Skip to main content
SpringerLink
Log in

Quantitative Localization and Comparison of Invariant Distances of Domains in \(\mathbb C^n\)

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

We obtain explicit bounds on the difference between “local” and “global” Kobayashi distances in a domain of \(\mathbb C^n\) as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension 1.

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. Andersson, M., Passare, M., Sigurdsson, R.: Complex Convexity and Analytic Functionals. Birkhäuser, Basel (2004)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bharali, G., Zimmer, A.: Goldilocks domains, a weak notion of visibility, and applications. Adv. Math. 310, 377–425 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bracci, F., Fornæss, J.E., Wold, E.F.: Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains. Math. Z. 292, 879–893 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bracci, F., Nikolov, N., Thomas, P.J.: Visibility of Kobayashi geodesics in convex domains and related properties. Math. Z. 301, 2011–2035 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fornaess, J.E.: Strictly pseudoconvex domains in convex domains. Am. J. Math. 98, 529–569 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Forstnerič, F., Rosay, J.-P.: Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings. Math. Ann. 279, 239–252 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fornæss, J.E., Wold, E.F.: An estimate for the squeezing function and estimates of invariant metrics. In: Bracci, F., Byun, J., Gaussier, H., Hirachi, K., Kim, K.-T., Shcherbina, N. (eds.) Complex Analysis and Geometry: KSCV10, Gyeongju, Korea. Springer Proceedings in Mathematics & Statistics, vol. 144, pp. 135–147. Springer, Tokyo (2014)

  9. Hörmander, L.: Notions of Convexity. Birkhäuser, Basel (1994)

    MATH  Google Scholar 

  10. Jacquet, D.: \(\mathbb{C} \)-convex domains with \(C^2\) boundary. Complex Var. Elliptic Equ. 51, 303–312 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, , vol. 9, 2nd edn. de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin/Boston (2013)

  12. Liu, J., Wang, H.: Localization of the Kobayashi metric and applications. Math. Z. 297, 867–883 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Nikolov, N.: Estimates of invariant distances on “convex’’ domains. Ann. Mat. Pura Appl. 193, 1595–1605 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nikolov, N.: Comparison of invariant functions on strongly pseudoconvex domains. J. Math. Anal. Appl. 421, 180–185 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nikolov, N., Andreev, L.: Estimates of the Kobayashi and quasi-hyperbolic distances. Ann. Mat. Pura Appl. 196, 43–50 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nikolov, N., Pflug, P., Thomas, P.J.: Upper bound for the Lempert function on smooth domains. Math. Z. 266, 425–430 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nikolov, N., Pflug, P., Thomas, P.J.: On different extremal bases for “convex domains’’. Proc. Am. Math. Soc. 141, 3223–3230 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nikolov, N., Pflug, P., Zwonek, W.: Estimates for invariant metrics on \(C\)-convex domains. Trans. Am. Math. Soc. 363, 6245–6256 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nikolov, N., Thomas, P.J.: Comparison of the real and the complex Green functions, and sharp estimates of the Kobayashi distance. Ann. Sc. Norm. Super. Pisa Cl. Sci. 18, 1125–1143 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Nikolov, N., Trybuła, M.: Estimates for the squeezing function near strictly pseudoconvex boundary points with applications. J. Geom. Anal. 30, 1359–1365 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nikolov, N., Trybuła, M., Andreev, L.: Boundary behavior of invariant functions on planar domains. Complex Var. Elliptic Equ. 61, 1064–1072 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  23. Venturini, S.: Comparison between the Kobayashi and Carathéodory distances on strongly pseudoconvex bounded domains in \(\mathbb{C} ^n\). Proc. Am. Math. Soc. 107, 725–730 (1989)

    MATH  Google Scholar 

  24. Zimmer, A.M.: Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. 365, 1425–1498 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the referee for his very careful reading of the paper and his detailed suggestions which helped us correct some inaccuracies and improve the exposition.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113, Sofia, Bulgaria

    Nikolai Nikolov

  2. Faculty of Information Sciences, State University of Library Studies and Information Technologies, 69A, Shipchenski Prohod Str., 1574, Sofia, Bulgaria

    Nikolai Nikolov

  3. Institut de Mathématiques de Toulouse; UMR5219 Université de Toulouse; CNRS UPS, 31062, Toulouse Cedex 9, France

    Pascal J. Thomas

Authors
  1. Nikolai Nikolov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pascal J. Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pascal J. Thomas.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Nikolai Nikolov is partially supported by the National Science Fund, Bulgaria under contract KP-06-N52/3.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nikolov, N., Thomas, P.J. Quantitative Localization and Comparison of Invariant Distances of Domains in \(\mathbb C^n\). J Geom Anal 33, 35 (2023). https://doi.org/10.1007/s12220-022-01086-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-01086-9

Keywords

Mathematics Subject Classification

Search

Navigation