Carleman Approximation of Maps into Oka Manifolds

Abstract

In this paper we obtain a Carleman approximation theorem for maps from Stein manifolds to Oka manifolds. More precisely, we show that under suitable complex analytic conditions on a totally real set $ M $ of a Stein manifold $X$, every smooth map $ X \rightarrow Y $ to an Oka manifold $Y$ satisfying the Cauchy-Riemann equations along $ M $ up to order $ k $ can be $ \mathscr{C}^k $-Carleman approximated by holomorphic maps $ X \rightarrow Y $. Moreover, if $ K $ is a compact $ \mathscr{O}(X) $-convex set such that $ K \cup M $ is $ \mathscr{O}(X) $-convex, then we can $ \mathscr{C}^k $-Carleman approximate maps which satisfy the Cauchy-Riemann equations up to order $ k $ along $ M $ and are holomorphic on a neighbourhood of $ K $, or merely in the interior of $K$ if the latter set is the closure of a strongly pseudoconvex domain.