Advertisement

Invariant metrics on pseudoconvex domains

  • David W. Catlin

Abstract

Let Ω be a bounded domain in ℂn. Each of the metrics of Bergman, Caratheodory, and Kobayashi assigns a positive number to a given non-zero tangent vector X above a point z in Ω. This assignment is invariant in the sense that if f is a biholomorphism of Ω onto another bounded domain Ω′, then the metric applied to X equals the value of the metric on Ω′ applied to the tangent vector df(X) at the point f(z). Although it is very difficult to calculate the precise value of the above metrics in all but a few special cases, it is sometimes possible to compute a formula for the asymptotic behavior of the metric as the point z approaches the boundary of Ω. When Ω is a smoothly bounded strongly pseudoconvex domain, asymptotic formulas for the Bergman metric were obtained by Diederich [3] and later in much more precise form by Fefferman [4]. Formulas for the asymptotic behavior of the Caratheodory and Kobayashi metric on the same domains were obtained by Graham [5]. In this note we shall consider the case of pseudoconvex domains of finite type in ℂ2 Instead of determining an asymptotic formula for the above metrics, we obtain only a formula that expresses the approximate size of the metrics. In a sense which we shall make precise, these metrics are all equivalent for the given class of domains. We also obtain a formula for the approximate size of the Bergman kernel K(z,\( \bar{\text z} \)) of the domain Ω

Keywords

Unit Ball Asymptotic Formula Finite Type Hyperbolic Manifold Pseudoconvex Domain 
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. 1.
    S. Bergman, The kernel function and conformal mapping, Math. Surveys, V. 2nd ed., Providence, 1970.Google Scholar
  2. 2.
    T. Bloom and I. Graham, A geometric characterization of points of type m on real submanifolds of ℂn, J. of Diff. Geometry 12 (1977), 171–182.MathSciNetGoogle Scholar
  3. 3.
    K. Diederich, Das Randverhalten der Bergmanschen Kernfunktion und Metrik in Streng pseudo-konvexen Gebieten, Math. Ann. 187 (1970), 9–36.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Inventiones Math. (1974), 1–65.Google Scholar
  5. 5.
    I. Graham, The boundary behavior of the Caratheodory and Kobayashi metrics on strictly pseudoconvex domains in ℂn with smooth boundary, Ph.D. dissertation, Princeton Univ., 1973.Google Scholar
  6. 6.
    L. Hörmander, Introduction to complex analysis in several variables, Van Nostrand, 1966.MATHGoogle Scholar
  7. 7.
    S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, 1970.MATHGoogle Scholar
  8. 8.
    J.J. Kohn, Boundary behavior of \( \bar \partial \) on weakly pseudoconvex manifolds of dimension two, J. Diff. Geometry 6 (1972), 523–542.MathSciNetMATHGoogle Scholar
  9. 9.
    A. Nagel, E.M. Stein, and S. Wainger, Proc. Nat’l. Acad. Sci., U.S.A., Vol. 78, pp. 6596–6599, 1981.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston, Inc. 1984

Authors and Affiliations

  • David W. Catlin

There are no affiliations available

Personalised recommendations