Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(2): 543-557

Online first article March 13, 2017      Printed March 31, 2017

https://doi.org/10.4134/BKMS.b160177

Copyright © The Korean Mathematical Society.

Geometric characterization of $q$-pseudoconvex domains in ${\mathbb{C}}^n$

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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