Families of Proper Holomorphic Embeddings and Carleman-type Theorems with parameters

We solve the problem of simultaneously embedding properly holomorphically into $\Bbb C^2$ a whole family of $n$-connected domains $\Omega_r\subset\Bbb P^1$ such that none of the components of $\Bbb P^1\setminus\Omega_r$ reduces to a point, by constructing a continuous mapping $\Xi\colon\bigcup_r\{r\}\times\Omega_r\to\Bbb C^2$ such that $\Xi(r,\cdot)\colon\Omega_r\hookrightarrow\Bbb C^2$ is a proper holomorphic embedding for every $r$. To this aim, a parametric version of both the Anders\'en-Lempert procedure and Carleman's Theorem is formulated and proved.


Introduction
Existence of proper holomorphic embeddings of Riemann surfaces R into 2-dimensional complex manifolds X, e.g., X = C 2 , with prescribed geometrical properties, e.g., being complete, has been an active area of research over the recent years. Various techniques have been developed, but in several cases, positive results have been obtained only at the cost of perturbing the complex structure of R (seeČerne-Forstnerič [4], Alarcón [1] and Alarcón-López [2]). It can be hoped, however, that if you let r be a local parameter on the moduli space of Riemann surfaces of a given type, and you perform various constructions continuously with the parameter r near a given point r 0 , then you will get a perturbation of the complex structure for each given r, but at least one perturbation will correspond to your initial r 0 . Indeed this is the philosophy behind the embedding results of Globevnik-Stensønes [7]. The purpose of this article is to take a first step towards results of this type that may be generalized to larger classes of Riemann surfaces.
We will consider the following. It is known that any n-connected domain Ω in the Riemann sphere may be mapped univalently onto a domain in the Riemann sphere whose complement consists of n parallel disjoint slits with a given inclination Θ to the real axis. The univalent map achieving this is uniquely determined by Θ and the choice of a certain normalization of the Laurent series expansion at a chosen point ζ ∈ Ω being sent to ∞ (see Goluzin,[8], page 213). Considering a continuous family of n-connected domains, we obtain a continuously varying family of uniformizing slit-maps.
Let C j ⊂ C be compact disks and I j ⊂ R >0 be compact intervals, j = 1, . . . , n. Set B j := C j × I j and B := B 1 × · · · × B n . Let r = ((a 1 , b 1 ), . . . , (a n , b n )) denote the coordinates on B, and letting l r,j denote the closed straight line segment which is parallel to the real axis with right end-point a j (r) and of length b j (r), we assume that L r := {l r,1 , . . . , l r,n } is a set of pairwise disjoint slits, and thus P 1 \ L r is an n-connected domain, none of whose boundary components are isolated points. After possibly having to apply the map z → (z −a 1 (r))/b 1 (r) we may assume that for all r we have that l r,1 = [−1, 0] ⊂ C.
The goal is to prove the following. Then there exists a continuous map Ξ : Ω → C 2 such that for each r ∈ B we have that Ξ(r, ·) : Ω r → C 2 is a proper holomorphic embedding.

The Setup
We will now introduce a setup to prove Theorem 1.1. First, we need the notion of a certain directed family of curves.
Let C > 0 and R > 1. Let Γ denote the half line Γ = {x ∈ R ⊂ C : x ≥ R − 1}, let B ⊂ R m be a compact set, and denote by (r, x) the coordinates on B × Γ. Let h, h ′ = ∂h ∂x ∈ C (B × Γ), and assume that Definition 2.1. Let θ ∈ [0, 2π). Then the set of curves is referred to as being θ-directed, and subordinate to R, C. A family of curves is said to be θ-directed if it is θ-directed subordinate to R, C for sufficiently large R, C.
With the notation in the previous section, set ψ(z) := 1 z + 1, λ r,j := ψ(l r,j ), c j (r) := ψ(a j (r)). Then Λ r := {λ r,1 , . . . , λ r,n } is a set of disjoint slits in P 1 , where λ r,1 is the negative real axis and λ r,j are circular slits (or possibly straight line segments along the real axis) for j = 2, . . . , n. We set e iθ r,j := ψ ′ (a j (r))/|ψ ′ (a j (r))|, i.e., we have that e iθ r,j is a unit tangent to the circle Λ r,j on which λ r,j lies at the point c j (r). Setting α r,j (z) := e −iθ r,j (z − c j (r)) we have that α r,j (Λ r,j ) is a circle which is tangent to the real axis at the origin, and we let κ r,j denote the signed curvature of this circle; positive if the circle is in the upper half plane, negative if the circle is in the lower half plane, and zero if the circle is the real axis.
