Several Complex Variables Problem List — Problem 78

Definitions and mathematical statement

Write points of ${\mathbb{C}}^2$ as $(\lambda,z)$, and let $\pi(\lambda,z)=\lambda$, $\mathbb D=\{\lambda\in{\mathbb{C}}:|\lambda|<1\}$, and $\mathbb T=\{\lambda\in{\mathbb{C}}:|\lambda|=1\}$. Interpret the source phrase "lies over the circle" as $Y\subseteq\pi^{-1}(\mathbb T)$. For a compact set $Y\subseteq{\mathbb{C}}^2$, define its polynomial hull by \[ \widehat Y=\left\{w\in{\mathbb{C}}^2: |p(w)|\leq\sup_{y\in Y}|p(y)|\text{ for every }p\in{\mathbb{C}}[\lambda,z]\right\}, \] with the usual convention $\widehat\varnothing=\varnothing$.

The question is whether the following assertion holds: for every compact $Y\subseteq\pi^{-1}(\mathbb T)$ and every open set $U\subseteq{\mathbb{C}}^2$ satisfying $\widehat Y\subseteq U$, there exists a Riemann surface \[ S\subseteq U\cap\pi^{-1}(\mathbb D) \] such that $\pi|_S:S\to\mathbb D$ is a finite-sheeted branched covering of positive degree.

For this formulation, the conventional embedded interpretation is that $S$ is a one-dimensional complex submanifold of ${\mathbb{C}}^2$, and that $\pi|_S$ is a proper surjective holomorphic map with finite fibers. Locally, at each point of $S$, the projection has the form $t\mapsto t^e$ in suitable coordinates, for an integer $e\geq1$.

These interpretations require review against the source. The supplied text does not specify whether $S$ must be connected or whether singular analytic curves represented by their normalizations are allowed. It imposes neither nonemptiness of $Y$ nor $\pi(Y)=\mathbb T$, and it states no boundary condition relating $S$ to $Y$. No such additional hypotheses are included here; whether any were intended remains unresolved.

Short English statement

Does every open neighborhood of the polynomial hull of a compact set lying over the unit circle contain a Riemann surface whose coordinate projection is a finite-sheeted branched covering of the entire unit disc? The supplied statement leaves possible restrictions on the compact set, the surface, and its boundary unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $800278$, exact source problem code catalogue entry, title "Several Complex Variables Problem List --- Problem 78." The registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 78. Supplied source URL: https://arxiv.org/abs/math/9509202. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access.
• The quoted problem is attributed to H. Alexander and J. Wermer. The source supplies the historical contact u22330@uicvm.bitnet for Alexander and identifies Wermer with Brown University. The registry separately lists Buzzard, Fornaess, and Krantz in its proposer field and gives the year 1995.