On the Kobayashi hyperbolicity of certain tube domains

TL;DR

This paper proves that specific families of tube domains in complex two-space are Kobayashi-hyperbolic, expanding understanding of their complex geometric properties.

Contribution

It establishes the Kobayashi hyperbolicity of three families of tube domains with particular automorphism groups and envelopes of holomorphy.

Findings

All domains in the studied families are Kobayashi-hyperbolic.

The automorphism group of these domains is isomorphic to R ⋉ R^2.

Their envelope of holomorphy is the entire complex plane C^2.

Abstract

In an earlier article the second author introduced three families of tube domains in ${\mathbf C}^2$ with holomorphic automorphism group isomorphic to ${\mathbf R}\ltimes{\mathbf R}^2$ and envelope of holomorphy equal to ${\mathbf C}^2$. In the present paper we show that every domain in each of these families is Kobayashi-hyperbolic.