An Essay on the Bergman Metric and Balanced Domains

  • Takeo Ohsawa
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)


Let Ω be a bounded pseudoconvex domain in ℂ n and let be the Bergman kernel function of Ω. The boundary behavior of K Ω reflects the mass distribution of L 2 holomorphic functions on Ω through the geometry of the boundary ∂Ω in a very natural way, as one can see it from [H1] and various subsequent works (cf. [D], [P], [D’A], [O-1,2], [D-H-O], [C], [D-H], [B-S-Y], etc.). From the viewpoint of biholomorphic geometry, the Bergman metric
is also a natural quantity attached to Ω. Being a Hermitian metric invariant under the biholomorphic transformations, the Bergman metric is of intrinsic nature, while the values of K Ω are not. It is well known that one can draw important information on proper holomorphic mappings from the asymptotics of K Ω and ds Ω 2 (see [F] and [B-N] for more precise statements). Besides such an application, the boundary behavior of the Bergman kernel and the metric is of considerable significance in the current complex analysis, because they supply questions that urge further developments of the so called L 2 method for the ∂-operator. For instance, it was asked in [O-1] whether or not if ∂Ωis C 2-smooth, δΩ (z) being the distance from z to ∂Ω,and this was answered affirmatively in [O-T] as a corollary of a very general extension theorem for L 2 holomorphic functions, which found even an application to algebraic geometry (cf. [A-S]).


Pseudoconvex Domain Boundary Behavior Bergman Kernel Plurisubharmonic Function Complete Orthonormal System 
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Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Takeo Ohsawa
    • 1
  1. 1.Graduate School of MathematicsNagoya UniversityJapan

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