AN EMBEDDING OF C IN C^{2} WITH HYPERBOLIC COMPLEMENT

GREGERY T. BUZZARD AND JOHN ERIK FORNÆSS

Abstract. Let X be a closed, 1-dimensional, complex subvariety of C^{2} and let \overline{B} be a closed ball in C^{2} − X. Then there exists a Fatou-Bieberbach domain Ω with X ⊆ Ω ⊆ C^{2} − \overline{B} and a biholomorphic map Φ : Ω → C^{2} such that C^{2} − Φ(X) is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in C^{2} whose complement is hyperbolic, and there is a nontrivial Fatou-Bieberbach domain containing any finite collection of complex lines.

1. Introduction

This paper was motivated by the question of whether there exists a proper holomorphic embedding Φ : C → C^{2} such that C^{2} − Φ(C) is Kobayashi hyperbolic.

For the case of a polynomial embedding P : C → C^{2}, Abhyankar and Moh [AM] and Suzuki [S], proved that there exists a polynomial automorphism F of C^{2} such that (F ∘ P)(C) = C × {0}. In contrast, Forstneric, Globevnik and Rosay [FGR] have shown that there exists a proper holomorphic embedding Φ : C → C^{2} such that for no automorphism H of C^{2} is it true that (H ∘ Φ)(C) = C × {0}. This leaves open the question of whether the complement of an embedded copy of C is always Kobayashi hyperbolic. We answer this question in the following theorem.

Theorem 1.1. There exists a proper holomorphic embedding Φ : C → C^{2} such that C^{2} − Φ(C) is Kobayashi hyperbolic.

This theorem is a corollary of theorem 4.1. As additional corollaries, we obtain the following theorems.

Theorem 1.2. There exists a proper holomorphic embedding of C into C^{2} such that any nonconstant holomorphic image of C intersects this embedding infinitely many times.

Theorem 1.3. Let X be any finite set of complex lines in C^{2}. Then there exists a Fatou-Bieberbach domain Ω containing X with \overline{Ω} ≠ C^{2}.

Research at MSRI is supported in part by NSF grant DMS-9022140

This last theorem answers a question of Rosay and Rudin [RR], who constructed a Fatou-Bieberbach domain containing the coordinate axes and asked if a Fatou-Bieberbach domain can contain more than 2 complex lines.

2. Controlling images of the unit disk

We first prove a lemma showing that certain collections of linear disks give control on certain (nonlinear) maps from the unit disk to C^{2}.

Given two concentric balls in C^{2}, we find a collection of linear disks near the boundary of the inner ball such that any map from the unit disk into the larger ball which maps 0 near the center of the two balls must map most of the disk into the smaller ball. We make this more precise in the following lemma. The proof uses a normal families argument and is similar in spirit to the construction of non-tame sets in [RR].

For notation, let π_{j} denote projection to the jth coordinate, j = 1, 2, let Δ(0, r) be the (open) disk of radius r centered at 0 in C, and let B(0, r) denote the (open) ball of radius r centered at the origin in C^{2}.

Lemma 2.1. Let k ∈ Z^{+}, 0 ≤ n_{1} < n_{2} < n_{3}. Let X be a closed, complex, 1-dimensional subvariety of C^{2}. Then there exist finitely many affine complex linear maps L_{j} : C → C^{2} with \overline{Δ_{j}} := L_{j}(\overline{Δ(0, 1)}) ⊆ B(0, n_{2}) − (\overline{B(0, n_{1})} ∪ X) having pairwise disjoint closures such that if

ϕ : Δ(0, 1) → B(0, n_{3}) − ∪_{j}Δ_{j},
ϕ(0) ∈ \overline{B(0, n_{1}/2)},

with dist(ϕ(0), X) ≥ 1/k, then ϕ(Δ(0, 1 − 1/2^{k})) ⊆ B(0, n_{2}).

In fact we prove a stronger result allowing small perturbations in place of the linear disks obtained above.

