Geometric characterization of $q$-pseudoconvex domains in ${\mathbb{C}}^n$
Bull. Korean Math. Soc. 2017 Vol. 54, No. 2, 543-557
Published online March 13, 2017
Printed March 31, 2017
Hedi Khedhiri
Rue Ibn Eljazzar
Abstract : In this paper, we investigate the notion of $q$-pseudoconvexity to discuss and describe some geometric characterizations of $q$-pseudo\-convex domains $\Omega\subset\C^n$. In particular, we establish that $\Omega$ is $q$-pseudo\-convex, if and only if, for every boundary point, the Levi form of the boundary is semipositive on the intersection of the holomorphic tangent space to the boundary with any $(n-q+1)$-dimensional subspace $E\subset\C^n$. Furthermore, we prove that the Kiselman's minimum principal holds true for all $q$-pseudoconvex domains in $\C^p\times\C^n$ such that each slice is a convex tube in $\C^n$.
Keywords : $q$-pseudoconvex domain, $q$-subharmonic function, exhaustion function, Levi form of the boundary
MSC numbers : 32T, 32U05, 32U10
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