Bergman metric

Summary

The Bergman metric of a bounded domain $\Omega \subseteq \C^n$ is the Kähler metric whose potential is the logarithm of the Bergman kernel on the diagonal. It is defined from the Hilbert space of square-integrable holomorphic functions, the Bergman space. Hence every biholomorphic map between bounded domains is an isometry for it. Stefan Bergman introduced it, and on the unit disc it is the hyperbolic metric up to a constant factor. It is a basic tool for the study of biholomorphic maps, for example in Fefferman's theorem on their boundary extension.