The Kähler–einstein Metric for Some Hartogs Domains over Bounded Symmetric Domains

We study the complete Kähler-Einstein metric of a Hartogs domain Ω̃ built on an irreducible bounded symmetric domain Ω, using a power N of the generic norm of Ω. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain Ω̃ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ∈ [0, 1[. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value μ0 of μ. We work out the details for the two exceptional symmetric domains. The special value μ0 seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains. Introduction Let D be a bounded domain in C. The complete (normalized) Kähler-Einstein metric on D is the Hermitian metric E Ez(u, v) = ∂u∂vg ∣∣ z , whose generating function g is the unique solution of the complex Monge-Ampère equation with boundary condition det ( ∂g ∂zi∂z ) = e (z ∈ D), g(z) → ∞ (z → ∂D) (see [1], [2], [3]). Let Ω be a bounded irreducible symmetric domain in V ≃ C; we will always consider such a domain in its circled realization. For a real positive number μ, let Ω̃ be the Hartogs type domain defined by Ω̃ = Ω̃k(μ) = { (z, Z) ∈ Ω × C | ‖Z‖ 2 < N(z, z) } , where N(z, z) denotes the generic norm of Ω (see Appendix A.3). The Bergman kernel of Ω̃ has been computed in [4] . Date: 12th January 2005. 2000 Mathematics Subject Classification. Primary: 32F45, 32M15; secondary: 32A25.