Complete complex hypersurfaces in the ball come in foliations

TL;DR

The paper proves that all smooth complete closed complex hypersurfaces in the unit ball can be realized as level sets of a noncritical holomorphic function, leading to a foliation of the ball by such hypersurfaces.

Contribution

It demonstrates that the unit ball admits a holomorphic foliation by complete closed complex hypersurfaces, and extends the result to submanifolds of arbitrary codimension.

Findings

Every such hypersurface is a level set of a noncritical holomorphic function.

Existence of a holomorphic foliation of the ball by complete closed complex hypersurfaces.

Generalization to submanifolds of arbitrary positive codimension.

Abstract

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q\in\{1,\ldots,n-1\}$.