Proof. It suffices to prove this for θ = 0. Then α r,j (Λ r,j ) is parametrized near the origin by Setg r,j (z) = g r,j (α −1 r,j (z)) We have that Since g r,j (z) is close to a constant when z is close to c j (r), the uniform bound in the definition of (−π)-directed holds. Now where v r,j (x) is bounded and scaling it to have almost unit length we see Choose δ > 0 small, and let a, b ∈ △ δ (1/ √ 2), and set A a,b (z, w) := (az + bw, −bz + aw). Write a = r a e iϑa , b = r b e iϑ b . Then the family Γ 1 defined by Γ 1 = {π 1 • A a,b • φ r (λ r,1 ) : r ∈ B} is (ϑ a − π)-directed, and each family Γ j , j = 2, . . . , n, defined by Proof. For j = 2, . . . , n this is just Proposition 2.1 since for any fixed j we have that π 1 • A a,b • φ r (λ r,j ) is parametrized by For j = 1 this is because π 1 • A a,b • φ r (λ r,j ) is parametrized by r a e iϑa z + g r (z) where g r (z) is uniformly comparable to 1 z .

Carleman approximation with parameters
We will start by introducing some notation. Afterwards, we present Theorem 3.1, a Carlemantype theorem (see e.g., [5]), which is the main result of the present section: families of smooth functions holomorphic on a disc can be approximated by entire functions on a smaller disc and on the union of several Lipschitz curves. The proof is obtained applying inductively Corollary 3.1, which in turn easily follows from Proposition 3.1, a tool that allows to approximate smooth functions on compact pieces of a Lipschitz curve; Corollary 3.1 extends the result to several curves. Proposition 3.1 relies on three technical lemmata that will be presented in Section 3.3.
for some C > 0, for every (r, x) ∈ B × Γ, and every k = 1, . . . , n. Then, setting l = 1/2, we have that so h k is l-Lipschitz and in this way we also call its graph. Let 0 = θ 1 < θ 2 < · · · < θ n < 2π and define the Lipschitz curves and their union If D ⊆ Ω ⊆ C are domains, a useful notation is given by setting  . Assume that f ∈ P(B, C, △ ρ+3+ 3C 2 ) for some ρ > R. Then for any ǫ ∈ C (C), ǫ > 0, there exists g ∈ P(B, C) such that 3.2. Proof of Theorem 3.1. Fix j ∈ N, j ≥ R and let b be some real number such that For ρ ≥ C set ψ(ρ) := arcsin C ρ and define Then S ρ is the wedge in the right half-plane bounded by the straight lines passing through the origin and the intersection between ∂△ ρ and the lines y = ±C. Up to consider a larger R, we assume e iθ j S ρ ∩ e iθ k S ρ = ∅ for all j = k for ρ ≥ R. We define further the following sets Given δ > 0, we will denote the open δ-neighborhood of D as The following proposition, or rather its corollary below, is the main technical ingredient in the proof of the Carleman Theorem 3.1. The proposition follows from Lemma 3.1, Lemma 3.2, and finally Lemma 3.3 below.
Proof. On e −iθ k Γ k,r define α k (r, z) := α(r, e iθ k z). Using the proposition we obtain approximations Q t,k (r, z). Then setting will yield the result for sufficiently small t.
Proof of Theorem 3.1: The proof is by induction on k ≥ 0, and the induction hypothesis is the following. For every j = 0, . . . , k there exist: We start by setting g 0 := f ; then in the case k = 0 we see that (i), (ii) hold, and (iii) is void. Assume now that the induction hypothesis holds for some k ≥ 0. Fix η > 0 such that and choose a cutoff function . Now g k may be approximated on △ η+ρ+k+3+ 3C 2 to arbitrary precision by h k ∈ C (B)[z] using Taylor series expansion, and so approximates g k to arbitrary precision. Hence it suffices to approximate α k to arbitrary precision by a suitable function. Multiplying α k by a suitable cutoff function so that Corollary 3.1 applies, we have that α k may be approximated to arbitrary precision on r Γ r ∩ △ ρ+k+2+3+ 3C 2 by a function Q k ∈ P(B, C) which is arbitrarily small on △ ρ+k . Setting then g k+1 , completes the induction step. We may finish the proof of Theorem 3.1 by setting g := lim j→∞ g j , which exists by (iii), and the approximation holds by (ii).
3.3. Lemmata: Mergelyan-type and Runge's Theorems with parameters. The three lemmata we present and prove in this section are fundamental ingredients to formulate a Mergelyan-type Theorem (see e.g., [5]). The first one of them generalizes a theorem proved by P. Manne in his Ph.D. thesis [11] and is about the holomorphic (entire) approximation of a family of smooth functions, each of which is defined on a Lipschitz curve in the complex plane.
Proof. Extend h 1 to a function h on the whole real line by setting h(r, is the Gaussian kernel. We start by proving (8). (8) follows. For any fixed η > 0 we split S r as we immediately get the following upper bound: which holds for every z ∈ Γ r , r ∈ B and t > 0. Similarly, one sees that for all ǫ > 0, η > 0 there exists t 0 > 0 such that for every z ∈ Γ r , r ∈ B, 0 < t < t 0 . We need one last property of the kernel, that is for all z ∈ Γ r , r ∈ B and t > 0. Let us consider the function which is holomorphic entire. Let z = x ∈ R and define for T > 0 and let ρ ± r (T ) be the straight line segment between ±T and ±T + ih(r, ±T ). Set which is a piecewise C 1 -smooth closed curve which is nullhomotopic, hence we get for every t > 0, r ∈ B, x ∈ R and T > 0. On the other hand Passing to the limit as T → +∞, the vertical contributions vanish (as h is bounded), while for every t > 0, r ∈ B and x ∈ R. This implies that the entire function F is identically 1 on the real line for every t > 0 and r ∈ B, so by the identity principle it is constantly 1 on the whole C; in particular (11) holds true.
We gathered all the ingredients to prove (7). Let ǫ > 0, let η > 0 such that |α(r, ζ)−α(r, z)| < ǫ for all z ∈ Γ r , ζ ∈ S (1) r , for all r ∈ B. Then where the second equality follows from (11) and the last inequality follows from (9) and (10). So we can conclude, since this last quantity can be taken arbitrarily small for ǫ small, uniformly in z ∈ Γ r and r ∈ B, for all 0 < t < t 0 , where t 0 comes from (10).
The following Lemma shows how to modify the approximation constructed in Lemma 3.1, so that, besides approximating the given smooth function, it becomes arbitrarily small on a suitable region. The price to pay is that the approximation obtained this way is no more entire; we will get "entireness" back with Lemma 3.3, that is a parametric version of Runge's Theorem (see e.g., [5]). Lemma 3.2. Assume n = 1, and let α be as in Proposition 3.1. Then for every ǫ > 0 there exists {ξ t } t>0 ⊂ P(B, Ω(δ)) such that for every r ∈ B, 0 < t < t 0 , and ξ t → 0 as t → 0 (13) uniformly on B × ω 2 (δ), for some δ > 0.
Lemma 3.3 (Runge-type Theorem with parameters). With the notation of the previous lemma, for all ǫ > 0, there is Q t ∈ C (B)[z] (polynomial with coefficients in C (B); in particular {Q t } t≥0 ⊂ P(B, C)) such that ξ t − Q t B×Ω < ǫ for all t > 0, r ∈ B.

Andersén-Lempert Theory
We will now apply Andersén-Lempert Theory in B × C 2 . We have that B ⊂ R N , and when talking about analytic properties of sets and functions on B × C 2 we will think of B × C 2 ⊂ C N × C 2 , C N = R N + iR N . For instance, by saying that K ⊂ B × C 2 is polynomially convex compact we mean polynomially convex in C N × C 2 ; this is, in fact, equivalent to K r being polynomially convex in {r} × C 2 for each r ∈ B.
With the setup introduced in Section 2 we now set s r,j := φ r (λ r,j ) and we set S r := n j=1 s r,j .
