J Geom AnalDOI 10.1007/s12220-012-9306-4

Proper Holomorphic Embeddings of Riemann Surfaceswith Arbitrary Topology into C

2

Antonio Alarcón · Francisco J. López

Received: 28 November 2011© Mathematica Josephina, Inc. 2012

Abstract We prove that given an open Riemann surface N of arbitrary (finite or in-finite) topology, there exists an open domain M ⊂ N homeomorphic to N whichproperly holomorphically embeds in C

2. Furthermore, M can be chosen with hyper-bolic conformal type.

In particular, any open orientable surface M admits a complex structure C suchthat (M, C) can be properly holomorphically embedded into C

2.

Keywords Riemann surfaces · Holomorphic embeddings

Mathematics Subject Classification 32C22 · 32H02

1 Introduction

It is classically known that any open Riemann surface properly holomorphically em-beds in C

3 and immerses in C2 [3, 12, 13, 15]. Bell–Narasimhan’s conjecture asserts

that any open Riemann surface can be properly holomorphically embedded in C2 [2,

Conjecture 3.7, p. 20]. Although this old embeddability problem has generated vastliterature, it still remains open.

Communicated by Marco Abate.

Both authors are partially supported by MCYT-FEDER research projects MTM2007-61775and MTM2011-22547, and Junta de Andalucía Grant P09-FQM-5088.

A. Alarcón (�) · F.J. LópezDepartamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spaine-mail: alarcon@ugr.es

F.J. Lópeze-mail: fjlopez@ugr.es

A. Alarcón, F.J. López

The first existence results for discs and annuli can be found in [16] and [1, 10],respectively. More recently, it has been proved that any finitely connected planar do-main without isolated boundary points properly holomorphically embeds into C

2 [9](see also [5]). Furthermore, any open orientable surface of finite topology admits acomplex structure properly holomorphically embedding in C

2 [4]. In the last fewyears, this area has experienced a great growth. Especially interesting are the worksby Wold [17, 18] and Forstneric and Wold [7]. These authors have shown that anybordered Riemann surface R whose closure R admits a (non-proper) holomorphicembedding f : R → C

2 actually properly holomorphically embeds into C2. (A bor-

dered Riemann surface is the interior of a compact one-dimensional complex man-ifold with smooth boundary consisting of finitely many closed Jordan arcs.) Theirmethod consists of deforming f (R) into an unbounded embedded complex curve� ⊂ C

2, such that � is biholomorphic to R and there exists a Fatou–Bieberbachdomain � ⊂ C

2 with � ⊂ � and ∂� ⊂ ∂�. Then �|� : � → C2 is a proper holo-

morphic embedding, where � : � → C2 is a Fatou–Bieberbach map. The embeddingproblem for infinitely connected circular planar domains has been treated in [8, 11].

In all these constructions, the topological type of the surface, and even its confor-mal structure, is not changed during the process.

The aim of this paper is to show that the topology of an open Riemann surfaceplays no role in this setting. We extend the above-mentioned result by Cerne andForstneric [4] to the case of surfaces with arbitrary (finite or infinite) topology, prov-ing the following topological version of Bell–Narasimhan’s conjecture:

Main Theorem Let N be an open Riemann surface. Then there exists an opendomain M ⊂ N homeomorphic to N carrying a proper holomorphic embeddingY : M → C

2. In particular, any open orientable surface M admits a complex struc-ture C such that the Riemann surface (M, C) properly holomorphically embeds in C

2.

The surface N (and so M) can have uncountably many ends and an arbitrary(possibly infinite) number of handles. For instance, M can be chosen homeomor-phic to an arbitrary compact surface minus a Cantor set. In this line, Orevkov [14]showed a Cantor set K ⊂ C whose complement C − K admits a proper holomorphicembedding into C

2.The proper holomorphic embedding Y : M → C

2 in the Main Theorem is ob-tained by a subtle modification of Forstneric’s and Wold’s ideas. The map Y is thelimit of a sequence of holomorphic embeddings {Yn : Mn → C

2}n∈N, where {Mn}n∈N

is a suitable expansive sequence of compact regions in N and M = ⋃n∈N

Mn. Thesequence is constructed by combining a bridge principle for holomorphic embeddingswith a refinement of the techniques in [7, 17, 18].

It is worth mentioning that the open Riemann surface M in the Main Theoremcan be chosen of hyperbolic conformal type. Finally, let us point out that the MainTheorem actually follows from a more general extension result for holomorphic em-beddings into C

2 (see Theorem 4.5).

