A CARLEMAN TYPE THEOREM

FOR PROPER HOLOMORPHIC EMBEDDINGS

Gregery T. Buzzard and Franc Forstneric

Abstract

In 1927, Carleman showed that a continuous, complex-valued function on the real line can be approximated in the Whitney topology by an entire function restricted to the real line. In this paper, we prove a similar result for proper holomorphic embeddings. Namely, we show that a proper C^{r} embedding of the real line into C^{n} can be approximated in the strong C^{r} topology by a proper holomorphic embedding of C into C^{n}.

&1. Introduction

We denote the complex numbers by C and the real numbers by R. Let C^{n} be the complex Euclidean space of dimension n with complex coordinates z = (z_{1}, …, z_{n}).

To motivate our main result we recall the Carleman approximation theorem [3], [10]: For each continuous function λ: R → C and positive continuous function η: R → (0, ∞) there exists an entire function f on C such that |f(t) − λ(t)| < η(t) for all t ∈ R. This extends the Weierstrass approximation theorem in which the approximation takes place on compact intervals in R. If λ is smooth, we can also approximate its derivatives by those of f. A more general result was proved by Arakelian [1] (see [13] for a simple proof).

Our main result is an extension of Carleman’s theorem to proper holomorphic embeddings of C into C^{n} for n > 1:

1.1 Theorem. Let n > 1 and r ≥ 0 be integers. Given a proper embedding λ: R ↪ C^{n} of class C^{r} and a continuous positive function η: R → (0, ∞), there exists a proper holomorphic embedding f: C ↪ C^{n} such that

|f^{(s)}(t) − λ^{(s)}(t)| < η(t), t ∈ R, 0 ≤ s ≤ r.

If in addition T = {t_{j}} ⊂ R is discrete, there exists f as above such that

f^{(s)}(t) = λ^{(s)}(t), t ∈ T, 0 ≤ s ≤ r.

Buzzard’s research at MSRI is supported in part by NSF grant DMS-9022140. Forstneric is supported in part by an NSF grant and by a grant from the Ministry of Science of the Republic of Slovenia.

Here f^{(s)}(t) denotes the derivative of f order s, and similarly for λ.

We emphasize that, in general, one can not expect to extend λ to a holomorphic embedding of C into C^{n}, even if λ is real-analytic. In this case λ will extend holomorphically to some open set in C, but in general not to all of C; and even if λ extends to C, the extension need not be a proper map into C^{n}. So our result is the best possible in this context.

Another motivation for Theorem 1.1 are the recent developments on embedding Stein manifolds in C^{n}; see the papers [2], [6], [7], [8]. (For Stein manifolds and other topics in several complex variables mentioned here we refer the reader to Hörmander [11].) In these papers it was shown that a Stein manifold M which admits a proper holomorphic embedding in C^{n} for some n > 1 also admits an embedding f: M ↪ C^{n} whose image f(M) contains a given discrete subset E ⊂ C^{n} (with prescribed finite order jet of f(M) at any point of E), and such that f(M) intersects the image of any entire map g: C^{d} → C^{n} of rank d = n − dim M at infinitely many points [6, Theorem 5.1]. This implies in particular that f(M) cannot be mapped into any hyperplane in C^{n} by an automorphism of C^{n}. In this context we recall that any Stein manifold M embeds in C^{n} for n > (3 dim M + 1)/2 according to Eliashberg and Gromov [4].

In light of these results it is a natural question whether one can in some sense prescribe the embedding M ↪ C^{n} on a positive dimensional submanifold N ⊂ M. Our result provides an affirmative answer in the simplest case when M = C and N = R × {i0} ⊂ C. The details of our construction are considerable even in this simplest case, and the full scope of the method remains to be seen. If λ is of class C^{∞}, our method can be modified so that we approximate to increasingly high order on complements of compact subsets of R. We shall not go into details of this. Another possible extension is to approximate a proper smooth embedding by a proper holomorphic embedding on a finite set of real lines (or certain other real curves) in C.

A seemingly more difficult question is whether a result of this type holds when N is a complex submanifold of positive dimension in a complex manifold M. For instance, does every proper holomorphic embedding λ: C ↪ C^{n} for n ≥ 3 extend to a proper holomorphic embedding f: C^{2} ↪ C^{n} ?

