Complete proper holomorphic embeddings of strictly pseudoconvex domains into balls

Abstract

We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with ${\mathcal{C}}^2$-boundary in ${\mathbb{C}}^n$ into the unit ball of ${\mathbb{C}}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.