Skip to main content
SpringerLink
Log in

Estimation on invariant distances on pseudoconvex domains of finite type in dimension two

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study the class of smooth bounded weakly pseudoconvex domains that are of finite type (in the sense of J. Kohn) and prove effective estimates on the invariant distances of Bergman and Kobayashi and also for the inner distance that is associated to the Caratheodory distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balogh, Z., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75, 504–533 (2000)

    Article  Google Scholar 

  2. Bergman, S.: The kernel function and conformal mapping. Mathematical Surveys, Vol. V, second edition, AMS, Providence, R.I., 1970

  3. Catlin, D.W.: Estimations of invariant metrics in dimension two. Math. Z. 200, 429–466 (1989)

    Article  Google Scholar 

  4. Cho, S.: Estimates on invariant metrics on some pseudoconvex domains in . J. Korean Math. Soc. 32, 661–678 (1995)

    Google Scholar 

  5. Diederich, K.: Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudokonvexen Gebieten. Math. Ann. 187, 9–36 (1970)

    Article  Google Scholar 

  6. Diederich, K.: Über die 1. und 2. Ableitungen der Bergmanschen Kernfunktion und ihr Randverhalten. Math. Ann. 203, 129–170 (1973)

    Google Scholar 

  7. Diederich, K. - Fornæss, J.E.: Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions. Inv. Math. 39, 129–141 (1977)

    Article  Google Scholar 

  8. Diederich, K., Fornæss, J.E.: Pseudoconvex domains: Existence of Stein neighborhoods. Duke Math. J. 44, 641–662 (1977)

    Article  Google Scholar 

  9. Diederich, K., Fornæss, J.E., Herbort, G.: Boundary behavior of the Bergman metric. Proc. Symp. Pure Math. Providence R.I., 41, 59–67 (1984)

    Google Scholar 

  10. Diederich, K., Herbort, G.: Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results. J. Geom. Anal. 3, 237–267 (1993)

    Google Scholar 

  11. Diederich, K., Herbort, G.: Pseudoconvex domains of semiregular type. Contributions to Complex Analysis and Analytic Geometry H. Skoda, J.M. Trépreau (eds.), Braunschweig: Vieweg. Aspects Math. E 26, 127–161 (1994)

    Google Scholar 

  12. Diederich, K., Ohsawa, T.: An estimate for the Bergman distance on pseudoconvex domains. Ann. Math. 141, 181–190 (1995)

    Google Scholar 

  13. Forstneric, F., Rosay, J.P.: Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mapppings. Math. Ann. 279, 239–252 (1987)

    Article  Google Scholar 

  14. Graham, I.: Boundary behavior of the Caratheodory and Kobayashi metrics for strongly pseudoconvex domains in with smooth boundary. Trans. Am. Math. Soc. 207, 219–240 (1975)

    Google Scholar 

  15. Herbort, G.: Invariant metrics and peak functions on pseudoconvex domains of homogeneous finite diagonal type. Math. Z. 209, 223–243 (1992)

    Google Scholar 

  16. Herbort, G.: On the invariant differential metrics near pseudoconvex boundary points where the Levi form has corank one. Nagoya Math. J. 130, 25–54 (1993); Nagoya Math. J. 135, 149–152 (1994)

    Google Scholar 

  17. Hörmander, L.: L2-estimates and existence theorems for the -operator. Acta Math. 113, 89–152 (1965)

    Google Scholar 

  18. Jarnicki, M., Pflug, P.: Invariant distances and metrics in complex analysis. De Gruyter Expositions in Mathematics Vol. 9, De Gruyter Verlag Berlin 1993

  19. Klimek, M.: Extremal plurisubharmonic functions and invariant pseudodistances. Bull. Soc. Math. France 113, 231–240 (1985)

    Google Scholar 

  20. Kohn, J.J.: Boundary behavior of on weakly pseudoconvex manifolds of dimension two. J. Differ. Geom. 6, 523–542 (1972)

    Google Scholar 

  21. Kobayashi, S.: Intrinsic distances, measures and geometric function theory. Bull. Am. Math. Soc. 82, 357–416 (1976)

    Google Scholar 

  22. Ma, Daowei: Boundary behavior of invariant metrics and volume forms on strongly pseudoconvex domains. Duke Math. J. 63,3, 673–697 (1991)

    Article  Google Scholar 

  23. Nagel, A., Stein, E., Wainger, S.: Boundary behavior of functions holomorphic in domains of finite type. Proc. Natl. Acad. Sci. 78(11), 6596–6599 (1981)

    Google Scholar 

  24. Reiffen, H.-J.: Die differentialgeometrischen Eigenschaften der invarianten Distanzfunktion von Caratheodory. Schr. Math. Inst. Univ. Münster, 26, (1963)

  25. Royden, H.: Remarks on the Kobayashi metric. In “Several Complex Variables, II” Springer LN in Mathematics 189, 125–137 (1971)

    Google Scholar 

  26. Vormoor, N.: Topologische Fortsetzung biholomorpher Funktionen auf dem Rande bei beschränkten streng-pseudokonvexen Gebieten im mit C∞ -Rand. Math. Ann. 204, 239–261 (1973)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bergische Universität Wuppertal, Fachbereich C - Mathematik und Naturwissenschaften, Gaußstraße 20, D-42097, Wuppertal, Germany

    Gregor Herbort

Authors
  1. Gregor Herbort
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gregor Herbort.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herbort, G. Estimation on invariant distances on pseudoconvex domains of finite type in dimension two. Math. Z. 251, 673–703 (2005). https://doi.org/10.1007/s00209-005-0829-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-005-0829-2

Keywords

Search

Navigation