On defining functions for unbounded pseudoconvex domains
  We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $\varphi \colon U \to \mathbb R$ defined on an open neighbourhood $U \subset \mathcal{M}$ of $\overline{\Omega}$ such that $\Omega = \{\varphi < 0\}$, $d\varphi \neq 0$ on $b\Omega$ and $\varphi$ is strictly plurisubharmonic near $b\Omega$. We then introduce the notion of the core $\mathfrak{c}(\Omega)$ of an arbitrary domain $\Omega \subset \mathcal{M}$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $\Omega$ fails to be strictly plurisubharmonic. If $\Omega$ is not relatively compact in $\mathcal{M}$, then in general $\mathfrak{c}(\Omega)$ is nonempty, even in the case when $\mathcal{M}$ is Stein. It is shown that every strictly pseudoconvex domain $\Omega \subset \mathcal{M}$ with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of $\mathfrak{c}(\Omega)$. We then investigate properties of the core. Among other results we prove 1-pseudoconcavity of the core, we show that in general the core does not possess an analytic structure, and we investigate Liouville type properties of the core. 
Tobias Harz, Nikolay Shcherbina and Giuseppe Tomassini
math.CV