Complete nonsingular holomorphic foliations on Stein manifolds

Abstract

Let X be a Stein manifold of complex dimension n > 1 endowed with a Riemannian metric g. We show that for every integer k with [ n 2 ] ≤ k ≤ n − 1 there is a nonsingular holomorphic foliation of dimension k on X all of whose leaves are closed and g-complete. The same is true if 1 ≤ k < [ n 2 ] provided that there is a complex vector bundle epimorphism T X → X × Cn−k . We also show that if F is a proper holomorphic foliation on Cn (n > 1) then for any Riemannian metric g on Cn there is a holomorphic automorphism Φ of Cn such that the image foliation Φ∗F is g-complete. The analogous result is obtained on every Stein manifold with Varolin’s density property.