Lemma 2.2. With the hypotheses of lemma 2.1, there exist disks Δ_{j} = L_{j}(Δ(0, 1)) as in that lemma, plus δ > 0, such that if Δ′_{j} is the image of a holomorphic map which is within δ of L_{j} on Δ(0, 1) for all j, then for any

ϕ : Δ(0, 1) → B(0, n_{3}) − ∪_{j}Δ′_{j},
ϕ(0) ∈ \overline{B(0, n_{1}/2)},

with dist(ϕ(0), X) ≥ 1/k, we have ϕ(Δ(0, 1 − 1/2^{k})) ⊆ B(0, n_{2}).

Proof of lemma 2.2: Let n_{1} < r < n_{2}, and let {a_{j}} and {b_{j}} be countable dense subsets of B(0, r) − \overline{B(0, n_{1})} and B(0, n_{2}) − \overline{B(0, r)}, respectively, such that π_{l}a_{j} ≠ π_{l}a_{m} if j ≠ m, l = 1, 2, and likewise for b_{j}.

For each j, let A_{j}(z) = (α_{j}z, 0)+a_{j} and B_{j}(w) = (0, β_{j}w)+b_{j}, where α_{j} > 0 is chosen maximal such that A_{j}(Δ(0, 2)) ⊆ B(0, r) − (\overline{B(0, n_{1})} ∪ X), and analogously for β_{j} with disks in the outer shell.

For notation, we let A′_{j,m} and B′_{j,m} denote functions holomorphic and within 1/m of A_{j} and B_{j} on Δ(0, 1). For such functions, let

D_{m} := ∪_{j=1}^{m}(A′_{j,m}(Δ(0, 1)) ∪ B′_{j,m}(Δ(0, 1))).

To reach a contradiction, assume that for each m there exists D_{m} as above and ϕ_{m} : Δ(0, 1) → B(0, n_{3}) − D_{m}, ϕ_{m}(0) ∈ \overline{B(0, n_{1}/2)} with dist(ϕ_{m}(0), X) ≥ 1/k, and ϕ(Δ(0, 1 − 1/2^{k})) ⊈ B(0, n_{2}).

The set {ϕ_{m}} is a normal family since the image of each ϕ_{m} lies inside a fixed ball, so we may assume that some subsequence converges to a map ϕ : Δ(0, 1) → B(0, n_{3}) with ϕ(0) ∈ \overline{B(0, n_{1}/2)}, dist(ϕ(0), X) ≥ 1/k, and ϕ(Δ(0, 1)) ⊈ B(0, n_{2}). Then there exist nonempty open sets Ω_{1}, Ω_{2} ⊆ Δ(0, 1) such that ϕ(Ω_{1}) ⊆ B(0, r) − \overline{B(0, n_{1})} and ϕ(Ω_{2}) ⊆ B(0, n_{2}) − \overline{B(0, r)}.

We claim that π_{2}ϕ is constant. If not, then since ϕ(0) ∉ X, there exists z_{1} ∈ Ω_{1} such that ϕ(z_{1}) ∉ X and (π_{2}ϕ)′(z_{1}) ≠ 0, and there exists a subsequence a_{j_{l}} → ϕ(z_{1}). Moreover, there exists c > 0 such that α_{j_{l}} ≥ c for all l.

By continuity, there exists a neighborhood V of z_{1} such that |π_{1}ϕ(z)− π_{1}ϕ(z_{1})| < c/2 for all z ∈ V and such that π_{2}ϕ(V) is a neighborhood of π_{2}ϕ(z_{1}). For l large, π_{2}a_{j_{l}} ∈ π_{2}ϕ(V), and hence by choice of V there exists z′_{1} ∈ V such that ϕ(z′_{1}) ∈ A_{j_{l}}(Δ(0, 1)). But then ϕ(z) − A_{j_{l}}(ζ) has an isolated zero in Δ^{2}(0, 1), which persists under small perturbations. Hence ϕ_{m} must intersect D_{m} for large m, a contradiction, so π_{2}ϕ ≡ const. A similar argument for π_{1}ϕ shows that ϕ ≡ const, which contradicts ϕ(0) ∈ \overline{B(0, n_{1}/2)} and ϕ(Δ(0, 1)) ⊈ B(0, n_{2}).

Thus, no such sequence {ϕ_{m}} exists, so we obtain the lemma by taking {L_{j}} to be some finite subset of {A_{j}} ∪ {B_{j}} and δ sufficiently small.

