Unbounded Kobayashi hyperbolic domains in $\mathbb C^n$

Abstract

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space C n to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in C 3 that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in C 3 and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.