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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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)
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)
Docquier, F., Grauert, H.: Levisches problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140, 94–123 (1960)
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)
Wermer, J.: An example concerning polynomial convexity. Math. Ann. 139, 147–150 (1959)
Wermer, J.: Addendum to “An example concerning polynomial convexity”. Math. Ann. 140, 322–323 (1960)
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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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DOI: https://doi.org/10.1007/s00209-019-02413-7