Proper Holomorphic Embeddings of Riemann Surfaces

2 Preliminaries

As usual, we denote by ‖ · ‖ the Euclidean norm in Cn, n ∈ N, and for any com-

pact topological space X and continuous map f : X → Cn we denote by ‖f ‖ =max{‖f (p)‖ |p ∈ X} the maximum norm of f on X.

Non-compact Riemann surfaces without boundary are said to be open.

Remark 2.1 Throughout this paper, N and ω will denote a fixed but arbitrary openRiemann surface and a complete smooth conformal metric on N , respectively.

For any S ⊂ N , S◦ and S will denote the interior and the closure of S in N ,respectively.

Given a Riemann surface M contained in N , we denote by ∂M the 1-dimensionaltopological manifold determined by its boundary points. Open connected subsets ofN will be called domains, and those proper topological subspaces of N being Rie-mann surfaces with boundary are said to be regions.

A subset S ⊂ N is said to be Runge if the inclusion map jS : S ↪→ N in-duces a group monomorphism (jS)∗ : H1(S,Z) → H1(N ,Z). In this case, we iden-tify the groups H1(S,Z) and (jS)∗(H1(S,Z)) ⊂ H1(N ,Z) via (jS)∗ and considerH1(S,Z) ⊂ H1(N ,Z). If S is a compact region, S is Runge if and only if N − S hasno relatively compact components in N .

Two Runge subsets S1, S2 ⊂ N are said to be isotopic if H1(S1,Z) = H1(S2,Z).Two Runge subsets S1, S2 ⊂ N are said to be homeomorphically isotopic if thereexists a homeomorphism σ : S1 → S2 such that σ∗ = IdH1(S1,Z), where σ∗ is theinduced group morphism on homology. In this case, σ is said to be an isotopicalhomeomorphism. Two Runge compact regions in N are isotopic if and only if theyare homeomorphically isotopic.

Definition 2.2 (Admissible set) A compact subset S ⊂ N is said to be admissible ifand only if

• MS := S◦ is a finite collection of pairwise disjoint compact regions in N with C 0

boundary,• CS := S − MS consists of a finite collection of pairwise disjoint analytical Jordan

arcs,• any component α of CS with an endpoint P ∈ MS admits an analytical extension

β in N such that the unique component of β − α with endpoint P lies in MS , and• S is Runge.

Let S be an admissible subset in N . An open Runge subset W in N is said to be atubular neighborhood of S if S ⊂ W , S is isotopic to W , and χ(W − S) = 0, whereχ(·) means the Euler characteristic. In addition, if W is a Runge compact (possiblynon-connected) surface with boundary, S is isotopic to W , and χ(W − S) = 0, thenW is said to be a compact tubular neighborhood of S.

For any subset S ⊂ N , a function f : S → Cn, n ∈ N, is said to be holomorphic if

there exists an open set U ⊂ N containing S and a holomorphic function h : U → C

such that h|S = f .

A. Alarcón, F.J. López

Definition 2.3 Let S ⊂ N be an admissible set. A function f : S → Cn, n ∈ N, is

said to be admissible if f |MSis holomorphic, and for any component α of CS and

any open analytical Jordan arc β in N containing α, f admits a smooth extension fβ

to β satisfying that fβ |U∩β = h|U∩β , where U ⊂ N is an open domain containingMS and h : U → C

n is a holomorphic extension of f .

Likewise, a complex 1-form θ of type (1,0) on S is said to be admissible if forany closed conformal disc (W, z) in N such that W ∩ S is admissible, then θ |W∩S =g(z)dz for an admissible function g : W ∩ S → C.

Given an admissible function f : S → Cn, we set df as the vectorial admissible 1-

form given by df |MS= d(f |MS

) and df |α∩W = (f ◦ α)′(x)dz|α∩W , where (W, z =x + ıy) is a conformal chart on N such that α ∩ W = z−1(R ∩ z(W)).

If f : S → Cn is admissible, then the C 1-norm of f on S is given by

‖f ‖1 = maxS

(‖f ‖ + ‖df/ω‖).

3 Main Lemma

Set π1 : C2 → C the projection π1(z,w) = z. We will need the following definition.

Definition 3.1 [7, 18] Let M ⊂ N be a Riemann surface possibly with boundary,and let X : M → C

2 be a proper holomorphic embedding. A point p = (p1,p2)

of the complex curve � := X(M) is said to be exposed (with respect to π1) if thecomplex line �p = π−1

1 (π1(p)) = {(p1,w) | w ∈ C} intersects � only at p and thisintersection is transverse, that is to say, �p ∩ � = {p} and Tp�p ∩ Tp� = {0}.

The proof of the following technical lemma is inspired by the ideas of Forstnericand Wold [7, 18]. Roughly speaking, Lemma 3.2 below asserts that an embeddedbordered Riemann surface in C

2 whose boundary lies outside a Euclidean ball can beperturbed near the boundary in such a way that the boundary of the arising surfacelies outside a bigger ball in C

2. The strength of this lemma is that embeddedness ispreserved in this process.

For any r > 0 we denote by B(r) = {z ∈ C2 | ‖z‖ < r} and B(r) = {z ∈ C

2 |‖z‖ ≤ r}.

Lemma 3.2 Let M be a Runge compact region in N , let X : M → C2 be a holomor-

phic embedding, and let r > 0 such that

X(∂M) ⊂ C2 − B(r). (3.1)

Then, for any ξ > 0 and any r > r , there exist a Runge compact region M on Nand a holomorphic embedding X : M → C

2 satisfying that:

(L.1) M is a compact tubular neighborhood of M ,(L.2) ‖X − X‖1 < ξ on M ,(L.3) X(∂M) ⊂ C

2 − B(r), and

Proper Holomorphic Embeddings of Riemann Surfaces

(L.4) X(M − M◦) ⊂ C2 − B(r).

The above lemma would be a straightforward consequence of [7, Corollary 1.2]except for property (L.4), which plays a crucial role in the proof of our main re-sult. Its proof consists of a suitable modification of the techniques in [7] for ensuringthis property. The proper embedding Y : M → C

2 in the Main Theorem will be ob-tained as a limit of a sequence of holomorphic embeddings {Yj : Nj → C

2}j∈N withYj (∂Nj ) ⊂ C

2 − B(rj ) and Yj (Nj − N◦j−1) ⊂ C

2 − B(rj − 1), where {Nj }j∈N is anexhaustion of M by Runge compact regions and {rj }j∈N ↗ +∞. The latter propertyof Yj , which is obtained from (L.4), is the key for guaranteeing the properness of Y(see the proof of Theorem 4.5 in Sect. 4).

Proof of Lemma 3.2 Fix ξ0 ∈ ]0, ξ [ so that

X(∂M) ⊂ C2 − B(r + ξ0); (3.2)

see (3.1). Take ε0 > 0 to be specified later.We begin exposing boundary points as in [7].Since we are assuming that X holomorphically extends beyond M , there exists a

Runge compact region N1 on N and a holomorphic embedding Y1 : N1 → C suchthat

(a.1) N1 is a compact tubular neighborhood of M ,(a.2) Y1|M = X, and(a.3) Y1(N1 − M◦) ⊂ C

2 − B(r + ξ0).

Write ∂M = ⋃mj=1 Cj , where {Cj }mj=1 are the connected components of ∂M .

Choose a point aj ∈ Cj and an analytic Jordan arc γj ⊂ N◦1 − M◦ with initial point

aj , otherwise disjoint from ∂M and such that the intersection of γj and Cj is trans-verse, ∀j = 1, . . . ,m. Take the arcs {γj }j=1,...,m so that M

⋃(∪m

j=1γj ) is admissible.Let bj denote the other endpoint of γj , and let U ⊂ N◦

1 be a compact tubular neigh-borhood of M such that bj /∈ U , U ∪ (

⋃mj=1 γj ) is admissible, and γj := γj ∩ U is a

Jordan arc with an endpoint at aj , j = 1, . . . ,m.On the other hand, consider pairwise disjoint smooth regular Jordan arcs {λj | j =

1, . . . ,m} in C2 such that

(b.1) Y1(aj ) is an endpoint of λj , Y1(γj ) ⊂ λj , and (λj − Y1(γj )) ∩ Y1(U) = ∅,(b.2) λj ⊂ C2 − B(r + ξ0),(b.3) the other endpoint pj of λj satisfies �pj

∩ (Y1(M) ∪ (⋃m

i=1 λi)) = {pj } andTpj

�pj∩ T C

pjλj = {0}, where T C

pjλj is the complexification of the real tangent

line to λj at pj , and(b.4) |π1(pj )| > r + ξ0,

for all j ∈ {1, . . . ,m}; see Figure 1. Notice that (b.3) is a generalization of Defini-tion 3.1. Item (b.2) is possible since (a.3) holds. Properties (b.2) and (b.4) will beused to guarantee (L.4) in the statement of the lemma.

