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Abstract: Let $X$ be a Stein manifold of complex dimension $n \ge 1$ 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 $\Phi$ of $\mathbb{C}^n$ such that the image foliation $\Phi_*\mathcal{F}$ is ${\mathfrak g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property.
Keywords: Stein manifolds, complete holomorphic foliations, density property
Published in DiRROS: 19.02.2024; Views: 167; Downloads: 59
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