One of the main tools in our construction is the following result from [9]. The case r = 0 was obtained earlier in [8]. This result can also be obtained by methods in [5].

1.2 Proposition. Let K ⊂⊂ C^{n} (n ≥ 2) be a compact, polynomially convex set, and let C ⊂ C^{n} be a smooth embedded arc of class C^{∞} which is attached to K in a single point of K. Given a C^{∞} diffeomorphism F: K ∪ C → K ∪ C′ ⊂ C^{n} such that F is the identity on (K ∪ C) ∩ U for some open neighborhood U of K, and given numbers r ≥ 0, ϵ > 0, there exist a neighborhood W of K and an automorphism Φ ∈ AutC^{n} satisfying

∥Φ − Id ∥_{C^{r}(W)} < ϵ, ∥Φ − F ∥_{C^{r}(C)} < ϵ.

(Here Id denotes the identity map.) Moreover, for each finite subset Z ⊂ K ∪ C we can choose Φ such that it agrees with the identity to order r at each point z ∈ Z ∩ K and Φ|_{C} agrees with F to order r at each point of Z ∩ C.

The same result holds with any finite number of disjoint hairs attached to K.

We explain our notation. Δ_{ρ} is the closed disk in C of radius ρ and center 0. B is the open unit ball in C^{n} with center 0, and RB is the ball of radius R. For a set A ⊂ C^{n} and ρ > 0, let A + ρ\overline{B} = \{a + z: a ∈ A, |z| ≤ ρ\}. We identify C and R with their images in C^{n} under the embedding ζ → (ζ, 0, …, 0). For 1 ≤ j ≤ n we denote by π_{j} the coordinate projection π_{j}(z_{1}, …, z_{n}) = z_{j}.

In the proof we shall use special automorphisms of C^{n} of the form

Ψ(z) = z + f(πz)v, z ∈ C^{n},

where v ∈ C^{n}, π: C^{n} → C^{k} is a linear map for some k < n with πv = 0 (in most cases k = 1), and f is an entire function on C^{k}. An automorphism of this form is called a shear; clearly Ψ^{−1}(z) = z − f(πz)v.

&2. Some lemmas.

The following is standard, e.g., [12, prop. 2.15.4].

2.1 Lemma. Let λ: R ↪ C^{n} be a C^{∞} proper embedding. Then there exists a continuous η: R → (0, ∞) such that if γ: R → C^{n} with |γ^{(s)}(t) − λ^{(s)}(t)| < η(t) for all t ∈ R, s = 0, 1, then γ is a proper embedding.

Recall that a compact set A ⊂⊂ C^{n} is polynomially convex if for each z ∈ C^{n}\A there is a holomorphic polynomial P on C^{n} such that |P (z)| > max{|P (w)|: w ∈ A}. We refer the reader to [11] for properties of such sets.

2.2 Lemma. Let A ⊂ C^{n} be compact and polynomially convex and ρ > 0. Let I ⊂ R be an interval whose endpoints lie in C^{n}\(A ∪ Δ_{ρ}), and let r, ϵ > 0. Then there exists an automorphism Ψ(z) = z + g(z_{1})e_{2} of C^{n} such that

(i) |Ψ(z) − z| < ϵ for z ∈ Δ_{ρ}, (ii) ∥Ψ|_{R}(t) − t∥_{C^{r}(I)} < ϵ, and

(iii) Ψ(t) ∉ A for t ∈ \overline{R\I}. If Z ⊂ I is finite, we can choose Ψ as above so that g^{(s)}(t) = 0 for t ∈ Z, 0 ≤ s ≤ r.

Proof. Let μ_{1} < μ_{2} denote the endpoints of I in R, and let Γ_{j} = {(μ_{j}, ζ, 0, …, 0): ζ ∈ C} for j = 1, 2. Let R > max{|μ_{1}|, |μ_{2}|} + 1 such that A ⊂ RB. Consider the set E_{j} = A ∩ Γ_{j}. Since A is polynomially convex, E_{j} is polynomially convex in Γ_{j} and hence Γ_{j}\E_{j} is connected. Since the endpoints of I lie in Γ_{j}\E_{j}, there exists a smooth curve γ_{j}: [0, 1] → Γ_{j}\E_{j} with γ_{j}(0) = (μ_{j}, 0, …, 0) and |π_{2}γ_{j}(1)| > R + 1 for j = 1, 2.

