Families of Proper Holomorphic Embeddings and Carleman-Type Theorem with parameters

The Journal of Geometric Analysis (2023) 33:75 https://doi.org/10.1007/s12220-022-01110-y Families of Proper Holomorphic Embeddings and Carleman-Type Theorem with parameters Giovanni Domenico Di Salvo1 · Tyson Ritter2 · Erlend Fornæss Wold1 Received: 4 January 2022 / Accepted: 5 October 2022 © The Author(s) 2022 Abstract We solve the problem of simultaneously embedding properly holomorphically into C2 a whole family of n-connected domains r ⊂ P1 such that none of the components of P1 \ r reduces to a point, by constructing a continuous mapping  : r {r } × r → C2 such that (r , ·) : r → C2 is a proper holomorphic embedding for every r . To this aim, a parametric version of both the Andersén–Lempert procedure and Carleman’s Theorem is formulated and proved. Keywords Proper holomorphic embedding · Approximation theory · Andersén–Lempert theory · Several complex variables 1 Introduction Existence of proper holomorphic embeddings of Riemann surfaces R into 2dimensional complex manifolds X , e.g., X = C2 , 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 (see Černe–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 r0 , then you will get a perturbation of the complex structure for each given r , but at least one perturbation will correspond to your initial r0 . Indeed B Giovanni Domenico Di Salvo Tyson Ritter Erlend Fornæss Wold 1 University of Oslo, Postboks 1337 Blindern, 0316 Oslo, Norway 2 University of Stavanger, Postboks 8600, 4036 Stavanger, Norway 0123456789().: V,-vol 123 75 Page 2 of 19 G. D. Di Salvo et al. this is the philosophy behind the embedding results of Globevnik–Stensønes [7]. The purpose of this article is to take a first step toward 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  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  to the real axis. The univalent map achieving this is uniquely determined by  and the choice of a certain normalization of the Laurent series expansion at a chosen point ζ ∈  being sent to ∞ (see Goluzin [8, p. 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 := B1 × · · · × Bn . Let r = ((a1 , b1 ), . . . , (an , bn )) denote the coordinates on B, and letting lr , 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 := {lr ,1 , . . . , lr ,n } is a set of pairwise disjoint slits, and thus P1 \ L r is an n-connected domain, none of whose boundary components are isolated points. After possibly having to apply the map z → (z − a1 (r ))/b1 (r ), we may assume that for all r we have that lr ,1 = [−1, 0] ⊂ C. The goal is to prove the following. Theorem 1.1 In B × P1 set  = (B × P1 ) \   {r } × L r . r ∈B Then, there exists a continuous map  :  → C2 such that for each r ∈ B, we have that (r , ·) : r → C2 is a proper holomorphic embedding. 2 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  denote the half line  = {x ∈ R ⊂ C : x ≥ R − 1}, let B ⊂ Rm be a compact set, and denote by (r , x) the coordinates on B × . Let h, h  = ∂∂hx ∈ C (B × ), and assume that |h(r , x)| <  1 C   , h (r , x) < . 2 2 Definition 2.1 Let θ ∈ [0, 2π ). Then, the set of curves eiθ · {x + i h(r , x) : r ∈ B, x ∈ } 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. 123 Families of Proper Holomorphic Embeddings Page 3 of 19 75 With the notation in the previous section, set ψ(z) := 1z + 1, λr , j := ψ(lr , j ), c j (r ) := ψ(a j (r )). Then, r := {λr ,1 , . . . , λr ,n } is a set of disjoint slits in P1 , where λr ,1 is the negative real axis and λr , j are circular slits (or possibly straight line segments along the real axis) for j = 2, . . . , n. We set eiθr , j := ψ  (a j (r ))/|ψ  (a j (r ))|, i.e., we have that eiθr , j is a unit tangent to the circle r , j on which λr , j lies at the point c j (r ). Setting αr , j (z) := e−iθr , j (z − c j (r )) we have that αr , j ( r , j ) is a circle which is tangent to the real axis at the origin, and we let κr , 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. Proposition 2.1 Fix j ∈ {2, . . . , n} and suppose that gr , j ∈ O( tinuous family of functions, for r ∈ B. Let θ ∈ [0, 2π ), and set ϕ j (r , z) := δ (c j (r ))) is a con- eiθ + gr , j (z). αr , j (z) Then, the family  j of curves ϕ(r , λr , j ) is (θ − π )-directed. Proof It suffices to prove this for θ = 0. Then αr , j ( origin by ηr , j (x) = x + i r , j ) is parametrized near the κr , j 2 x + O(x 4 ). 2 Set g̃r , j (z) = gr , j (αr−1 , j (z)) We have that 1 ϕ j (r , x) = + g̃r , j (ηr , j (x)) κ x + i r2, j x 2 + O(x 4 ) κ x − i r2, j x 2 + O(x 4 ) + g̃r , j (ηr , j (x)) = 2 4 x + O(x )  κr , j 1 −i + O(x 2 ) (1 + O(x 2 )) + g̃r , j (ηr , j (x)) = x 2 κr , j 1 = −i + O(x) + g̃r , j (ηr , j (x)). x 2  Since gr , 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 ϕ j (r , x) = −1 + vr , j (x), x2 where vr , j (x) is bounded and scaling it to have almost unit length we see x 2 ϕ j (r , x) = −1 + x 2 vr , j (x). 123 75 Page 4 of 19 G. D. Di Salvo et al. Proposition 2.2 Fix θ2 , . . . , θn ∈ (0, 2π ). Define φr : C \ {c2 (r ), . . . , cn (r )} → C2 by ⎛ ⎞ n iθ j e ⎠. φr (z) := ⎝z, αr , j (z) j=2 √ Choose δ > 0 small, and let a, b ∈ δ (1/ 2), and set Aa,b (z, w) := (az+bw, −bz+ aw). Write a = ra eiϑa , b = rb eiϑb . Then, the family 1 defined by 1 = {π1 ◦ Aa,b ◦ φr (λr ,1 ) : r ∈ B} is (ϑa − π )-directed, and each family  j , j = 2, . . . , n, defined by  j = {π1 ◦ Aa,b ◦ φr (λr , j ) : r ∈ B} is (ϑb + θ j − π )-directed. Proof For j = 2, . . . , n this is just Proposition 2.1 since for any fixed j we have that π1 ◦ Aa,b ◦ φr (λr , j ) is parametrized by rb ei(ϑb +θ j ) + αr , j (z)  k= j rb ei(ϑb + (...truncated)