Complete domains with respect to the Carathéodory distance

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.