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 ${\mathbb{C}}^n$ 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.

18 pages, 3 figures