The Kähler-Einstein metric for some Hartogs domains over bounded symmetric domains
We study the complete Kähler-Einstein metric of a Hartogs domain
tilde 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 Kähler-Einstein metric
satisfies a complex Monge-Ampère equation with boundary condition. The
domain tilde Omega is in general not homogeneous, but it has a
subgroup of automorphisms, the orbits of which are parameterized by
X \in [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 \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.