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On the Gromov Hyperbolicity of Convex Domains in \({\mathbb {C}}^n\)

  • Hervé Gaussier
  • Harish Seshadri
Article

Abstract

We prove that if a \({\mathcal {C}}^\infty \)-smooth bounded convex domain in \({\mathbb {C}}^n\) contains a holomorphic disc in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also give examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.

Keywords

Gromov hyperbolicity Kobayashi distance Convex domain 

Mathematics Subject Classification

32F45 32Q45 53C23 

Notes

Acknowledgements

We would like to thank the referee for valuable comments and suggestions that considerably improved the exposition.

References

  1. 1.
    Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics, Complex Analysis and Geometry. Mediterranean Press, Rende (1989)Google Scholar
  2. 2.
    Azukawa, K., Suzuki, M.: The Bergman metric on a Thullen domain. Nagoya Math. J. 89, 1–11 (1983)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balogh, Z.M., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75, 504–533 (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Balogh, Z., Buckley, S.: Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261–301 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Barth, T.: Convex domains and Kobayashi hyperbolicity. Proc. Am. Math. Soc. 79, 556–558 (1980)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bedford, E., Pinchuk, S.: Convex domains with non-compact groups of automorphisms. Mat. Sb. 185, 3–26 (1994)zbMATHGoogle Scholar
  7. 7.
    Benoist, Y.: Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Études Sci. 9, 181–237 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Benoist, Y.: Convexes hyperboliques et quasiisométries. Geom. Dedicata 122, 109–134 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bland, J.: The Einstein–Kähler metric on \(\{\vert { z}\vert ^2+\vert w\vert ^{2p}<1\}\). Michigan Math. J. 33, 209–220 (1986)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bland, J., Duchamp, T.: Moduli for pointed convex domains. Invent. Math. 104, 61–112 (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10, 266–306 (2000)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bonk, M., Heinonen, J., Koskela, P.: Uniformizing Gromov hyperbolic spaces. In: Astérisque vol. 270. Amer Mathematical Society (2001)Google Scholar
  13. 13.
    Cheeger, J., Ebin, D.: Comparison Theorems in Riemannian Geometry. AMS Chelsea Publishing, Providence (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    de La Harpe, P., Ghys, E.: Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, 83. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  15. 15.
    Frankel, S.: Complex geometry of convex domains that cover varieties. Acta Math. 163, 109–149 (1989)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gaussier, H.: Characterization of convex domains with noncompact automorphism group. Mich. Math. J. 44, 375–388 (1997)CrossRefzbMATHGoogle Scholar
  17. 17.
    Graham, I.: Boundary behavior of the Carathéodory, Kobayashi, and Bergman metrics on strongly pseudoconvex domains in \({ C}^{n}\) with smooth boundary. Bull. AMS 79, 749–751 (1973)CrossRefzbMATHGoogle Scholar
  18. 18.
    Gromov, M.: Hyperbolic Groups. Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8. Springer, New York (1987)Google Scholar
  19. 19.
    Hästö, P., Lindén, H., Portilla, A., Rodríguez, J., Tourís, E.: Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics. J. Math. Soc. Jpn. 64, 247–261 (2012)CrossRefzbMATHGoogle Scholar
  20. 20.
    Herbig, A.-K., McNeal, J.D.: Convex defining functions for convex domains. J. Geom. Anal. 22, 433–454 (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kobayashi, S.: Invariant distances on complex manifolds and holomorphic mappings. J. Math. Soc. Jpn. 19, 460–480 (1967)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehren der Mathematischen Wissenschaften, vol. 318. Springer, Berlin (1998)CrossRefGoogle Scholar
  23. 23.
    Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. An Introduction, 2nd edn. World Scientific, New Jersey (2005)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lempert, L.La: Métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. Fr. 109, 427–474 (1981)CrossRefzbMATHGoogle Scholar
  25. 25.
    Nikolov, N.: Estimates of invariant distances on “convex” domains. Ann. Mat. Pura Appl. 193, 1595–1606 (2014)CrossRefzbMATHGoogle Scholar
  26. 26.
    Nikolov, N., Pflug, P.: Estimates for the Bergman kernel and metric of convex domains in \({{\mathbb{C}}}^n\). Ann. Polon. Math. 81, 73–78 (2003)CrossRefzbMATHGoogle Scholar
  27. 27.
    Nikolov, N., Pflug, P., Zwonek, W.: Estimates for invariant metrics on \({{\mathbb{C}}}\)-convex domains. Trans. Am. Math. Soc. 363, 6245–6256 (2011)CrossRefzbMATHGoogle Scholar
  28. 28.
    Royden, H.L., Wong, P.M.: Carathéodory and Kobayashi metrics on convex domains (preprint) Google Scholar
  29. 29.
    Yau, S.-T.: A general Schwarz Lemma for Kähler manifolds. Am. J. Math. 100, 197–203 (1978)CrossRefzbMATHGoogle Scholar
  30. 30.
    Zimmer, A.: Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. 365, 142–198 (2016)CrossRefzbMATHGoogle Scholar
  31. 31.
    Zimmer, A.: Gromov hyperbolicity, the Kobayashi metric, and \({{\mathbb{C}}}\)-convex sets. Trans. Am. Math. Soc. 369, 8437–8456 (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Univ. Grenoble Alpes, CNRS, IFGrenobleFrance
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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