Bergman-Calabi diastasis and Kähler metric of constant holomorphic sectional curvature

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.

17 pages. Final revised version to appear in Pure and Applied Mathematics Quarterly (Special Issue in honor of Joseph J. Kohn)

References in corpus (1)

• Sharp pointwise and uniform estimates for $\bar{\partial }$