Skip to main content
SpringerLink
Log in

The Bergman metric on complete Kähler manifolds

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

We use the existence of a bounded uniformly Hölder continuous plurisubharmonic exhaustion function to characterize the Bergman completeness of a complete Kähler manifold. As an application, we proved that any simply-connected complete Kähler manifold with sectional curvature bounded above by a negative constant is Bergman complete.

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. Berndtsson, B.: The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman. Ann. Inst. Fourier (Grenoble) 46, 1083–1094 (1996)

    MathSciNet  Google Scholar 

  2. Berndtsson, B., Charpentier, Ph.: A Sobolev mapping property of the Bergman kernel. Math. Z. 235, 1–10 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blocki, Z., Pflug, P.: Hyperconvexity and Bergman completeness. Nagoya J. Math. 151, 221–225 (1998)

    MATH  Google Scholar 

  4. Chen, B.Y., Zhang, J.H.: Bergman exhaustivity, completeness and stability. Adv. Math. (Chinese) 29, 397–410 (2000)

    MATH  Google Scholar 

  5. Demailly, J.P.: Estimations L 2 pour l'opérateur d'un fibré vectoriel holomorphe semi-positiv au dessus d'une variété kählérienne complète. Ann. Sci. Ec. Norm. Sup. 15, 457–511 (1982)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Donnelly, H., Fefferman, C.: L 2–cohomology and index theorem for the Bergman metric. Ann. Math. 118, 593–618 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Farkas, H.M., Kra, I.: Riemann surfaces. GTM 71, Springer-Verlag, 1980

  9. Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole. Lecture Notes in Mathematics 699, Springer-Verlag, 1979

  10. Herbort, G.: The Bergman metric on hyperconvex domains. Math. Z. 232, 183–196 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Hörmander, L.: An introduction to Complex Analysis in Several Variables. North Holland, 1990

  12. Kerzman, N., Rosay, J.P.: Fonctions plurisousharmonique d'exhaustion borées et domaines taut. Math. Ann. 257, 171–184 (1981)

    Google Scholar 

  13. Kobayashi, S.: Geometry of bounded domains. Trans. Amer. Math. Soc. 92, 267–290 (1959)

    MATH  Google Scholar 

  14. Kobayashi, S.: Hyperbolic Complex spaces. A Series of Comprehensive Studies in Mathematics 318, Springer-Verlag, 1998

  15. Ohsawa, T.: Boundary behavior of the Bergman kernel function on pseudoconvex domains. Publ. RIMS, Kyoto Univ. 20, 897–902 (1984)

    Google Scholar 

  16. Zwonek, W.: Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions. Dissertations. Math. 388, 103 pp. (2000)

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Bo-Yong Chen

    Present address: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

Authors and Affiliations

  1. Department of Applied Mathematics, Tongji University, Shanghai, 200092, P.R. China

    Bo-Yong Chen

Authors
  1. Bo-Yong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo-Yong Chen.

Additional information

Mathematics Subject Classification (2000): 32H10

Supported by NSFC grant no. 10271089

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, BY. The Bergman metric on complete Kähler manifolds. Math. Ann. 327, 339–349 (2003). https://doi.org/10.1007/s00208-003-0456-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-003-0456-3

Keywords

Search

Navigation