On defining functions and cores for unbounded domains I

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 $$\varphi :U \rightarrow {\mathbb {R}}$$ defined on an open neighbourhood $$U \subset {\mathcal {M}}$$ of $$\overline{\Omega }$$ such that $$\Omega = \{\varphi < 0\}$$ , $$d\varphi \ne 0$$ 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. In particular, we prove that the core is always 1-pseudoconcave.