Embedded complex curves in the affine plane

This paper brings several contributions to the classical Forster-Bell-Narasimhan conjecture and the Yang problem concerning the existence of proper and almost proper (hence complete) injective holomorphic immersions of open Riemann surfaces in the affine plane $\mathbb C^2$ satisfying interpolation and hitting conditions. We also show that in every compact Riemann surface there is a Cantor set whose complement admits a proper holomorphic embedding in $\mathbb C^2$. The focal point is a lemma saying the following. Given a compact bordered Riemann surface, $M$, a closed discrete subset $E$ of its interior $\mathring M=M\setminus bM$, a compact subset $K\subset \mathring M\setminus E$ without holes in $\mathring M$, and a $\mathscr C^1$ embedding $f:M\hookrightarrow \mathbb C^2$ which is holomorphic in $\mathring M$, we can approximate $f$ uniformly on $K$ by a holomorphic embedding $F:M\hookrightarrow \mathbb C^2$ which maps $E\cup bM$ out of a given ball and satisfies some interpolation conditions.


INTRODUCTION AND MAIN RESULTS
This paper contributes to the following three interesting topics in global complex geometry, having the main focus on the interrelationship between them: • The classical Forster-Bell-Narasimhan conjecture (see [19,11]) asking whether every open Riemann surface admits a proper holomorphic embedding in C 2 .The general case is still open; for positive results, see the surveys in [23,] and [16].• Yang's problem [50,51] concerning the existence of complete bounded complex submanifolds of C n ; see the up-to-date comprehensive survey [2].• The existence of dense holomorphic curves in complex manifolds; see Forstnerič and Winkelmann [47,21] and [6,Section 10].
The focal point of the paper is the following lemma, which is proved in Section 2.
Lemma 1.1.Let M be a compact bordered Riemann surface with boundary of class C s for some s > 1.Given a C 1 embedding f : M ֒→ C 2 which is holomorphic on M = M \ bM, a compact set K ⊂ M without holes, a compact polynomially convex set L ⊂ C 2 such that f (M \ K) ∩ L = ∅, finite sets A = {α 1 , . . ., α l } ⊂ M \ K and B = {β 1 , . . ., β l } ⊂ C 2 \ L, a closed discrete set E ⊂ M such that E ∩ (A ∪ K) = ∅, and numbers ǫ > 0 and r > 0, there is a holomorphic embedding F : M ֒→ C 2 satisfying the following conditions: (d) F agrees with f to a given finite order at a given finite set of points in K \ f −1 (B).
We may assume that the surface M in the lemma is a closed domain in a compact Riemann surface, R, whose boundary bM consists of real analytic Jordan curves, and M has no connected component without boundary.(See Stout [43,Theorem 8.1] and note that any conformal diffeomorphism of M onto such a domain is of class C 1 (M); see [7,Theorem 1.10.10].)A map F : M → C 2 is said to be holomorphic if it extends to a holomorphic map from an open neighbourhood of M in the ambient Riemann surface.A hole of a compact set in an open surface is a relatively compact connected component of its complement.See Remark 1.11 concerning the validity of the hypotheses of the lemma.Lemma 1.1 is based on techniques developed by Wold [48,49] and Forstnerič and Wold [28] for constructing proper holomorphic embeddings of bordered Riemann surfaces in C 2 .In their constructions, some boundary points of the given surface M are sent to infinity where they remain at all subsequent steps.Our proof of Lemma 1.1 uses a similar construction with additional precision, but the points at infinity are finally brought back to C 2 .Thus, the main novelty of Lemma 1.1, and of the related Lemma 4.2, is that we keep the entire Riemann surface M as an embedded complex curve in C 2 while pushing its boundary and the discrete set E ⊂ M (or a countable family of discs in Lemma 4.2) arbitrarily far towards infinity.
These two lemmas lead to new existence, approximation, interpolation, and hitting theorems for complete injectively immersed complex curves in C 2 , or in domains of C 2 , satisfying additional global conditions such as being proper, almost proper, or dense, which are presented in the sequel.At the same time, they give a simpler proof of a number of known results.Our lemmas reduce proofs of these applications to formal induction schemes without the need of dealing with the technical issues at every step.A similar role is played by the Riemann-Hilbert method (see [24,17] and [7,Chapter 6]), but when the target is a complex surface that technique, unlike ours, tends to introduce self-intersections which cannot be removed since a generic immersion from a Riemann surface has transverse double points.
As a first illustration of the constructions that can be carried out using Lemma 1.1, we establish the following interpolation result for almost proper injective holomorphic immersions of bordered Riemann surfaces in C 2 .It is proved in Section 3.
Theorem 1.2.Let M be a compact bordered Riemann surface with C s boundary for some s > 1, and let f : M ֒→ C 2 be a C 1 embedding which is holomorphic on M .Given a compact set K ⊂ M without holes, a discrete sequence α j ∈ M \ K without repetition which clusters only on bM, a sequence β j ∈ C 2 without repetition, and a number ǫ > 0, there is an almost proper injective holomorphic immersion F : M ֒→ C 2 satisfying the following conditions: (ii) F agrees with f to a given finite order at a given finite set in K\f −1 ({β j : j = 1, 2, . ..}). (iii) F (α j ) = β j for all j = 1, 2, . ... In particular, F can be chosen such that F ( M) is every dense in C 2 .
If the sequence β j ∈ C 2 is closed and discrete, then there is a proper holomorphic embedding F : M ֒→ C 2 satisfying conditions (i)-(iii).
Recall that a continuous map f : X → Y of topological spaces is said to be almost proper if for every compact set K ⊂ Y the connected components of f −1 (K) are all compact.Given a compact bordered Riemann surface M as in Theorem 1.2, one cannot hit an arbitrary countable subset of C 2 by proper holomorphic maps M → C 2 , but this can be done by almost proper maps.In fact, almost proper maps are in some sense the best class of holomorphic maps M → C 2 that can hit any given countable subset of C 2 .
The last part of Theorem 1.2 implies the following known result concerning the Forster-Bell-Narasimhan Conjecture; see Globevnik [30] in the case of the disc, and [28, Corollaries 1.2 and 1.3] by Forstnerič and Wold and [35,Theorem 1] by Kutzschebauch et al. for an arbitrary M. We state it with additional precision concerning approximation and interpolation.
Corollary 1.3.Given a holomorphic embedding f : M ֒→ C 2 of a compact bordered Riemann surface M, a compact set K ⊂ M without holes, and closed discrete sequences α j ∈ M and β j ∈ C 2 without repetitions such that K ∩ {α j : j ∈ N} = ∅, we can approximate f uniformly on K by proper holomorphic embeddings F : M ֒→ C 2 satisfying F (α j ) = β j for all j = 1, 2, . ... We give a unified proof of Theorem 1.2 and Corollary 1.3, based on Lemma 1.1, and we supply some details related to [35,Lemma 2.2]; see Remark 2.1.The analogous result for algebraic curves in C 2 is a special case of [25,Theorem 1.3]; see also [23,Theorem 4.17.1].
Recall that an immersed submanifold ϕ : Z → R n is said to be complete if the Riemannian metric on Z, obtained by pulling back the Euclidean metric on R n via ϕ, is a complete metric; equivalently, for every proper path γ : [0, 1) → Z the path ϕ • γ : [0, 1) → R n has infinite Euclidean length.It is obvious that every almost proper immersion ϕ : Z → R n is complete, any hence the immersions F provided by Theorem 1.2 are complete.It seems that this gives the first examples of a specific bordered Riemann surface, other than the disc, admitting a complete nonproper injective holomorphic immersion into C 2 .The construction of complete injectively immersed complex lines C ֒→ C 2 was given by the authors in [5].
The technique of bringing back the points at infinity also applies in conjunction with [29,Lemma 3.1], thereby yielding an analogue of Lemma 1.1 for circle domains with countably many boundary components in the Riemann sphere CP 1 ; see Lemma 4.2.This gives a simpler proof of the theorem of Forstnerič and Wold [29,Theorem 1.1] saying that every circle domain in CP 1 embeds properly holomorphically in C 2 ; see Theorem 4.1.(For domains with finitely many boundary components, this was proved by Globevnik and Stensøness [31].)As indicated in [29, p. 500], the analogous result likely holds for circle domains in tori.Nothing seems to be known about this problem for domains in compact Riemann surfaces of genus > 1, where the main problem is to find a suitable initial embedding of the uniformized surface into C 2 .On the other hand, in Section 5 we use Lemma 1.1 to prove the following hitting theorem for almost proper injective holomorphic immersions in C 2 from domains obtained by removing countably many pairwise disjoint closed discs from any compact Riemann surface.
Theorem 1.4.Let R be a compact Riemann surface and Ω = R \ ∞ i=0 D i be an open domain in R whose complement is the union of countably many pairwise disjoint closed discs D i with C s boundaries for some s > 1.Given a C 1 embedding f : (ii) F agrees with f to a given finite order at a given finite set of points in Ω, and (iii) B ⊂ F (Ω).
In particular, there exists an almost proper (hence complete) injective holomorphic immersion Ω ֒→ C 2 with everywhere dense image.Lemma 1.1 can also be combined with the method developed by Forstnerič [22] for constructing complete bounded embedded holomorphic null curves in C 3 with Cantor ends, as well as complete bounded minimal surfaces in R 3 and some other related types of surfaces with Cantor ends.In this way we obtain the following result proved in Section 6. whose complements embed properly in C 2 were given by Orevkov [37], and Di Salvo [15] showed that Orevkov's construction also yields examples having Hausdorff dimension zero.
So far, we have been talking about (almost) proper injective holomorphic immersions in C 2 .However, Lemma 1.1 can also be applied to construct complete injectively immersed holomorphic curves in more general domains in C 2 , at the cost of losing control of their conformal structure and in some case of almost properness.
To motivate this line of developments, we recall that Yang [50,51] asked in 1977 whether there exist complete bounded complex submanifolds of a complex Euclidean space C n of dimension > 1.The Yang problem has been a focus of interest in the last decades; we refer to the recent survey [2].It is an open problem whether for every compact bordered Riemann surface M as in Lemma 1.1 there is a complete holomorphic embedding M ֒→ C 2 with bounded image; see [8,Problem 1.5].In fact, given such a Riemann surface M other than the closed disc, all known complete holomorphic embeddings M ֒→ C 2 are proper in C 2 (see [28,Corollary 1.2]) or else the complex structure of the embedded surface may change.By using Lemma 1.1, we construct complete embedded complex curves with a given smooth structure in any pseudoconvex Runge domain of C 2 as in the following theorem.In this case one clearly cannot control the complex structure of the examples.Theorem 1.6 is proved in Section 7. The special case when D = C 2 and B = ∅, guaranteeing the existence of properly embedded complex curves in C 2 with arbitrary topology, was established in 2013 by Alarcón and López [9,Theorem 4.5].This showed that there is no topological restriction to the Forster-Bell-Narasimhan conjecture.(For embeddings in C × C * and (C * ) 2 , see Ritter [38,39], Lárusson and Ritter [36], and Remark 2.3.)In the special case when D = B is the open unit ball and B ⊂ B is closed and discrete, Theorem 1.6 was proved by Alarcón and Globevnik [8].Likewise, when D = B or D = C 2 and S is of finite topology, it was established by the authors in [5], except for the almost properness condition.For arbitrary pseudoconvex Runge domains D in C 2 , Theorem 1.6 also generalizes and simplifies the proofs of some hitting results for (not necessarily complete) properly embedded complex curves in D due to Forstnerič, Globevnik, and Stensøness [20] and Alarcón [1].Adapting the arguments in [9,8,5] to the use of labyrinths of compact sets in pseudoconvex Runge domains, constructed by Charpentier and Kosiński in [13], leads to a proof of Theorem 1.6 in the case when B is closed in D and discrete, or S is finitely connected.The proof of Theorem 1.6 that we give here, based on Lemma 1.1 and using the labyrinths from [13], is considerably simpler and provides the general case of the theorem.
In Section 8 we establish the following analogue of Theorem 1.6 in which we do not impose any condition whatsoever on the given connected domain in C 2 ; the cost being not to guarantee almost properness of the obtained immersion.
Theorem 1.7.Let X ⊂ C 2 be a connected domain and B be a countable subset of X.Given an open Riemann surface S, there are a domain M ⊂ S, which is diffeotopic to S, and a complete injective holomorphic immersion F : M ֒→ X such that B ⊂ F (M).In particular, F can be chosen to have everywhere dense image in X.
All similar results in the literature pertain to special domains in C 2 ; see the discussion below Theorem 1.6 and the survey [2].On the other hand, omitting the injectivity condition in dimension two, it was shown by Forstnerič and Winkelman [47,21] that every connected complex manifold X with dim X > 1 admits an immersed holomorphic disc D = {ζ ∈ C : |ζ| < 1} → X with dense image; if dim X > 2 then the immersion can be chosen injective.Recently the analogous result was obtained by the authors for any bordered Riemann surface and for some other classes of open Riemann surfaces [6,Theorem 10.1].
Let d H denote the Hausdorff distance between subsets of Euclidean spaces.The following corollary, which follows by inspecting the proof of Theorem 1.7, shows that every embedded holomorphic disc is arbitrarily close to a complete one in the Hausdorff distance.
Corollary 1.8.Given a C 1 embedding G : D ֒→ C 2 which is holomorphic on D, a compact set K ⊂ D, and a number ǫ > 0, there is a complete injective holomorphic immersion We do not know whether the injective immersion F in Corollary 1.8 can be chosen to extend continuously to D or to satisfy |F − G| < ǫ on D. In particular, it remains an open question whether F can be chosen such that F (D) is bounded by a Jordan curve.All these tasks can be carried out if one allows the map F to have double points; see [3,4,45].
Insisting on the almost properness condition, we also establish the following result which is obtained by a slight modification of the proof of Theorem 1.7.Again, we do not impose any condition whatsoever on the given connected domain X in C 2 .Theorem 1.9.Let X ⊂ C 2 be a connected domain and B be a countable subset of X.On every open Riemann surface S there exists a connected, relatively compact domain M such that M has the same topological genus as S and there is a complete almost proper, injective holomorphic immersion F : M → X with B ⊂ F (M).In particular, F can be chosen to have everywhere dense image in X.
Our method of proof does not allow to ensure that the domain M in Theorem 1.9 is homeomorphic to the given open Riemann surface S. In particular, we cannot control its ends set, which could be more complicated than that of S. Remark 1.10 (On completeness).By a minor modification of the proofs, using that any two metrics on a compact space are comparable, we can ensure that the injective holomorphic immersions F obtained in Theorems 1.2, 1.4, 1.5, 1.6, 1.7, and 1.9 are complete with respect to any given Riemannian metric (not necessarily complete or the Euclidean one) in the target domain C 2 , D ⊂ C 2 , or X ⊂ C 2 , respectively.Remark 1.11 (On the hypotheses in Lemma 1.1).It is not known whether every compact bordered Riemann surface embeds holomorphically in C 2 .Here is a way to obtain such surfaces.Any compact Riemann surface, R, admits a holomorphic immersion f : R → CP 2 in the projective plane with finitely many simple double points f (a j ) = f (b j ), where a j = b j for j = 1, . . ., m (see Griffiths and Harris [32]).Given a complex line Λ ⊂ CP 2 , the punctured Riemann surface and hence any compact domain in R ′ is embedded in C 2 .There is considerable freedom in the above choices, showing that most domains with smooth boundary in any compact Riemann surface satisfy Lemma 1.1.However, we are not aware of suitable results in the literature on controlling the location of double points in an immersed compact Riemann surface in CP 2 , and it seems an open problem whether one could put all punctures in the above construction in an arbitrarily small disc around any given point of R. If this were true, then one could embed the interior of any finite bordered Riemann surface properly holomorphically in C 2 .