Since A is compact, there exists δ > 0 such that γ_{j}([0, 1]) + 3δ\overline{B} ⊂ C^{n}\A. Let π_{2}(z) = z_{2}. Let K = {x + iy ∈ C: μ_{1} − δ/2 ≤ x ≤ μ_{2} + δ/2, |y| ≤ ρ + 1}. Define a function h: K ∪ [−R, R] → C by

h(t) = \{ \begin{matrix}π_{2}γ_{1}(1) & if t ∈ [−R, μ_{1} − 2δ]; \\ π_{2}γ_{1}((μ_{1} − δ − t)/δ) & if t ∈ [μ_{1} − 2δ, μ_{1} − δ]; \\ π_{2}γ_{2}((t − μ_{2} − δ)/δ) & if t ∈ [μ_{2} + δ, μ_{2} + 2δ]; \\ π_{2}γ_{2}(1) & if t ∈ [μ_{2} + 2δ, R]; \\ 0 & otherwise. \end{matrix}

Choose η, 0 < η < min{ϵ, δ}. By Mergelyan’s theorem [14, p.386] there is an entire function g on C such that |h(z) − g(z)| < η for z ∈ K ∪ [−R, R]. The shear Ψ(z) = z + g(z_{1})e_{2} then satisfies (i) and (iii). Since I ⊂ IntK, Cauchy’s estimates imply that it also satisfies (ii) provided that η > 0 is chosen sufficiently small. The last condition on g is a trivial addition to Mergelyan’s theorem. ♠

2.3 Lemma. Let λ: R ↪ C^{n} be a proper, C^{∞} embedding, K ⊂ C^{n} compact, ϵ > 0, and r ∈ Z_{+}. Let Z ⊂ R be finite, and suppose λ(t) ∈ C = C × {0}^{n−1} for each t ∈ Z. Then there exists a shear Ψ(z) = z + h(z_{1})v for some v ∈ C^{n} with π_{1}v = 0 such that

(i) Ψ(C) ∩ λ(R) = λ(Z), (ii) |Ψ(z) − z| < ϵ for z ∈ K, and (iii) Ψ(z) = z + O(|z − λ(t)|^{r+1}) as z → λ(t), for all t ∈ Z.

Proof. Let Z = {t_{j}}_{j=1}^{s}. Define a polynomial on C by h(ζ) = Π_{1≤j≤s}(ζ − π_{1}λ(t_{j}))^{r+1}. Consider the map Φ: C × C^{n−1} → C^{n} given by

Φ(z_{1}, α_{2}, …, α_{n}) = (z_{1}, 0, …, 0) + h(z_{1})(0, α_{2}, …, α_{n}).

Clearly Φ is an automorphism of (C\λ(Z)) × C^{n−1}. Let Δ_{R,j} denote the closed disk of radius R in C with center π_{1}λ(t_{j}) for j = 1, 2, …, s. Choose R > 0 such that the discs Δ_{R,j} for 1 ≤ j ≤ s are pairwise disjoint. Choose a ρ, 0 < ρ < R, such that ρ^{2} is a regular value of μ_{j}(t) = |π_{1}λ(t) − π_{1}λ(t_{j})|^{2} (t ∈ R) for each j = 1, 2, …, s.

Let M_{ρ} = C\∪_{1≤j≤s} intΔ_{ρ,j}. Let Φ_{ρ} = Φ|_{M_{ρ}×C^{n−1}} and ∂Φ_{ρ} = Φ|_{∂M_{ρ}×C^{n−1}}. A simple check shows that Φ_{ρ} and ∂Φ_{ρ} are transverse to λ(R). Hence by the transversality theorem, there exists a set A_{ρ} ⊂ C^{n−1} of full measure such that for each α = (α_{2}, …, α_{n}) ∈ A_{ρ}, Φ(M_{ρ}, α) = {Φ(z_{1}, α): z_{1} ∈ M_{ρ}} and λ(R) are transverse, hence disjoint by dimension considerations.

