On the characterization of Grauert tubes covered by the ball
type: Item
instance of: scholarly article
title: On the characterization of Grauert tubes covered by the ball
zbMATH Open document ID: 0892.32011
author: Su-Jen Kan
published in: Mathematische Annalen
publication date: 1997-09-09T00:00:00Z
review text: A real analytic manifold \(X\) admits a complexification \(X_C\), a complex manifold which contains \(X\) as the real points, or fixed points of an antiholomorphic involution, or complex conjugation \(\sigma: X_C \to X_C\). If \(X\) is compact there is a neighborhood \(M\) of \(X\) in \(X_C\) and a smooth strictly plurisubharmonic function \(\rho: M\to [0,1)\) such that \(X\) is the zero set of \(\rho\). Under two extra data the function \(\rho\) is unique. Thus, the set \(X^r_C= \{\rho<r\}\) is called the Grauert tube of radius \(r\) over the center \(X= \{\rho=0\}\). The author proves the following Theorem: Let \(\Omega= X^r_C\) be a Grauert tube covered by the unit ball \(B^n\), then \(X= H^n/ \Gamma\) for some discrete subgroup \(\Gamma\) of \(O(n,1)\) and \(\Omega= B^n/ \Gamma\), where \(H^n =\{x\in R^n;\;| x |^2 <1\}\) is the real hyperbolic space with a certain complete Riemannian metric with constant sectional curvature.
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32C05
zbMATH DE Number: 1058818
zbMATH Keywords: real analytic manifold
zbMATH Keywords: complexification
zbMATH Keywords: Grauert tube
reviewed by: Akihiko Morimoto
MaRDI profile type: Publication
full work available at URL: s002080050102
OpenAlex ID: W1968953617
DOI: 10.1007/S002080050102
description: scientific article; zbMATH DE number 1058818