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Pure and Applied Mathematics Quarterly
Volume 18 (2022)
Number 2
Special issue in honor of Joseph J. Kohn on the occasion of his 90th birthday
Guest Editors: J.E. Fornaess, Stanislaw Janeczko, Duong H. Phong, and Stephen S.T. Yau
Bergman–Calabi diastasis and Kähler metric of constant holomorphic sectional curvature
Pages: 481 – 502
DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a6
Authors
Abstract
We prove that for a bounded domain in $\mathbb{C}^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman–Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman–Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set.
Keywords
Bergman metric, Bergman representative coordinate, holomorphic sectional curvature, hyperconvex domain, Lu Qi-Keng domain, $L^2$-domain of holomorphy, pluripolar set
2010 Mathematics Subject Classification
Primary 32F45. Secondary 32D20, 32Q05, 32T05.
Received 25 June 2021
Accepted 24 August 2021
Published 13 May 2022