The first two coefficients of the Bergman function expansions for Cartan-Hartogs domains

TL;DR

This paper derives explicit formulas for the first two coefficients of the Bergman function expansion on Cartan-Hartogs domains, characterizes when these coefficients are constant, and classifies invariant metrics with constant coefficients.

Contribution

It provides explicit formulas for Bergman coefficient expansions and classifies invariant metrics with constant coefficients on Cartan-Hartogs domains.

Findings

Explicit formulas for  and  coefficients of Bergman expansion.

Necessary and sufficient conditions for these coefficients to be constant.

Complete classification of invariant metrics with constant coefficients.

Abstract

Let $\phi$ be a globally defined real K\"{a}hler potential on a domain $\Omega\subset \mathbb{C}^d$, and $g_{F}$ be a K\"{a}hler metric on the Hartogs domain $ M=\{(z,w)\in \Omega\times\mathbb{C}^{d_0}: \|w\|^2<e^{-\phi(z)}\}$ associated with the K\"{a}hler potential $\Phi_{F}(z,w)=\phi(z)+F(\phi(z)+\ln\|w\|^2)$. Firstly, we obtain explicit formulas of the coefficients $\mathbf{a}_j\;(j=1,2)$ of the Bergman function expansion for the Hartogs domain $( M,g_F)$ in a momentum profile $\varphi$. Secondly, using explicit expressions of $\mathbf{a}_j\;(j=1,2)$, we obtain necessary and sufficient conditions for the coefficients $\mathbf{a}_j\;(j=1,2)$ to be constants. Finally, we obtain all the invariant complete K\"{a}hler metrics on Cartan-Hartogs domains such that their the coefficients $\mathbf{a}_j\; (j=1,2)$ of the Bergman function expansions are constants.