Bergman theory of certain generalized Hartogs triangles

TL;DR

This paper derives explicit formulas for the Bergman kernel on certain generalized Hartogs triangles in complex analysis, providing new insights into the Lu Qi-Keng problem and boundary behavior of the kernel.

Contribution

It offers a closed form expression for the Bergman kernel on specific Hartogs triangles, advancing understanding of their complex geometric properties.

Findings

Explicit Bergman kernel formulas for specified domains

New observations on the Lu Qi-Keng problem

Analysis of boundary behavior of the kernel

Abstract

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{\gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_{\gamma}(z,z)$.