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 definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain k j=1 Ω j B d 0 (µ) is defined as the Hartogs type domain constructed over the product k j=1 Ω j of irreducible bounded symmetric domains Ω j (1 ≤ j ≤ k), with the fiber over each point (z 1 , · · · , z k ) ∈ k j=1 Ω j being a ball in C d 0 of the radius k j=1 N Ω j (z j , z j ) µ j 2 of the product of positive powers of their generic norms. Any such domain k j=1 Ω j B d 0 (µ) (k ≥ 2) is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric αg(µ) (α > 0) on the domain k j=1 Ω j B d 0 (µ) to be a balanced metric, where g(µ)is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.
Using the theory of classical domains and Jack polynomials, we generalize several integrals over the unitary group, due to Fyodorov and Khoruzhenko, to integrals over the circular orthogonal and circular symplectic ensembles. While in the unitary case a fundamental role is played by complex matrices, it is anti-symmetric and symmetric matrices which play the same role in the latter two cases. In our workings, a generalization of the evaluation of the integral of a Jack polynomial times the Selberg weight sought by Fyodorov and Khoruzhenko is identified.
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