An Example on $s$-H-Convexity in $\mathbb{C}^2$

TL;DR

This paper constructs a specific bounded domain in complex two-dimensional space with smooth boundary that has a Stein neighborhood basis but is not $s$-H-convex for any $s \,\geq 1$, highlighting limitations of $s$-H-convexity.

Contribution

It provides a counterexample demonstrating that having a Stein neighborhood basis does not imply $s$-H-convexity for any $s\geq 1$ in complex analysis.

Findings

Constructed a bounded domain with smooth boundary in $\,\mathbb{C}^2$

Showed the domain has a Stein neighborhood basis

Proved the domain is not $s$-H-convex for any $s\geq 1$

Abstract

We construct a bounded domain $\Omega$ in $\mathbb{C}^2$ with boundary of class $\mathcal{C}^{1,1}$, such that $\overline{\Omega}$ has a Stein neighborhood basis, but is not $s$-H-convex for any real number $s\geq{1}$.