An embedding of the unit ball that does not embed into a Loewner chain
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Mathematische Zeitschrift
An embedding of the unit ball that does not embed into a Loewner chain J. E. Fornæss1 · E. F. Wold2 Received: 5 July 2019 / Accepted: 14 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We construct a holomorphic embedding φ : B3 → C3 such that φ(B3 ) is not Runge in any strictly larger domain. As a consequence, S = S 1 for n = 3. Mathematics Subject Classification 32E20 · 32E30 · 32H02
1 Introduction Recall that a Loewner chain is a family f t : Bn → Cn of holomorphic injections, f t (0) = 0, f (0) = et · id, t ∈ [0, ∞), with f t (Bn ) ⊆ f s (Bn ) for t ≤ s. We let S denote the set of all univalent maps f : Bn → Cn with f (0) = 0, f (0) = id, we let S 1 denote the set of all f ∈ S such that f embeds into a Loewner chain, i.e., f = f 0 where ( f t )t≥0 is a Loewner chain, and finally we let S 0 denote the set of all f ∈ S 1 for whom we require that the family (e−t f t )t≥0 is normal. In one variable, the three sets coincide, and they are all compact. On the other hand, in higher dimensions, the sets S and S 1 are certainly not compact, as can by seen as a consequence of the automorphism group of Cn being huge for n ≥ 2. On the other hand, it is known that S 0 is compact, and so we get the chain of inclusions S0 S1 ⊆ S.
(1.1)
However, if f ∈ there exist ψ ∈ (the set of entire injective maps), and g ∈ S 0 such that f = ψ ◦ g, and so we may say that S 1 splits (see e.g. [2, Theorem 2.6]), S 1,
I (Cn )
S 1 = I (Cn ) ◦ S 0 .
(1.2)
The background for this article is that it has been unknown whether it is also the case that S = I (Cn )◦ S 0 , or equivalently, whether S = S 1 (this problem was mentioned and discussed
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E. F. Wold [email protected] J. E. Fornæss [email protected]
1
Department of Mathematics, NTNU, 7491 Trondheim, Norway
2
Department of Mathematics, University of Oslo, Postboks 1053 Blindern, 0316 Oslo, Norway
123
J. E. Fornæss, E. F. Wold
in [1]). In this context, the following closely related problem was recently posed by Bracci: Let f ∈ S . Does there exist a Fatou–Bieberbach domain ⊂ Cn such that f (Bn ) is Runge in ? This turns out not to be the case. 3
Theorem 1.1 For any > 0 there exists a continuous injective map φ : B → C3 with φ ∈ O(B3 ), and such that (i) φ − id B3 < , and (ii) if φ(B3 ) ⊂ is a Runge pair, then φ(B3 ) = . Since the conditions in Docquier–Grauert [3] (Definition 20) are satisfied for the increasing family ( f t (Bn ))0≤t≤t0 for any fixed t0 , and for any Loewner chain, it follows from [3] (Satz 17–19) that each pair ( f 0 (Bn ), f t (Bn )) is a Runge pair, and we get our second theorem as a corollary: Theorem 1.2 For n = 3 we have that S = S 1 .
2 Preliminaries The problem mentioned above was recently studied by Gaussier and Joi¸ta [4]. In particular, they studied the map f (z 1 , z 2 , z 3 ) = (z 1 , z 1 z 2 + z 3 − 1, z 1 z 22 − z 2 + 2z 2 z 3 ).
(2.1)
This map was constructed by Wermer [5,6] to produce a non Runge embedded polydisk in C3 . We will study the same map, but
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