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

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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)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  4. 4.
    Balogh, Z., Buckley, S.: Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261–301 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Barth, T.: Convex domains and Kobayashi hyperbolicity. Proc. Am. Math. Soc. 79, 556–558 (1980)CrossRefMATHGoogle Scholar
  6. 6.
    Bedford, E., Pinchuk, S.: Convex domains with non-compact groups of automorphisms. Mat. Sb. 185, 3–26 (1994)MATHGoogle Scholar
  7. 7.
    Benoist, Y.: Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Études Sci. 9, 181–237 (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Benoist, Y.: Convexes hyperboliques et quasiisométries. Geom. Dedicata 122, 109–134 (2006)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  10. 10.
    Bland, J., Duchamp, T.: Moduli for pointed convex domains. Invent. Math. 104, 61–112 (1991)CrossRefMATHGoogle Scholar
  11. 11.
    Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10, 266–306 (2000)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  14. 14.
    de La Harpe, P., Ghys, E.: Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, 83. Birkhäuser, Boston (1990)MATHGoogle Scholar
  15. 15.
    Frankel, S.: Complex geometry of convex domains that cover varieties. Acta Math. 163, 109–149 (1989)CrossRefMATHGoogle Scholar
  16. 16.
    Gaussier, H.: Characterization of convex domains with noncompact automorphism group. Mich. Math. J. 44, 375–388 (1997)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  20. 20.
    Herbig, A.-K., McNeal, J.D.: Convex defining functions for convex domains. J. Geom. Anal. 22, 433–454 (2010)CrossRefMATHGoogle Scholar
  21. 21.
    Kobayashi, S.: Invariant distances on complex manifolds and holomorphic mappings. J. Math. Soc. Jpn. 19, 460–480 (1967)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  25. 25.
    Nikolov, N.: Estimates of invariant distances on “convex” domains. Ann. Mat. Pura Appl. 193, 1595–1606 (2014)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle Scholar
  30. 30.
    Zimmer, A.: Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. 365, 142–198 (2016)CrossRefMATHGoogle Scholar
  31. 31.
    Zimmer, A.: Gromov hyperbolicity, the Kobayashi metric, and \({{\mathbb{C}}}\)-convex sets. Trans. Am. Math. Soc. 369, 8437–8456 (2017)CrossRefMATHGoogle 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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