Wild holomorphic foliations of the ball
Antonio Alarcon

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 of admits a nonsingular holomorphic foliation by closed complex hypersurfaces such that both the union of the complete leaves of and the union of the incomplete leaves of are dense subsets of . In particular, every leaf of is both a limit of complete leaves of and a limit of incomplete leaves of . This gives the first example of a holomorphic foliation of 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 with .
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Taxonomy
TopicsHolomorphic and Operator Theory · Meromorphic and Entire Functions · Geometric and Algebraic Topology
