An embedding of the unit ball that does not embed into a Loewner chain

Abstract

We construct a holomorphic embedding \(\phi :\mathbb B^3\rightarrow {\mathbb {C}}^3\) such that \(\phi ({\mathbb {B}}^3)\) is not Runge in any strictly larger domain. As a consequence, \(\mathcal S\ne {\mathcal {S}}^1\) for \(n=3\).

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References

  1. 1.

    Arosio, L., Bracci, F., Wold, E.F.: Embedding univalent functions in filtering Loewner chains in higher dimension. Proc. Am. Math. Soc. 143(4), 1627–1634 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bracci, F., Graham, I., Hamada, H., Kohr, G.: Variation of Loewner chains, extreme and support points in the class \(S^0\) in higher dimensions. Constr. Approx. 43(2), 231–251 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Docquier, F., Grauert, H.: Levisches problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140, 94–123 (1960)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Gaussier, H., Joiţa, C.: On Runge neighbourhoods of closures of domains biholomorphic to a ball. In: Geometric Function Theory in Higher Dimensions. Springer INdAM Series (2017)

  5. 5.

    Wermer, J.: An example concerning polynomial convexity. Math. Ann. 139, 147–150 (1959)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Wermer, J.: Addendum to “An example concerning polynomial convexity”. Math. Ann. 140, 322–323 (1960)

    MathSciNet  Article  Google Scholar 

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Correspondence to E. F. Wold.

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Fornæss, J.E., Wold, E.F. An embedding of the unit ball that does not embed into a Loewner chain. Math. Z. 296, 73–78 (2020). https://doi.org/10.1007/s00209-019-02413-7

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Mathematics Subject Classification

  • 32E20
  • 32E30
  • 32H02