Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 8 (2016), 2031-2050.

A long $\mathbb{C}^2$ without holomorphic functions

Luka Boc Thaler and Franc Forstnerič

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We construct for every integer n > 1 a complex manifold of dimension n which is exhausted by an increasing sequence of biholomorphic images of n (i.e., a long n), but does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new holomorphic invariants of a complex manifold X, the stable core and the strongly stable core, which are based on the long-term behavior of hulls of compact sets with respect to an exhaustion of X. We show that every compact polynomially convex set B n such that B = B ¯ is the strongly stable core of a long n; in particular, holomorphically nonequivalent sets give rise to nonequivalent long n’s. Furthermore, for every open set U n there exists a long n whose stable core is dense in U. It follows that for any n > 1 there is a continuum of pairwise nonequivalent long n’s with no nonconstant plurisubharmonic functions and no nontrivial holomorphic automorphisms. These results answer several long-standing open problems.

Article information

Anal. PDE, Volume 9, Number 8 (2016), 2031-2050.

Received: 4 March 2016
Revised: 19 July 2016
Accepted: 28 August 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32E10: Stein spaces, Stein manifolds 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions

holomorphic function Stein manifold long $\mathbb C^n$ Fatou–Bieberbach domain Chern–Moser normal form


Boc Thaler, Luka; Forstnerič, Franc. A long $\mathbb{C}^2$ without holomorphic functions. Anal. PDE 9 (2016), no. 8, 2031--2050. doi:10.2140/apde.2016.9.2031.

