A uniformization theorem for Stein spaces
type: Item
instance of: scholarly article
title: A uniformization theorem for Stein spaces
zbMATH Open document ID: 1470.32025
author: Xiaojun Huang
author: Ming Xiao
published in: Complex Analysis and its Synergies
publication date: 2021-05-07T00:00:00Z
review text: About five years ago, the authors of this article proved Cheng's conjecture: the Bergman metric of a bounded smoothly, bounded, strongly pseudoconvex domain \(\Omega\) in \({\mathbb C}^n\) is Kähler-Einstein if and only if \(\Omega\) is biholomorphic to the ball. See [the authors, ``Bergman-Einstein metrics, hyperbolic metrics and Stein spaces with spherical boundaries'', Preprint, \url{arxiv:1604.07065}]. The present advance contains a similar result for normal Stein spaces \(X\) with isolated singularities. Specifically, the authors assume that the Bergman metric \(\omega\) on \(X\) is complete at infinity, and the induced Bergman-Bochner map is injective on the regular part \(X_0\) of \(X\). Then \(X\) is biholomorphic to the ball if and only if \((X_0,\omega)\) has constant holomorphic sectional curvature. Whether the holomorphic sectional curvature in the above statement can be replaced by Ricci's curvature is stated as a conjecture.
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32Q20
zbMATH DE Number: 7346307
zbMATH Keywords: Bergman metric
zbMATH Keywords: Stein space
zbMATH Keywords: uniformization
zbMATH Keywords: Kähler-Einstein metric
MaRDI profile type: Publication
full work available at URL: s40627-020-00044-x
OpenAlex ID: W3015610415
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DOI: 10.1007/S40627-020-00044-X
description: scientific article; zbMATH DE number 7346307