Complete hyperbolicity of Hartogs domains (Q1416708)

Let \(X\) be a \(\mathbb{C}\)-analytic space and let \(\varphi: X\to [-\infty,\infty)\) be an upper semicontinuous function on \(X\). Then the Hartogs domain is defined as \[ \Omega_\varphi(X):= \{(z,w)\in X\otimes \mathbb{C}:| w|< e^{-\varphi(z)}\} \] and the following result is established. Theorem 3.2: If \(\Omega_\varphi(X)\) is complete hyperbolic, then so is \(X\) and furthermore \(\varphi\) is real-valued, continuous and plurisubharmonic. As shown by a construction due to Jarnicki and Pflug, the above necessary condition is not sufficient. In this direction, the authors prove the following Lemma 3.8: Assume that \(\varphi\) has the property (S) in Definition 3.7 (see the paper), for any point of \(X\). Then the converse of Theorem 3.2 holds. Also it was shown that this property (S) is indeed stronger than the plurisubharmonic condition. As a corollary, the authors prove the following. Theorem 3.6: Assume that \(\varphi\) is strictly plurisubharmonic. Then \(\Omega_\varphi(X)\) is complete hyperbolic provided \(X\) is. Finally, it is shown that the property (S) is not necessary for the complete hyperbolicity of \(\Omega_\varphi(X)\). The paper is well written.