Let A = ∩_{j=1}^{∞}A_{1/j}. Then A ⊂ C^{n−1} has full measure, and for each α ∈ A we see that Φ(C\λ(Z), α) and λ(R) are disjoint. Finally, choose α ∈ A such that |h(z_{1})α| < ϵ for z_{1} ∈ π_{1}(K), and let Ψ(z) = z + h(z_{1})α. Then Ψ(z_{1}, 0, …, 0) = Φ(z_{1}, α_{2}, …, α_{n}), and Ψ satisfies the conclusions of the lemma. ♠

2.4 Lemma. Let λ: R ↪ C^{n} be a C^{∞} embedding, f: C ↪ C^{n} a proper holomorphic embedding, and I ⊂ R a closed interval with f|_{I} = λ|_{I}. Let K ⊂ C^{n} be compact and polynomially convex, a, r, ϵ > 0, and T ⊂ R discrete. Suppose that λ(t), f(t) ∉ K for t ∈ \overline{R\I}. Then there exists Φ ∈ AutC^{n} such that if g = Φ ∘ f, then

(i) |g^{(s)}(t) − λ^{(s)}(t)| < ϵ for t ∈ [−a, a], 0 ≤ s ≤ r, (ii) g^{(s)}(t) = λ^{(s)}(t) for t ∈ T ∩ [−a, a], 0 ≤ s ≤ r, and (iii) |Φ(z) − z| < ϵ for z ∈ K.

Proof. We may assume that I ⊂ (−a, a). Let I_{1}, I_{2} be the two connected components of {ζ ∈ I: f(ζ) ∈ C^{n}\K} containing the respective endpoints of I, and let I_{0} = I\(I_{1} ∪ I_{2}). Let A be the polynomial hull of K ∪ f(I_{0}). Then A is the union of K ∪ f(I_{0}) and the bounded connected components of f(C)\(K ∪ f(I_{0})). Note that f(I_{1}) and f(I_{2}) lie in f(C)\A since f(t) ∉ K for all t ∈ \overline{R\I}.

Let L = A ∪ f([−a, a]). Then C = \overline{L\A} is the union of two embedded arcs, each containing an endpoint of f([−a, a]). Define F on L by F (z) = z if z ∈ A, and F (z) = λf^{−1}(z) if z ∈ f([−a, a]). Then F is a C^{∞} diffeomorphism of L which extends as the identity map on (A ∪ C) ∩ U for some neighborhood U of A. Apply proposition 1.2 to get Φ ∈ AutC^{n} such that |Φ(z) − z| < ϵ for z ∈ K and such that g = Φ ∘ f satisfies (i) and (ii). ♠

&3. Proof of theorem 1.1.

Choose a smooth cutoff function χ on R such that χ(t) = 1 for |t| small and supp χ ⊂ (−1, 1). Define the constant C = C_{r} > 1 such that ∥χh∥_{C^{r}} ≤ C∥h∥_{C^{r}} for each h ∈ C^{r}(R). We fix such C for the entire proof.

By approximation we may assume that λ: R ↪ C^{n} in theorem 1.1 is a proper C^{∞} embedding. Decreasing η if necessary we may also assume that η satisfies lemma 2.1 for λ and η(t) < 1/2 for all t ∈ R.

We use an inductive procedure to obtain a sequence of proper holomorphic embeddings f_{k}: C ↪ C^{n} such that f = lim_{k→∞} f_{k} exists on C and satisfies theorem 1.1. Each f_{k} will be a restriction to C = C × {0}^{n−1} of a holomorphic automorphism of C^{n}. The next map f_{k+1} will be of the form f_{k+1} = Φ_{k+1} ∘ f_{k} ∘ Ψ_{k+1} for suitably chosen Φ_{k+1}, Ψ_{k+1} ∈ AutC^{n}.

We will describe the case k = 1 after the inductive step is given. Recall that Δ_{k} is the closed disc in C = C × {0}^{n−1} with center 0 and radius k, and B is the unit ball in C^{n}. For the induction at step k, suppose we have the following:

(a) closed balls B_{j} = R_{j}\overline{B} ⊂ C^{n} with R_{j} ≥ max\{j + 1, R_{j−1} + 1\}, j = 1, …, k,

(b) automorphisms Φ_{1}, …, Φ_{k} of C^{n} with |Φ_{j}(z) − z| < 2^{−j} for z ∈ B_{j−1}, j = 2, …, k,

