X$ is $ c$-complete if and only if $ \tilde X$ is $ c$-complete. We also show the following $ D$ to be domains of bounded holomorphy: let $ (X,A;\alpha )$ be a Riemann domain and $ D$ a domain in $ X$ with $ \alpha (D)$ bounded in $ {{\mathbf{C}}^n}$. Let $ B(D)$ separate the points of $ D$. Suppose there is a compact set $ K$ such that for any $ x \in D$ there is an analytic automorphism $ \sigma \in \operatorname{Aut} (D)$ and a point $ a \in K$ such that $ \sigma (x) = a$. Then $ D$ is a domain of bounded holomorphy.

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Keywords: $ c$-complete, $ c$-hyperbolic, domain of bounded holomorphy, boundedly holomorphic convex, envelope of bounded holomorphy, Stein manifold of bounded type, analytic automorphism, bounded homogeneous domain
Article copyright: © Copyright 1975 American Mathematical Society

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