Oka Domains in Euclidean Spaces

Abstract In this paper, we find surprisingly small Oka domains in Euclidean spaces ${\mathbb {C}}^n$ of dimension $n>1$ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set $E$ in ${\mathbb {C}}^n$, we show that ${\mathbb {C}}^n\setminus E$ is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces $\Sigma _t\subset {\mathbb {C}}^n$ for $t\in {\mathbb {R}}$ dividing ${\mathbb {C}}^n$ in an unbounded hyperbolic domain and an Oka domain such that at $t=0$, $\Sigma _0$ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if $E$ is a closed set in ${\mathbb {C}}^n$ for $n>1$ whose projective closure $\overline E\subset \mathbb {C}\mathbb {P}^n$ avoids a hyperplane $\Lambda \subset \mathbb {C}\mathbb {P}^n$ and is polynomially convex in $\mathbb {C}\mathbb {P}^n\setminus \Lambda \cong {\mathbb {C}}^n$, then ${\mathbb {C}}^n\setminus E$ is an Oka domain.