PROOF OF LEMMA 1.1
The proof involves four main steps: (1) using a holomorphic automorphism of C 2 to satisfy the interpolation conditions in (e), (2) exposing and sending to infinity a point in each boundary component of M, (3) pushing the boundary bM and the discrete set E out of a given ball by a holomorphic automorphism of C 2 (see condition (a)), and (4) bringing back the points at infinity.The first three steps are obtained by following and augmenting the proofs of [28, Corollary 1.2] and [35, Lemma 2.2], while the last step uses a new idea.For the sake of readability we give a complete exposition, beginning with preliminaries.
Denote the coordinates on C 2 by z = (z 1 , z 2 ), and let Let f : M ֒→ C 2 and the sets B, L ⊂ C 2 be as in Lemma 1.1, and let c 1 , . . ., c l ′ ∈ K \ f −1 (B) denote the points at which we must fulfil the interpolation condition (d).By Mergelyan theorem (see [18,Theorem 16]) we can approximate f in the C 1 (M) topology by a holomorphic map f : U → C 2 from an open neighbourhood U ⊂ R of M which agrees with f to a given finite order k at every point c 1 , . . ., c l ′ .Assuming that the approximation is close enough and up to shrinking U around M, we may assume that f : U ֒→ C 2 is a holomorphic embedding satisfying f (U \ K) ∩ L = ∅.By a standard transversality argument we can also ensure that B ∩ f (U) = ∅.We replace f by f and drop the tilde.
Finally, by a small perturbation of the map f , keeping the above conditions in place, we can ensure that the set L ∪ f (M) is polynomially convex in C 2 .Indeed, by Stolzenberg's theorem [41] the polynomial hull of L ∪ f (bM) is the union of this set with complex curves having their boundaries in L∪f (bM), and we can arrange that there are no such curves besides f (M).Here is an explicit way of doing this.Choose a compact domain M ′ ⊂ U containing M in its interior such that M is a strong deformation retract of M ′ .We may assume that the function with interpolation at the points c 1 , . . ., c l ′ ) by a smooth function f2 on M ′ which is holomorphic on M′ and does not extend holomorphically across any boundary point of M ′ .Set f = (f 1 , f2 ).Take a point p ∈ bM ′ at which df 1 (p) = 0. Locally at the image point q = f (p) ∈ C 2 we can represent the complex curve Σ ′ = f (M ′ ) with smooth boundary bΣ ′ = f (bM ′ ) as a graph over the first coordinate such that the graphing function is holomorphic on the local projection of Σ ′ but does not extend holomorphically past the point q 1 = π 1 (q) = f 1 (p).It follows that Σ ′ is not contained in any complex curve containing q in the interior, since such a curve would provide a holomorphic extension of the graphing function to a neighbourhood of q 1 .We claim that L ∪ Σ ′ is polynomially convex.Indeed, by Stolzenberg [41] and the boundary uniqueness theorem (see Chirka [14, Proposition 1, p. 258]) shows that Σ ′ ∪ Λ is a complex curve in C 2 containing C, contradicting the choice of f .This proves the claim.Since M is holomorphically convex in M ′ , Rossi's local maximum modulus principle (see Rosay [40] for a simple proof) implies that L ∪ f (M) is also polynomially convex.Furthermore, for every compact set K ⊂ M without holes such that f (M \ K) ∩ L = ∅ the set L ∪ f (K) is polynomially convex.We now replace f by a map satisfying all the stated conditions.
(1) Fulfilling condition (e) in the lemma.We have arranged above that the set is also polynomially convex (see the argument above).Recall that By the choice of f we have that B ∩ f (U) = ∅.By [23,Proposition 4.15.3]there is a holomorphic automorphism Φ ∈ Aut(C 2 ) which approximates the identity map on L ′ , it agrees with the identity to a given finite order k at the points f (c 1 ), . . ., f (c l ′ ) ∈ f (K) ⊂ L ′ , and it satisfies Φ(f (α j )) = β j for j = 1 . . ., l. Replacing f by Φ • f we may thus assume that f fulfills condition (e) in the lemma and the other properties remain in place.We now add to A the given finite set of points in M at which we shall interpolate the map to a given finite order in the subsequent steps of the proof, and we suitably enlarge the set K ⊂ M so that it has no holes and contains this new bigger set A, while the discrete set E ⊂ M remains in M \ (A ∪ K).This will ensure that the final map F will satisfy condition (e).
(2) Exposing boundary points.We follow the exposition in [28,Sect. 4].On each boundary curve C j ⊂ bM we choose a point a j and attach to M a smooth embedded arc γ j ⊂ U such that γ j ∩ M = {a j }, the intersection of bM and γ j is transverse at a j , and the arcs γ 1 , . . ., γ m are pairwise disjoint.Let b j denote the other endpoint of γ j .On the image side, we choose smoothly embedded pairwise disjoint arcs λ 1 , . . ., λ m ⊂ C 2 \ L ′ , where L ′ is given by (2.1), such that for every j = 1, . . ., m we have that λ j ∩ f (M) = f (a j ), λ j agrees with f (γ j ) near the endpoint f (a j ), the other endpoint p j of λ j satisfies only at the point p j , and the tangent vector to λ j at p j has nonvanishing second component.
(See [28, Fig. 2, p. 109] where the projection π 1 is used in place of π 2 .) We now modify f , keeping it fixed on a neighbourhood U 1 ⊂ U of M and extending it to a smooth diffeomorphism γ j → λ j for every j = 1, . . ., m such that f (b j ) = p j .Set Applying Mergelyan theorem (see [18,Theorem 16]), we can approximate f as closely as desired in C 1 (S) by a holomorphic map f : V → C 2 on an open neighbourhood V ⊂ R of S such that f agrees with f to a given finite order k at the points in the finite set A ⊂ M defined in the previous step, and f agrees with f to the second order at the endpoints a j and b j of γ j for j = 1, . . ., m.If the approximation is close enough and up to shrinking V around S, the map f : V ֒→ C 2 is a holomorphic embedding satisfying Furthermore, for every j = 1, . . ., m the complex line Λ j (2.2) intersects the embedded complex curve f (V ) only at the point p j = f (b j ) = f (b j ) and the intersection is transverse.
We have now arrived at the main point of the exposing of points technique.By [28, Theorem 2.3] there is a conformal diffeomorphism such that for every j = 1, . . ., m we have that τ (a j ) = b j , τ maps a small neighbourhood U j ⊂ M of the point a j ∈ bM in a thin tube around the arc γ j , τ agrees with the identity map to a given order k at the points of the finite set A ⊂ M , and τ is arbitrarily C 1 close to the identity map on M \ m j=1 U j .We can choose τ to have any finite order of smoothness on M; for technical reasons which will become apparent in the next step we shall assume that it is of class C 3 (M).The map is then a C 3 embedding which is holomorphic on M , its image h(M) is a compact domain with C 3 boundary in the embedded complex curve f (V ) ⊂ C 2 , and τ can be chosen such that for each j = 1, . . ., m the complex line Λ j (2.2) intersects h(M) only at p j = f (b j ).
Assuming as we may that τ is close enough to the identity on K, (2.3) implies Let g be a rational shear on C 2 of the form where ρ > 0, θ j ∈ R, and P (z 2 ) is a holomorphic polynomial chosen such that the function m j=1 ρ e iθ j z 2 −π 2 (p j ) + P (z 2 ) vanishes to order k at the point π 2 (f (a)) for every a ∈ A, and P vanishes at every point π 2 (p j ) for j = 1, . . ., m.By taking the constant ρ > 0 in (2.6) sufficiently small, the polynomial P can be chosen such that |P | is as small as desired on the compact set π 2 (L ′ ) where L ′ is given by (2.1).This gives a holomorphic embedding with simple poles at the points b 1 , . . ., b m .Similarly, we have a which is holomorphic on M , it approximates the embedding h (2.5) as closely as desired on K provided that the constant ρ > 0 in (2.6) is chosen small enough, it satisfies (2.9) it agrees with h (and hence with f ) to order k at the points of A, and g • h sends the points a 1 , . . ., a m to infinity.More precisely, for every j = 1, . . ., m the set (2.10) is a properly embedded curve of class C 3 , diffeomorphic to R, which is asymptotic to a line at every end (see [49,Lemma 2] for the details), the first coordinate projection π 1 : C 2 → C maps σ j to a proper curve σj = π 1 (σ j ) ⊂ C, and π 1 : σ j → σj is a diffeomorphism near infinity.Furthermore, the numbers θ j ∈ R in (2.6) can be chosen such that the projected curves σj have different asymptotic directions, and for every sufficiently big number s > 0 the set C\ sD ∪ m j=1 σj has no bounded connected components.These choices are independent of the number ρ > 0, which can be chosen arbitrarily small.With the curves σ j given by (2.10) and the discrete set E ⊂ M as in the lemma, we define Since E only clusters on bM, the set E ′ only clusters on Γ, so E ′ ∪ Γ is closed and the projection π 1 : E ′ ∪ Γ → C is proper.Furthermore, since the curves σ j are asymptotic to lines at infinity, for any C-linear projection π ′ 1 : By a linear change of coordinates on C 2 we may assume that this holds for π 1 .Furthermore, if π 1 was not changed much, then by our conditions on Γ there is a number s 0 ≥ 0 such that (2.13) the domain C \ (sD ∪ π 1 (Γ)) has no holes for s ≥ s 0 . Remark ∅ and numbers r > 0 (big) and ǫ > 0 (small), there is an automorphism Φ ∈ Aut(C 2 ) satisfying the following conditions: and (iii) Φ agrees with the identity map to a given order k at given points q 1 , . . ., q l ∈ L.
Proof.We shall obtain Φ as a composition Φ = φ • ψ of two automorphisms, where φ will do the main job and ψ will be a shear (2.16) taking care of things at infinity.
Choose a compact polynomially convex set L ′ ⊂ C 2 containing L in its interior such that L ′ ∩ (E ∪ Γ) = ∅ and a number ǫ ′ > 0 to be specified later.Let s 0 ≥ 0 be as in (2.13).Pick s ≥ s 0 such that L ′ ⊂ sD × C and set We move Γ out of the ball rB by an isotopy of embeddings of class C 3 within the set C 2 \ L ′ .Since Γ is a union of curves, the union of L ′ with the image of Γ at every stage of the isotopy is polynomially convex by Stolzenberg [42].Hence, [26, Theorem 2.1] due to Forstnerič and Løw furnishes an automorphism (The proof of [26, Theorem 2.1] relies on the Andersén-Lempert theorem in the version given by Forstnerič and Rosay [27]; see also [ We now correct φ 2 • φ 1 so that the above conditions are preserved and the interpolation condition (iii) holds.Since φ 2 • φ 1 is close to the identity on L ′ , its k-jet at each point q j is close to the k-jet of the identity map.To satisfy (iii) we take φ = φ 3 • φ 2 • φ 1 where a suitable automorphism φ 3 ∈ Aut(C 2 ) is obtained by [23, proof of Theorem 4.9.2, p. 140], which relies on the jet-interpolation theorem for holomorphic automorphisms [23,Corollary 4.15.5,p. 174].Note that φ 3 can be chosen arbitrarily close to the identity map on any given compact set in C 2 if the k-jet of φ 2 • φ 1 at each point q j is close enough to the k-jet of the identity map.Hence, assuming that ǫ ′ > 0 is chosen small enough, the automorphism and condition (iii).Recall the notation (2.14) and set The problem now is that φ(E ′′ ∪Γ ′ ) may intersect the ball rB.These intersections are removed by precomposing φ with a shear ψ ∈ Aut(C 2 ) of the form for a suitably chosen entire function ξ : C → C which is close to 0 on the disc sD.The idea is explained in [35, proof of Lemma 2.2] (based on [12, Lemma 2.2]); however, some additions to their argument are necessary in light of Remark 2.1.Note that φ −1 (rB) is a compact polynomially convex set.Let s ≥ s 0 be as above, and pick s 1 > s such that (2.17) The shear ψ of the form (2.16) should be chosen such that By (2.17) this does not impose any condition on the function ξ on C \ s 1 D, while on sD it suffices to take ξ close enough to 0 in view of (2.15).It is explained in [49, Lemma 1] how to determine ξ on the curves π 1 (Γ) ∩(s 1 D \ sD) such that ψ(Γ) ∩φ −1 (rB) = ∅.To find an entire function ξ on C such that the above holds, one uses Mergelyan theorem (see [18,Theorem 16]) and condition (2.13).Since the set E ′ only clusters on Γ, there are at most finitely many points Q = {e 1 , . . ., e i } ⊂ E ′ such that ψ(e j ) ∈ φ −1 (rB) for j = 1, . . ., i, i.e., condition (2.18) fails only at these points.In view of the last two conditions in (2.12) we can redefine ξ at the points π 1 (e 1 ), . . ., π 1 (e i ) ∈ s 1 D \ sD, approximating the previously chosen function sufficiently closely on the polynomially convex set π 1 (( so that the new shear map ψ satisfies (2.18).The interpolation of the identity at the given points q 1 , . . ., q l ∈ L is achieved by choosing ξ such that it vanishes to order k at every point π 1 (q j ) ∈ π 1 (L) ⊂ sD for j = 1, . . ., l.The automorphism Φ = φ • ψ ∈ Aut(C 2 ) then satisfies Lemma 2.2.
(4) Bringing back the points at infinity.Recall that f : V ֒→ C 2 is a holomorphic embedding satisfying (2.3) and g • f : ) and h = f • τ (2.5).The compact set K ⊂ M is holomorphically convex in M , and in view of (2.9) the compact set is polynomially convex (for the details, see [48] or [23, proof of Theorem 4.14.6,p. 168]).
Let Φ ∈ Aut(C 2 ) be given by Lemma 2.2 with L replaced by L, the set E ′ = (g • h)(E), and with interpolation to order k (see condition (iii)) on the finite set Consider the holomorphic embedding The image of F is an embedded complex curve in C 2 containing the image of the embedding By the construction, F • τ satisfies Lemma 1.1 except that the points a 1 , . . ., a m ∈ bM are sent to infinity.We now bring them back to C 2 as follows.On every curve τ (C j ) ⊂ τ (bM) we choose a proper closed arc I j τ (C j ) containing the point τ (a j ) = b j in its interior.Let v be a smooth vector field on R along the set I = m j=1 I j which is transverse to τ (bM) and points to the interior of τ (M).Mergelyan theorem allows us to approximate v by a holomorphic vector field on a neighbourhood of τ (M) in V , still denoted v, which vanishes to order k at every point of A. (Note that the tangent bundle of V is trivial, so we may think of v as a function.)The flow ψ t of v for small values of |t|, with ψ 0 = Id, exists on a neighbourhood of τ (M) in V and consists of biholomorphic maps whose k-jet at every point of A agrees with the k-jet of the identity map.Since v points to the interior of τ (M) along I, for small t > 0 the closed domain ψ t (τ (M)) ⊂ V (which is conformally diffeomorphic to M) does not contain any of the points b 1 , . . ., b m , and hence for such t the embedding