3. Approximation of linear disks by complex subvarieties

In the following lemma, we use the notation reg(X) to denote the set of regular points of a subvariety.

Lemma 3.1. Let K ⊆ C^{2} be polynomially convex and let X be a closed, 1-dimensional, complex subvariety of C^{2}. Let L ⊆ X be compact such that for each p ∈ reg(X) − (K ∪ L) there exists a curve η : [0, 1) → reg(X) − (K ∪ L) with η(0) = p and lim_{t→1} ∥η(t)∥ = ∞.

Then K ∪ L is polynomially convex.

Proof: Although the ideas in the proof are standard, we include it for convenience.

Let p ∈ C^{2} − (K ∪ L). If p ∉ X, then there exists an entire g such that g ≡ 0 on X but g(p) ≠ 0. Also, there exists f entire such that |f| < 1 on K but |f(p)| > 1. Then for some m, we have |f^{m}g(p)| > 1 and |f^{m}g| < 1 on K ∪L, and we can approximate f^{m}g by a polynomial with the same properties.

If p ∈ X, then the preceding argument shows that if K_{p} := K ∪ L ∪ {p}, then K̂_{p} ⊆ K ∪ X, where K̂_{p} is the polynomial hull of K_{p}. The hypotheses on L imply that X ∩ K_{p} is Runge in X, so we can approximate functions holomorphic on X ∩ K_{p} by functions holomorphic on X.

Let f(p) = 2 and f ≡ 1/2 on (X ∩ K_{p}) − {p}, and approximate f by a function g holomorphic on X such that |f − g| < 1/4 on the domain of f. By [GR, theorem 18, ch. VIII], we may extend g to be holomorphic on C^{2}, and then we may restrict g to a neighborhood V of X such that |g| < 1 on V ∩ K. Composing with a convex function σ, we obtain a psh function σ ∘ |g| which is 0 on a neighborhood of (X ∩ K_{p}) − {p} and larger than 1 at p. We can extend this function by 0 so that it is psh in a neighborhood of K ∪ X, 0 on K ∪ L, and larger than 1 at p, then restrict it to a Runge neighborhood of K̂_{p}.

Using standard \overline{∂} techniques, it follows that there exists a polynomial P on C^{2} such that |P (p)| > 1 and |P | ≤ 1 on K ∪ L, so K ∪ L is polynomially convex.

In the following lemma, we start with a closed, 1-dimensional, complex subvariety X of C^{2}, an automorphism Ψ, and some linear disks contained outside Ψ(X) and outside the ball of radius n. We construct an automorphism of C^{2} such that the image of Ψ(X) under this automorphism contains pieces approximating each of the linear disks, and such that this automorphism is near the identity on B(0, n) and on some large piece of Ψ(X). We do this in such a way that given some polynomially convex set contained in a ball disjoint from Ψ(X) and B(0, n + 1), the image of this set is disjoint from B(0, n + 2).

For notation, let ϵ > 0, R, n ≥ 0, and let Ψ be an automorphism of C^{2}. Let X be a closed, 1-dimensional, complex subvariety of C^{2}, and let L_{j} : C → C^{2}, j = 1, …, N be finitely many affine complex linear maps with Δ_{j} := L_{j}(Δ(0, 1)) having pairwise disjoint closures such that \overline{Δ_{j}} ∩ Ψ(X) = ∅ and \overline{Δ_{j}} ∩ \overline{B(0, n)} = ∅ for all j. Let B_{0} = B(p_{0}, r_{0}) be a ball with \overline{B_{0}} ∩ (\overline{B(0, n + 1)} ∪ Ψ(X) ∪ ∪_{j}\overline{Δ_{j}}) = ∅, and let K_{0} ⊆ B_{0} be polynomially convex and compact.

Lemma 3.2. There exists H : C^{2} → C^{2} an automorphism such that ∥H −I∥ < ϵ on B(0, n)∪Ψ(B(0, R)∩X) and such that for each j, there is a submanifold σ_{j} of HΨ(X) such that the orthogonal projection of σ_{j} onto Δ_{j} is a diffeomorphism and such that σ_{j} is near Δ_{j} in the sense of lemma 2.2. Moreover, there exists a ball B′_{0} with \overline{B′_{0}} ∩ (\overline{B(0, n + 2)} ∪ HΨ(X)) = ∅ and H(K_{0}) ⊆ B′_{0}.

