Some geometric properties of special domains in a Banach space (Q1873624)

The author gives some explicit sufficient conditions for the complete hyperbolicity of Hartogs domains and Reinhardt domains in Banach analytic spaces. Let \(X\) be a Banach analytic space and \(\varphi\) an upper-semicontinuous function on \(X\). Define \(\Omega_{\varphi(X)}= \{(x,\lambda)\in x\times \mathbb{C}:|\lambda|< e^{-\varphi(x)}\}\) and we call it a Banach Hartogs domain. In this paper, the author first studies \(\Omega_{\varphi(X)}\). By making use of the properties of \(\varphi\), he gives a sufficient condition under which \(\Omega_{\varphi(X)}\) is complete hyperbolic. He also gives a necessary and sufficient condition for which \(\Omega_{\varphi(X)}\) has the holomorphic extension property through closed pluripolar sets. He next studies a balanced pseudoconvex Reinhardt domain in \(X\) and gives an extension of results due to \textit{M. Jarnicki} and \textit{P. Pflug} [Trans. Am. Math. Soc. 304, 385-404 (1987; Zbl 0649.32011), see also \textit{P. Pflug}, North-Holland Math. Stud. 88, 331-337 (1984; Zbl 0536.32001)].