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

Abstract

Let $\varphi$ be a globally defined real Kähler potential on a domain $\Omega \subset {\mathbb{C}}^d$, and $g_F$ be a Kähler metric on the Hartogs domain $M=\{(z,w)\in \Omega \times {\mathbb{C}}^{d_0}:\Vert w{\Vert }^2<e^{-\varphi (z)}\}$ associated with the Kähler potential ${\Phi }_F(z,w)=\varphi (z)+F(\varphi (z)+\ln \Vert w{\Vert }^2)$. 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 $\phi$. 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 Kähler metrics on Cartan-Hartogs domains such that their the coefficients ${\mathbf{a}}_j(j=1,2)$ of the Bergman function expansions are constants.