Consider an admissible embedding Y1 : U ∪ (⋃m

j=1 γj ) → C2 such that

(c.1) Y1|U = Y1, and

A. Alarcón, F.J. López

Fig. 1 The arcs λj

(c.2) Y1(γj ) = λj , j = 1, . . . ,m. In particular, Y1(bj ) = pj .

By Mergelyan’s Theorem (see, for instance, [6, Theorem 3.2]), we can find aRunge compact region N2 and a holomorphic embedding Y2 : N2 → C

2 such that

(d.1) N2 is a compact tubular neighborhood of U (hence, of M), with γj ⊂ N2 ⊂ N◦1

and bj ∈ ∂N2, j = 1, . . . ,m,(d.2) ‖Y2 − Y1‖1 < ε0 on U ∪ (

⋃mj=1 γj ),

(d.3) Y2(N2 − M◦) ⊂ C2 − B(r + ξ0), and(d.4) Y2(bj ) = Y1(bj ) = pj is an exposed point for Y2(N2).

Notice that (d.3) can be guaranteed from (a.3), (b.2), (c.1), and (c.2). It is needed toensure (L.4). Property (d.4) is possible thanks to (b.3) (see [7, Theorem 4.2] for moredetails).

The second step in the proof of Lemma 3.2 consists of pushing Y2(∂N2) out ofB(r). Now we are inspired by [18] and [7, Theorem 5.1].

Write ∂N2 = ⋃mj=1 �j , where {�j }mj=1 are the connected components of ∂N2. Set

g : C2 → C × C, g(z,w) =

⎛

⎝z,w +m∑

j=1

αj

z − π1(Y2(bj ))

⎞

⎠ , (3.3)

where the coefficients αj ∈ C−{0} are chosen so that the following assertions hold:

(e.1) π2 maps the curve μj := (g ◦ Y2)(�j − {bj }) ⊂ C2 into an unbounded curve

δj ⊂ C, and π2 : μj → δj is a diffeomorphism near infinity, where π2 : C2 → C

is given by π2(z,w) = w.(e.2) D(ρ) ∪ (

⋃mj=1 δj ) is Runge in C for any large enough ρ ∈ R, where D(ρ) =

{z ∈ C | |z| ≤ ρ}.(e.3) ‖g ◦ Y2 − Y2‖1 < ε0 on M .(e.4) (g ◦ Y2)((N2 − {bj }j=1,...,m) − M◦) ⊂ C

2 − B(r + ξ0).

This can be guaranteed by a careful choice of the argument of the complex numberαj , while |αj | must be chosen as small as needed, j = 1, . . . ,m. To achieve properties(e.3) and (e.4), we argue as follows. First, fix pairwise disjoint small open discs Wj ⊂N , j = 1, . . . ,m, such that bj ∈ Wj and

∣∣π1

(Y2(Wj ∩ N2)

)∣∣ > r + ξ0 for all j ; (3.4)

Proper Holomorphic Embeddings of Riemann Surfaces

see (b.4). Then choose |αj |, j = 1, . . . ,m, small enough so that (g ◦Y2)(N2 − (M◦ ∪j

Wj )) ⊂ C2 − B(r + ξ0) (see (d.3)) and ‖g ◦ Y2 − Y2‖1 < ε0 on M . Since π1 ◦ g =

π1, (3.4) gives that (g ◦ Y2)(N2 ∩ Wj) ⊂ C2 − B(r + ξ0) as well. Property (e.4) is

important to obtain (L.4).Label W := N2 −{bj }j=1,...,m, set Z : W → C

2, Z := g ◦Y2|W , and note that Z isa well-defined holomorphic embedding thanks to (d.4). Furthermore, Z has the fol-lowing property: there exists a compact polynomially convex K0 ⊂ Z(W) in C

2 suchthat K := K0 ∪ B(r + ξ0) is polynomially convex and Z(M) ⊂ K ⊂ C

2 − Z(∂W)

(see (e.4) and the proof of Theorem 5.1 in [7]). Moreover, there exists a holomorphicautomorphism φ of C

2 such that

(f.1) Z(M) ∪ B(r + ξ0) ⊂ K ⊂ C2 − Z(∂W); notice that ∂W = ∂N2 − {bj }j=1,...,m,

(f.2) ‖φ − IdC2‖ < ε0 on K , and ‖φ ◦ Z − Z‖1 < ε0 on M , and(f.3) (φ ◦ Z)(∂W) ⊂ C

2 − B(r).