PROOF OF THEOREM 1.2
We begin with some reductions.As in Lemma 1.1, we assume as we may that M is a closed smoothly bounded domain in a compact Riemann surface R. Fix K, α j , β j , and ǫ as in the statement of Theorem 1.2.Also, fix finitely many points c 1 , . . ., c l in K \ f −1 ({β j : j = 1, 2, . ..}) for the interpolation condition (ii), as well as a number k ∈ N for the interpolation order.By Mergelyan theorem in the C 1 topology with interpolation at the points c 1 , . . ., c l (see [18,Theorem 16]), we may assume that f is given by a holomorphic embedding f : Choose a smoothly bounded compact domain K 0 ⊂ M which is a strong deformation retract of M such that K ⊂ K0 .By renumbering the points α j and β j , we may assume that α 1 , . . ., α j ∈ K 0 and α i / ∈ K 0 for all i > j.Applying Lemma 1.1 we find a holomorphic embedding f 0 : M ֒→ C 2 which approximates f as closely as desired on K, it agrees with f to order k at the points c 1 , . . ., c l , and it satisfies f 0 (α i ) = β i for i = 1, . . ., j.We now add the points {α 1 , . . ., α j } to the finite set {c 1 , . . ., c l } and drop them from the list of α-points in the theorem.Likewise, we drop β 1 , . . ., β j from the list of β-points.
We take f 0 as our new starting map and K 0 as the new initial compact set in the theorem.Set K −1 = ∅ and ǫ 0 = ǫ/2.We may assume without loss of generality that β j = 0 for all j = 1, 2, . .., and that there is r 0 > 0 such that Choose a sequence 0 < r 0 < r 1 < r 2 < • • • with lim j→∞ r j = +∞.Fix a point α 0 ∈ K 0 \ f −1 0 ({β j : j = 1, 2, . ..}) and set β 0 = f 0 (α 0 ).We shall inductively construct a sequence of triples X j = {f j , K j , ǫ j }, j ∈ N, where • K j ⊂ M is a smoothly bounded compact domain which is a strong deformation retract of M, and and the following conditions hold for all j ∈ N: (3 j ) ǫ j < ǫ j−1 /2 and every holomorphic map ϕ : M → C 2 with |ϕ − f j−1 | < 2ǫ j everywhere on K j−1 is an embedding on K j−2 .
(5 j ) f j agrees with f j−1 to order k at c i for all i ∈ {1, . . ., l}.
Assuming the existence of such a sequence, the proof of Theorem 1.2 is completed as follows.Conditions (1 j ), (2 j ), (3 j ), and (3.2) ensure that there exists a limit map which is an injective holomorphic immersion and satisfies (3.3) sup This implies condition (i) in the theorem; recall that f 0 = f and 2ǫ 1 < ǫ 0 < ǫ.Conditions (4 j ) and (5 j ) ensure that F (α i ) = β i for all i = 1, 2, . . .and F agrees with f to order k at c i for all i ∈ {1, . . ., l}; so, (ii) and (iii) hold as well.Now, (3.3) and condition (6 j ) guarantee that Since the increasing sequence of compact sets K j exhausts M (see (3.2)) and we have that lim j→∞ r j = +∞, it follows that F : M ֒→ C 2 is an almost proper map.
Likewise, (3.3) and (7 j ) give that (3.5) inf If the sequence β j ∈ C 2 is closed and discrete, we have that lim j→∞ |β j | = +∞, and hence This, (3.2), (3.5), and conditions (1 j ) imply that in this special case the injective immersion F : M ֒→ C 2 which we constructed is a proper map, hence an embedding, thereby proving the final assertion of the theorem.
Let us explain the induction.The basis is given by the triple X 0 = {f 0 , K 0 , ǫ 0 }; observe that it meets conditions (1 0 ), (4 0 ), and (6 0 ), see (3.1), while (2 0 ), (3 0 ), (5 0 ), and (7 0 ) are void.For the inductive step we fix j ∈ N and assume that we have a triple X j−1 = {f j−1 , K j−1 , ǫ j−1 } enjoying (1 j−1 ), (4 j−1 ), and (6 j−1 ).By the Cauchy estimates, there is a number ǫ j > 0 so small that (3 j ) holds true.Recall that β j = 0 and set By (6 j−1 ), we have that Choose a number r > r j so large that (3.6) Lemma 1.1 then applies to the embedding f j−1 , the compact set K j−1 (it has no holes since it is a strong deformation retract of M), the compact polynomially convex set L ⊂ C 2 , the singletons {α j } ⊂ M \ K j−1 (see (1 j−1 )) and ), and the numbers ǫ j > 0 and r > 0, furnishing us with a holomorphic embedding f j : M ֒→ C 2 satisfying (2 j ), (5 j ), (7 j ), and the following conditions: Note that (b) and (4 j−1 ) ensure (4 j ).Finally, in view of (a), (c), and the fact that r > r j , we can choose a smoothly bounded compact domain K j ⊂ M , being a strong deformation retract of M, such that (1 j ) and (6 j ) hold true.Indeed, by (a) we can first take a smoothly bounded compact domain K ′ j ⊂ M which is a strong deformation retract of M such that K j−1 ∪ {α i : i = 0, . . ., j} = K j−1 ∪ {α j } ⊂ K′ j and (3.7) Otherwise, E ′ is a finite set (recall that the sequence α i ∈ M only clusters on bM) and we can choose K ′ j so that E ′ ⊂ K′ j \ K j−1 ; see (1 j−1 ).Let Ω 1 , . . ., Ω m denote the connected components of K ′ j \ Kj−1 ; these are smoothly bounded compact annuli since K j−1 ⊂ K′ j is a strong deformation retract of K ′ j .Fix i ∈ {1, . . ., m}.Conditions (a), (c), and (3.7) imply that the set Since Ω ′ i is open in Ω i , these properties and the maximum principle show that Ω ′ i path connected, and hence we can choose a smooth Jordan arc It is clear that every sufficiently large smoothly bounded compact domain K j ⊂ K′ j \ γ which is a strong deformation retract of M satisfies conditions (1 j ) and (6 j ).This closes the induction.
Note that at each step of the induction we are allowed to choose the compact domain K j ⊂ M as large as desired under the only restriction imposed by the second part of condition (1 j ).So, since the sequence α j ∈ M is closed and discrete, we can proceed in such a way that condition (3.2) is satisfied.This completes the proof of Theorem 1.2.We shall use the following analogue of Lemma 1.1 adapted to this situation.

PROPER EMBEDDINGS OF CIRCLE DOMAINS
and a number r > 0, we can approximate f as closely as desired uniformly on K by a holomorphic embedding F : M k ֒→ C 2 which agrees with f at finitely many given points in K and satisfies This lemma is obtained by combining [29, proof of Lemma 3.1] with the proof of Lemma 1.1 in Section 2. The only difference in [29,Lemma 3.1] when compared to the technique used in the earlier papers [28,48,49] is that the conformal diffeomorphism τ : M → τ (M) ⊂ V in (2.4) is chosen such that it maps M onto a domain with piecewise smooth boundary in the ambient Riemann surface V .(In the context of Lemma 4.2, we apply this argument to the bordered Riemann surface M = M k .)The finitely many corner points of τ (bM) are mapped to infinity by the embedding f ; see (2.5).The main point of this change is to ensure that the first coordinate projection π 1 : C 2 → C, restricted to the image of M, is injective near infinity; this enables one to find a shear ψ (2.16) in Step (3) of the proof of Lemma 1.1 such that the resulting automorphism Φ = φ • ψ ∈ Aut(C 2 ) maps all the discs D i out of the given ball rB.It is clear that the method in step (4) of the proof of Lemma 1.1, using the flow ψ t of a suitably chosen holomorphic vector field on V , still applies to a domain with corners, so we can bring the points at infinity back to C 2 .Finally, we approximate the new conformal diffeomorphism ψ t • τ sufficiently closely by a smooth conformal diffeomorphism from M onto its image in the ambient surface V , thereby removing the corners.We leave further details to the reader.
Proof of Theorem 4.1.Let B denote the unit ball of C 2 .Set M 0 = CP 1 \ D0 ; this is a closed disc.Choose a holomorphic embedding f 0 : M 0 ֒→ C 2 and a compact, smoothly bounded, O(Ω)-convex set K 0 ⊂ Ω.Its O( M0 )-convex hull is the union of K 0 with at most finitely many smoothly bounded open discs in M0 , each containing a disc from the family {D i } i∈N .Hence, there are finitely many discs D j(1) , . . ., D j(k 1 ) such that, setting 2), . . ., j(k 1 )}.By Lemma 4.2 we can approximate the embedding f 0 | M 1 as closely as desired uniformly on K 0 by a holomorphic embedding By the same argument as above, we can find finitely many discs D j(k 1 +1) , . . ., D j(k 2 ) from the given family such that, setting (1), j(2), . . ., j(k 2 )}.By Lemma 4.2 applied to the embedding f 1 | M 2 and the polynomially convex set L = B ⊂ C 2 we can approximate f 1 on K 1 by a holomorphic embedding Hence, there is a smoothly bounded Continuing inductively, we obtain the following: (1) an increasing sequence

