Several Complex Variables Problem List — Problem 63

Definitions and mathematical statement

A domain in ${\mathbb{C}}^2$ is a nonempty connected open subset. A Fatou--Bieberbach domain is a domain $\Omega\subsetneq{\mathbb{C}}^2$ for which there exists a biholomorphic map $\Phi:{\mathbb{C}}^2\to\Omega$; that is, $\Phi$ is bijective and both $\Phi$ and $\Phi^{-1}$ are holomorphic.

Write $\mathcal O(\Omega)$ for the holomorphic functions on $\Omega$. A domain $\Omega\subset{\mathbb{C}}^2$ is Runge in ${\mathbb{C}}^2$ if, for every $f\in\mathcal O(\Omega)$, every nonempty compact set $K\subset\Omega$, and every $\varepsilon>0$, there exists a polynomial $p\in{\mathbb{C}}[z_1,z_2]$ such that \[ \sup_{z\in K}|f(z)-p(z)|<\varepsilon. \]

Question (source Problem 63). Does there exist a Fatou--Bieberbach domain $\Omega\subsetneq{\mathbb{C}}^2$ that is not Runge in the ambient space ${\mathbb{C}}^2$? Equivalently, does there exist such a domain together with $f\in\mathcal O(\Omega)$, a nonempty compact set $K\subset\Omega$, and $\varepsilon>0$ such that \[ \forall p\in{\mathbb{C}}[z_1,z_2],\qquad \sup_{z\in K}|f(z)-p(z)|\geq\varepsilon? \] The supplied registry labels the problem open; this label is supplied source metadata, not an independently verified assessment of its current status.

Short English statement

Does a proper domain in ${\mathbb{C}}^2$, biholomorphic to all of ${\mathbb{C}}^2$, exist on which polynomial approximation of holomorphic functions fails on some compact subset? The question concerns the domain's inclusion in the ambient ${\mathbb{C}}^2$.

Sources

• Local UnsolvedMath registry, permanent ID $800264$, source code catalogue entry, title Several Complex Variables Problem List --- Problem 63. The supplied record labels the problem open and lists Buzzard, Fornaess, and Krantz under proposer metadata, with year 1995.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 63. Supplied source URL: https://arxiv.org/abs/math/9509202. The source statement attributes the question to F. Forstneric and gives the contact address forstner@math.wisc.edu.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; no independent source visit or literature verification is asserted here.