A Fatou–Bieberbach domain in $$\mathbb {C}^2$$ which is not Runge

Abstract

Since a paper by Rosay and Rudin (Trans. Am. Math. Soc. 310, 47-86, 1988) there has been an open question whether all Fatou-Bieberbach domains are Runge. We give an example of a Fatou-Bieberbach domain in C 2 which is not Runge. The domain provides (yet) a negative answer to a problem of Bremermann.

A long ℂ2 which is not Stein

“…The construction of the set Y is in Section 2 of [6], and (ii) is Lemma 3.1 of [6]. The existence of Ω then follows since C * ×C admits Fatou-Bieberbach domains.…”

“…The construction relies on the following fact, which is the content of [6]. The construction of the set Y is in Section 2 of [6], and (ii) is Lemma 3.1 of [6].…”

“…Fornaess [2] gave a negative answer to the latter question in dimension 3, and we will use the same idea of proof. Whereas in [2] a main ingredient was a construction by Wermer [4] of a non-Runge polydisk in C 3 we will use the construction of a non-Runge Fatou-Bieberbach domain in C 2 , [6].…”

A long ℂ2 without holomorphic functions

“…Theorem 1.6 is proved in Section 4. We construct manifolds with these properties by improving the recursive procedure devised by Wold [37,38]. The following key ingredient was introduced in [37]; it will henceforth be called the Wold process (see Remark 3.2):…”

“…To simplify the notation, we consider the case n = 2; it will be obvious that the same proof applies in any dimension n ≥ 2. We shall follow Wold's construction from [37,38] up to a certain point, adding a new twist at the end.…”