(c) numbers ϵ_{j} > 0 such that ϵ_{1} < 2^{−1} and ϵ_{j} < ϵ_{j−1}/2 < 2^{−j} for j = 1, …, k,

(d) automorphisms Ψ_{1}, …, Ψ_{k} of C^{n} of the form Ψ_{j}(z) = z + g_{j}(z_{1})e_{2} + h_{j}(z_{1})v_{j}, where π_{1}(v_{j}) = 0 and |Ψ_{j}(z) − z| < ϵ_{j} for |z| ≤ j,

(e) closed intervals I_{j} = [−a_{j}, a_{j}], j = 1, …, k, with a_{j} > max{a_{j−1} + 2, j + 2}, and

(f) numbers 0 < δ_{j} < C^{−1} inf{η(t): t ∈ I_{j}}, j = 1, …, k, such that the automorphism

f_{k} = Φ_{k} ∘ ⋯ Φ_{1} ∘ Ψ_{1} ∘ ⋯ ∘ Ψ_{k} ∈ AutC^{n}

(whose restriction to C = C × {0}^{n−1} provides an embedding C ↪ C^{n}) satisfies: (1_{k}) f_{k}(Δ_{j} + ϵ_{k}\overline{B}) ⊂ IntB_{j} for j = 1, …, k,

(2_{k}) |f_{k}^{(s)}(t) − λ^{(s)}(t)| < η(t) for t ∈ I_{k} and 0 ≤ s ≤ r,

(3_{k}) |f_{k}^{(s)}(t) − λ^{(s)}(t)| < δ_{k} for t ∈ I_{k}\(−a_{k} + 1, a_{k} − 1) and 0 ≤ s ≤ r,

(4_{k}) f_{k}^{(s)}(t) = λ^{(s)}(t) for t ∈ T ∩ I_{k}, 0 ≤ s ≤ r, (5_{k}) f_{k}(C) ∩ λ(R) = λ(T ∩ I_{k}), (6_{k}) |λ(t)| > R_{k} + 1 for |t| ≥ a_{k} − 1, (7_{k}) |f_{k}(t)| > R_{k} for |t| ≥ a_{k} − 1.

We will now show how to obtain these hypotheses at step k + 1. Let I_{k}^{1} and I_{k}^{2} be the two connected components of the set {ζ ∈ I_{k}\Δ_{k+1}: |f_{k}(ζ)| > R_{k}} containing the respective endpoints of the interval I_{k}. Let I_{k}^{0} = I_{k}\(I_{k}^{1} ∪ I_{k}^{2}) be the middle interval. By (7_{k}) we have I_{k}^{0} ⊂ (−a_{k} + 1, a_{k} − 1).

Let K_{k} be the polynomial hull of the set B_{k} ∪f_{k}(Δ_{k+1} ∪I_{k}^{0}). Since f_{k}(C) is a complex submanifold of C^{n} and B_{k} is polynomially convex, it is seen easily that K_{k} is contained in B_{k} ∪ f_{k}(C), and it is the union of B_{k} ∪ f_{k}(Δ_{k+1} ∪ I_{k}^{0}) and the bounded connected components of the complement f_{k}(C)\(B_{k} ∪ f_{k}(Δ_{k+1} ∪ I_{k}^{0})) (see Lemma 5.4 in [6]). Note that (7_{k}) and (e) imply that f_{k}(R\(−a_{k} + 1, a_{k} − 1)) ⊂ C^{n}\K_{k}.

Choose R_{k+1} > R_{k} + 1 such that K_{k} ⊂ (R_{k+1} − 1)B, and let B_{k+1} = R_{k+1}\overline{B}. Choose a_{k+1} > a_{k} + 2 to get (6_{k+1}). We now want to approximate λ on the larger interval I_{k+1} = [−a_{k+1}, a_{k+1}] by the image of the next embedding C ↪ C^{n} (to be constructed). In order to apply lemma 2.4 we first approximate λ as follows:

3.1 Lemma. There exists a proper C^{∞} embedding λ_{k}: R ↪ C^{n} satisfying

(i) λ_{k} = f_{k} on [−a_{k} + 1, a_{k} − 1], (ii) λ_{k} = λ on R\I_{k},

(iii) |λ_{k}^{(s)}(t) − λ^{(s)}(t)| < η(t) for t ∈ I_{k}\(−a_{k} + 1, a_{k} − 1), 0 ≤ s ≤ r,

(iv) λ_{k}^{(s)}(t) = λ^{(s)}(t) for t ∈ T, 0 ≤ s ≤ r, and

(v) λ_{k}(t) ∉ K_{k} when |t| ≥ a_{k} − 1.

Proof. We define the cutoff function χ_{k} on R using χ, so that χ_{k} = 1 on [−a_{k} + 1, a_{k} − 1], χ_{k} = 0 on R\I_{k}, and ∥χ_{k}h∥_{C^{r}} < C∥h∥_{C^{r}} as before. Let

λ̂_{k}(t) = f_{k}(t)χ_{k}(t) + λ(t)(1 − χ_{k}(t)), t ∈ R.

By lemma 2.1, (3_{k}), (4_{k}), and choice of η and δ_{k}, we see that (i)–(iv) are satisfied for λ̂_{k} in place of λ_{k}.

To obtain (v) we use a transversality argument to perturb λ̂_{k} on I_{k}\(−a_{k} + 1, a_{k} − 1). First note that if |t| > a_{k}, then |λ̂_{k}(t)| = |λ(t)| > R_{k} + 1 by (6_{k}), so λ̂_{k}(t) ∉ B_{k}. Also, by (5_{k}), we see that λ̂_{k}(t) ∉ f_{k}(C), so λ̂_{k}(t) ∉ K_{k}. Next, if t ∈ T ∩(I_{k}\(−a_{k} +1, a_{k} −1)), then by (4_{k}), (7_{k}), and (e) we see that λ̂_{k}(t) = f_{k}(t) ∉ K_{k}. Hence there exists a neighborhood V of T ∩ (I_{k}\(−a_{k} + 1, a_{k} − 1)) such that λ̂_{k}(\overline{V}) ∩ K_{k} = ∅.

Thus we need only perturb λ̂_{k} on I_{k}\(V ∪ (−a_{k} + 1, a_{k} − 1)) to get (v). Note that if t ∈ I_{k}\(−a_{k} + 1, a_{k} − 1), then from (6_{k}) and (2_{k}) we see that |λ̂_{k}(t)| > R_{k} + 1/2, so λ̂_{k}(t) ∉ B_{k}. Finally, a simple transversality argument implies that we can make an arbitrarily small C^{∞} perturbation of λ̂_{k} to avoid f_{k}(C), and hence we get λ_{k} with λ_{k} = λ̂_{k} outside I_{k}\(V ∪ (−a_{k} + 1, a_{k} − 1)) and λ_{k} satisfying (i)–(v). ♠

Now we can use lemma 2.4 to approximate λ_{k}, hence to approximate λ. Set

\begin{aligned}δ_{k+1} & = min\{η(t): t ∈ I_{k+1}\}/(2C), \\ σ_{k+1} & = min\{η(t) − |λ_{k}^{(s)}(t) − λ^{(s)}(t)|: t ∈ I_{k+1}, 0 ≤ s ≤ r\} > 0. \end{aligned}

Choose ϵ > 0 so small that

ϵ < min\{2^{−(k+1)}, δ_{k+1}, σ_{k+1}\}, f_{k}(Δ_{j} + ϵ_{k}\overline{B}) + ϵ\overline{B} ⊂ IntB_{j}, 1 ≤ j ≤ k.

Apply lemma 2.4 with λ = λ_{k}, f = f_{k}, I = [−a_{k} + 1, a_{k} − 1], K = K_{k}, a = a_{k+1}, r and T unchanged, and ϵ as above. This provides Φ_{k+1} ∈ AutC^{n} and G = Φ_{k+1} ∘ f_{k} ∈ AutC^{n} satisfying

|Φ_{k+1}(z) − z| < ϵ  for z ∈ K_{k}, hence on B_{k};
|G^{(s)}(t) − λ_{k}^{(s)}(t)| < ϵ for t ∈ I_{k+1}, 0 ≤ s ≤ r;
G^{(s)}(t) = λ_{k}^{(s)}(t)  for t ∈ T ∩ I_{k+1}, 0 ≤ s ≤ r.

In particular, (2_{k+1})–(4_{k+1}) hold with G in place of f_{k+1}.

