Oka Domains in Euclidean Spaces

In this paper, we find surprisingly small Oka domains in Euclidean spaces 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 C-n, we show that C-n \ 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 E-t subset of C-n for t is an element of R dividing 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 C-n for n > 1 whose projective closure (sic) subset of CPn avoids a hyperplane A subset of CPn and is polynomially convex in CPn\ A similar to= C-n, then C-n \ E is an Oka domain.