Proper holomorphic embeddings with small limit sets

Abstract

Let $X$ be a Stein manifold of dimension $n\ge 1$. Given a continuous positive increasing function $h$ on ${\mathbb{R}}_{+}=[0,\infty )$ with ${\lim }_{t\to \infty }h(t)=\infty$, we construct a proper holomorphic embedding $f=(z,w):X\hookrightarrow {\mathbb{C}}^{n+1}\times {\mathbb{C}}^n$ satisfying $|w(x)|<h(|z(x)|)$ for all $x\in X$. In particular, $f$ may be chosen such that its limit set at infinity is a linearly embedded copy of ${\mathbb{CP}}^n$ in ${\mathbb{CP}}^{2n}$.

References in corpus (3)

• Recent developments on Oka manifolds
• Oka domains in Euclidean spaces
• Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets