Complete domains with respect to the Carathéodory distance. II

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.