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.


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


Zbl 0227.32008
Full Text: DOI arXiv


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