., and
(3) a sequence of holomorphic embeddings f j : M j ֒→ C 2 such that for every j ∈ N, the map f j approximates f j−1 uniformly on K j−1 as closely as desired and it satisfies (4.2) Here, J j ⊂ N is such that Mj \ i∈J j D i = Ω.The second condition in (4.2) gives B for all j = 1, 2, . . . .
Assuming as we may that the approximation of f j−1 by f j is sufficiently close at every step, these conditions clearly imply that the sequence f j converges uniformly on compacts in Ω to a proper holomorphic embedding f = lim j→∞ f j : Ω ֒→ C 2 .
Remark 4.3.To prove Theorem 1.4 in this special case, we modify the above construction so that for each j = 1, 2, . . .we first pick a point α j ∈ Ω \ K j−1 and then choose the next embedding f j : M j ֒→ C 2 so that it approximates f j−1 on K j−1 , it satisfies and f j agrees with f j−1 at the previously chosen points α 1 , . . ., α j−1 ∈ K j−1 so that f j (α i ) = β i ∈ B holds for i = 1, . . ., j.We then pick the next compact, smoothly bounded, (However, we cannot fulfil condition (4.3) due to the interpolation condition f j (α j ) = β j , since there is no assumption on the set B = {β j } ⊂ C 2 .)By choosing the set K j ⊂ Ω big enough at every step to ensure that ∞ j=1 K j = Ω, condition (4.4) ensures that the limit holomorphic embedding f = lim j→∞ f j : Ω ֒→ C 2 is almost proper.
The general case of Theorem 1.4 is proved in the following section.

