Oka domains in Euclidean spaces

TL;DR

This paper demonstrates the existence of surprisingly small Oka domains in complex Euclidean spaces, constructed via convex sets and geometric conditions, revealing new examples of Oka domains near minimal size limits.

Contribution

It introduces new classes of Oka domains in ^n, especially those close to halfspaces, based on geometric and convexity conditions, expanding the known landscape of Oka theory.

Findings

^n minus certain convex sets is an Oka domain

Existence of Oka domains slightly larger than a halfspace

Construction of smooth hypersurfaces dividing ^n into hyperbolic and Oka regions

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 which are only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives families of smooth real hypersurfaces $\Sigma_t\subset \mathbb C^n$ $(t\in\mathbb R)$ dividing $\mathbb C^n$ in an unbounded hyperbolic domain and an Oka domain such that at the threshold value $t=0$ the hypersurface $\Sigma_0$ 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.