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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 Poincaré 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) \(l_{D} \equiv 0\).

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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