Bergman metric


Let Gn be a domain and let K(z,w) be the Bergman kernel on G. We define a Hermitian metric on the tangent bundle Tzn by


for zG. Then the length of a tangent vector ξTzn is then given by


This metric is called the Bergman metric on G.

The length of a (piecewise) C1 curve γ:[0,1]n is then computed as


The distance dG(p,q) of two points p,qG is then defined as

dG(p,q):=inf{(γ) all piecewise C1 curves γ such that γ(0)=p and γ(1)=q}.

The distance dG is called the Bergman distance.

The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain. More importantly, the distance dG is invariant under biholomorphic mappings of G to another domain G. That is if f is a biholomorphism of G and G, then dG(p,q)=dG(f(p),f(q)).


  • 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Bergman metric
Canonical name BergmanMetric
Date of creation 2013-03-22 15:04:49
Last modified on 2013-03-22 15:04:49
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 32F45
Related topic BergmanKernel
Defines Bergman distance