Gromov hyperbolicity, the Kobayashi metric, and $\mathbb {C}$-convex sets
HTML articles powered by AMS MathViewer
- by Andrew M. Zimmer PDF
- Trans. Amer. Math. Soc. 369 (2017), 8437-8456 Request permission
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
- Mats Andersson, Mikael Passare, and Ragnar Sigurdsson, Complex convexity and analytic functionals, Progress in Mathematics, vol. 225, Birkhäuser Verlag, Basel, 2004. MR 2060426, DOI 10.1007/978-3-0348-7871-5
- 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, DOI 10.1007/s000140050138
- Theodore J. Barth, Convex domains and Kobayashi hyperbolicity, Proc. Amer. Math. Soc. 79 (1980), no. 4, 556–558. MR 572300, DOI 10.1090/S0002-9939-1980-0572300-3
- Yves Benoist, Convexes hyperboliques et fonctions quasisymétriques, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 181–237 (French, with English summary). MR 2010741, DOI 10.1007/s10240-003-0012-4
- 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
- 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, DOI 10.1007/978-3-662-12494-9
- S. Buckley, Gromov hyperbolicity of invariant metrics, http://www.uma.es/investigadores/grupos/cfunspot/research/0806pBuckley.pdf, 2008. Accessed: 2016-01-12.
- 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, DOI 10.1090/S0002-9947-1988-0930081-7
- 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, DOI 10.1006/jfan.1998.3317
- H. Gaussier and H. Seshadri, On the Gromov hyperbolicity of convex domains in $\mathbb {C}^n$, ArXiv e-prints (2013).
- Hervé Gaussier, Characterization of convex domains with noncompact automorphism group, Michigan Math. J. 44 (1997), no. 2, 375–388. MR 1460422, DOI 10.1307/mmj/1029005712
- William Goldman, Geometric structures on manifolds, http://www.math.umd.edu/~wmg/gstom.pdf, 2015. Accessed: 2016-01-12.
- 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, DOI 10.1090/pspum/041/740874
- 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, DOI 10.1515/9783110253863
- Anders Karlsson, Non-expanding maps and Busemann functions, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1447–1457. MR 1855841, DOI 10.1017/S0143385701001699
- Anders Karlsson and Guennadi A. Noskov, The Hilbert metric and Gromov hyperbolicity, Enseign. Math. (2) 48 (2002), no. 1-2, 73–89. MR 1923418
- 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 445016
- Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. An introduction. MR 2194466, DOI 10.1142/5936
- 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, DOI 10.1007/BF01848107
- Peter R. Mercer, Complex geodesics and iterates of holomorphic maps on convex domains in $\textbf {C}^n$, Trans. Amer. Math. Soc. 338 (1993), no. 1, 201–211. MR 1123457, DOI 10.1090/S0002-9947-1993-1123457-0
- Nikolai Nikolov, Pascal J. Thomas, and Maria Trybuła, Gromov (non-)hyperbolicity of certain domains in $\Bbb C^n$, Forum Math. 28 (2016), no. 4, 783–794. MR 3518388, DOI 10.1515/forum-2014-0113
- Nikolai Nikolov, Peter Pflug, and Włodzimierz Zwonek, Estimates for invariant metrics on $\Bbb C$-convex domains, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6245–6256. MR 2833552, DOI 10.1090/S0002-9947-2011-05273-6
- Nikolai Nikolov and Maria Trybuła, The Kobayashi balls of ($\Bbb {C}$-)convex domains, Monatsh. Math. 177 (2015), no. 4, 627–635. MR 3371366, DOI 10.1007/s00605-015-0746-3
- 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
- Sergio Venturini, Pseudodistances and pseudometrics on real and complex manifolds, Ann. Mat. Pura Appl. (4) 154 (1989), 385–402. MR 1043081, DOI 10.1007/BF01790358
- 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, DOI 10.1007/s00208-015-1278-9
Additional Information
- Andrew M. Zimmer
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 831053
- Email: aazimmer@uchicago.edu
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 8437-8456
- MSC (2010): Primary 32F45, 53C23, 32F18
- DOI: https://doi.org/10.1090/tran/6909
- MathSciNet review: 3710631