Proof: Choose p_{1}, …, p_{N} ∈ Ψ(reg(X) − \overline{B(0, R)}) − \overline{B(0, n)} and choose pairwise disjoint C^{2} curves γ_{j} : [0, 1] → C^{2}, j = 0, …, N which are disjoint from Ψ(X ∩ \overline{B(0, R)}) ∪ \overline{B(0, n)} such that γ_{j}(0) = p_{j} and such that if j ≥ 1, then γ_{j}(1) = L_{j}(0), while ∥γ_{0}(1)∥ > n + 2. Let W_{j} be a neighborhood of γ_{j} such that the W_{j} have pairwise disjoint closures which are also disjoint from Ψ(X ∩ \overline{B(0, R)}) ∪ \overline{B(0, n)}, and such that \overline{Δ_{j}} ⊆ W_{j} if j ≥ 1 and \overline{B(p_{0}, r_{0})} ⊆ W_{0}. For j ≥ 1, let r_{j} > 0 such that Ψ(X) ∩ \overline{B(p_{j}, r_{j})} ⊆ W_{j} and such that there is an embedding g_{j} : Δ(0, 2) → C^{2} with g_{j}(\overline{Δ(0, 1)}) = Ψ(X) ∩ \overline{B(p_{j}, r_{j})}.

Let K = Ψ(X∩\overline{B(0, R)})∪\overline{B(0, n)}∪\overline{B_{0}}∪(∪_{j=1}^{N}g_{j}(\overline{Δ(0, 1)})). In order to apply the approximation result in [FR, theorem 2.1], we will construct a family Φ_{t} of maps which are biholomorphic in a neighborhood of K and C^{2} in t with Φ_{0} = I. We will construct Φ_{t} so that Φ_{1} is the identity on B(0, n) ∪ Ψ(X ∩ B(0, R)) and maps each g_{j}(Δ(0, 1)) to a disk near Δ_{j}, and such that Φ_{t}(K) is polynomially convex for all t, then approximate Φ_{1} by a global automorphism.

We show how to construct Φ_{t} around a neighborhood of p_{1}. For simplicity, assume p_{1} = 0 and g_{1}(0) = p_{1}. Now, given any complex linear map L such that L(\overline{Δ(0, 1)}) ⊆ W_{1} is tangent to g_{1}(Δ(0, 1)) at p_{1}, we can first use the family g_{1}(χ_{1}(t)g_{1}^{−1}(p)) with χ_{1}(0) = 1, χ_{1}(1) small and positive, to shrink g_{1}(Δ(0, 1)) to a nearly linear disk tangent to L(Δ(0, 1)), then use the family Λ_{1}(t)g_{1}(χ_{1}(1)g_{1}^{−1}(p)) to expand the small disk to approximate L(Δ(0, 1)).

Note that by choosing χ_{1}(1) very small, we can make the diameter of g_{1}(χ_{1}(1)Δ(0, 1)) as small as we like. In particular, we can translate this disk along γ_{1} and use a one-parameter family of rotations to make the image tangent to Δ_{j}, then expand as before to make the new image approximate Δ_{j}. Note that all of this can be done within W_{j}, and that at each stage, the image of Δ(0, 1) is contained in an affine linear image of X. Moreover, we can make the family C^{2} in t. Reparametrizing, and using a similar construction for each p_{j}, we can define Φ_{t} on ∪_{j=1}^{N}g_{1}(\overline{Δ(z_{j}, r_{j})}). We can use a similar procedure to shrink \overline{B_{0}} and move it along γ_{0} inside W_{0} so that the image is \overline{B′_{0}} as in the statement.

Finally, each g_{j}, j ≥ 1 can be extended to be a biholomorphic embedding of \overline{Δ^{2}(0, 1)} into W_{j}, so we can use the same argument to define Φ_{t} on a neighborhood of ∪_{j=1}^{N}g_{j}(\overline{Δ(0, 1)}), and similarly extend it to a neighborhood of \overline{B_{0}}. Define Φ_{t} ≡ I on a neighborhood of Ψ(X ∩ \overline{B(0, R)}) ∪ \overline{B(0, n)} for all t.

