On the hyperbolicity of tube domains in ${\mathbb C}^2$

We will discuss the problem of finding an easily verifiable criterion for the Kobayashi-hyperbolicity of tube domains in ${\mathbb C}^2$. The problem naturally splits into two cases: (1) the envelope of holomorphy of the domain is strictly less than ${\mathbb C}^2$, (2) the envelope of holomorphy of the domain is all of ${\mathbb C}^2$. Mostly, we will be interested in the first case. In this situation one can assume that the base of the tube domain lies in the upper half-plane. Surprisingly, even in this apparently simple case there is no easily verifiable criterion for hyperbolicity. In the talk, we will discuss known necessary and sufficient conditions.