The Embedding Conjecture and the Approximation Conjecture in Higher Dimension

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

Abstract

In this paper we show the equivalence among three conjectures (and related open questions), namely, the embedding of univalent maps of the unit ball into Loewner chains, the approximation of univalent maps with entire univalent maps and the immersion of domain biholomorphic to the ball in a Runge way into Fatou-Bieberbach domains.

Keywords

Loewner theory Embedding problems Approximation of univalent maps 

Mathematics Subject Classification

32E30 32A10 

Notes

Acknowledgements

The author would like to express his gratitude to Prof. Filippo Bracci for his availability and for introducing him into Loewner theory.

References

  1. 1.
    Andersen, E., Lempert, L.: On the group of holomorphic automorphisms of \({\mathbb {C}^{n}}\). Invent. Math. 110, 371–388 (1992)Google Scholar
  2. 2.
    Arosio, L., Bracci, F., Fornaess Wold, E.: Embedding univalent functions in filtering Loewner chains in higher dimension. Proc. Am. Math. Soc. 143, 1627–1634 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Docquier, F., Grauert, H.: Levisches problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140, 94–123 (1960)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fefferman, C.: On the Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math. 26, 667–669 (1974)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. CRC Press, Boca Raton (2003)MATHGoogle Scholar
  6. 6.
    Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Can. J. Math. 54, 324–351 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kohr, G.: Using the method of Loewner chains to introduce some subclasses of biholomorphic mappings in \({\mathbb {C}^{n}}\). Rev. Roum. Math. Pures Appl. 46, 743–760 (2001).Google Scholar
  8. 8.
    Pommerenke, C., Jensen, G.: Univalent functions. Vandenhoeck und Ruprecht, Göttingen (1975)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento Di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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