The Kähler-Einstein metric for some Hartogs domains over symmetric domains

Abstract

We study the complete K\"{a}hler-Einstein metric of a Hartogs domain $\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of $\Omega$. The generating function of the K\"{a}hler-Einstein metric satisfies a complex Monge-Amp\`{e}re equation with boundary condition. The domain $\widetilde {\Omega}$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This allows to reduce the Monge-Amp\`{e}re equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $\mu_{0}$ of $\mu$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.