On Hyperbolicity of Spirallike Circularlike domain

TL;DR

This paper establishes that spirallike circularlike domains are Kobayashi hyperbolic precisely when their core is empty, linking hyperbolicity to boundedness and providing a new perspective on these complex domains.

Contribution

It proves a necessary and sufficient condition for hyperbolicity of spirallike circularlike domains based on the emptiness of their core, connecting geometric properties to hyperbolic behavior.

Findings

Kobayashi hyperbolicity is equivalent to the domain being bounded.

Such domains are biholomorphic to bounded domains if and only if they are hyperbolic.

The paper proposes an open problem related to these domains.

Abstract

In this paper, we prove that a spirallike circularlike domain is Kobayashi hyperbolic if and only if its core is empty. In particular, we show that such a domain is Kobayashi hyperbolic if and only if it is (biholomorphic to) a bounded domain. We also propose a problem in this area.