Since f_{k}(Δ_{k+1}) ⊂ K_{k} ⊂ (R_{k+1} − 1)B, we can choose ϵ′_{k+1} < ϵ_{k} small enough that (1_{k+1}) holds with G in place of f_{k+1} and ϵ′_{k+1} in place of ϵ_{k+1}, and such that if ψ ∈ AutC^{n} with ∥ψ(t) − t∥_{C^{r}(I_{k+1})} < ϵ′_{k+1}, then (2_{k+1}) and (3_{k+1}) hold with G ∘ ψ in place of f_{k+1}. Let ϵ_{k+1} = ϵ′_{k+1}/2. Then with G in place of f_{k+1}, we have (1_{k+1})–(4_{k+1}), (6_{k+1}), and G(−a_{k+1}), G(a_{k+1}) ∉ B_{k+1} by (6_{k+1}) and (2_{k+1}).

Next we want to obtain (7_{k+1}). We do this using lemma 2.2 to change the embedding so that the image of R\I_{k+1} misses B_{k+1} while leaving the embedding essentially unchanged on Δ_{k+1} ∪ I_{k+1}. Apply lemma 2.2 with A = G^{−1}(B_{k+1}), ρ = k + 1, I = I_{k+1}, r unchanged, Z = T ∩ I_{k+1} and ϵ = ϵ_{k+1}/2. This gives a shear

ψ_{k+1}(z) = z + g_{k+1}(z_{1})e_{2}

with

\{ \begin{matrix}|ψ_{k+1}(z) − z| < ϵ_{k+1}/2, z ∈ Δ_{k+1}; \\ ∥ψ_{k+1}|_{R}(t) − t∥_{C^{r}(I_{k+1})} < ϵ_{k+1}/2; \\ g_{k+1}^{(s)}(t) = 0, t ∈ T ∩ I_{k+1}, 0 ≤ s ≤ r; \\ ψ_{k+1}(t) ∉ G^{−1}(B_{k+1}), t ∈ \overline{R\I_{k+1}}. \end{matrix}

Let H = Gψ_{k+1}. Then with H in place of f_{k+1}, we have (1_{k+1})–(4_{k+1}), (6_{k+1}), and (7_{k+1}).

For the final correction, we use lemma 2.3 to obtain (5_{k+1}) while maintaining the other properties. Let R > a_{k+1} be such that A = G^{−1}(B_{k+1}) ⊂ RB. Let δ > 0 be such that

ψ_{k+1} ([−R, R]\(−a_{k+1}, a_{k+1}) + δ\overline{B}) ∩ A = ∅,

and such that if θ ∈ AutC^{n}, with |θ(z) − z| < δ on R\overline{B}, then

∥ψ_{k+1}θ|_{R}(t) − t∥_{C^{r}(I_{k+1})} < ϵ_{k+1}.  (1)

Apply lemma 2.3 with λ replaced by H^{−1}λ, K = R\overline{B}, r unchanged, Z = T ∩ I_{k+1}, and ϵ = min{δ, ϵ_{k+1}/2}. This gives a shear θ_{k+1}(z) = z + h_{k+1}(z_{1})v_{k+1} with π_{1}v_{k+1} = 0 such that

\{ \begin{matrix}|θ_{k+1}(z) − z| < min\{δ, ϵ_{k+1}/2\}, z ∈ Δ_{k+1}; \\ θ_{k+1}(C) ∩ H^{−1}λ(R) = H^{−1}λ(T ∩ I_{k+1}); \\ h_{k+1}^{(s)}(t) = 0, t ∈ T ∩ I_{k+1}, 0 ≤ s ≤ r, \end{matrix}

and such that (1) holds with θ = θ_{k+1}. Also, by the choice of R and δ,

ψ_{k+1}θ_{k+1}(\overline{R\I_{k+1}}) ∩ A = ∅.

Taking Ψ_{k+1} = ψ_{k+1}θ_{k+1} and

f_{k+1} = H ∘ θ_{k+1} = Φ_{k+1} ∘ f_{k} ∘ Ψ_{k+1}

we obtain (5_{k+1}) and preserve the remaining hypotheses. Hence we obtain (1_{k+1})–(7_{k+1}). Note that (k + 1)B ⊂ B_{k+1} so we also obtain (a)–(f), thus finishing the inductive step.