Proposition 4.1. Let K ⊂ (B × C 2 ) \ S be a compact set such that K is polynomially convex. Let T > 0, and let ǫ > 0. Then there exists a continuous map g : B × C 2 → C 2 such that the following hold for all r ∈ B.
Then η N,t is an isotopy of injective holomorphic maps near the real line in the z-plane, and leaves the real line invariant, fixing 0. We see that Note that for any s ′ > s we have that lim N →∞ η N,t = Id uniformly on {Re(z) ≥ s ′ }. Now let σ N,t be the inverse isotopy to η N,t , i.e., σ N,t = η N,1−t •η −1 N,1 ; it is injective holomorphic near the real line in the z-plane, and by choosing N large, may be extended, arbitrarily close to the identity, to any set {Re(z) ≥ s ′ } for s ′ > s.
We may now define ψ r,t (·) on s r,j by ψ r,t := F r,j • σ N,t • F −1 r,j . The claims of the lemma are satisfied by choosing N large, and δ sufficiently small. Remark 4.1. If δ is sufficiently small and ǫ further sufficiently small we get that is polynomially convex. Observe first that K r ∪ S r (T ′ , T ′′ ) is polynomially convex, since K r is, and S r (T ′ , T ′′ ) is a collection of disjoint arcs. For a sufficiently small δ ′ it is known that the tube S r (T ′ , T ′′ )(δ ′ ) is polynomially convex, and for sufficiently small δ ′ we have that K r ∪S r (T ′ , T ′′ )(δ ′ ) is polynomially convex. Then if δ < δ ′ and we consider ψ t,r as in the lemma with δ ′ instead of δ, if ǫ is small enough we get our claim, since the δ, δ ′ may be chosen independently of r.
Proof of Proposition 4.1: Fix 0 < δ << 1. For each η ∈ δB 2 we set v η = (0, 1) + η, and we let π η denote the orthogonal projection onto the orthogonal complement of v η . After applying the linear transformation it follows from Proposition 2.2 that the family π η (s r,1 ) is (ϑ 1,η −π)-directed and that the families π η (s r,j ) are (ϑ 2,η + θ j − π)-directed, where the ϑ j,η 's vary continuously with η. From now on we will assume that have applied the transformation A without changing the notation for all sets considered above.
By increasing T > 0 we may assume that K r ⊂ T B 2 for all r, and we fix R as in Theorem 3.1 such that π η (T B 2 ) ⊂ △ R , and choose T ′ < T ′′ such that π η (S r (T ′ , T ′′ )) ⊂ C \ △ R+3+ 3 2 C for all r and all η.
Let ψ t be the isotopy from Lemma 4.1, extended to be the identity on some neighborhood of K which we regard as being included in U . On ψ t 0 (U ) we define the vector field X t 0 (ζ) = d dt t=t 0 ψ t (ψ −1 t 0 (ζ)) (here ζ = (r, x) = (r, z, w)). The goal is to follow the standard Andersén-Lempert procedure parametrically for approximating the flow of the time dependent vector field X t by compositions of flows of complete fields, but to modify these so that they do not move S \ S(0, T ′′ ). The proof is the same as the corresponding proof in [10] where this was done without parameters, but we include here a sketch and some additional details. The reader is assumed to be familiar with the Andersén-Lempert-Forstnerič-Rosay construction.
Step 1: We will find flows σ r,j (t, x), j = 1, . . . , m, such that the composition σ r,m • · · · • σ r,1 (t, x) approximates ψ t . The flows are of two forms: Step 2: The plan is then roughly to find a family of cutoff functions χ j ∈ C ∞ (C 2 ), 0 ≤ χ j ≤ 1, such that χ j ≡ 1 near T ′ B 2 and χ j ≡ 0 near C 2 \ T ′′ B 2 , and definẽ in such a way that all compositionsσ (j) r :=σ r,j • · · · •σ r,1 are as close to the identity as we like in C 1 -norm on S r (T ′ , T ′′ )(δ/2). Note thatσ r,j = σ r,j on T ′ B 2 andσ r,j = Id outside T ′′ B 2 . In particular the families π j (σ(j) r (S r )) are as close as we like to the original families π j (S r ) and identical outside some compact set.
Step 4: Approximate the coefficients c r,j in the sense of Carleman using Theorem 3.1.
