Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The first two coefficients of the Bergman function expansions for
  Cartan-Hartogs domains   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.

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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: q-bio