The Bergman Kernel on some Hartogs domains

TL;DR

This paper derives explicit formulas for the Bergman kernel on specific Hartogs domains, enabling detailed analysis of their boundary behavior and advancing understanding of complex analysis in several variables.

Contribution

The authors provide new explicit formulas for the Bergman kernel on two families of Hartogs domains, using slice computations and differential operators, which were not previously available.

Findings

Explicit formulas for the Bergman kernel on specified Hartogs domains.

Analysis of the boundary behavior of the kernel functions.

Enhanced understanding of complex structures in these domains.

Abstract

We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function on $$\{(z_1,z_2,w)\in\mathbb C^3:e^{|w|^2}|z_1|^2+|z_2|^2<1\}$$ and on $$\{(z_1,z_2,w)\in\mathbb C^3:|z_1|^2+|z_2|^2+|w|^2<1+|z_2w|^2\;{\rm and} \;|w|<1\}.$$ We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal.