Gromov hyperbolicity and the Kobayashi metric

It is well known that the unit ball endowed with the Kobayashi metric is isometric to complex hyperbolic space and in particular is an example of a negatively curved Riemannian manifold. One would then suspect that when Ω ⊂Cd is a domain close to the unit ball then (Ω ,KΩ ) should be negatively curved (in some sense). Unfortunately, for general domains the Kobayashi metric is no longer Riemannian and thus will no longer have curvature in a local sense. Instead one can ask if the Kobayashi metric satisfies a coarse notion of negative curvature from geometric group theory called Gromov hyperbolicity. Gromov hyperbolic metric spaces have been intensively studied and have a number of remarkable properties. Thus it seems natural to determine the domains for which the Kobayashi metric is Gromov hyperbolic and then to use the theory of such metric spaces to prove new results in several complex variables. The first major result in this direction is due to Balogh and Bonk: