Wild holomorphic foliations of the ball

Abstract

We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$. In particular, every leaf of $\mathcal F$ is both a limit of complete leaves of $\mathcal F$ and a limit of incomplete leaves of $\mathcal F$. This gives the first example of a holomorphic foliation of $\mathbb{B}_n$ by connected closed complex hypersurfaces having a complete leaf that is a limit of incomplete ones. We obtain an analogous result for foliations by complex submanifolds of arbitrary pure codimension $q$ with $1\le q<n$.