PROOF OF THEOREM 1.4
We shall need the following generalization of [29,Lemma 2.2] to domains in an arbitrary compact Riemann surface R.
= Ω be an exhaustion by smoothly bounded compact connected sets with K j ⊂ Kj+1 for all j ∈ N. By choosing K 1 big enough, every set K j has the same genus ν as R, and hence R \ K j is the union of finitely many open discs U j = {U j 1 , . . ., U j m(j) } with pairwise disjoint closures.We claim that for j large enough there are discs ∆ 1 , . . ., ∆ m ∈ U j satisfying (5.1).Indeed, if this is not the case, there is a decreasing sequence of closed discs Given a compact subset L ⊂ R \ Ω and open smoothly bounded discs ∆ i ⊂ R (i = 1, . . ., m) with pairwise disjoint closures satisfying (5.1), we shall say that the set → Ω is a path such that δ(0) ∈ c(Γ) and δ has a limit point in L, then δ([0, 1)) ∩ Γ = ∅.
Proof of Theorem 1.4.Let R and Ω = R \ ∞ i=0 D i be as in the theorem, so D i are closed pairwise disjoint closed discs.Note that Ω has the same topological genus ν as R. Set (5.3) Di for j = 0, 1, . . . .For all j ≥ 0 we have that Ω ⊂ Mj , M j+1 ∪ Dj+1 = M j , bM j+1 = bM j ∪ bD j+1 , and Ω = ∞ j=0 Mj .By a surrounding system Γ ⊂ Ω for M j , we shall mean a surrounding system for R and B ⊂ C 2 be as in the statement of the theorem.We shall assume without loss of generality that the set B = {β 1 , β 2 , . ..} is infinite, and for simplicity of notation we assume that k = 0 (the same argument will apply in the general case).Set f 0 = f : M 0 = R \ M0 ֒→ C 2 .Fix points c 1 , . . ., c l ∈ Ω at which we wish to interpolate (see condition (ii) in the theorem).We may assume without loss of generality that f 0 ({c 1 , . . ., c l }) ∩ B = ∅.Using Lemma 5.1 we find a smooth Jordan curve Γ 0 ⊂ Ω surrounding the disc D 0 = R \ M0 such that Γ 0 ∩ {c 1 , . . ., c l } = ∅ and f 0 vanishes nowhere on Γ 0 ; the last condition is easily arranged by a small deformation.Then, choose a smoothly bounded compact connected domain K 0 ⊂ Ω with genus ν such that We shall inductively construct a sequence of tuples T j = {f j , K j , Γ j , ǫ j , α j }, j ∈ N, where • K j ⊂ Ω is a smoothly bounded compact connected domain of genus ν, • Γ j ⊂ Kj is a surrounding system for M j , • ǫ j > 0 is a number, and and the following conditions hold for all j = 1, 2, . . .: ( (6 j ) f j agrees with f j−1 to a given order at c i ∈ K 0 for i = 1, . . ., l.
The basis of the induction is given by T 0 = {f 0 , K 0 , Γ 0 , ǫ 0 , α 0 }.It meets conditions (1 0 ), (5 0 ), and (7 0 ), while the remaining conditions are void for j = 0.For the inductive step, assume that we have tuples T 0 , . . ., T j−1 satisfying the required conditions for some j ≥ 1, and let us construct T j .Choose ǫ j > 0 so small that (3 j ) holds.Next, choose a compact set , and the number ǫ j > 0 furnishes a holomorphic embedding f j : M j ֒→ C 2 satisfying (4 j )-(6 j ) and By Lemma 5.1 there is a surrounding system Γ j = j i=0 γ i ⊂ Ω∩V for M j such that γ i = b∆ i , where ∆ i ⊂ V is a disc containing D i and the closed discs ∆ i for i = 0, 1, . . ., j are pairwise disjoint.Hence, Γ j ∩ K j−1 = ∅, K j−1 ∪ {α j } ⊂ c(Γ j ) (this is the second condition in (2 j )), and (7 j ) holds.Finally, choose any smoothly bounded compact connected domain K j ⊂ Ω containing K j−1 ∪ {α j } ∪ Γ j in its interior.Hence, K j is of genus ν (the same as the genus of R) and conditions (1 j ) and (2 j ) hold.The induction may now proceed.Note that condition (5.4) can be fulfilled since we may choose K j ⊂ Ω as large as desired at each step.
As in the proof of Theorem 1.2, there is a limit map F = lim j→∞ f j : ∞ j=0 K j = Ω ֒→ C 2 which is an injective holomorphic immersion and satisfies conditions (i), (ii), and (iii) in the statement of the theorem; note that B = F ({α j : j ∈ N}) ⊂ F (Ω). Finally, conditions (2 j )-(4 j ) and (7 j ) guarantee that inf is an exhaustion of Ω by connected, smoothly bounded compact domains in view of (5.4), (1 j ), and (2 j ), this inequality shows that the map F : Ω → C 2 is almost proper.

