Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Gromov hyperbolicity, the Kobayashi metric, and $ \mathbb{C}$-convex sets

Author: Andrew M. Zimmer
Journal: Trans. Amer. Math. Soc. 369 (2017), 8437-8456
MSC (2010): Primary 32F45, 53C23, 32F18
Published electronically: June 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $ \mathbb{C}$-convex sets with $ C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $ \mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

References [Enhancements On Off] (What's this?)

  • [1] Mats Andersson, Mikael Passare, and Ragnar Sigurdsson, Complex convexity and analytic functionals, Progress in Mathematics, vol. 225, Birkhäuser Verlag, Basel, 2004. MR 2060426
  • [2] Zoltán M. Balogh and Mario Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000), no. 3, 504-533. MR 1793800,
  • [3] Theodore J. Barth, Convex domains and Kobayashi hyperbolicity, Proc. Amer. Math. Soc. 79 (1980), no. 4, 556-558. MR 572300,
  • [4] Yves Benoist, Convexes hyperboliques et fonctions quasisymétriques, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 181-237 (French, with English summary). MR 2010741,
  • [5] Yves Benoist, A survey on divisible convex sets, Geometry, analysis and topology of discrete groups, Adv. Lect. Math. (ALM), vol. 6, Int. Press, Somerville, MA, 2008, pp. 1-18. MR 2464391
  • [6] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • [7] S. Buckley, Gromov hyperbolicity of invariant metrics,, 2008. Accessed: 2016-01-12.
  • [8] Chin-Huei Chang, M. C. Hu, and Hsuan-Pei Lee, Extremal analytic discs with prescribed boundary data, Trans. Amer. Math. Soc. 310 (1988), no. 1, 355-369. MR 930081,
  • [9] Siqi Fu and Emil J. Straube, Compactness of the $ \overline \partial $-Neumann problem on convex domains, J. Funct. Anal. 159 (1998), no. 2, 629-641. MR 1659575,
  • [10] H. Gaussier and H. Seshadri, On the Gromov hyperbolicity of convex domains in $ \mathbb{C}^n$, ArXiv e-prints (2013).
  • [11] Hervé Gaussier, Characterization of convex domains with noncompact automorphism group, Michigan Math. J. 44 (1997), no. 2, 375-388. MR 1460422,
  • [12] William Goldman, Geometric structures on manifolds,, 2015. Accessed: 2016-01-12.
  • [13] Robert E. Greene and Steven G. Krantz, Stability of the Carathéodory and Kobayashi metrics and applications to biholomorphic mappings, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 77-93. MR 740874,
  • [14] Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, Second extended edition, de Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter GmbH & Co. KG, Berlin, 2013. MR 3114789
  • [15] Anders Karlsson, Non-expanding maps and Busemann functions, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1447-1457. MR 1855841,
  • [16] Anders Karlsson and Guennadi A. Noskov, The Hilbert metric and Gromov hyperbolicity, Enseign. Math. (2) 48 (2002), no. 1-2, 73-89. MR 1923418
  • [17] Shoshichi Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 129-135. MR 0445016
  • [18] Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings: An introduction, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2194466
  • [19] László Lempert, Complex geometry in convex domains, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 759-765. MR 934278,
  • [20] Peter R. Mercer, Complex geodesics and iterates of holomorphic maps on convex domains in $ {\bf C}^n$, Trans. Amer. Math. Soc. 338 (1993), no. 1, 201-211. MR 1123457,
  • [21] Nikolai Nikolov, Pascal J. Thomas, and Maria Trybuła, Gromov (non-)hyperbolicity of certain domains in $ \mathbb{C}^n$, Forum Math. 28 (2016), no. 4, 783-794. MR 3518388,
  • [22] Nikolai Nikolov, Peter Pflug, and Włodzimierz Zwonek, Estimates for invariant metrics on $ \mathbb{C}$-convex domains, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6245-6256. MR 2833552,
  • [23] Nikolai Nikolov and Maria Trybuła, The Kobayashi balls of ( $ \mathbb{C}$-)convex domains, Monatsh. Math. 177 (2015), no. 4, 627-635. MR 3371366,
  • [24] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 125-137. MR 0304694
  • [25] Sergio Venturini, Pseudodistances and pseudometrics on real and complex manifolds, Ann. Mat. Pura Appl. (4) 154 (1989), 385-402. MR 1043081,
  • [26] Andrew M. Zimmer, Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type, Math. Ann. 365 (2016), no. 3-4, 1425-1498. MR 3521096,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32F45, 53C23, 32F18

Retrieve articles in all journals with MSC (2010): 32F45, 53C23, 32F18

Additional Information

Andrew M. Zimmer
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Received by editor(s): September 28, 2014
Received by editor(s) in revised form: January 21, 2016
Published electronically: June 27, 2017
Additional Notes: This material is based upon work supported by the National Science Foundation under grant number NSF 1400919.
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society