An embedding of the unit ball that does not embed into a Loewner chain

Abstract

We construct a holomorphic embedding $\varphi :{\mathbb{B}}^3\to {\mathbb{C}}^3$ such that $\varphi ({\mathbb{B}}^3)$ is not Runge in any strictly larger domain. As a consequence, $\mathcal{S}\ne {\mathcal{S}}^1$ for $n=3$.