On defining functions and cores for unbounded domains I

We show that every strictly pseudoconvex domain Omega with smooth boundary in a complex manifold M admits a global defining function, i.e., a smooth plurisubharmonic function phi : U -> Rdefined on an open neighbourhoodU subset of M of (Omega) over bar such that Omega = {phi < 0}, d phi not equal 0 on b Omega and phi is strictly plurisubharmonic near b Omega. We then introduce the notion of the core c(Omega) of an arbitrary domain Omega subset of 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 M, then in general c(Omega) is nonempty, even in the case whenMis Stein. It is shown that every strictly pseudoconvex domain Omega subset of M with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of c(Omega). We then investigate properties of the core. In particular, we prove that the core is always 1-pseudoconcave.