Since the union of two disjoint closed balls is polynomially convex, and since the remaining hypotheses of the previous lemma are satisfied for all t, we see that Φ_{t}(K) is polynomially convex for each t. Hence by [FR, theorem 2.1], there is a neighborhood V of K such that Φ_{1} can be approximated uniformly on V by automorphisms H of C^{2}.

Choosing Φ_{t} such that Φ_{1}g_{j}(Δ(0, 1)) is close to Δ_{j} for all j and Φ_{1}(K_{0}) ⊆ B′_{0}, then choosing an automorphism H close to Φ_{1}, we obtain the lemma.

4. Main theorem

Theorem 4.1. Let X be a closed, 1-dimensional, complex subvariety of C^{2} and B_{0} a ball with \overline{B_{0}} ∩ X = ∅. Then there exists a domain Ω ⊆ C^{2} −\overline{B_{0}} containing X and a biholomorphic map Φ from Ω onto C^{2} such that C^{2}−Φ(X) is Kobayashi hyperbolic. Moreover, all nonconstant images of C in C^{2} intersect Φ(X) in infinitely many points.

Remark 4.2. Since Φ in the statement of the theorem is biholomorphic, it is necessarily a proper holomorphic embedding of X into C^{2}.

Proof: The first step is to construct Φ as the limit of automorphisms of C^{2} which are constructed inductively. We will define a sequence Φ_{m} of automorphisms, numbers R_{m} ↗ ∞, and finitely many affine complex linear disks Δ_{j}^{m} such that (1_{m}) ∥Φ_{m}(p)∥ ≥ m + 1 if p ∈ X and ∥p∥ ≥ R_{m}, (2_{m}) ∥Φ_{m}(p)∥ ≥ m + 1 if p ∈ \overline{B_{0}}, (3_{m}) ∥Φ_{m+1} − Φ_{m}∥ ≤ 1/2^{m} on X ∩ \overline{B(0, R_{m})}, (4_{m}) ∥Φ_{m+1} ∘ Φ_{m}^{−1} − I∥ ≤ 1/2^{m} on \overline{B(0, m)} and (5_{m}) if l < m and ϕ : Δ(0, 1) → B(0, l + 2) − ∪_{j}(Δ_{j}^{l})′ as in lemma 2.2

with ϕ(0) ∈ \overline{B(0, l/2)} and dist(ϕ(0), Φ_{l}(X)) ≥ 1/l, then ϕ(Δ(0, 1− 1/2^{l})) ⊆ B(0, l + 1).

Changing coordinates by a translation, we may assume \overline{B_{0}}∩\overline{B(0, 1)} = ∅, so we can take Φ_{0} = I, R_{0} = 1, and choose a ball B_{0}^{0} such that \overline{B_{0}} ⊆ B_{0}^{0} and \overline{B_{0}^{0}} ∩ (\overline{B(0, 1)} ∪ X) = ∅.

For the inductive construction, suppose we have m ≥ 0, Φ_{l} and R_{l} for l ≤ m and linear discs Δ_{j}^{l} for l < m satisfying (1_{l}) and (2_{l}) for l ≤ m and (3_{l}), (4_{l}) and (5_{l}) for l < m. Suppose also that there exist balls B_{0}^{l}, l ≤ m, with Φ_{l}(\overline{B_{0}}) ⊆ B_{0}^{l} and \overline{B_{0}^{l}} ∩ (\overline{B(0, l + 1)} ∪ Φ_{l}(X)) = ∅, l ≤ m. Note that Φ_{m}(\overline{B_{0}}) is polynomially convex since Φ_{m} is a global automorphism. Suppose also that Φ_{m}(X ∩ B(0, R_{m})) has submanifolds approximating each Δ_{j}^{l} as in lemma 2.2.

We use lemma 2.2 with m, m + 1 and m + 2 in place of n_{1}, n_{2} and n_{3}, respectively, with Φ_{m}(X) in place of X, and with k = m. This gives finitely many affine complex linear L_{j}^{m} : C → C^{2} with \overline{Δ_{j}^{m}} := L_{j}^{m}(\overline{Δ(0, 1)}) ⊂ B(0, m + 1) − \overline{B(0, m)} having pairwise disjoint closures such that if ϕ : Δ(0, 1) → B(0, m + 2) − ∪_{j}(Δ_{j}^{m})′ as in lemma 2.2, and ϕ(0) ∈ \overline{B(0, m/2)} with dist(ϕ(0), Φ_{m}(X)) ≥ 1/m, then ϕ(Δ(0, 1 − 1/2^{m})) ⊆ B(0, m + 1).

