On necessary and sufficient conditions for the Kobayashi hyperbolicity of tube domains in ${\mathbb C}^2$

TL;DR

This paper investigates the conditions under which certain tube domains in complex two-space are Kobayashi hyperbolic, demonstrating that affine obstructions are not sufficient and providing new insights into the geometric criteria involved.

Contribution

The authors construct examples showing that the known affine necessary condition for hyperbolicity is not sufficient, clarifying the precise geometric obstructions for Kobayashi hyperbolicity in these domains.

Findings

Affine obstructions are not sufficient for hyperbolicity.

Constructed examples demonstrate the limitations of existing conditions.

The obstructions are more complex than previously understood.

Abstract

This note concerns tube domains in ${\mathbb C}^2$ with the envelope of holomorphy not equal to the entire space. We construct examples showing that for such domains the sufficient condition for Kobayashi hyperbolicity due to M. Jarnicki and P. Pflug cannot be replaced by its weaker "affine" variant, which is known to be a necessary condition for hyperbolicity. Thus, we arrive at the somewhat unexpected conclusion that the obstructions for a domain in the above class to be Kobayashi hyperbolic are not just "affine".