On canonical metrics on Cartan-Hartogs domains - Feng, Zhiming; Tu, Zhenhan (arXiv.org, 2014)
The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain \(Ω^{B^{d_0}}(μ)\) endowed with the canonical metric \(g(μ)\), we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal{H}_{α}\) of square integrable holomorphic functions on \((Ω^{B^{d_0}}(μ), g(μ))\) with the weight \(\exp\{-α φ\}\) (where \(φ\) is a globally defined Kähler potential for \(g(μ)\)) for \(α>0\), and, furthermore, we give an explicit expression of the Rawnsley's \(ε\)-function expansion for \((Ω^{B^{d_0}}(μ), g(μ)).\) Secondly, using the explicit expression of the Rawnsley's \(ε\)-function expansion, we show that the coefficient \(a_2\) of the Rawnsley's \(ε\)-function expansion for the Cartan-Hartogs domain \((Ω^{B^{d_0}}(μ), g(μ))\) is constant on \(Ω^{B^{d_0}}(μ)\) if and only if \((Ω^{B^{d_0}}(μ), g(μ))\) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.