Such φ is constructed in [17] from (e.1) and (e.2); see also the proof of Theorem 5.1in [7].

Define X : W → C2, X := φ ◦ Z, and let us check that X almost satisfies the

conclusion of Lemma 3.2.

• W ◦ is an open tubular neighborhood of M ; see (d.1) and the definitions of U andW .

• ‖X − X‖1 < ξ on M . Indeed, use (a.2), (d.2), (e.3), (f.1), and (f.2), and assumethat ε0 was chosen small enough from the beginning.

• X(∂W) ⊂ C2 − B(r); see (f.3).

• X(W −M◦) ⊂ C2 −B(r). Indeed, if ε0 is chosen small enough from the beginning,

then taking into account (f.1), (f.2), and that φ is bijective, we conclude that φ(C2 −B(r + ξ0)) ⊂ C

2 − φ(B(r)). Then use (e.4).

Taking into account these properties of X, we finish by setting M as a suitableshrinking of W satisfying (L.1). The proof is done. �

4 Main Theorem

Theorem 4.5 below follows from Lemma 3.2 and the existence of good exhaustionsof N by compact sets in the following sense.

Definition 4.1 An exhaustion {Mj }j∈N of N by Runge compact regions is said tobe good if Mj−1 ⊂ M◦

j and the Euler characteristic χ(Mj − M◦j−1) ∈ {−1,0} for all

j ≥ 2 (if N has finite topology then χ(Mj − M◦j−1) = 0 for any j large enough).

Lemma 4.2 Given a Runge compact region N ⊂ N , there exists a good exhaustion{Mj }j∈N of N with M1 = N .

Proof Let {Kj }j∈N be an exhaustion of N by compact regions with K1 = N andKj−1 ⊂ K◦

j for all j ≥ 2. Without loss of generality, we can suppose that Kj isRunge for all j . Indeed, otherwise substitute Kj for the smallest compact Runge set

A. Alarcón, F.J. López

in N containing it (just add to Kj the relatively compact components of N − Kj ),and then remove the repeated terms of the new sequence.

We need the following.

Claim 4.3 Let S1 and S2 be two compact bordered surfaces with S1 ⊂ S2 − ∂S2, andassume that S1 is Runge in S2 − ∂S2 and n := −χ(S2 − S◦

1) > 0.Then, there exists a sequence R1 = S1, R2, . . . ,Rn+1 = S2 such that Rj is a

compact Runge region in S2 − ∂S2, Rj ⊂ R◦j+1, and −χ(Rj+1 − R◦

j ) = 1 for allj ∈ {1, . . . , n}.

The claim trivially holds if n = 1. To prove it, reason by induction. As-sume that the statement holds when −χ(S2 − S◦

1) ≤ n − 1, n > 1, and sup-pose that −χ(S2 − S◦

1) = n. Since −χ(S2 − S◦1) > 0, there exists a Jordan curve

α ∈ H1(S2,Z) − H1(S1,Z) contained in S2 − ∂S2 and intersecting S2 − S◦1 in a Jor-

dan arc α with endpoints in ∂S1 and otherwise disjoint from S1. Let S1 be a compacttubular neighborhood of S1 ∪ α in S2 − ∂S2, and notice that S1 is Runge in S2 − ∂S2,−χ(S1 −S◦

1) = 1, and −χ(S2 − S◦1) = n−1. To finish, apply the induction hypothesis

to the couple (S1, S2) and add S1 to the arising sequence.The lemma follows by applying Claim 4.3 to any two consecutive Kj , Kj+1 with

χ(Kj+1 −K◦j ) > 0, and then inserting the generated finite sequences into the original

one. �

We will also need the following.

Definition 4.4 Let K be a compact subset of N , let f : K → C2 be a topological

embedding, and let n ∈ N. We define

�(K,f,n) := 1

2n2inf

{∥∥f (p) − f (q)

∥∥

∣∣∣∣ p,q ∈ K, d(p, q) >

1

n

}

,

where d(·, ·) means distance in the Riemannian surface (N ,ω); see Remark 2.1. No-tice that �(K,f,n) > 0.

