Unbounded Kobayashi Hyperbolic Domains in $${\mathbb {C}}^n$$Cn

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