Gromov Hyperbolicity of Bounded Convex Domains

Abstract

In this chapter we describe some results on Gromov hyperbolicity in several complex variables. In particular, we prove that the Kobayashi metric on a smoothly bounded convex domain is Gromov hyperbolic if and only if the domain has finite type and we provide a proof that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain. We will also describe connections between the Kobayashi metric and Hilbert metric and then describe recent work of Benoist concerning the Gromov hyperbolicity of the Hilbert metric. Finally, we list a number of open problems.