Advertisement

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)

Abstract

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]).

Keywords

Pseudoconvex Domain Boundary Behavior Bergman Kernel Plurisubharmonic Function Complete Orthonormal System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A-S]
    Angehrn, U. and Siu, Y. T. Effective freeness and separation of points for disjoint bundles, Invent. Math. 122 (1995), 291–308.MathSciNetMATHGoogle Scholar
  2. [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
  3. [B-S-Y]
    Boas, H. P, Straube, E. J. and Yu, J. Boundary limits of the Bergman kernal and metric, Michigan Math. J. 42 (1995), 449–461.MathSciNetMATHGoogle Scholar
  4. [C]
    Catlin, D. Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), 429–466.MathSciNetCrossRefMATHGoogle Scholar
  5. [D’A]
    D’Angelo, J. A note on the Bergman kernel, Duke. Math. J. 45 (1978), 259–265.MathSciNetMATHGoogle Scholar
  6. [D]
    Diederich, K. Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudokonvexen Gebieten, Math. Ann. 187 (1970), 9–36.MathSciNetCrossRefMATHGoogle Scholar
  7. [D-F]
    Diederich, K. and Fornaess, J. E. Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions, Inv. Math. 39 (1977), 129–141.MathSciNetMATHGoogle Scholar
  8. [D-H]
    Diederich, K. and Herbort, G. Extension of holomorphic L 2 -functions with weighted growth conditions, Nagoya Math. J. 126 (1992), 141–157.MathSciNetMATHGoogle Scholar
  9. [D-H-O]
    Diederich, K, Herbort, G. and Ohsawa, T. The Bergman kernel on uniformly extendable pseudoconvex domains, Math. Ann. 273 (1986), 471–478.MathSciNetCrossRefMATHGoogle Scholar
  10. [D-0]
    Diederich, K. and Ohsawa, T. An estimate for the Bergman distance on pseudoconvex domains, Ann. Math. 141 (1995), 181–190.MathSciNetCrossRefMATHGoogle Scholar
  11. [E]
    Ermine, J.-L. Conjecture de Sérre et espaces hyperconvexes, Lecture Notes in Mathematics 670, Springer, 1978, 124–139.Google Scholar
  12. [F]
    Fefferman, C. The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.MathSciNetMATHGoogle Scholar
  13. [H-1]
    Hörmander, L. L 2 estimates and existence theorems for the 5-operator, Acta Math. 113 (1965), 89–152.Google Scholar
  14. [H-2] Hörmander, L. An introduction to complex analysis in several variables, North Holland, 1990. [J-P-1]
    Jarnicki, M. and Pflug, P. Bergman completeness of complete circular domains, Ann. Pol. Math. 50 (1989), 219–222.MathSciNetMATHGoogle Scholar
  15. [J-P-2]
    Hörmander, L. Invariant distances and metrics in complex analysis, de Gruyter expositions in math. 9, 1993.Google Scholar
  16. [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
  17. [O-1]
    Ohsawa, T. A remark on the completeness of the Bergman metric, Proc. Jap. Acad. 57 (1981), 238–240.Google Scholar
  18. [O-2]
    Ohsawa, T. Boundary behavior of the Bergman kernel function on pseudoconvex domains, Publ. RIMS, Kyoto Univ. 20 (1984), 897–902.MathSciNetCrossRefMATHGoogle Scholar
  19. [O-3]
    Ohsawa, T. On the Bergman kernel of hyperconvex domains, Nagoya. Math. J. 129 (1993), 43–52.MathSciNetMATHGoogle Scholar
  20. [0-4]
    Ohsawa, T. Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya. Math. J. 137 (1995), 145–148.MathSciNetMATHGoogle Scholar
  21. [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
  22. [0-S-2]
    Ohsawa, T. and Sibony, N. Nonexistence of Levi-flat hypersurfaces in P~,preprint.Google Scholar
  23. [O-T]
    Ohsawa, T. and Takegoshi, K. On the extension of L 2 holomorphic functions, Math. Z. 195 (1987), 197–204.MathSciNetCrossRefMATHGoogle Scholar
  24. [P]
    Pflug, P. Quadratintegrable holomorphe Funktionen und die Serre Vermutung, Math. Ann. 216 (1975), 285–288.MathSciNetCrossRefMATHGoogle Scholar
  25. [St]
    Stehlé, J.-L. Fonctions plurisousharmoniques et convexité holomorphe de certain fibrés analytiques, Lecture Notes in Mathematics 474, 155–180.Google Scholar
  26. [Su]
    Suita, N. Capacities and kernels on Riemann surfaces, Arch. Rational Mech. Anal. 46 (1972), 212–217.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

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

Personalised recommendations