Export citation


  • A. Abbondandolo, L. Arosio, J. E. Fornæss, P. Majer, H. Peters, J. Raissy, and L. Vivas, “A survey on non-autonomous basins in several complex variables”, 2013.
  • E. Andersén, “Volume-preserving automorphisms of ${\bf C}\sp n$”, Complex Variables Theory Appl. 14:1-4 (1990), 223–235.
  • E. Andersén and L. Lempert, “On the group of holomorphic automorphisms of ${\bf C}\sp n$”, Invent. Math. 110:2 (1992), 371–388.
  • L. Arosio, F. Bracci, and E. F. Wold, “Solving the Loewner PDE in complete hyperbolic starlike domains of $\mathbb{C}\sp N$”, Adv. Math. 242 (2013), 209–216.
  • M. S. Baouendi, P. Ebenfelt, and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series 47, Princeton University Press, 1999.
  • H. Behnke and K. Stein, “Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität”, Math. Ann. 116:1 (1939), 204–216.
  • H. Behnke and P. Thullen, “Zur Theorie der Funktionen mehrerer komplexer Veränderlichen”, Math. Ann. 109:1 (1934), 313–323.
  • L. Boc Thaler, Fatou components, Ph.D. thesis, University of Ljubljana, 2016, hook \posturlhook.
  • M. Brown, “The monotone union of open $n$-cells is an open $n$-cell”, Proc. Amer. Math. Soc. 12 (1961), 812–814.
  • S. S. Chern and J. K. Moser, “Real hypersurfaces in complex manifolds”, Acta Math. 133 (1974), 219–271.
  • P. G. Dixon and J. Esterle, “Michael's problem and the Poincaré–Fatou–Bieberbach phenomenon”, Bull. Amer. Math. Soc. $($N.S.$)$ 15:2 (1986), 127–187.
  • F. Docquier and H. Grauert, “Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten”, Math. Ann. 140 (1960), 94–123.
  • C. Fefferman, “The Bergman kernel and biholomorphic mappings of pseudoconvex domains”, Invent. Math. 26 (1974), 1–65.
  • J. E. Fornæss, “An increasing sequence of Stein manifolds whose limit is not Stein”, Math. Ann. 223:3 (1976), 275–277.
  • J. E. Fornæss, “A counterexample for the Levi problem for branched Riemann domains over ${\bf C}^{n}$”, Math. Ann. 234:3 (1978), 275–277.
  • J. E. Fornæss, “Short $\mathbb C^k$”, pp. 95–108 in Complex analysis in several variables–-Memorial Conference of Kiyoshi Oka's Centennial Birthday, edited by K. Miyajima et al., Adv. Stud. Pure Math. 42, Math. Soc. Japan, Tokyo, 2004.
  • J. E. Fornæss and N. Sibony, “Increasing sequences of complex manifolds”, Math. Ann. 255:3 (1981), 351–360.
  • J. E. Fornæss and E. L. Stout, “Polydiscs in complex manifolds”, Math. Ann. 227:2 (1977), 145–153.
  • F. Forstnerič, “Stability of polynomial convexity of totally real sets”, Proc. Amer. Math. Soc. 96:3 (1986), 489–494.
  • F. Forstnerič, “An elementary proof of Fefferman's theorem”, Exposition. Math. 10:2 (1992), 135–149.
  • F. Forstnerič, “Proper holomorphic mappings: a survey”, pp. 297–363 in Several complex variables (Stockholm, 1987/1988), edited by J. E. Fornæss, Math. Notes 38, Princeton University Press, 1993.
  • F. Forstnerič, Stein manifolds and holomorphic mappings, Ergebnisse der Mathematik $($3$)$ 56, Springer, Heidelberg, 2011.
  • F. Forstnerič, “Holomorphic families of long $\mathbb{C}\sp {2}$'s”, Proc. Amer. Math. Soc. 140:7 (2012), 2383–2389.
  • F. Forstnerič and E. Løw, “Global holomorphic equivalence of smooth submanifolds in ${\bf C}\sp n$”, Indiana Univ. Math. J. 46:1 (1997), 133–153.
  • F. Forstnerič and J.-P. Rosay, “Approximation of biholomorphic mappings by automorphisms of ${\bf C}\sp n$”, Invent. Math. 112:2 (1993), 323–349.
  • S. Kaliman and F. Kutzschebauch, “On the present state of the Andersén–Lempert theory”, pp. 85–122 in Affine algebraic geometry, edited by D. Daigle et al., CRM Proc. Lecture Notes 54, American Mathematical Society, Providence, RI, 2011.
  • F. Lárusson, “What is $\ldots$ an Oka manifold?”, Notices Amer. Math. Soc. 57:1 (2010), 50–52.
  • S. I. Pinchuk and S. V. Khasanov, “Asymptotically holomorphic functions and their applications”, Mat. Sb. $($N.S.$)$ 134(176):4 (1987), 546–555, 576. In Russian; translated in Math. USSR-Sb. 62:2 (1989), 541–550.
  • H. Poincaré, “Les fonctions analytiques de deux variables et la représentation conforme”, Rend. Circ. Mat. Palermo 23 (1907), 185–220.
  • T. Ritter, “A strong Oka principle for embeddings of some planar domains into ${\mathbb C}\times{\Bbb C}\sp \ast$”, J. Geom. Anal. 23:2 (2013), 571–597.
  • J.-P. Rosay and W. Rudin, “Holomorphic maps from ${\bf C}\sp n$ to ${\bf C}\sp n$”, Trans. Amer. Math. Soc. 310:1 (1988), 47–86.
  • K. Stein, “Überlagerungen holomorph-vollständiger komplexer Räume”, Arch. Math. $($Basel$)$ 7 (1956), 354–361.
  • G. Stolzenberg, “The analytic part of the Runge hull”, Math. Ann. 164 (1966), 286–290.
  • E. L. Stout, The theory of uniform algebras, Bogden & Quigley, Tarrytown-on-Hudson, NY, 1971.
  • E. L. Stout, Polynomial convexity, Progress in Mathematics 261, Birkhäuser, Boston, 2007.
  • D. Varolin, “The density property for complex manifolds and geometric structures”, J. Geom. Anal. 11:1 (2001), 135–160.
  • J. Wermer, “An example concerning polynomial convexity”, Math. Ann. 139 (1959), 147–150.
  • E. F. Wold, “A Fatou–Bieberbach domain in $\mathbb C\sp 2$ which is not Runge”, Math. Ann. 340:4 (2008), 775–780.
  • E. F. Wold, “A long $\mathbb C\sp 2$ which is not Stein”, Ark. Mat. 48:1 (2010), 207–210.