The case k = 1 is similar to the general step. First apply proposition 1.2 with K = ∅, C = [−3, 3] ⊂ C, F = λ, ϵ = C^{−1} inf{η(t): t ∈ [−3, 3]}, and Z = T ∩ [−3, 3] to get ϕ_{1}^{1} ∈ AutC^{n} satisfying the conclusions of that proposition. Choose R_{1} ≥ 2 such that ϕ_{1}^{1}(Δ_{1}) ⊂ (R_{1} −1)B, choose a_{1} > 4 to get (6_{1}), and let I_{1} = [−a_{1}, a_{1}]. Choose δ_{1} to satisfy (f) for j = 1.

Define a proper C^{∞} embedding λ_{0} as in lemma 3.1 so that (i)–(v) are satisfied with λ_{0} in place of λ_{k}, ϕ_{1}^{1} in place of f_{k}, 3 in place of a_{k}, [−3, 3] in place of I_{k}, and ϕ_{1}^{1}(Δ_{1}) in place of K_{k}. Apply lemma 2.4 with λ = λ_{0}, f = ϕ_{1}^{1}, I = [−2, 2], K = ϕ_{1}^{1}(Δ_{1}), a = a_{1}, ϵ = δ_{1}, and T and r unchanged. This gives ϕ_{1}^{2} ∈ AutC^{n} such that

|ϕ_{1}^{2}(z) − z| < δ_{1} ≤ 1/2, z ∈ ϕ_{1}^{1}(Δ_{1}),

and such that Φ_{1} = ϕ_{1}^{2}ϕ_{1}^{1} satisfies

\{ \begin{matrix}∥Φ_{1} − λ_{0}∥_{C^{r}(I_{1})} < ϵ; \\ Φ_{1}^{(s)}(t) = λ_{0}^{(s)}(t), t ∈ T ∩ I_{1}, 0 ≤ s ≤ r. \end{matrix}

As before, we can apply lemmas 2.2 and 2.3 to obtain ϵ_{1} > 0 and a shear Ψ_{1} such that the hypotheses (1_{1})–(7_{1}) hold for f_{1} = Φ_{1}Ψ_{1}, and (a)–(f) hold for k = 1. This completes the base case.

To finish the proof of theorem 1.1, note that

Ψ_{1} ⋯ Ψ_{k}(z) = z + ∑_{j=1}^{k}(g_{j}(z_{1})e_{2} + h_{j}(z_{1})v_{j})

and that (c) implies

|g_{j}(z_{1})e_{2} + h_{j}(z_{1})v_{j}| < 2^{−j}, |z_{1}| < j.

Hence this sum converges uniformly on compacts to a shear Ψ(z) = z + G(z_{1}) for some holomorphic map G: C → {0} × C^{n−1}.

By proposition 4.2 in [6], the composition Φ_{k} ∘ ⋯ ∘ Φ_{1} converges locally uniformly to a biholomorphic map from a domain Ω onto C^{n}, and

Ω = ∪_{k=1}^{∞}(Φ_{k} ⋯ Φ_{1})^{−1}(B_{k−1}).

We claim that Ψ(C × {0}) ⊂ Ω. Let k > 1. By (1_{k}) we have

Ψ_{1} ⋯ Ψ_{k}(Δ_{k−1} + ϵ_{k}\overline{B}) ⊂ (Φ_{k} ⋯ Φ_{1})^{−1}(B_{k−1}).

Since |Ψ_{j}(z) − z| < ϵ_{j} on Δ_{k−1} for j ≥ k, and ∑_{j=k+1}^{∞} ϵ_{j} < ϵ_{k} by (c), we see that

lim_{m→∞} Ψ_{k+1} ⋯ Ψ_{m}(z) ∈ Δ_{k−1} + ϵ_{k}\overline{B}

for each z ∈ Δ_{k−1}. Hence

Ψ(Δ_{k−1}) ⊂ (Φ_{k} ⋯ Φ_{1})^{−1}(B_{k−1}) ⊂ Ω, k > 1,

so the claim holds. In particular, ΦΨ: C → C^{n} is a proper holomorphic embedding.

Finally, using the conditions (1_{k})–(7_{k}), we see that ΦΨ: C ↪ C^{n} is a proper holomorphic embedding with the desired properties. ♠

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Gregery T. Buzzard, Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Franc Forstneric, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA