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.