On defining functions and cores for unbounded domains. I. (English) Zbl 1386.32030
Authors’ abstract: 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 \(\varphi: U\to\mathbb R\) defined on an open neighbourhood \(U\subset M\) 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 \(c(\Omega)\) of an arbitrary domain \(\Omega\subset M\) 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 when \(M\) is Stein. It is shown that every strictly pseudoconvex domain \(\Omega\subset M\) 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.
For Part II, see [the authors, J. Geom. Anal. 30, No. 3, 2293–2325 (2020; Zbl 1462.32038)].
For Part II, see [the authors, J. Geom. Anal. 30, No. 3, 2293–2325 (2020; Zbl 1462.32038)].
Reviewer: Alexandr Yu. Rashkovsky (Stavanger)
Citations:
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