Moscow-Petersburg seminar on low-dimensional mathematics
October 29, 2004
G. Roos (SPb)
Invariant metrics on bounded complex domains
Abstract:
The Kaehler-Einstein metric of a bounded complex domain is an invariant
metric, generated by a solution of a complex Monge-Amp\`ere equation with
boundary condition. For some bounded complex non-homogeneous domains, built
over bounded symmetric domains, it is possible to reduce the complex
Monge-Amp\`ere equation, which is a highly non-linear PDE, to an ordinary
differential equation.
In special cases, the solution is explicit (and has a simple form); in
the same cases, the Bergman kernel of the domain, which generates another
invariant metric, seems also to have special properties. This relies on a
combinatorial conjecture on the coefficients of some special polynomials; up
to now, this conjecture has been verified in many significant cases
(especially, in the exceptional "low dimensions" 16 and 27). If this
conjecture is true, it would allow to compare on some domains the
Kaehler-Einstein metric and the Bergman metric, two different invariant
metrics for which few comparison theorems are known.
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