2014/05/09 by Tobias Harz, Harz, Tobias, Nikolay Shcherbina +3
Mathematics · #32U05 #Complex Variables (math.CV) #FOS: Mathematics #Primary 32T15 #Secondary 32C15 #math.CV #msc:32C15 #msc:32T15 #msc:32U05
paper · pdf · doi:10.48550/arxiv.1405.2250
86 pages, Comments are welcome
arxiv created 2014/08/10 · arxiv updated 2014/08/12
We show that every strictly pseudoconvex domain Ω with smooth boundary in a complex manifold M admits a global defining function, i.e., a smooth plurisubharmonic function φ\colon U → \mathbb R defined on an open neighbourhood U ⊂ M of Ω such that Ω= \φ< 0\, dφ≠ 0 on bΩ and φ is strictly plurisubharmonic near bΩ. We then introduce the notion of the core \mathfrakc(Ω) of an arbitrary domain Ω⊂ M as the set of all points where every smooth and bounded from above plurisubharmonic function on Ω fails to be strictly plurisubharmonic. If Ω is not relatively compact in M, then in general \mathfrakc(Ω) is nonempty, even in the case when M is Stein. It is shown that every strictly pseudoconvex domain Ω⊂ M with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of \mathfrakc(Ω). 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.