Geometric properties of semitube domains

TL;DR

This paper investigates the geometric properties of semitube domains in complex two-dimensional space, extending previous results to non-smooth cases and exploring the relationship between pseudoconvexity and domain slices.

Contribution

It generalizes existing theorems on semitube domains by removing smoothness constraints and analyzes pseudoconvexity in non-smooth settings.

Findings

Extended Burgués and Dwilewicz's results to non-smooth semitube domains.

Established a link between pseudoconvexity and the number of connected components of slices.

Provided an example of a non-convex domain with pseudoconvex images under isometries.

Abstract

In the paper we study the geometry of semitube domains in $\mathbb C^2$. In particular, we extend the result of Burgu\'es and Dwilewicz for semitube domains dropping out the smoothness assumption. We also prove various properties of non-smooth pseudoconvex semitube domains obtaining among others a relation between pseudoconvexity of a semitube domain and the number of connected components of its vertical slices. Finally, we present an example showing that there is a non-convex domain in $\mathbb C^n$ such that its image under arbitrary isometry is pseudoconvex.