We then apply lemma 3.2 with R = R_{m}, n = m, Ψ = Φ_{m}, X unchanged, \{L_{j}^{m}\} in place of \{L_{j}\}, B_{0}^{m} in place of B_{0}, and K_{0} = Φ_{m}(\overline{B_{0}}). This gives an automorphism H : C^{2} → C^{2} such that ∥H − I∥ ≤ ϵ on \overline{B(0, m)} ∪ Φ_{m}(X ∩ \overline{B(0, R_{m})}) and such that there are submanifolds σ_{j}^{m} of Hϕ_{m}(X) which are close enough to the Δ_{j}^{m} for lemma 2.2 and such that there is a ball B_{0}^{m+1} with \overline{B_{0}^{m+1}} ∩ (\overline{B(0, m + 2)} ∪ HΦ_{m}(X)) = ∅ and HΦ_{m}(\overline{B_{0}}) ⊆ B_{0}^{m+1}.

By taking ϵ sufficiently small, (1_{m}) together with the fact that \overline{Δ_{j}^{l}} ⊆ B(0, m) for l < m implies that we can make HΦ_{m}(X) pass close to all of these disks, in the sense of lemma 2.2. Thus, taking Φ_{m+1} = HΦ_{m} and R_{m+1} ≥ R_{m} + 1 large, we obtain the inductive condition on submanifolds of Φ_{m+1}(X), as well as (1_{m+1}), (2_{m+1}), (3_{m}), (4_{m}) and (5_{m}).

To show that {Φ_{m}} converges to give a map Φ : Ω → C^{2}, let Ω_{m} := Φ_{m}^{−1}(B(0, m)). If p ∈ Ω_{m}, then ∥Φ_{m+1}(p) − Φ_{m}(p)∥ = ∥(Φ_{m+1} ∘ Φ_{m}^{−1})(Φ_{m}(p))−I(Φ_{m}(p))∥ < \frac{1}{2^{m}} . Hence ∥Φ_{m+1}(p)∥ < m+1/2^{m} < m+1, so p ∈ Ω_{m+1}. It follows that the sequence {Φ_{m}} converges locally uniformly on Ω := ∪Ω_{m} to a map Φ.

Suppose p ∈ Ω_{m+1} − Ω_{m}. Then

∥Φ(p) − Φ_{m+1}(p)∥ = ∥ lim_{k→∞}(Φ_{k}(p) − Φ_{m+1}(p))∥

≤ ∑_{k≥m+1} ∥Φ_{k+1}(p) − Φ_{k}(p)∥

< \frac{1}{2^{m}}

Hence Φ(p) ∈ B(0, m + 1 + 1/2^{m}) − B(0, m − 1/2^{m}). It follows that Φ is a proper map from Ω to C^{2}. In particular, Φ must have maximal rank 2. Since Φ is a limit of automorphisms, it follows that Φ has nonvanishing Jacobian everywhere in Ω. A standard argument implies that Φ is injective on Ω, and (4_{m}) implies that B(0, m − 1) ⊆ Φ_{k}(Ω_{m}) for k ≥ m, so that Φ is surjective. Hence Φ is a biholomorphism from Ω onto C^{2}.

From (2_{m}) we see that \overline{B_{0}} ∩ Ω_{m} = ∅ for all m, hence \overline{B_{0}} ∩ Ω = ∅.

If p ∈ X, then p ∈ X ∩ \overline{B(0, R_{m})} for some m. It follows from (3_{m}) that ∥Φ_{k}(p)∥ ≤ ∥Φ_{m}(p)∥ + ∑_{j=m}^{k−1} 1/2^{j} < ∥Φ_{m}(p)∥ + 1 for all k > m. Hence if k is large enough, ∥Φ_{k}(p)∥ < k and so p ∈ Ω_{k}. Therefore X ⊂ Ω.

