Wild holomorphic foliations of the ball

TL;DR

This paper constructs a holomorphic foliation of the complex unit ball in multiple dimensions, where both complete and incomplete leaves are dense, illustrating complex limit behaviors of the leaves.

Contribution

It provides the first example of a holomorphic foliation with dense sets of both complete and incomplete leaves in the unit ball of complex space.

Findings

Dense union of complete leaves in the foliation

Dense union of incomplete leaves in the foliation

Existence of leaves as limits of both complete and incomplete leaves

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 $1\le q<n$.