## 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

Zbl 0227.32008
Full Text:

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