A Mergelyan Theorem for Mappings to ℂ^{2} \ ℝ^{2}

by

Jörg Winkelmann*

The classical Mergelyan theorem states that a holomorphic function on a compact set K in ℂ can be approximated by global holomorphic functions provided that the complement K^{c} is connected. We are interested in such approximation phenomena for holomorphic mappings with values in complex manifolds. Of course the target manifold can not be arbitrary. E.g. Liouville’s theorem excludes bounded domains as possible target manifolds. More generally it is easy to see that a Mergelyan theorem can not hold unless the Kobayashi pseudodistance on the target manifold vanishes identically.

Of course the Mergelyan theorem holds for maps into ℂ^{n}. Furthermore if the Hausdorff dimension of a closed set A in ℂ^{n} is less than 2n − 2, then the theorem remains true for maps into ℂ^{n} \ A by a density argument (see prop. 2). However this density argument breaks down in the case where A is a submanifold of real dimension 2n − 2. Thus we have to use different methods to prove a Mergelyan theorem in the case codim_{ℝ}(A) = 2. Theorem. Let K be a compact subset of ℂ with ℂ \ K connected, ϵ > 0 and f : K → ℂ^{2} \ ℝ^{2} a continuous map which is holomorphic in the interior of K.

Then there exists a holomorphic map g : ℂ → ℂ^{2} \ ℝ^{2} with ||f − g||_{K} < ϵ.

Remarks.

i) The only property of ℂ^{2} \ ℝ^{2} we will use is the following: ℂ^{2} \ ℝ^{2} = ℂ^{2} \ (A_{1} × A_{2}) with A_{i} ⊂ ℂ where the sets A_{i} are real–analytic Weierstraß sets (e.g. A_{i} = {h_{i} = 0} where h_{i} are real harmonic functions).

ii) If A is a closed subset in ℂ^{n} such that the Hausdorff dimension of A is less than 2n−2, then Hol(ℂ, ℂ^{n} \ A) is dense in Hol(ℂ, ℂ^{n}) (see prop. 2). Consequently a Mergelyan theorem holds for maps to ℂ^{n} \ A.

iii) If A is a real submanifold of real dimension 2n − 2 in ℂ^{n}, then Hol(ℂ, ℂ^{n} \ A) is not

dense in Hol(ℂ^{n}) (see prop. 3). iv) If A is a complex hypersurface in ℂ^{n} of ”sufficiently high degree”, e.g. a union of n+1

hyperplanes in general position, then a Mergelyan theorem for mappings to ℂ^{n} \ A can not hold, because ℂ^{n} \ A has non–trivial Kobayashi–pseudodistance.

v) Königsberger [4] has shown that a Mergelyan theorem holds for mappings to complex Lie groups. This result has been generalized by Dietmair [2] for mappings to complex manifolds on which a complex Lie group acts transitively. Before beginning the proof, let us fix some notation: If A is a closed set then A(A)

(resp. A(A, Y) where Y is a complex manifold) denotes the set of all continuous functions (resp. maps to Y) which are holomorphic in the interior of A. A^{c} denotes the complement of A.

* Part of this work was carried out during a visit at the mathematical institute of the University of Lille. The author wishes to thank the colleagues there for their hospitality.

1991 AMS Mathematics Subject Classification 30E10, 32E30, 32H99

Our first step in the proof is to reduce the problem from arbitrary compact sets to closure of domains with smooth boundary. Lemma 1. Let K be a compact subset of ℂ with K^{c} connected, and U an open subset of ℂ containing K.

Then there exists an open domain G in ℂ with smooth boundary ∂G such that K ⊂ G ⊂⊂ U. Furthermore Ḡ^{c} = ℂ \ Ḡ is connected.

Proof. For any x ∈ K^{c} and any real curve γ linking x to ∞ inside K^{c} define Ω(x, γ) = inf_{t}(d(γ(t), K)). For x ∈ K^{c} let ω(x) be the supremum of Ω(x, γ) over all such curves γ. Define ω(x) = 0 for x ∈ K. It is easy to check that the function ω has the following properties:

1) ω(x) = 0 if and only if x ∈ K;

2) |ω(x − a) − ω(x)| ≤ |a| for all x, a ∈ ℂ (in particular ω is continuous);

3) ω(x) = d(x, K) for all x which are not contained in the linear-convex hull of K. (Therefore the set {ω ≤ c} is compact for all c).

4) The sets {ω(x) > c} are connected for all c.

