The Bergman kernel for intersection of two complex ellipsoids

TL;DR

This paper derives explicit formulas for the Bergman kernel functions of the intersection of two complex ellipsoids in three dimensions, providing insights into their geometric and function-theoretic properties.

Contribution

It presents the first explicit closed-form expressions for Bergman kernels on these intersections and explores the Lu Qi-Keng problem for specific parameter values.

Findings

Explicit Bergman kernel formulas for specific ellipsoid intersections

Analysis of the Lu Qi-Keng problem in these domains

Connections between hypergeometric functions and complex geometry

Abstract

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for intersection of two complex ellipsoids $\{z \in \mathbb{C}^3 \colon |z_1|^p + |z_2|^q < 1, \quad |z_1|^p + |z_3|^r < 1\}$. We consider cases $p=6, q= r= 2$ and $p=q=r=2$. We also investigate the Lu Qi-Keng problem for $p=q=r=2$.