Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains

Abstract

We give an estimate for the distance function related to the Kobayashi metric on a bounded strictly pseudoconvex domain with C2-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.