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

TL;DR

This paper provides a pluripotential theory-based sufficient condition for unbounded domains in complex space to be Kobayashi hyperbolic and constructs a specific example illustrating the condition's limitations.

Contribution

It introduces a new sufficient condition for unbounded domains to be Kobayashi hyperbolic and presents an example showing this condition is not necessary.

Findings

A pluripotential theory-based sufficient condition for hyperbolicity.

Construction of a Kobayashi hyperbolic domain with a nonempty core.

Demonstration that the condition is not necessary for hyperbolicity.

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.