2018/02/06 by Antonio Alarcon, Alarcon, Antonio
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #math.CV #math.DG
paper · pdf · doi:10.48550/arxiv.1802.02004
To appear in the Journal of Differential Geometry
arxiv created 2020/09/03 · arxiv updated 2020/09/04
In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball \mathbbBn of ℂn (n≥ 2) is a level set of a noncritical holomorphic function on \mathbbBn all of whose level sets are complete. This shows that \mathbbBn admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in \mathbbBn 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 \mathbbBn by smooth complete closed complex submanifolds of any pure codimension q∈\1,…,n-1\.