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Fatou-Bieberbach domains in \(\mathbb C^n\setminus\mathbb R^k\). (English) Zbl 1333.32015

A proper subdomain \(\Omega\) in \(\mathbb{C}^n\) is called a Fatou-Bieberbach domain if \(\Omega\) is biholomorphic to \(\mathbb{C}^n\). In contrast to the Riemann mapping in the complex plane \(\mathbb{C}\), it is known that there exist infinitely many different Fatou Bieberbach domains in \(\mathbb{C}^n\) (see, e.g., [K. Kodaira, J. Differ. Geom. 6, 33–46 (1971; Zbl 0227.32008)]), provided \(n>1\). Hence throughout we will assume that \(n>1\). One of the main result of this paper is the following.
Theorem 1.1. Let \(L\) be a totally real affine subset of \(\mathbb{C}^n\) such that \(\dim_RL<n\) and let \(K\subset\mathbb{C}^n\setminus L\) be a compact subset. If \(K\cup L\) is polynomially convex, then there exists a Fatou-Bieberbach domain \(\Omega\subset\mathbb{C}^n\) which separates \(K\) from \(L\), i.e., such that \(K\subset\Omega\subset\mathbb{C}^n\setminus L\).
The authors show that the assumption \(\dim_R L<n\) is crucial by exhibiting a counterexample. It is known that \(\mathbb{C}^n\setminus\mathbb{R}^n\) is a union of Fatou-Bieberbach domains, in view of which the authors raise the following problem.
Problem 1.2. Assume that \(K\subset\mathbb{C}^n\setminus\mathbb{R}^n\) is a compact subset such that \(K\cup\mathbb{R}^n\) is polynomially convex and \(K\) is contractible to a point. Does there exist a Fatou-Bieberbach domain \(\Omega\) in \(\mathbb{C}^n\) which separates \(K\) from \(\mathbb{R}^n\)?
Using Theorem 1.1, they deduce the following result.
Theorem 1.3. Assume that \(L\) is an affine totally real subspace of \(\mathbb C^n\) with \(\dim_RL<n\). Let \(X\) be a Stein manifold with \(\dim X<n\), \(E\subset X\) be a holomorphically convex compact subset and \(U\subset X\) be an open subset containing \(E\). Let \(f: U\to\mathbb{C}^n\) be a holomorphic mapping such that \(f(E)\cap L=\emptyset\). Then \(f\) can be approximated uniformly on \(E\) by holomorphic maps \(X\to\mathbb{C}^n\setminus L\).
As observed by the authors, this result is a first step to understand the following problem.
Problem 1.5. Is \(\mathbb{C}^n\setminus\mathbb{R}^k\) an Oka manifold for arbitrary pairs of integers \((k,n)\) provided \(1\leq k\leq n\)?
Equivalently, does the conclusion of Theorem 1.3 remain valid for maps \(X\to\mathbb{C}^n\setminus \mathbb{R}^k\) and for Stein manifolds \(X\) of arbitrary dimension?
As remarked by the authors, this problem is motivated by:
a) the search for the existence of concrete examples of Oka manifolds as open subsets in \(\mathbb{C}^n\) and
b) the open question whether the set of Oka fibers is open in any holomorphic family of compact complex manifolds.

MSC:

32E10 Stein spaces
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32Q28 Stein manifolds

Citations:

Zbl 0227.32008
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References:

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