Let ν = min{ω(x) : x ∉ U}. (The above listed properties of ω ensure that this minimum exists and is non–zero.) Now let ρ be a convolution of ω with a C^{∞} smoothing kernel χ. We require that supp(χ) is contained in a ball of radius \frac{1}{4}ν around the origin. Due to property 2 this implies |ω(x) − ρ(x)| ≤ \frac{1}{4}ν for all x. The theorem of Sard implies that there exists a δ with \frac{1}{4}ν < δ < \frac{1}{2}ν such that ρ′(x) ≠ 0 for all x with ρ(x) = δ. Now let G_{0} = {ρ < δ} and G be the union of G_{0} with all bounded connected components of ℂ \ G_{0}. Then K ⊂ G_{0} and ω(x) < \frac{3}{4}ν for all x ∈ G_{0}. Since \{ω > \frac{3}{4}ν\} is unbounded and connected, it follows that ω(x) ≤ \frac{3}{4}ν for all x ∈ Ḡ. Hence Ḡ ⊂ U.

Corollary. Let K be a compact subset of ℂ with K^{c} connected, f ∈ A(K, ℂ^{∈} \ ℝ^{∈}) and ϵ > 0.

Then there exists an open domain G in ℂ with smooth boundary ∂G and g ∈ A(⪯ G, ℂ^{∈}\ ℝ^{∈}) such that K ⊂ G and ||f − g||_{K} < ϵ. Furthermore Ḡ^{c} = ℂ \ Ḡ is connected.

Proof. Let g ∈ Hol(ℂ, ℂ^{2}) with ||f − g||_{K} < ϵ. Now apply the lemma with U = g^{−1} (ℂ^{2} \ ℝ^{2}).

Lemma 2. Let G be a bounded open domain in ℂ with smooth boundary, ϵ > 0 and f a holomorphic function on ℂ.

Then there exists a holomorphic function f_{1} on ℂ such that ||f − f_{1}||Ḡ < ϵ and A = f_{1}^{−1}(ℝ) intersects ∂G transversally at any point of intersection.

Proof. Let Θ be a (real differentiable) vector field on ∂G which is nowhere vanishing. Define a map Φ : ℝ^{3} × ∂K → ℝ^{2} by (a, b, c, x) ↦ (Θℜf_{a,b,c}(x), ℜf_{a,b,c}(x)) with f_{a,b,c}(x) = f(x) + (a + ib)z + c. Φ has everywhere maximal rank, hence Φ^{−1}(0, 0) is a two-dimensional submanifold whose projection on ℝ^{3} is a set of measure zero. Thus for appropriate a, b, c the function f_{1} = f_{a,b,c} has the required properties.

Lemma 3. Let G be a bounded domain with smooth boundary and a defining function ρ. Assume that {ρ ≤ a} is compact for all a.

Then there exists a positive number c > 0 such that for all ϵ with 0 < ϵ < c the domain G_{ϵ} = {ρ < ϵ} has smooth boundary.

Proof. Let C be the set of critical points of ρ. C is a closed set. Since {ρ ≤ a} is compact for all a, it follows that ρ(C) is a closed set.

Next we need some auxiliary results on Weierstrass sets. A closed subset A of ℂ is called a Weierstrass set if any function f ∈ A(A) can be approximated uniformly on A by global holomorphic functions. The theorem of Arakelyan [1] states that a closed subset A ⊂ ℂ is a Weierstrass set if and only if the following two conditions are fulfilled:

i) ℂ̄ \ A is connected and ii) ℂ̄ \ A is locally connected at ∞.

Here ℂ̄ = ℂ ∪ {∞} ≃ ℙ_{1} is the one–point compactification of ℂ.

The latter condition is equivalent to each of the following two conditions:

ii’) For every open neighbourhood U of ∞ in ℂ \ A there exists a smaller open neighbourhood V of ∞ such that for all points x ∈ V there exists a real curve inside U linking x with ∞.

ii”) For every compact subset K of ℂ the union B of all bounded connected components of ℂ \ (A ∪ K) is bounded.

Furthermore the criterion of Eilenberg guarantees that condition i) is fulfilled if A is a simply–connected and locally pathwise–connected closed set. Lemma 4. The set A = f^{−1}(ℝ) is simply–connected for all holomorphic maps f : ℂ → ℂ.

Proof. Let G be an arbitrary bounded open domain. The maximum principle implies that ∂G is not contained in A unless f is constant. From this it follows that A is simply– connected.

Lemma 5. Let f : ℂ → ℂ be a holomorphic map. Then the complement A^{c} of A = f^{−1}(ℝ) is locally connected at infinity.

