Complete hyperbolicity of Hartogs domains
type: Item
instance of: scholarly article
title: Complete hyperbolicity of Hartogs domains
zbMATH Open document ID: 1043.32019
author: Nguyen Quang Dieu
author: Do Duc Thai
published in: Manuscripta Mathematica
publication date: 2003-12-16T00:00:00Z
review text: 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.
reviewed by: Tan Vovan
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32Q45
zbMATH DE Number: 2018260
MaRDI profile type: Publication
full work available at URL: s00229-003-0388-y
OpenAlex ID: W2068934612
DOI: 10.1007/S00229-003-0388-Y
description: scientific article; zbMATH DE number 2018260