An Essay on the Bergman Metric and Balanced Domains
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]).
KeywordsPseudoconvex Domain Boundary Behavior Bergman Kernel Plurisubharmonic Function Complete Orthonormal System
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- [B-N]Bell, S. R. and Narasimhan, R. Proper holomorphic mappings of complex spaces, Complex manifolds, S. R. Bell et al. Springer, 1997.Google Scholar
- [E]Ermine, J.-L. Conjecture de Sérre et espaces hyperconvexes, Lecture Notes in Mathematics 670, Springer, 1978, 124–139.Google Scholar
- [H-1]Hörmander, L. L 2 estimates and existence theorems for the 5-operator, Acta Math. 113 (1965), 89–152.Google Scholar
- [H-2] Hörmander, L. An introduction to complex analysis in several variables, North Holland, 1990. [J-P-1]
- [J-P-2]Hörmander, L. Invariant distances and metrics in complex analysis, de Gruyter expositions in math. 9, 1993.Google Scholar
- [K]Kobayashi, S. Geometry of bounded domains,Trans. Amer. Math. Soc. 92 (1959), 267–290. [M-Y] Maitani, F. and Yamaguchi, H. Variation of three metrics on the moving Riemann surfaces,preprint.Google Scholar
- [O-1]Ohsawa, T. A remark on the completeness of the Bergman metric, Proc. Jap. Acad. 57 (1981), 238–240.Google Scholar
- [0-S-1]Ohsawa, T. and Sibony, N. Bounded P. S. H. functions and pseudoconvexity in Köhler manifolds,to appear in Nagoya Math. J.Google Scholar
- [0-S-2]Ohsawa, T. and Sibony, N. Nonexistence of Levi-flat hypersurfaces in P~,preprint.Google Scholar
- [St]Stehlé, J.-L. Fonctions plurisousharmoniques et convexité holomorphe de certain fibrés analytiques, Lecture Notes in Mathematics 474, 155–180.Google Scholar
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