Characterizations of the unit ball \(B^n\) in complex Euclidean space
type: Item
instance of: scholarly article
title: Characterizations of the unit ball \(B^n\) in complex Euclidean space
zbMATH Open document ID: 0547.32013
DOI: 10.1007/BF01168151
author: I. Graham
author: Hung-Hsi Wu
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173592
review text: Two characterizations are given. It is shown that if \(M\) is a complex manifold such that at one point of \(M\) the Eisenman-Kobayashi and Eisenman- Carathéodory volume forms coincide and are nonzero, then \(M\) is biholomorphic tothe unit ball \(B^n\). This generalizes previous results of B. Wong, Rosay, and Dektyarev. It is also shown, using results of Lempert and Royden - P.M. Wong, that if \(\Omega\) is a strictly convex domain in \({\mathbb{C}}^ n\) such that either (a) there exists a point \(p\in\Omega\) and a holomorphic map \(\tau: B^n\to\Omega\), \(\tau(0)=p,\) such that \(d\tau(0)\) carries the unit ball onto the indicatrix of the Kobayashi metric; or (b) there exists a point \(p\in\Omega \) and a holomorphic map \(\sigma:\Omega\to B^n\), \(\sigma(p)=0\), such that \(d\sigma(p)\) carries the indicatrix of the Carathéodory metric onto the unit ball, then \(\Omega\) is biholomorphic to \(B^n\).
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32A07
zbMATH DE Number: 3871768
zbMATH Keywords: biholomorphic characterizations of the unit ball
zbMATH Keywords: volume form
zbMATH Keywords: strictly convex domain
zbMATH Keywords: Kobayashi metric
zbMATH Keywords: Carathéodory metric
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3871768