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 Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a Cartan-Hartogs domain Ω B (µ) endowed with the natural Kähler metric g(µ), Zedda conjectured that the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B (µ), g(µ)) is constant on Ω B (µ) if and only if (Ω B (µ), g(µ)) is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan-Hartogs domain (Ω B (µ), g(µ)).

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In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by $$\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}$$ Ω n = { ( z 1 , … , z n ) ∈ C n : | z 1 | < | z 2 | < ⋯ < | z n | < 1 } which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric $$g(\nu )$$ g ( ν ) associated with the Kähler potential $$\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)-\nu _n\ln (1-\vert z_n\vert ^2)$$ Φ n ( z ) : = - ∑ k = 1 n - 1 ν k ln ( | z k + 1 | 2 - | z k | 2 ) - ν n ln ( 1 - | z n | 2 ) on $$\varOmega _n$$ Ω n . As main contributions, on one hand we compute the explicit form for Bergman kernel of weighted Hilbert space, and then, we obtain the necessary and sufficient condition for the metric $$g(\nu )$$ g ( ν ) on the domain $$\varOmega _n$$ Ω n to be a balanced metric. On the other hand, by using the Calabi’s diastasis function, we prove that the Hartogs triangles admit a Berezin quantization.

The purpose of this paper is twofold.