Smooth Stein Neighborhood Domains Beyond One H Convexity

Definitions and mathematical statement

A domain is a nonempty connected open set. For a bounded domain $\Omega\subset{\mathbb{C}}^2$, put $K=\overline{\Omega}$. Pseudoconvexity means that $\Omega$ admits a continuous plurisubharmonic exhaustion function. For $r\in\{2,\infty\}$, boundary regularity of class $C^r$ means that near each boundary point there is a real-valued $C^r$ defining function $\rho$ with $\Omega=\{\rho<0\}$ locally and ${\,\mathrm{d}}\rho\ne0$ on the boundary.

Using the usual interpretation that the neighborhood basis concerns the closure, $K$ has a Stein neighborhood basis if, for every open set $U\subset{\mathbb{C}}^2$ containing $K$, there is a Stein open set $V\subset{\mathbb{C}}^2$ such that \[ K\subset V\subset U. \] The supplied record does not explicitly specify that the basis concerns $\overline{\Omega}$; this interpretation requires confirmation against the cited report.

Existence question. For each of the regularity classes $C^2$ and $C^\infty$, does there exist a bounded pseudoconvex domain $\Omega\subset{\mathbb{C}}^2$ whose boundary has that regularity, whose closure has a Stein neighborhood basis, and which is not $1$-$H$-convex in the sense intended by the cited source?

Unresolved qualification. The supplied record gives no definition of $1$-$H$-convexity, including its underlying object and any quantitative neighborhood conditions. That term is retained as a source-dependent predicate; its negation cannot be expanded into a precise mathematical condition from the supplied evidence alone. Thus the existence question above remains conditional on recovering that definition. The source writes $C^2$ "or" $C^\infty$; the two regularity variants are displayed separately here. An example with $C^\infty$ boundary would also satisfy the $C^2$ requirement, while the converse does not follow from the stated regularity.

The supplied dated assessment reports a $C^{1,1}$ counterexample and labels the $C^2$ and $C^\infty$ existence question unresolved. These are supplied literature claims, not independently verified mathematical conclusions.

Short English statement

The question asks whether a bounded pseudoconvex domain in ${\mathbb{C}}^2$ can have a Stein neighborhood basis and fail the source's $1$-$H$-convexity condition when its boundary is $C^2$, or even $C^\infty$. The missing definition of $1$-$H$-convexity prevents a fully explicit formulation. The supplied curation reports only a $C^{1,1}$ counterexample and labels the smoother cases open.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30003925$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied source citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. Supplied source and literature link, labeled "Oberwolfach Report 37/2018": https://doi.org/10.4171/owr/2018/37. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature reference: Strong Stein neighborhood bases, https://arxiv.org/abs/0705.0507. The record attributes a $C^{1,1}$ counterexample to cited work without identifying a theorem, page, or precise construction.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the $C^2$ or $C^\infty$ existence question remains unresolved despite a $C^{1,1}$ counterexample. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Neither linked source has been independently consulted for this draft.