Wild holomorphic foliations of the ball

Abstract

We prove that the open unit ball Bn of C n (n ≥ 2) admits a nonsingular holomorphic foliation F by closed complex hypersurfaces such that both the union of the complete leaves of F and the union of the incomplete leaves of F are dense subsets of Bn. In particular, every leaf of F is both a limit of complete leaves of F and a limit of incomplete leaves of F . This gives the first example of a holomorphic foliation of Bn 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 ≤ q < n.