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On the Analytic Curve of C^{2} which is not Omitted by Every Fatou-Bieberbach Domain

Yukinobu Adachi, Kurakuen, Nishinomiya, Hyogo, Japan
### Abstract

Let *C *be an irreducible (may be transendental) analytic curve whose genus is geater than 1. Then every Fatou-Bieberbach domain does not omit *C*.

**Keywords:** fatou-bieberbach domain, hyperbolic cuve, transcendental algebraic type curve, kobayashi hyperbolic

*American Journal of Mathematical Analysis*, 2015 3 (1),
pp 19-20.

DOI: 10.12691/ajma-3-1-4

Received December 17, 2014; Revised January 27, 2015; Accepted February 06, 2015

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- ADACHI, YUKINOBU. "On the Analytic Curve of C
^{2}which is not Omitted by Every Fatou-Bieberbach Domain."*American Journal of Mathematical Analysis*3.1 (2015): 19-20.

- ADACHI, Y. (2015). On the Analytic Curve of C
^{2}which is not Omitted by Every Fatou-Bieberbach Domain.*American Journal of Mathematical Analysis*,*3*(1), 19-20.

- ADACHI, YUKINOBU. "On the Analytic Curve of C
^{2}which is not Omitted by Every Fatou-Bieberbach Domain."*American Journal of Mathematical Analysis*3, no. 1 (2015): 19-20.

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### 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 **C**^{2} does not omit every Fatou-Bieberbach domain is unchanged one by a transformation of *Aut*(**C**^{2}) (Theorem 3.7). Therefore every Fatou-Bieberbach domain does not omit *T*(*A*) with *T **∈ **Aut*(**C**^{2}). And we prove also that there is an algebraic type curve of **C**^{2} 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*(**C**^{2}) (Proposition 3.3).

### 2. Preliminary

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

Remark 2.2 (Theorem 2.2 in ^{[2]}). Let Ω_{0} be every Fatou-Bieberbach domain in **C**^{2} arising from a polynomial automorphism, namely a polynomial basin. Then Ω_{0} does not omit every algebraic curve of **C**^{2}.

Theorem 2.3 (Theorem 4.1 in Buzzard and Fornaess ^{[4]}).* Let **X **be an** **arbitrary analytic curve of ***C**^{2}*. Then there is a Fatou-Bieberbach domain Ω** **with** ** **and** ** **where some** ** **is biholomorphic such** **that every nonconstant holomorphic map **f **: ***C ***→ ***C**^{2}* intersects with Φ(**X**)** **at in_nite points and ***C**^{2}* **− **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 ***C**^{2}*. Then every Fatou-Bieberbach domain does not omit C or some (not all) Fatou-Bieberbach do-main Ω*_{0}* omits C*.

### 3. Conclusion

Proposition 3.1. *Let** ** **be a transendental hyperbolic curve of ***C***2 which is** **isomorphic to a hyperbolic algebraic curve **A**. If** ** **is transformed to an alge-braic hyperbolic curve **A **of ***C**^{2}* by **T **∈ **Aut**(***C**^{2}*), 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 **C**^{2} by *T **◦ *Φ, where Φ is a biholomorphic map of Ω to **C**^{2}. 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**(***C**^{2}*)*.

Proof. We assume that Φ(*X*) is transformed to an algebraic curve *A *by some *T **∈ **Aut*(**C**^{2}). As *T*(Ω) is a Fatou-bieberbach domain with *T **◦*Φ(*X*) = *A*. Let *f *: **C ***→ ***C**^{2} 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 ***C**^{2}* 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**(***C**^{2}*). 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} : **C**^{2} *→ *Ω*′ ***C**2*−*Φ(*X*), where Φ*′ *is a biholomorphic map of Ω*′ *to **C**^{2}. Since **C**^{2}*−*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 ***C**^{2}* which is transformed** **by **T **∈ **Aut**(***C**^{2}*) to some analytic curve **C**′ **of ***C**^{2}* 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 Ω_{0} with Ω_{0} ∩ C = . Then T(Ω_{0}) ∩ T(C) = . Since T(Ω_{0}) 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 ***C**^{2}* which does not omit** **every Fatou-Bieberbach domain. Then **T**(**C**) with every **T **∈ **Aut**(***C**^{2}*) 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 ***C**^{2}* does not** **omit every Fatou-Bieberbach domain is unchanged one by a transformation** **of **Aut**(***C**^{2}*)*.

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 ***C**^{2}* **which is transformed by** **som**e **T **∈ **Aut**(***C**^{2}*) **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 **C**^{2} or Φ(*X*), which is the same notation of Theorem 2.3, by some *T **∈ **Aut*(**C**^{2})?

Proposition 3.10. *Let **C **be an irreducible analytic curve of ***C**^{2}* with** **g**(**C**) **> **1. Then **D **= ***C**^{2}* **− **C **is not biholomorphic to ***C**^{2}*, that is, **D **is not** **a Fatou-Bieberbach domain*.

Proof. We assume that there is a biholomorphic map Φ of *D *to **C**^{2}(*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 *α*1*, **· · · **, α**n*. 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 *α*0 by well known Thullen’s theorem.

Since {P(ξ, η) = α, α α_{0}, α_{1}, · · · , α_{n}} is considered as a Riemann surface R_{0} = R − {p_{1}, · · · , p_{m}} where R is a compact Riemann surface, that is an algebraic type Riemann surface such as π : R_{0} → {P(ξ, η) = α} **C**^{2} is the normalization. And at some p_{i} ∈ {p_{1}, · · · , p_{m}} the cluster set of π in** ****C**^{2}^{ }is C. Because it is a pseudoconcave set of** ****C**^{2}^{ }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*) *∈ *(**C**^{2}), *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** ****C**^{2}, *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*(**C**^{2}).

Then *T*_{|}_{D}* *= Φ. It contradicts to the assumption. Then *D *is not biholomorphic to **C**^{2}.

Proposition 3.11. *Let **C **be an irreducible analytic curve of ***C**^{2}* such that** **D **= ***C**^{2}* **− **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 **C**^{2} to D′. Let I be an inclusion map of D′ to D and {f = 0} = C where f ∈ (**C**^{2}). Since g = f ◦ I ◦ Φ is an entire function of **C**^{2} and g ≠ 0. There is a complex line L of **C**^{2} 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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