1. Introduction

We call that a domain is a Fatou-Bieberbach domain if is biholomorphic to

We call that an irreducible algebraic curve A of is hyperbolic when or else and the set consists more than two different points, where is the genus of A and is the line at infinity.

In the previous paper ^[2], we proved the case A (theorem 2.1). In this paper, every Fatou-Bieberbach domain does not omit C which is the same in Abstract (Theorem 3.12). Moreover we prove that the property that an analytic curve of C² does not omit every Fatou-Bieberbach domain is unchanged one by a transformation of Aut(C²) (Theorem 3.7). Therefore every Fatou-Bieberbach domain does not omit T(A) with T ∈ Aut(C²). And we prove also that there is an algebraic type curve of C² which is isomorphic to an algebraic curve of any topological type such that it can not be transformed to an algebraic curve by any T ∈ Aut(C²) (Proposition 3.3).

2. Preliminary

Theorem 2.1 (Theorem 2.1 in ^[2]). Let A be a hyperbolic algebraic curve of C². Then every Fatou-Bieberbach domain Ω does not omit A.

Remark 2.2 (Theorem 2.2 in ^[2]). Let Ω₀ be every Fatou-Bieberbach domain in C² arising from a polynomial automorphism, namely a polynomial basin. Then Ω₀ does not omit every algebraic curve of C².

Theorem 2.3 (Theorem 4.1 in Buzzard and Fornaess ^[4]). Let X be an arbitrary analytic curve of C². Then there is a Fatou-Bieberbach domain Ω with and where some is biholomorphic such that every nonconstant holomorphic map f : C → C² intersects with Φ(X) at in_nite points and C² − Φ(X) is Kobayashi hyperbolic.

Example 2.4 (Example 9.7 in ^[6]). There is a Fatou-Bieberbach domain Ω0 such that namely the closure of omits an algebraic complex line L.

Proposition 2.5 (Proposition 3.1 in ^[2]). For some transendental complex line such that there is a Fatou-Bieberbach domain Ω0 with

From Theorem 2.3, Example 2.4 and Proposition 2.5, following theorem is easy to see.

Theorem 2.6. Let C be an analytic curve of C². Then every Fatou-Bieberbach domain does not omit C or some (not all) Fatou-Bieberbach do-main Ω₀ omits C.

3. Conclusion

Proposition 3.1. Let be a transendental hyperbolic curve of C2 which is isomorphic to a hyperbolic algebraic curve A. If is transformed to an alge-braic hyperbolic curve A of C² by T ∈ Aut(C²), then every Fatou-Bieberbach domain does not omit .

Proof. We assume that there is a Fatou-Bieberbach domain Ω with Then and T(Ω) is biholomorphic to C² by T ◦ Φ, where Φ is a biholomorphic map of Ω to C². Therefore T(Ω) is a Fatou-Bieberbach domain and it contradicts to Theorem 2.1.

Proposition 3.2. Let X and Ω be the same of Theorem 2.3. Then Φ(X) can not be transformed to any algebraic curve A which is hyperbolic or non by any T ∈ Aut(C²).

Proof. We assume that Φ(X) is transformed to an algebraic curve A by some T ∈ Aut(C²). As T(Ω) is a Fatou-bieberbach domain with T ◦Φ(X) = A. Let f : C → C² be a map to an algebraic complex line. Then (T ◦ Φ(X)) ∩ f(C) is a finite set of points at most and Φ(X) ∩ (T−1 ◦ f(C)) is also. It is a contradiction to Theorem 2.3.

Proposition 3.3 (cf. Proposition 3.12 in ^[2]). There is a transcendental analytic curve of C² which is isomorphic to an algebraic curve of any topo-logical type, that is an algebraic type curve of any type, such that it can not be transformed to an algebraic curve by any T ∈ Aut(C²). If we take X an algebraic type curve of any topological type, Φ(X) which is the same notation of Theorem 2.3 is such one.

Proposition 3.4. Let X, Ω and Φ be the same of Theorem 2.3. Then every Fatou-Bieberbach domain Ω′ does not omit Φ(X).

Proof. If some Ω′ omits Φ(X), Φ′⁻¹ : C² → Ω′ C2−Φ(X), where Φ′ is a biholomorphic map of Ω′ to C². Since C²−Φ(X) is Kobayashi hyperbolic, Φ′⁻¹ is a constant map. It is a contradiction.