PROOF OF THEOREM 1.5
We begin by recalling a construction of a Cantor set in a domain Ω 0 ⊂ C. In the first step, we choose a smoothly bounded compact convex domain ∆ 0 ⊂ Ω 0 .Removing from ∆ 0 an open neighbourhood Υ 0 of the vertical straight line segment divides ∆ 0 in two smoothly bounded compact convex subsets ∆ 1 0 and ∆ 2 0 of the same width.Finally, for j = 1, 2 we remove from ∆ j 0 an open neighborhood Υ j 0 of the horizontal straight line segment dividing ∆ j 0 in two convex subsets of the same height, making sure that the two connected components of ∆ j 0 \ Υ j 0 are smoothly bounded compact convex domains.This gives a compact set (6. 1) ) ⊂ Ω0 which is the union of four pairwise disjoint, smoothly bounded compact convex domains Ω j 1 , j = 1, . . ., 4. In the second step, we repeat the same procedure for each convex compact domain Ω j 1 from the first generation, thereby getting four pairwise disjoint smoothly bounded compact convex domains in its interior.This gives a compact set Ω 2 ⊂ Ω1 which is the union of sixteen smoothly bounded compact convex domains.Continuing inductively, we obtain a decreasing sequence of smoothly bounded compact domains (6.2) such that for each i ≥ 1 the domain Ω i consists of 4 i pairwise disjoint smoothly bounded compact convex domains.The intersection is then a Cantor set in C.Moreover, choosing the separating neighbourhoods sufficiently small at each step of the construction, we may ensure that µ(C) > µ(∆ 0 ) − δ for any given δ > 0, where µ denotes the 2-dimensional Lebesgue measure on C.
We now explain the proof of Theorem 1.5.Let R be a compact Riemann surface and B ⊂ C 2 be a countable set.Assume that B = {β 1 , β 2 , . ..} is infinite and 0 / ∈ B. Let Ω 0 be a smoothly bounded compact convex domain in a holomorphic coordinate chart on R such that there is a holomorphic embedding f 0 : R \ Ω0 ֒→ C 2 .Such a set and embedding f 0 always exists; we may for instance choose Ω to be the complement of a small open neighbourhood of the curves in a suitable homology basis of R. Fix ǫ 0 > 0, set K 0 = R \ Ω0 and K −1 = ∅, and assume without loss of generality that there is a number r 0 > 0 satisfying (6.4) f 0 (K 0 ) ∩ r 0 B = ∅.
Let ∆ 0 ⊂ Ω0 be a smoothly bounded compact convex domain so large that f 0 extends to a holomorphic embedding f 0 : R\ ∆0 ֒→ C 2 satisfying f 0 (R\ ∆0 )∩r 0 B = ∅.Choose a positive number ǫ 1 < ǫ 0 /2.An application of Mergelyan theorem in two steps (see [18,Theorem 16]) furnishes a compact set Ω 1 as in (6.1), consisting of four pairwise disjoint smoothly bounded compact convex domains, and a holomorphic embedding f0 : Indeed, we first extend f 0 to a smooth embedding , where E 0 is the vertical straight line segment dividing ∆ 0 in two convex subsets of the same width, such that f 0 (E 0 ) ∩ r 0 B = ∅.Note that E 0 intersects R \ ∆0 only at its endpoints and the intersections are transverse.By Mergelyan theorem (see [18,Theorem 16]) we can then approximate f 0 in the C 1 topology on (R \ ∆0 ) ∪ E 0 by a holomorphic embedding f ′ 0 : (R \ ∆0 ) ∪ Υ 0 ֒→ C 2 , where Υ 0 is a neighborhood of E 0 in ∆ 0 as explained above, such that We then repeat the process simultaneously in the two components ∆ 1 0 and ∆ 2 0 of ∆ 0 \ Υ 0 : we suitably extend , where E j 0 , j = 1, 2, is the horizontal straight line segment dividing ∆ j 0 in two convex subsets of the same height, and apply Mergelyan theorem to approximate f ′ 0 in the C 1 topology on (R \ ∆0 ) ∪ Υ 0 ∪ E 1 0 ∪ E 2 0 by a holomorphic embedding f0 : Ω 1 ֒→ C 2 which satisfies the conditions in (6.5) for K 1 = R \ Ω1 .Note that K 0 ⊂ K1 and choose a point α 1 ∈ K1 \ K 0 .Arguing as in the proof of Theorem 1.2, we may use Lemma 1.1 to obtain a holomorphic embedding f 1 : K 1 ֒→ C 2 satisfying the following conditions: (b) f 1 (α i ) = β i for i = 0, 1.
We repeat this procedure inductively, following the recursive construction of a Cantor set C in Ω described above; see (6.2) and (6.3).In this way, we construct a sequence of tuples T j = {f j , K j , ǫ j , α j }, j ∈ N, where where Ω j is a domain in Ω 0 consisting of 4 j pairwise disjoint smoothly bounded compact convex domains, • f j : K j ֒→ C 2 is a holomorphic embedding, • ǫ j > 0 is a number, and such that T j satisfies K j−1 ⊂ Kj and conditions (2 j )-(4 j ), (6 j ), and (7 j ) in the proof of Theorem 1.2 for all j ∈ N (with M replaced by K j−1 in (3 j )), and we have that Assume for a moment that such a sequence exists.Conditions (7.1) and (1 j ) imply that is a domain in S that is diffeotopic to S. In particular, there is a complex structure J on S such that the open Riemann surface (S, J) is biholomorphic to M. (For details in a similar setting, see [3, proof of Theorem 1.4 (b) and Corollary 1.5].)By (1 j ), (2 j ), (3 j ), and the maximum principle, there is a limit map which is an injective holomorphic immersion and satisfies (7.5) sup We claim that if each ǫ j > 0 is chosen sufficiently small then F satisfies the conclusion of Theorem 1.6.Indeed, if every ǫ j > 0 is small enough then (7.4), (7.5), and conditions (1 j ) and (7 j ) guarantee that length(F • γ) = +∞ for every proper path γ : [0, 1) → M, and hence F is complete.Likewise, (7.5) and conditions (5 j ) imply that F (bK j ) ∩ D j = ∅ for all j ∈ N whenever the ǫ j 's are sufficiently small, and hence F : M → D is an almost proper map in view of (7.2), (7.4), and conditions (1 j ).Indeed, if Q ⊂ D is compact and we take m ∈ N so large that Q ⊂ D m , then F −1 (Q) ∩ bK j = ∅ for every j > m, hence all components of Finally, if each ǫ j > 0 is chosen sufficiently small then F (K j \ Kj−1 ) ∩ D j−1 ∩ D m j = ∅ for all j ∈ N by (7.5) and conditions (6 j ), while if the given set B ⊂ D is closed and discrete then lim j→∞ min{j −1, m j } = +∞; see (7.2) and (7.3).These conditions imply that F : M ֒→ D is a proper map, and hence a proper holomorphic embedding, provided that B is closed in D and discrete.Therefore, F satisfies the conclusion of the theorem.
Case 1: The Euler characteristic of S j \ Sj−1 is 0. In this case, (7.1) and (1 j−1 ) imply that K j−1 ⊂ Sj and K j−1 is a strong deformation retract of S j .Choose an integer d > max{j, m j + 1} so large that f j−1 (K j−1 ) ⊂ D d and set d ′ = min{j − 1, m j } < d.Assume without loss of generality that β j / ∈ f j−1 (K j−1 ).By (5 j−1 ) and ( 7.3) we have that Pick a point a ∈ bK j−1 and attach to K j−1 a smooth embedded arc η ⊂ Sj such that η ∩ K j−1 = {a} and the intersection of bK j−1 and η is transverse at a. On the image side, choose a smoothly embedded arc such that λ agrees with f j−1 (η) near the endpoint f j−1 (a), λ ∩ f j−1 (K j−1 ) = f j−1 (a), and the other endpoint of λ equals β j .Let α j denote the other endpoint of η and extend f j−1 to a smooth diffeomorphism η → λ such that f j−1 (α j ) = β j .By Mergelyan theorem (see [18, Theorem 16]) we may assume in view of (5 j−1 ), (7.6), and (7.7) that there is a connected, smoothly bounded, compact domain K ⊂ S without holes such that It follows that K is a strong deformation retract of S j as well.By Charpentier and Kosiński is also polynomially convex, and f j−1 (K \ K j−1 ) ∩ L = ∅ by (iii).Lemma 1.1 furnishes a holomorphic embedding f j : K ֒→ C 2 satisfying the following conditions: . ., j} (see (iii) and (4 j−1 )).
Here, ǫ j > 0 is so small that condition (3 j ) holds and (see (iii) and (c)).Note that (7.8) ensures that the point α j ∈ η lies in the connected component of f −1 j (D d ) containing K j−1 .This, the maximum principle, and conditions (i) and (a) guarantee the existence of a connected, smoothly bounded, compact domain K j ⊂ K without holes in S satisfying (1 j ), (7.9) α j ∈ Kj , f j (bK j ) ∩ D d+1 = ∅, and f j (K j ) ⊂ D.
Case 2: The Euler characteristic of S j \ Sj−1 equals −1.In this case, there is a smooth Jordan arc E ⊂ Sj \ Kj−1 , transversely attached with its two endpoints to bK j−1 and otherwise disjoint from K j−1 , such that K j−1 ∪ E is a strong deformation retract of S j .Given ǫ > 0, an application of Mergelyan theorem (see [18,Theorem 16]) furnishes a connected, smoothly bounded, compact domain K ⊂ Sj , without holes in S, such that K j−1 ∪ E ⊂ K and K is a strong deformation retract of S j , and a holomorphic embedding g : K ֒→ D such that sup x∈K j−1 |g(x) − f j−1 (x)| < ǫ, g(α i ) = β i for all i ∈ {0, . . ., j − 1} (see (4 j−1 )), g(bK i ) ∩ D i = ∅ for all i ∈ {0, . . ., j − 1}, and g(K \ Kj−1 ) ∩ D j−1 = ∅ (see (5 j−1 )).See [1, p. 216, Case 1] for the details in a very similar situation.This reduces the proof of the inductive step to Case 1.This closes the induction and completes the proof of Theorem 1.6.
8. PROOF OF THEOREMS 1.7 AND 1.9 For simplicity of exposition we shall prove these results in the case when the open Riemann surface S is a disc, say, S = 2D = {ζ ∈ C : |ζ| < 2}.In particular, the domain M ⊂ S in Theorem 1.7 must be a disc, while the one in Theorem 1.9 must be a planar domain.The general cases are seen by combining the proof in these special cases with the procedure to prescribe the topology in the proof of Theorem 1.6; we leave the details to interested readers.
Proof of Theorem 1.7.Let X ⊂ C 2 and B = {β j } j∈N ⊂ X be as in the statement.Let us assume that S = 2D.Set D 0 = D and choose a holomorphic embedding f 0 : D 0 → X. Assume that f 0 (0) / ∈ B and set α 0 = 0, β 0 = f (α 0 ), and D −1 = ∅.Fix a number ǫ 0 > 0. We shall inductively construct a sequence of smoothly bounded closed discs D j ⊂ 2D, points α j ∈ Dj , holomorphic embeddings f j : D j → X, and numbers ǫ j > 0 satisfying the following conditions for all j ∈ N = {1, 2, . ..}: (iv j ) dist f j (0, bD j ) > j, where dist f j denotes the distance function on D j associated to the Riemannian metric induced on D j by the Euclidean one in C 2 via the immersion f j .(v j ) ǫ j < ǫ j−1 /2 and every holomorphic map ϕ : The basis of the induction is provided by the already chosen disc D 0 = D, point α 0 = 0 ∈ D0 , holomorphic embedding f 0 : D 0 → X, and number ǫ 0 > 0. They meet conditions (i 0 ), (iii 0 ), and (iv 0 ), while (ii 0 ) and (v 0 ) are void.For the inductive step, fix j ∈ N and assume that we have suitable objects D k , α k , f k , and ǫ k satisfying (i k ), (iii k ), and (iv k ) for all k ∈ {0, . . ., j − 1}.By (iv j−1 ) we can choose a number ǫ j > 0 so small that (v j ) is satisfied.
Reasoning as in Case 1 in the proof of Theorem 1.6, we may assume that there are a smoothly bounded closed disc Σ ⊂ 2D and a point α j ∈ Σ such that f j−1 extends to a holomorphic embedding f j−1 : Σ → X with (8.1) Moreover, Σ can be chosen as close to D j−1 as desired.
Note that M = j∈N D j ⊂ 2D is an open disc which is diffeotopic to S = 2D.By conditions (i j ), (ii j ), and (v j ), there is a limit holomorphic map F = lim j→∞ f j : M → C 2 such that |F − f j−1 | < 2ǫ j for all j ∈ N. So, by (v j ), F has range in X and is a complete injective immersion.Finally, conditions (iii j ) ensure that B ⊂ F (M).
Proof of Corollary 1.8.Let G, K, and ǫ be as in the statement.By Mergelyan theorem (see [18,Theorem 16]) there is a holomorphic embedding f 0 : D → C 2 with |f 0 − G| < ǫ/2 on D, hence d H (f 0 (D), G(D)) < ǫ/2.Let X ⊂ C 2 be an ǫ/2-tubular neighbourhood of f 0 (D).It follows that d H (X, G(D)) < ǫ.By the proof of Theorem 1.7 there are an open disc M with D ⋐ M ⋐ 2D and a complete injective holomorphic immersion h : M → X such that h(M) is everywhere dense in X, i.e., h(M) = X.We then have that d H (h(M), G(D)) < ǫ.Furthermore, an inspection of the proof of Theorem 1.7 shows that we can choose M so close to D that there is a holomorphic diffeomorphism φ : D → M satisfying |h • φ − f 0 | < ǫ/2 on the compact set K ⊂ D. It is clear that F = h • φ satisfies the conclusion of the corollary.
Proof of Theorem 1.9.Let B = {β j } j∈N ⊂ X ⊂ C 2 be as in the statement and S = 2D.Let D 0 , f 0 , α 0 , β 0 , D −1 , and ǫ 0 be as in the proof of Theorem 1.7.Choose an exhaustion (8.3) of X by compact domains.We shall inductively construct an increasing sequence of connected, smoothly bounded compact domains D j ⊂ 2D, as well as sequences of points α j ∈ Dj , holomorphic embeddings f j : D j → X, and numbers ǫ j > 0 satisfying conditions (i j )-(v j ) in the proof of Theorem 1.7 and also the following one for all j ∈ N = {1, 2, . ..}: (vi j ) f j (bD j ) ∩ K j = ∅.
(Unlike in the proof of Theorem 1.7, D j need not be a disc.)Note that (vi 0 ) holds true.For the inductive step, fix j ∈ N and assume that we have suitable objects D k , a k , f k , and ǫ k satisfying (i k ), (iii k ), (iv k ), and (vi k ) for all k ∈ {0, . . ., j − 1}.Choose ǫ j > 0 so small that (v j ) holds.Reasoning as in the proof of Theorem 1.7 we may assume that f j−1 extends to a holomorphic embedding f j−1 : Σ → X on a connected, smoothly bounded compact domain Σ ⊂ 2D such that D j−1 ⊂ Σ, D j−1 is a strong deformation retract of Σ, and there is a point α j ∈ Σ with f j−1 (α j ) = β j .By a small perturbation of the map f j−1 keeping the above conditions in place, we can ensure in addition that f j−1 (Σ) is polynomially convex in C 2 ; see the argument in the first part of the proof of Lemma 1.1 based on Stolzenberg's theorem [41].We can therefore choose a pair of smoothly bounded, relatively compact, pseudoconvex domains U 1 ⋐ U 2 ⋐ X which are Runge in C 2 such that f j−1 (Σ) ⊂ U 1 .We place a suitable labyrinth Γ = Γ j in U 2 \ U 1 and choose a holomorphic embedding f j : Σ → C 2 as in the proof of Theorem 1.7 with condition (a) replaced by (a') f j (bΣ) ∩ (K j ∪ U 2 ) = ∅.
In particular, conditions (b)-(d) and (8.2) in the proof of Theorem 1.7 are satisfied (in this case Σ ′ is a smoothly bounded compact domain in Σ containing D j−1 ∪ {α j } in its interior and such that D j−1 is a strong deformation retract of Σ ′ ).In view of (a') and the mentioned conditions, there is a connected, smoothly bounded compact domain D j ⊂ Σ ⊂ 2D such that D j−1 ∪ {α j } ⊂ Dj , f j (D j ) ⊂ X, and f j (bD j ) ∩ (K j ∪ U 2 ) = ∅.
(The last condition will be the key to ensure almost properness of the limit map; compare with the condition f j (bD j ) ∩ U 2 = ∅ in the proof of Theorem 1.7.In general, since the domain X need not be pseudoconvex and Runge in C 2 , we cannot choose D j so that D j−1 is a strong deformation retract of D j ; possibly D j has more boundary components than D j−1 .)Then, conditions (i j )-(vi j ) hold true, which closes the induction.
Set M = j∈N D j ⊂ 2D, a connected relatively compact domain in C. By the reasoning in the proof of Theorem 1.7, there is a limit map F = lim j→∞ f j : M → X which is a complete injective holomorphic immersion such that B ⊂ F (M). Finally, condition (vi j ) ensures that F is an almost proper map provided that each number ǫ j > 0 in the inductive process is chosen sufficiently small.Indeed, by such a choice (similar to that in (v j )) we can manage to grant that F (bD j ) ∩ K j = ∅ for all j ∈ N; see (ii j ) and (vi j ).This, (i j ), and (8.3) imply the almost properness of F : M → X.

