Universal functions on Stein manifolds

Abstract

We study universal holomorphic functions on a Stein manifold $M$ with projective compactification. Let $\left\{{\varphi }_n\right\}$ be a sequence of holomorphic automorphisms of $M$. We prove that if $\left\{{\varphi }_n^{-1}\right\}$ is $A$ run-away, then the set of all universal functions with respect to $\left\{{\varphi }_n\right\}$ in $\mathcal{A}\left(K\right)$ for all compact subsets $K$ with a certain property is the intersection of countable number of open dense subsets in the space of all holomorphic functions on $M$. We also note that there is a close connection between the direction of run-awayness and a family of compact sets for which there exists a universal function.