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Concave domains with trivial biholomorphic invariants. (English) Zbl 1019.32012
Let $$D$$ be a domain in $${\mathbb C}^n$$. Denote by $$O({\mathbb C}, D)$$ and $$O(\Delta, D)$$ the spaces of all holomorphic mappings from $$\mathbb C$$ to $$D$$ and from the unit disc $$\Delta\subseteq{\mathbb C}$$ to $$D$$, respectively. Let $$z, w \in D$$ and $$X \in {\mathbb C}^n$$. The Kobayashi metric and Lempert function are defined by $K_{D}(z, X) = \inf \{|\alpha|^{-1} : \exists f \text{ in }O(\Delta, D),\;f(0) = z,\;f'(0) = \alpha X \},$
$l_{D}(z, w)= \inf \{\tanh^{-1}|\alpha|: \exists f \text{ in }O(\Delta, D),\;f(0) = z,\;f(\alpha) = w \}.$ These invariants can be characterized as the largest metric and function which decrease under holomorphic mappings and coincide with the PoincarĂ© metric and distance on $$\Delta$$. A set $$D \subset{\mathbb C}^n$$ is called concave if its complement $${\mathbb C}^{n}\setminus D$$ is a convex set. The purpose of this note is to characterize the concave domains in $${\mathbb C}^n$$, $$n>1$$, whose Kobayashi metrics and Lempert functions identically vanish.
Theorem 1. Let $$D$$ be a concave domain in $${\mathbb C}^n$$, $$n>1$$. Then the following statements are equivalent : 1) $${\mathbb C}^{n}\setminus D$$ contains at most one $$(n - 1)$$-dimensional complex hyperplane; 2) for any $$z \in D$$, $$X \in {\mathbb C}^n\setminus 0$$ there is an injective $$f \in O({\mathbb C}, D)$$ such that $$f(0) = z$$, $$f'(0)=X$$; 3) for any $$z, w \in D$$, $$z \neq w$$ there is an injective $$f \in O({\mathbb C}, D)$$ such that $$f(0) = z$$, $$f(1) = w$$; 4) $$K_{D} \equiv 0$$; 5) $$l_{D} \equiv 0$$.

##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables 32F17 Other notions of convexity in relation to several complex variables 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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