Wild Holomorphic Foliations of the Ball

We prove that the open unit ball B-n 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 B-n. 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 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 <= q < n.