Bergman-Calabi diastasis and K\"ahler metric of constant holomorphic sectional curvature

TL;DR

This paper characterizes when a bounded domain in complex space with a constant holomorphic sectional curvature Bergman metric is biholomorphic to a ball, linking geometric properties with the Bergman-Calabi diastasis and providing explicit formulas.

Contribution

It establishes the equivalence between biholomorphism to a ball and properties of the Bergman-Calabi diastasis, extending Lu's theorem to incomplete cases and connecting with the Bergman representative coordinate.

Findings

Biholomorphic to a ball iff the Bergman-Calabi diastasis is exhaustive or hyperconvex.

Explicit formulas for the Bergman-Calabi diastasis involving the Bergman representative coordinate.

Domains with constant holomorphic sectional curvature Bergman metric are Lu Qi-Keng.

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.