Complete nonsingular holomorphic foliations on Stein manifolds

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