Proposition 3.5. Let C be an analytic curve of C² which is transformed by T ∈ Aut(C²) to some analytic curve C′ of C² which does not omit every Fatou-Bieberbach domain. Then every Fatou-Bieberbach domain does not omit C.

Proof. We assume that there is a Fatou-Bieberbach domain Ω₀ with Ω₀ ∩ C = . Then T(Ω₀) ∩ T(C) = . Since T(Ω₀) is a Fatou-Bieberbach domain which omits C′ = T(C). It contradicts to the property of C.

Corollary 3.6. Let C be an analytic curve of C² which does not omit every Fatou-Bieberbach domain. Then T(C) with every T ∈ Aut(C²) does not omits every Fatou-Bieberbach domain.

Proof. Since T⁻¹ ◦ T(C) = C, T(C) does not omit every Fatou-Bieberbach domain also by Proposition 3.5.

From Proposition 3.5 and Corollary 3.6 it is easy to see the following theorem.

Theorem 3.7. The property that the analytic curve of C² does not omit every Fatou-Bieberbach domain is unchanged one by a transformation of Aut(C²).

From Proposition 3.4 and Theorem 3.7 it is easy to see the following corollary.

Corollary 3.8. Let C be an analytic curve of C² which is transformed by some T ∈ Aut(C²) to Φ(X) where X, Ω and Φ are the same of Proposition 3.4. Then every Fatou-Bieberbach domain Ω′ does not omit C.

Problem 3.9. Is there which is not transformed to an algebraic hyperbolic curve A of C² or Φ(X), which is the same notation of Theorem 2.3, by some T ∈ Aut(C²)?

Proposition 3.10. Let C be an irreducible analytic curve of C² with g(C) > 1. Then D = C² − C is not biholomorphic to C², that is, D is not a Fatou-Bieberbach domain.

Proof. We assume that there is a biholomorphic map Φ of D to C²(x, y) such as Φ : x = ξ(z,w), y = η(z,w) where (z,w) ∈ D. Let P(x, y) be a nonconstant primitive polynomial, that is all level curve of {P(x, y) = α} is irreducible except at most finite number of α1, · · · , αn. Then P(ξ(z,w), η(z,w)) ∈ (D). If C is an essential singular curve of P(ξ, η), every level curve of P(ξ, η) can not be analytically continued to every point of C except at most one value α0 by well known Thullen’s theorem.

Since {P(ξ, η) = α, α α₀, α₁, · · · , α_n} is considered as a Riemann surface R₀ = R − {p₁, · · · , p_m} where R is a compact Riemann surface, that is an algebraic type Riemann surface such as π : R₀ → {P(ξ, η) = α} C² is the normalization. And at some p_i ∈ {p₁, · · · , p_m} the cluster set of π in C² is C. Because it is a pseudoconcave set of C² by Theorem 3.4 in ^[1] and it is a degeneration locus of Kobayashi pseudodistance by Theorem 3.6 in ^[1]. It is a contradiction to g(C) > 1 by Theorem 4 in ^[3].

Therefore C is not an essential singular curve of P(ξ, η), that is, P(ξ, η) is at most meromorphically continued to C. Since P(x, y) ∈ (C²), P(ξ, η) is holomorphically continued to C. We set such function as F(z,w).

Since every level curve of F(z,w) is holomorphically isomorphic to algebraic type Riemann surface and an analytic cuve of C², F(z,w) is an algebraic type entire function of Nishino’s sence ^[5], namely in the class (A). Then by principal theorem of ^[5], F = φ ◦ Q ◦ T where φ is a polynomial function of one complex variable because P(x, y) is primitive, Q is a primitive polynomial and T ∈ Aut(C²).

Then T_|D = Φ. It contradicts to the assumption. Then D is not biholomorphic to C².

Proposition 3.11. Let C be an irreducible analytic curve of C² such that D = C² − C is not a Fatou-Bieberbach domain. Then every subdomain of D′ of D is not a Fatou-Bieberbach domain.

Proof. If D′ is a Fatou-Bieberbach domain, there is a biholomorphic map Φ of C² to D′. Let I be an inclusion map of D′ to D and {f = 0} = C where f ∈ (C²). Since g = f ◦ I ◦ Φ is an entire function of C² and g ≠ 0. There is a complex line L of C² and g|_L is considered as a transendental entire function of C, it is a contradiction by little Picard theorem because I ◦ Φ|_L is an one to one map.

From Proposition 3.10 and 11, following theorem is easy to see.

Theorem 3.12. Let C be the same of Proposition 3.10. Then every Fatou-Bieberbach domain does not omit C.

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