Carleman Approximation of Maps into Oka Manifolds

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.