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 $1\le q<n$.

To appear in Indiana Univ. Math. J

References in corpus (1)

• A foliation of the ball by complete holomorphic discs

Cited by in corpus (1)

• Complete nonsingular holomorphic foliations on Stein manifolds