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 H α of square integrable holomorphic functions on (Ω B d 0 (µ), g(µ)) with the weight exp{−αϕ} (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.
The Fock-Bargmann-Hartogs domain D n,m (µ) (µ > 0) in C n+m is defined by the inequality w 2 < e −µ z 2 , where (z, w) ∈ C n × C m , which is an unbounded non-hyperbolic domain in C n+m . This paper introduces a Kähler metric αgThe 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 (µ), g(µ; ν)) with the weight exp{−αΦ} for α > 0. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric αg(µ; ν) (α > 0) on the domain D n,m (µ) to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.
The Fock-Bargmann-Hartogs domain Dn,m(µ) (µ > 0) in C n+m is defined by the inequality w 2 < e −µ z 2 , where (z, w) ∈ C n × C m , which is an unbounded nonhyperbolic domain in C n+m . Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain Dn,m(µ). In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock-Bargmann-Hartogs domain Dn,m(µ) with m ≥ 2 must be an automorphism.
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