No abstract

We consider the domain Ω of the Bergman-Hartogs type which bases on any bounded symmetric domain D, Ω = {(w (1) ,. .. , w (r) , z) ∈ C m 1 × • • • × C mr × D : ∥w (1) ∥ 2p 1 + • • • + ∥w (r) ∥ 2pr < KD(z, z) −q }, where KD(z, z) denotes the Bergman kernel on D, r is a positive integer, p1,. .. , pr > 1 and q > 0 are real parameters. We give the holomorphic automorphism group of Ω, and prove that the given holomorphic automorphism group is the full holomorphic automorphism group of Ω for r = 1. In addition, when r > 1, if the holomorphic automorphism mapping F on Ω maps (0, z) ∈ {0} × D to (0, z *) ∈ {0} × D, then F belongs to the given holomorphic automorphism group. Keywords domain of the Bergman-Hartogs type, holomorphic automorphism group, bounded symmetric domain MSC(2010) 32A07, 32M05

We study the complete Kähler–Einstein metric of certain Hartogs domains ${\rm\Omega}_{s}$ over bounded homogeneous domains in $\mathbb{C}^{n}$. The generating function of the Kähler–Einstein metric satisfies a complex Monge–Ampère equation with Dirichlet boundary condition. We reduce the Monge–Ampère equation to an ordinary differential equation and solve it explicitly when we take the parameter $s$ for some critical value. This generalizes previous results when the base is either the Euclidean unit ball or a bounded symmetric domain.