Complete Nonsingular Holomorphic Foliations on Stein Manifolds

Let X be a Stein manifold of complex dimension \(n>1\) $n>1$ endowed with a Riemannian metric \({\mathfrak {g}}\) $\mathfrak{g}$ . We show that for every integer k with \(\left[ \frac{n}{2}\right] \le k \le n-1\) $\left[\frac{n}{2}\right]\le k\le n-1$ there is a nonsingular holomorphic foliation of dimension k on X all of whose leaves are closed and \({\mathfrak {g}}\) $\mathfrak{g}$ -complete. The same is true if \(1\le k<\left[ \frac{n}{2}\right] \) $1\le k<\left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism \(TX\rightarrow X\times {\mathbb {C}}^{n-k}\) $TX\to X\times {\mathbb{C}}^{n-k}$ . We also show that if \({\mathcal {F}}\) $\mathcal{F}$ is a proper holomorphic foliation on \({\mathbb {C}}^n\) ${\mathbb{C}}^n$ \((n>1)\) $(n>1)$ then for any Riemannian metric \({\mathfrak {g}}\) $\mathfrak{g}$ on \({\mathbb {C}}^n\) ${\mathbb{C}}^n$ there is a holomorphic automorphism \(\Phi \) $\mathrm{\Phi }$ of \({\mathbb {C}}^n\) ${\mathbb{C}}^n$ such that the image foliation \(\Phi _*{\mathcal {F}}\) ${\mathrm{\Phi }}_{\ast }\mathcal{F}$ is \({\mathfrak {g}}\) $\mathfrak{g}$ -complete. The analogous result is obtained on every Stein manifold with Varolin’s density property.