On the Gromov Hyperbolicity of Convex Domains in $${\mathbb {C}}^n$$
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On the Gromov Hyperbolicity of Convex Domains in Cn Hervé Gaussier1 · Harish Seshadri2
Received: 3 October 2016 / Revised: 18 October 2017 / Accepted: 6 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We prove that if a C ∞ -smooth bounded convex domain in Cn 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
1 Introduction and Main Result The notion of Gromov hyperbolicity (or “δ-hyperbolicity”) of a metric space, introduced by Gromov in [18], can be loosely described as “negative curvature at large scales”. The prototype of a Gromov hyperbolic space is a simply connected complete Riemannian manifold with sectional curvature bounded above by a negative constant. One of the reasons for studying Gromov hyperbolic spaces is that they inherit many of the features of the prototype, even though the underlying space may not be a smooth manifold and the distance function may not arise from a Riemannian metric.
Communicated by Mario Bonk.
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Hervé Gaussier [email protected] Harish Seshadri [email protected]
1
Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France
2
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
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H. Gaussier, H. Seshadri
There are extensive studies of interesting classes of Gromov hyperbolic spaces in the literature, the class of word-hyperbolic discrete groups perhaps being the most studied (see, for instance, the reference [14] by Ghys and de la Harpe for an exposition of the theory of hyperbolic groups). In a different direction, domains in Euclidean spaces endowed with certain natural Finsler metrics have also been analyzed from this point of view. For instance, Bonk, Heinonen and Koskela [12] studied the Gromov hyperbolicity of planar domains endowed with the quasihyperbolic metric and Hästö, Lindén, Portilla, Rodriguez and Touris [19] obtained the Gromov hyperbolicity of infinitely connected Denjoy domains, equipped either with the hyperbolic metric or with the quasihyperbolic metric, from conditions on the Euclidean size of the complement of the domain. In higher dimensions, Balogh and Buckley [4] gave conditions equivalent to the Gromov hyperbolicity for domains contained in Rn and endowed with the inner spherical metric (or with the inner Euclidean metric if the domain is bounded). Benoist [7,8] gave, among other results, a necessary and sufficient condition, called quasisymmetric convexity, for a bounded convex domain of Rn endowed with its Hilbert metric to be Gromov hyperbolic. In this paper, we study domains in Cn endowed with the Kobayashi metric. The Kobayashi metric was introduced by Kobayashi as a tool to study geometric and dynamical properties of complex manifolds. It has proved to
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