Proof. Due to the preceding lemma A^{c} has no bounded connected components. Hence for any compact set K every bounded connected component of ℂ \ (K ∪ A) meets K. Thus in order to prove the boundedness of the union of all such components, it suffices to show that only finitely many connected components of ℂ \ (K ∪ A) meet K. By enlarging K we may assume that the boundary ∂K is real–analytic. Again by the maximum principle ℑ(f) is not constant on ∂K. The identity principle for real–analytic functions implies that ℑ(f) has only finitely many zeroes. This finishes the proof.

Corollary. For every holomorphic function f the set of real values f^{−1}(ℝ) is a Weierstrass set.

Remark: For the above corollary it is essential that ℝ is a real–analytic Weierstrass set. In general the preimage of an arbitrary Weierstrass set under a holomorphic map is not a Weierstrass set. For instance let H : ℝ_{0}^{+} → ℝ^{2} ≃ ℂ be given by t ↦ (\frac{−t}{t+1}sin^{2}t + 1, e^{−t}) and let A be the union of [0, 1] × {0} with the image of H. Then A is a closed Weierstrass set, but exp^{−1}(A) is not locally connected at infinity. Proposition 1. Let G be a bounded domain with smooth boundary and G^{c} connected, f ∈ A(Ḡ), and ϵ > 0.

Then there exists a holomorphic function f ∈ O(ℂ) such that ||f −f_{1}||Ḡ< ϵ and Ḡ∪A is a Weierstrass set for A = {ℜf_{1} = 0}. Furthermore one can require that each connected component of A \ G intersects ∂G in at most one point.

Proof. Let ρ be a defining function such that G_{δ}^{c} is connected for all 0 < δ. (This is possible, because there is a diffeomorphism of ℂ mapping G onto the unit disk.) Due to the Mergelyan theorem and Lemma 2 there is a function f_{1} ∈ O(ℂ) with ||f − f_{1}||_{G} < \frac{1}{2}ϵ such that A = ℑf_{1} = 0 intersects ∂G tranversally. Let δ_{0} > 0 be such that A intersects {ρ = c} tranversally for all 0 ≤ c ≤ δ_{0}. Let p_{1}, …, p_{m} ∈ S = {ρ = δ_{0}} be choosen such that the connected components of S \{p_{1}, …, p_{m}} contain at most one point x with ℑf_{1}(x) = 0 each. Now let γ_{i} be real disjoint curves in G_{δ_{0}}^{c} linking p_{i} with ∞. We may assume that in a small neighbourhood of ∞ the image of each γ_{i} equals {re^{iθ} : θ = θ_{i}} for some θ_{i}. This implies that ℂ \ L is locally connected at infinity, where L is the union of G_{δ_{0}} with the images of the γ_{i}. Since L is also simply–connected, it follows that L is a Weierstrass set. Let F (x) defined by F (x) = f_{1}(x) for x ∈ G_{δ_{0}} and F (x) = f_{1}(p_{i}) if x ∈ γ_{i}. Let g_{n} be a sequence of holomorphic functions on ℂ which converge uniformly on L to F. Uniform convergence of holomorphic functions implies the uniform convergence of their derivatives. Since F is holomorphic in the interior of L, it follows that for a sufficiently large n the set A_{n} = {ℑg_{n} = 0} intersects ∂G transversally and in as many points as {ℑf_{1} = 0}. Furthermore for n sufficiently large A_{n} has empty intersection with all γ_{i}. From this we can deduce that each connected component of A_{n} \ G has at most one point of intersection with ∂G. This together with Lemma 4. in turn implies that A_{n} ∪ Ḡ is simply–connected. Finally note that A ∪ Ḡ is locally connected at infinity due to Lemma 5.

Proof of the theorem. By Lemma 1 we may restrict our considerations to the case where K = Ḡ for a bounded open domain G with smooth boundary. By the classical Mergelyan theorem there is a holomorphic map F : ℂ → ℂ^{2} with ||f − F ||Ḡ < \frac{1}{2}ϵ and ||f − F ||Ḡ < d(f(Ḡ), ℝ^{2}); the latter inequality implying F (Ḡ) ⊂ ℂ^{2} \ℝ^{2}. Let A = {ℑF_{1} = 0}\G, A_{i} the connected components of A and L the union of Ḡ with A. By the preceding proposition we may assume that L is a Weierstrass set and that each A_{i} intersects ∂G in at most one point. We define a function h ∈ A(L) as follows:

h(x) = \{ \begin{matrix}0 & if x ∈ Ḡ \\ 1 − ℑF_{2}(x) & if x ∈ A_{i} and A_{i} ∩ ∂G = ∅ \\ −ℑF_{2}(x) + ℑF_{2}(q_{i}) & if x ∈ A_{i} and A_{i} ∩ ∂G = \{q_{i}\} \end{matrix}