Theorem 1 . 5 .
If R is a compact Riemann surface and B ⊂ C 2 is a countable subset, there exist a Cantor set C ⊂ R and an almost proper injective holomorphic immersion F : R \ C ֒→ C 2 whose image contains B. If B is closed and discrete in C 2 then F can be chosen to be a proper holomorphic embedding.Hence, every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in C 2 .The Cantor sets which arise in the proof of Theorem 1.5 are small modifications of the standard Cantor set in the plane, and they have almost full measure in a surrounding domain.The last statement in Theorem 1.5 generalizes a recent result by Di Salvo and Wold [16, Theorem 1.1], who constructed a Cantor set of large measure in CP 1 whose complement admits a proper holomorphic embedding in C 2 .The first examples of Cantor sets in CP 1

Theorem 1 . 6 .
Let D ⊂ C 2 be a pseudoconvex Runge domain and B be a countable subset of D. On every open Riemann surface S there are a domain M, which is diffeotopic to S, and a complete, almost proper, injective holomorphic immersion F : M ֒→ D such that B ⊂ F (M).If in addition the set B is closed in D and discrete, then F : M ֒→ D can be chosen to be a complete proper holomorphic embedding.

satisfies the conclusion of Lemma 1. 1 .
Remark 2.3.(A) The same proof shows that Lemma 1.1 also holds if the holomorphic map f : M → C 2 has finitely many branch points in M ; see [28, Theorem 1.1] for the details.(B) The proof of Lemma 1.1 can be adapted to give an analogous result for embeddings of bordered Riemann surfaces in C × C * .In this case, the compact set L ⊂ C × C * should be holomorphically convex in C × C * , and condition (i) in the lemma should be replaced by asking that F (E ∪ bM) lies outside the cylinder {(z 1 , z 2 ) ∈ C 2 : |z 1 | ≤ r, 1/r ≤ |z 2 | ≤ r} for a given r > 1.An inductive application of this lemma yields the analogue of Corollary 1.3 for proper holomorphic embeddings M ֒→ C × C * (see Ritter [38, Theorem 4]).
The first part of condition (b) is ensured by Lemma 1.1-(e), while the second part is granted by Lemma 1.1-(d).Condition (c) is implied by (2 j ) and(3.6).

Theorem 4 . 1 .
IN CP 1 INTO C 2 Recall that a circle domain in CP 1 is an open domain of the form (4.1) Ω = CP 1 \ ∞ i=0 D i where D i are pairwise disjoint closed round discs.By the uniformization theorem of He and Schramm [33], every domain of the form (4.1), where D i are pairwise disjoint closed topological discs (homeomorphic images of D), is conformally equivalent to a circle domain.We give a simpler proof of the following result [29, Theorem 1.1] due to Forstnerič and Wold.Every circle domain in CP 1 embeds properly holomorphically into C 2 .

Lemma 4 . 2 .
Let Ω be a circle domain (4.1) in CP 1 , and let k

Remark 4 . 4 .
A geometric disc in a Riemann surface R is the image of a round disc in the universal covering space R ∈ {CP 1 , C, D} of R. A circle domain in R is a domain all of whose complementary connected components are closed geometric disks and points (punctures).By He and Schramm [33, Theorem 0.2], every open Riemann surface with finite genus and at most countably many ends is conformally equivalent to a circle domain Ω in a compact Riemann surface R. If Lemma 4.2 can be proved for such domains without punctures, it would follow that any such domain embeds properly holomorphically in C 2 .

Lemma 5 . 1 .
Assume that R is a compact Riemann surface of genus ν and Ω is a connected open domain in R of the same genus ν.Given a closed set L in R which is a union of connected components of R \ Ω and an open set V ⊂ R containing L, there exist finitely many pairwise disjoint, smoothly bounded closed discs (a) sup x∈K 0 |f 1 (x) − f0 (x)| < ǫ 1 /2.Hence, sup x∈K 0 |f 1 (x) − f 0 (x)| < ǫ 1 by (6.5).

[ 13 ,
Lemma 2.4] there is a compact polynomially convex set Γ ⊂ D d+1 \ D d whose connected components are holomorphically contractible (for example, convex) such that D d ∪ Γ is polynomially convex and length(σ) > 1 for every path σ : [0, 1] → D \ Γ with σ(0) ∈ D d and σ(1) ∈ D \ D d+1 .It follows that the compact set [35,The last two conditions in (2.12) are not discussed in[35, proof of Lemma 2.2].Without them, we are unable to complete the proof of Lemma 2.2 (the problem appears in the construction of a shear ψ in (2.16)).Indeed, we do not know how to prove[35, Lemma 2.2]without assuming that the linear projections C 2 → C sufficiently close to π 1 are proper on Γ.(3) Pushing E ′ ∪ Γ out of the ball rB.We shall find an automorphism Φ ∈ Aut(C 2 ) sending the set E ′ ∪ Γ out of the given ball rB.This is accomplished by the following lemma based on [49,Lemma 1]and[35, Lemma 2.2].In light of Remark 2.1, we include a proof.Lemma 2.2.Let E ′ and Γ be as in(2.11),satisfying conditions (2.12) and (2.13) for the projection π 1