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


[1] Andersén, E.; Lempert, L., On the group of holomorphic automorphisms of \(\mathbb{C}^{n}\), Invent. Math., 110, 371-388, (1992) · Zbl 0770.32015
[2] Andrist, R., Shcherbina, N. and Wold, E. F., The Hartogs extension theorem for holomorphic vector bundles and sprays, Preprint, 2014. arXiv:1410.2578. · Zbl 1364.32010
[3] Andrist, R. and Wold, E. F., The complement of the closed unit ball in \(\mathbb{C}^{3}\) is not subelliptic, Preprint, 2013. arXiv:1303.1804. · Zbl 0297.32019
[4] Dloussky, G., From non-Kählerian surfaces to Cremona group of \(\mathbb{P}^{2}(\mathbb{C})\), Preprint, 2012. arXiv:1206.2518. · Zbl 1310.32013
[5] Drinovec Drnovšek, B.; Forstnerič, F., Strongly pseudoconvex Stein domains as subvarieties of complex manifolds, Amer. J. Math., 132, 331-360, (2010) · Zbl 1216.32009
[6] Fornæss, J. E. and Wold, E. F., Non-autonomous basins with uniform bounds are elliptic, to appear in Proc. Amer. Math. Soc. · Zbl 1355.32012
[7] Forstnerič, F., Complements of Runge domains and holomorphic hulls, Michigan Math. J., 41, 297-308, (1994) · Zbl 0811.32007
[8] Forstnerič, F., Stein Manifolds and Holomorphic Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 56, Springer, Berlin-Heidelberg, 2011. · Zbl 1247.32001
[9] Forstnerič, F., Oka manifolds: from Oka to Stein and back, Ann. Fac. Sci. Toulouse Math., 22, 747-809, (2013) · Zbl 0201.35602
[10] Forstnerič, F.; Lárusson, F., Survey of Oka theory, New York J. Math., 17a, 1-28, (2011) · Zbl 1225.32019
[11] Forstnerič, F.; Lárusson, F., Holomorphic flexibility properties of compact complex surfaces, Int. Math. Res. Not. IMRN, 13, 3714-3734, (2014) · Zbl 0524.65006
[12] Forstnerič, F.; Løw, E., Holomorphic equivalence of smooth submanifolds in \(\mathbb{C}^{n}\), Indiana Univ. Math. J., 46, 133-153, (1997) · Zbl 0883.32014
[13] Forstnerič, F.; Ritter, T., Oka properties of ball complements, Math. Z., 277, 325-338, (2014) · Zbl 0671.10031
[14] Forstnerič, F.; Rosay, J.-P., Approximation of biholomorphic mappings by automorphisms of \(\mathbb{C}^{n}\), Invent. Math., 112, 323-349, (1993) · Zbl 0792.32011
[15] Gromov, M., Convex integration of differential relations, I, Izv. Akad. Nauk SSSR Ser. Mat., 37, 329-343, (1973) · Zbl 0281.58004
[16] Gromov, M., Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete 9, Springer, Berlin-New York, 1986. · Zbl 0651.53001
[17] Hamm, H., Zum Homotopietyp Steinscher Räume, J. Reine Angew. Math.338 (1983), 121-135. · Zbl 0491.32010
[18] Hanysz, A., Oka properties of some hypersurface complements, Proc. Amer. Math. Soc., 142, 483-496, (2014) · Zbl 1297.32015
[19] Kutzschebauch, F. and Wold, E. F., Carleman approximation by holomorphic automorphisms of \(\mathbb{C}^{n}\), Preprint, 2014. arXiv:1401.2842. · Zbl 1402.32024
[20] Lárusson, F., What is … an Oka manifold, Notices Amer. Math. Soc., 57, 50-52, (2010) · Zbl 1191.32004
[21] Løw, E.; Wold, E. F., Polynomial convexity and totally real manifolds, Complex Var. Elliptic Equ., 54, 265-281, (2009) · Zbl 1165.32303
[22] Nakamura, I., Complex parallelisable manifolds and their small deformations, J. Differential Geom., 10, 85-112, (1975) · Zbl 0297.32019
[23] Rosay, J.-P.; Rudin, W., Holomorphic maps from \(\mathbb{C}^{n}\) to \(\mathbb{C}^{n}\), Trans. Amer. Math. Soc., 310, 47-86, (1988) · Zbl 1268.53006
[24] Stout, E. L., Polynomial Convexity, Birkhäuser, Boston, 2007. · Zbl 0119.37502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.