Two-dimensional slices of non-pseudoconvex open sets

TL;DR

This paper investigates the structure of non-pseudoconvex open sets in complex three-dimensional space, providing conditions under which certain unions of planes are contained in complex lines and characterizing the set of such planes in smooth cases.

Contribution

It introduces new conditions for the union of planes intersecting non-pseudoconvex sets and characterizes the union in smooth cases, advancing understanding of pseudoconvexity in complex analysis.

Findings

Conditions for $ ext{C}^3 ackslash S$ to be a complex line

In $ ext{C}^2$-smooth case, $S$ equals $ ext{C}^n$

Characterization of unions of planes intersecting non-pseudoconvex sets

Abstract

Let $D$ be a non-pseudoconvex open set in $\C^3$ and $S$ be the union of all two-dimensional planes with non-empty and non-pseudoconvex intersection with $D.$ Sufficient conditions are given for $\C^3\setminus S$ to belong to a complex line. Moreover, in the $\mathcal C^2$-smooth case, it is shown that $S=\C^n$.