Now we choose δ > 0 such that δ < d(F (Ḡ), ℂ^{2} \ ℝ^{2}) and δ < |ℑF_{2}(q)| for all q ∈ A ∩ ∂G and δ < min(\frac{1}{2}ϵ, 1). Let H ∈ O(ℂ) with ||H − h||_{L} < δ. Then g = (F_{1}, F_{2} + iH) maps ℂ to ℂ^{2} \ ℝ^{2} with ||f − g||Ḡ < ϵ.

Complements of small sets

Assume that a Mergelyan theorem holds for maps to a given complex manifold X. Let A be a closed subset of X which is small enough to ensure that Hol(ℂ, X \ A) is dense in Hol(ℂ, X). Then obviously there is also a Mergelyan theorem for maps with values in X \ A. Therefore we will now prove such a density result. Proposition 2. Let X be a complex manifold on which a real Lie group G acts transitively by biholomorphic transformations.

Let A be a closed subset of X. Assume that the Hausdorff dimension of A is less than 2n − 2, where n denotes the complex dimension of X. Then Hol(ℂ, X \ A) is dense in Hol(ℂ, X).

Proof. For f ∈ Hol(ℂ, X) and g ∈ G define f_{g}(x) = g · f(x). Define Φ : G × ℂ → X by Φ(g, x) = f_{g}(x). Now Φ is surjective and has everywhere maximal rank. Hence the Hausdorff codimension of Φ^{−1}(A) equals the codimension of A, i.e. it is larger than two. Let p : G × ℂ → G denote the natural projection. Then the Hausdorff dimension of p(Φ^{−1}(A)) is smaller than dim(G) which implies that the complement is dense. Hence for any f ∈ Hol(ℂ, X) there are arbitrarily small g ∈ G with f_{g}(ℂ) ⊂ X \ A.

Remark. The homogeneity assumption is quite essential, e.g. consider X = {(z, w) : |zw| < 1, |z| < 1} and A = {(0, 0), (0, 1)}.

However the Mergelyan theorem for ℂ^{2} \ ℝ^{2} can not be deduced in this way. Proposition 3. Let A be a closed subset in ℂ^{n}. Assume that there exists a point a ∈ A with an open neighbourhood U in ℂ^{n} such that U ∩ A is a real submanifold of real codimension two.

Then Hol(ℂ, ℂ^{n} \ A) is not dense in Hol(ℂ, ℂ^{n}).

Proof. We may replace U by a smaller open subset. Hence we may assume that π_{1}(U \A) ≃ ℤ. Furthermore there exists a complex–linear map λ : ℂ → ℂ^{n} with λ(0) = a and λ(Δ̄_{1}) ⊂ U such that λ_{∗} : π_{1}(∂Δ_{1}) → π_{1}(U \ A) is an isomorphism. If a sequence f_{n} ∈ Hol(ℂ, ℂ^{n}\A) converges to λ, then f_{n}(Δ̄_{1}) ⊂ U for almost all n. Thus f_{n}(Δ̄_{1}) ⊂ U \A for almost all n. This however implies that f_{n}(∂Δ_{1}) is contractible inside U \ A which leads to a contradiction.

Remark. For the proof we only need the existence of a holomorphic map λ with prescribed values λ(0) and dλ(0). References

[1] Arakelyan, N.U.: Uniform approximation on closed sets by entire functions. Izv. Akad. Nauk SSSR Ser. Math. 28, 1187–1206 (1964)

[2] Dietmair, Th: The Mergelyan–Bishop theorem in the case of mappings into homogeneous manifolds. Math. Ann. 295, 505–512 (1993)

[3] Gaier, D.: Lectures on complex approximation. Birkhäuser Boston Basel Stuttgart. (1987)

[4] Königsberger, K.: Der Satz von Mergelyan für Abbildungen in eine Liegruppe. Festschrift zum 60. Geburtstag von Prof. Dr. E. Thoma 1986

[5] Mergelyan, S.N.: On the representation of functions by series of polynomials on closed sets. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 78, 405–408 (1952) English translation: Transl. A.M.S. 3, 287–293 (1952)

Jörg Winkelmann

Math. Institut NA 4/64 Ruhr-Universität Bochum 44780 Bochum Germany JW@CPLX.RUHR-UNI-BOCHUM.DE