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

Robert Xin Dong (Department of Mathematics, University of Connecticut, Storrs, Ct., U.S.A.)

Bun Wong (Department of Mathematics, University of California, Riverside, Calif., U.S.A.)

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.

Full Text (PDF format)

Received 25 June 2021

Accepted 24 August 2021

Published 13 May 2022