Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains

Abstract

Abstract. We give an estimate for the distance function related to the Kobayashi metric on a bounded strictly pseudoconvex domain with C 2 -smooth boundary. Our formula relates the distance function on the domain with the Carnot-Carathéodory metric on the boundary. The estimate is precise up to a bounded additive term. As a corollary we conclude that the domain equipped with this distance function is hyperbolic in the sense of Gromov. Mathematics Subject Classification (2000). Primary 32H15, Secondary 32F15.

Nonpositive curvature and complex analysis

“…Its importance is widely appreciated. Gromov hyperbolicity was introduced by Gromov in the setting of geometric group theory [32], [33], [31], [25], but has played an increasing role in analysis on general metric spaces [12], [13], [7], with applications to the Martin boundary, invariant metrics in several complex variables [6] and extendability of Lipschitz mappings [42]. Here we survey the basics of Gromov hyperbolicity.…”

“…The outstanding result in this context is the following result of Balogh and Bonk; for proofs and definitions of terms used, see [5], [6], and [4, 4.1].…”

On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case

“…This construction was extended in [1] to obtain a pseudometric on a C 2 -smooth bounded strictly pseudoconvex domain in C n . Here we follow the same idea.…”

“…Notice that the case n = 2 is due to Bertrand [3], and that the link between strict pseudoconvexity and Gromov hyperbolicity was first pointed out for domains in C n (that is, when J is the standard complex structure) by Balogh and Bonk [1]. In these two papers, the proofs are based on sharp estimates of the Kobayashi metric.…”

Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

“…In recent years several investigators have been interested in showing that metrics used in geometric function theory are Gromov hyperbolic. For instance, the Klein-Hilbert and Kobayashi metrics are Gromov hyperbolic (under particular conditions on the domain of definition, see [8,19] and [5]); the Gehring-Osgood j-metric is Gromov hyperbolic; and the Vuorinen j-metric is not Gromov hyperbolic except in the punctured space (see [15]). Also, in [20] the hyperbolicity of the conformal modulus metric μ and the related so-called Ferrand metric λ * , is studied.…”