Balanced metrics on the Fock-Bargmann-Hartogs domains

TL;DR

This paper derives explicit formulas for the Bergman kernel on Fock-Bargmann-Hartogs domains equipped with a specific Kähler metric and characterizes when these metrics are balanced, contributing to complex geometry and metric theory.

Contribution

It provides explicit Bergman kernel formulas and necessary and sufficient conditions for the existence of balanced metrics on Fock-Bargmann-Hartogs domains.

Findings

Explicit Bergman kernel formula derived.

Necessary and sufficient conditions for balanced metrics established.

Existence of balanced metrics on a class of Fock-Bargmann-Hartogs domains confirmed.

Abstract

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper introduces a K\"{a}hler metric $\alpha g(\mu;\nu)$ $(\alpha>0)$ on $D_{n,m}(\mu)$, where $g(\mu;\nu)$ is the K\"{a}hler metric associated with the K\"{a}hler potential $\Phi(z,w):=\mu\nu{\Vert z\Vert}^{2}-\ln(e^{-\mu{\Vert z\Vert}^{2}}-\Vert w\Vert^2)$ ($\nu>-1$) on $D_{n,m}(\mu)$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on $(D_{n,m}(\mu), g(\mu;\nu))$ with the weight $\exp\{-\alpha \Phi\}$ for $\alpha>0$. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric $\alpha g(\mu;\nu)$ $(\alpha>0)$ on the domain $D_{n,m}(\mu)$ to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.