Several Complex Variables Problem List — Problem 37

Definitions and mathematical statement

A domain $\Omega\subset{\mathbb{C}}^2$ is a nonempty connected open set. It is pseudoconvex if it admits a continuous plurisubharmonic exhaustion function $\rho:\Omega\to{\mathbb{R}}$: for every $c\in{\mathbb{R}}$, the set $\{z\in\Omega:\rho(z)\le c\}$ is compact in $\Omega$. Here plurisubharmonic means that the restriction to each complex affine line is subharmonic on every connected component of its intersection with $\Omega$.

Let $S^2=\{x\in{\mathbb{R}}^3:\|x\|=1\}$. Does there exist a pseudoconvex domain $\Omega\subset{\mathbb{C}}^2$ homotopy equivalent to $S^2$? Explicitly, do there exist such an $\Omega$ and continuous maps \[ f:\Omega\longrightarrow S^2,\qquad g:S^2\longrightarrow\Omega \] with \[ g\circ f\simeq\operatorname{id}_{\Omega},\qquad f\circ g\simeq\operatorname{id}_{S^2}, \] where $\simeq$ denotes homotopy through continuous maps? The supplied question specifies neither boundedness nor any boundary regularity condition.

Short English statement

Does a nonempty connected pseudoconvex open subset of ${\mathbb{C}}^2$ have the homotopy type of the two-dimensional sphere? The question asks for existence, with no specified boundedness or boundary regularity.

Sources

• Local UnsolvedMath registry: permanent ID $800240$, source code catalogue entry, titled "Several Complex Variables Problem List --- Problem 37." The supplied registry labels the problem open; this status has not been independently verified.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 37. Supplied source URL: https://arxiv.org/abs/math/9509202. The supplied provenance records access on 2026-10-16 and extraction from arXiv LaTeX; these are source-record assertions, not a claim of independent access or verification.
• The quoted problem is signed F. Forstneric, with contact forstner@math.wisc.edu. The registry's proposer field lists Buzzard, Fornaess, and Krantz, while the item itself attributes the question to F. Forstneric.