Before continuing, note that the approximation of Δ_{j}^{k} by each Φ_{m}(X∩ B(0, R_{m})) implies that Φ(X) has submanifolds which approximate each Δ_{j}^{k} in the sense of lemma 2.2.

We show next that C^{2} − Φ(X) is Kobayashi hyperbolic. If not, there is a point p in C^{2} − Φ(X), a nonzero tangent vector ξ ∈ T_{p}C^{2}, and a sequence of holomorphic maps ϕ_{k} : Δ(0, k + 1) → C^{2} − Φ(X) such that ϕ_{k}(0) = p_{k}, ϕ′_{k}(0) = ξ_{k}, and (p_{k}, ξ_{k}) → (p, ξ).

Since p ∉ Φ(X), there is an integer m_{0} > 1 such that p ∈ B(0, m_{0}/2) and dist(p, Φ(X)) > 1/m_{0}. From (1_{m}) and (3_{m}), there exist k_{1} > 1, m_{1} > m_{0} such that dist(p_{k}, Φ_{m}(X)) ≥ 1/m_{0} for k ≥ k_{1}, m ≥ m_{1}.

Fixing k ≥ k_{1}, we see that ϕ_{k}(Δ(0, k)) ⊆ B(0, m + 2) for some m ≥ m_{1} large. Since ϕ_{k} misses Φ(X), we see that ψ_{k}(z) := ϕ_{k}(z/k) maps Δ(0, 1) into B(0, m+2)−∪_{j}(Δ_{j}^{m})′, ψ_{k}(0) = p_{k} and dist(ψ_{k}(0), Φ_{m}(X)) ≥ 1/m. Thus ψ_{k}(Δ(0, 1−1/2^{m})) ⊂ B(0, m+1) by (5_{m}). By induction, we see that ψ_{k}(Δ(0, Π_{m≥m_{1}}(1−1/2^{m}))) ⊂ B(0, m_{1} +1). Hence there exists r > 0 independent of k such that ∥ψ_{k}(z)∥ < m_{1} + 1 for all |z| < r. This implies that ∥ϕ_{k}(z)∥ < m_{1} + 1 for all |z| < kr and hence that ϕ′_{k}(0) → 0, a contradiction. Thus, C^{2} − Φ(X) is Kobayashi hyperbolic.

Finally, if g : C → C^{2} is a nonconstant holomorphic map which intersects Φ(X) only finitely many times, then these points of intersection are contained in some large ball. By reparametrizing, we may assume that g(0) ∉ Φ(X), then use an argument like the one just given to show that the image of g must be contained in some large ball, a contradiction.

The theorems in section 1 now follow immediately by taking X to be the z-axis for the first two theorems and to be any finite collection of complex lines for the third theorem. In Theorem 1.3 one can replace X by any countable union of closed subvarieties of C^{2} avoiding a fixed ball. The proof of Theorem 4.1 still goes through. Moreover, if X is a dense, countable union of closed subvarieties of C^{2} and S is any discrete set of points in the complement of X, then there exists a Fatou-Bieberbach domain Ω, X ⊂ Ω ⊂ C^{2} \ S.

Note that theorem 4.1 can be applied to any Riemann surface which admits a proper holomorphic embedding into C^{2}, so in particular, there is an embedding of the disk whose complement is Kobayashi hyperbolic.

References

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[FR] Forstneric, F., and Rosay, J.-P., Approximation of biholomorphic mappings by automorphisms of C^{n}, Invent. Math., 112 (1993), 323-349.

[GR] Gunning, R., and Rossi, C., Analytic functions of several complex variables, Prentice Hall, Englewood Cliffs, N.J., 1965.

[RR] J.-P. Rosay and W. Rudin, Holomorphic maps from C^{n} to C^{n}, Trans. Amer. Math. Soc., 310 (1988), no. 1, 47-86.

[S] Suzuki, M., Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace C^{2}, J. Math. Soc. Japan, 26 (1974), 241-257.

Gregery T. Buzzard Department of Mathematics The University of Michigan Ann Arbor, MI 48109, USA and MSRI 1000 Centennial Drive Berkeley, CA 94720, USA

John Erik Fornæss Department of Mathematics The University of Michigan Ann Arbor, MI 48109, USA