We now include some estimates explaining why the above scheme works (see also [10] where the construction is done without dependence of parameters).
Choose χ ∈ C ∞ (C 2 ) nonnegative such that χ ≡ 1 near T ′ B 2 and χ ≡ 0 near C 2 \ T ′′ B 2 . We may assume that the vector fields X r,t satisfy X r,t < α on S(T ′ , T ′′ )(δ) for any small α > 0. Thus, freezing the vector field at time i/N to obtain a vector field X i with a flow γ i r,t , we have that γ i r (t/N, x) − x ≤ (t/N )α on S r (T ′ , T ′′ )(2δ/3). Thus, we may assume that the compositions γ i r,t/N • · · · • γ 1 r,t/N exist and remain arbitrarily close to the identity on S r (T ′ , T ′′ )(δ/2) for i < N .
By Remark 4.1 we may approximate each vector field X i r to arbitrary precision by a polynomial vector field, which we will still denote by X i r . By the parametric Andersén-Lempert observation, see Lemma 4.9.9 in [6] and the proof of Theorem 2.3 in [9], the family of vector fields X i r may be written as a sum of shear and over-shear vector fields . It is known that the composition (Θ N r,t/nN ) n • · · · • (Θ 1 r,t/nN ) n (21) converges to the flow of X r,t on K r ∪ S r as N and then n tends to infinity. Our first goal is to replace the maps σ i r,j (t, x) by maps x + χ j (x)b i r,j (t, x)v j with χ j (x) = 1 for |x| ≤ T ′ and χ j (x) = 0 for |x| ≥ T ′′ in the composition (21) or partial compositions of it, and show that we still get maps that are close to the identity on S r (T ′ , T ′′ )(δ/2). The compositions thus obtained will remain the same on {|x| < T ′ } and be the identity map outside {|x| ≤ T ′′ }.
We will now modify the flows on S r (T ′ , T ′′ )(δ). We have that f i r (t, x) ≤ 2αt for t sufficiently small. If we setΘ We decompose in a natural way then the h r,j (t, ·) go to zero as t → 0. Now for large n definẽ σ r,1 (t/n, x) = x + χ 1 (x)h r,1 (t/n, x)v 1 , and by inductioñ σ r,j+1 (t/n, x) = x + χ j+1 (σ(j) −1 r (t/n, x))h r,j+1 (σ(j) −1 r (t/n, x))v j+1 . Thenσ i r,m,t/n • · · · •σ i r,1,t/n (x) =Θ i r (t/n, x), and we get that (Θ N r,t/N n ) n • · · · • (Θ 1 r,t/N n ) n − Id ≤ 2αt, and corresponding estimates hold for partial compositions. Note that this shows that S r (T ′ , T ′′ )(δ/2) remains in S r (T ′ , T ′′ )(2δ/3) where we may assume that the C 1 -norms of the f i r 's are arbitrarily small, and so by arguments similar to those above we get that all partial compositions are close to the identity in C 1 -norm, which will allow us to use the implicit function theorem to rewrite as in Step 3 above.
Finally, Step 4 is carried out exactly as in [10].
Let S r and S be the sets defined in Section 4 and set X r := φ r • ψ(P 1 \ L r ) = φ r (P 1 ) \ S r which is a 1-dimensional complex manifold with boundary ∂X r = S r . Define C r j := P 1 \ L r (1/j), such that {C r j } ∞ j=1 is a normal exhaustion of P 1 \ L r by O(P 1 \ L r )-convex compact sets. It follows that K r j := φ r • ψ(C r j ), j ≥ 1 is a normal exhaustion of X r by O(X r )-convex compact sets. The proof of Proposition 1 in [12] ensures the following two crucial facts: (i) K r j are polynomially convex, and (ii) given any K ⊂ C 2 \ S r compact polynomially convex, the set K ∪ K r j is polynomially convex for any j large enough . We construct now inductively a sequence of continuous mappings, whose continuous limit h : r∈B ({r} × X r ) → C 2 will be a fiberwise proper holomorphic embedding that we will compose with suitable mappings to prove the statement. Proposition 4.1 provides a continuous g 1 : B × C 2 → C 2 such that, for every r ∈ B • g 1 (r, ·) ∈ Aut C 2 , and • g 1 (r, S r ) ⊂ C 2 \ 1B 2 .