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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. II


Author: Dong S. Kim
Journal: Proc. Amer. Math. Soc. 53 (1975), 141-142
MSC: Primary 32H15
DOI: https://doi.org/10.1090/S0002-9939-1975-0382731-0
MathSciNet review: 0382731
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Abstract: In [1] we have obtained the following result: Let $ D$ be a bounded domain in $ {{\text{C}}^n}$. Suppose there is a compact subset $ K$ of $ D$ such that for every $ x\epsilon D$ there is an analytic automorphism $ f\epsilon \operatorname{Aut} (D)$ and a point $ a\epsilon K$ such that $ f(x) = a$. Then $ D$ is a domain of bounded holomorphy, in the sense that $ D$ is the maximal domain on which every bounded holomorphic function on $ D$ can be continued holomorphically (cf. Narasimhan [2, Proposition 7, p. 127]). Here we shall give a stronger result: Under the same assumptions, $ D$ is $ c$-complete. We note that a $ c$-complete domain is a domain of bounded holomorphy, in particular, a domain of holomorphy. A domain of bounded holomorphy, however, need not be $ c$-complete.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0382731-0
Keywords: $ c$-complete, analytic automorphism, bounded homogeneous domain
Article copyright: © Copyright 1975 American Mathematical Society