Skip to main content
SpringerLink
Log in

Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in \({\mathbb {C}}\) in terms of the asymptotic behaviour of quotients of invariant metrics.

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. Diederich, K., Fornæss, J.E., Wold, E.F.: Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type. J. Geom. Anal. 24(4), 2124–2134 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Diederich, K., Fornæss, J.E., Wold, E.F.: A Characterisation of the ball. Int. J. Math. 279 (2016)

  3. Forstnerič, F.: Noncritical holomorphic functions on Stein manifolds. Acta Math. 191, 143–189 (2003)

  4. Fu, S.: Asymptotic expansions of invariant metrics of strictly pseudoconvex domains. Can. Math. Bull. 38(2), 196–206 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Graham, I.: Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in \({\mathbb{C}}^n\) with smooth boundary. Trans. Am. Math. Soc. 207, 219–240 (1975)

    MATH  Google Scholar 

  6. Greene, R.E., Wu, H.: On a new gap phenomenon in Riemannian geometry. Proc. NatL. Acad. Sci. USA 79(2), 714–715 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ma, D.: Sharp Estimates of the Kobayashi Metric Near Strongly Pseudoconvex Points. The Madison Symposium on Complex Analysis (Madison, WI, 1991), 329 –338, Contemp. Math. 137, Amer. Math. Soc., Providence, RI (1992)

  8. Nikolov, N., Trybula, M., Andreev, L.: Boundary behaviour of invariant functions on planar domains. Complex Variables Elliptic Equ. 61(8), 1064–1072 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Tsuji, M.: Potential theory in modern function theory. Reprinting of the: original. Chelsea Publishing Co., New York (1959)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Matematisk Institutt, Universitetet i Oslo, PO-BOX 1053, Blindern, 0316, Oslo, Norway

    Erlend F. Wold

Authors
  1. Erlend F. Wold
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erlend F. Wold.

Additional information

Part of this work was done during the international research program “Several Complex Variables and Complex Dynamics” at the Center for Advanced Study at the Academy of Science and Letters in Oslo during the academic year 2016/2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wold, E.F. Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk. Math. Z. 288, 875–887 (2018). https://doi.org/10.1007/s00209-017-1917-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1917-9

Search

Navigation