A uniformization theorem for Stein spaces
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.
scientific article; zbMATH DE number 7346307
scientific article