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 M of a Stein manifold X X , every smooth map X → Y X \rightarrow Y to an Oka manifold Y Y satisfying the Cauchy-Riemann equations along M M up to order k k can be C k \mathscr {C}^k -Carleman approximated by holomorphic maps X → Y X \rightarrow Y . Moreover, if K K is a compact O ( X ) \mathscr {O}(X) -convex set such that K ∪ M K \cup M is O ( X ) \mathscr {O}(X) -convex, then we can C k \mathscr {C}^k -Carleman approximate maps which satisfy the Cauchy-Riemann equations up to order k k along M M and are holomorphic on a neighbourhood of K K or merely in the interior of K K if the latter set is the closure of a strongly pseudoconvex domain.