Now we can state and prove the main result of this paper.

Theorem 4.5 Let N be a Runge compact region on N and let Y : N → C2 be a

holomorphic embedding. Assume that

Y(∂N) ⊂ C2 − B(s) (4.1)

for a positive s.Then, for any ε > 0, there exist an open domain M ⊂ N and a proper holomor-

phic embedding X : M → C2 satisfying

(T.1) N ⊂ M, M is Runge and isotopic to N ,(T.2) ‖X − Y‖1 < ε on N , and(T.3) X (M − N◦) ⊂ C

2 − B(s).

Proper Holomorphic Embeddings of Riemann Surfaces

Proof Consider a good exhaustion {Mj }j∈N of N by Runge compact regions so thatM1 = N ; see Lemma 4.2.

Since Y(∂N) is compact, equation (4.1) guarantees the existence of s0 > s suchthat

Y(∂N) ⊂ C2 − B(s0). (4.2)

Take ε0 > 0 with

ε0 < min{ε, s0 − s}, (4.3)

to be specified later.

Claim 4.6 There exists a sequence {�j }j∈N := {(Nj , σj , Yj , εj )}j∈N, where

• Nj is a Runge compact region on N isotopic to Mj ,• σj : Nj → Mj is an isotopic homeomorphism,• Yj : Nj → C

2 is a holomorphic embedding, and• εj > 0, j ∈ N,

such that

(Aj ) Nj−1 ⊂ N◦j and σj |Nj−1 = σj−1,

(Bj ) εj < min{ε0/2j , �(Nj−1, Yj−1, j) , εj−1 , �j−1}, where

�j−1 := 1

2jmin

{

minNk

∥∥∥∥dYk

ω

∥∥∥∥

∣∣ k = 1, . . . , j − 1

}

> 0,

(Cj ) ‖Yj − Yj−1‖1 < εj on Nj−1,(Dj ) Yj (∂Nj ) ⊂ C

2 − B(s0 + j − 1), and(Ej ) Yj (Nj − N◦

j−1) ⊂ C2 − B(s0 + j − 2).

Proof The sequence is constructed inductively. Set �1 := (N, Id|N,Y, ε1), whereε1 < ε0/4. Equation (4.2) gives property (D1), whereas properties (A1), (B1), (C1),and (E1) do not make sense.

To prove the inductive step, assume that �1, . . . ,�j−1 are already constructedsatisfying the required properties, and let us construct �j , j ≥ 2.

We need to distinguish two cases, depending on χ(Mj − M◦j−1).

• Case 1. Assume χ(Mj − M◦j−1) = 0. Apply Lemma 3.2 to the data

M = Nj−1, X = Yj−1, r = s0 + j − 2, ξ = εj , and r = s0 + j − 1,

where εj is any positive number satisfying (Bj ). Observe that the lemma can beapplied thanks to property (Dj−1). Then we set �j := (Nj = M,σj , Yj = X, εj ),where M and X are the data arising from the lemma and σj : Nj → Mj is anyhomeomorphism with σj |Nj−1 = σj−1. Properties (Aj ), (Cj ), (Dj ), and (Ej ) di-rectly follow from (L.1), (L.2), (L.3), and (L.4) of Lemma 3.2, respectively.

• Case 2. Assume χ(Mj − M◦j−1) = −1. First of all, fix εj > 0 satisfying (Bj ).

Take a Runge compact region R1 and a holomorphic embedding Z1 : R1 → C2

such that

A. Alarcón, F.J. López

(a.1) R1 is a compact tubular neighborhood of Nj−1,(a.2) Z1|Nj−1 = Yj−1, and

(a.3) Z1(R1 − N◦j−1) ⊂ C

2 − B(s0 + j − 2).

(Recall that Yj−1 extends holomorphically beyond Nj−1 in N .) Consider a smoothJordan curve α ∈ H1(Mj ,Z)− H1(Mj−1,Z) contained in M◦

j and intersecting Mj −M◦

j−1 in a Jordan arc α with endpoints a, b in ∂Mj−1 and otherwise disjoint fromMj−1. Since Mj−1 and Mj are Runge and χ(Mj − M◦

j−1) = −1, H1(Mj ,Z) =H1(Mj−1 ∪ α,Z) and Mj−1 ∪ α is Runge as well. Take an analytic Jordan arc γ ⊂N −N◦

j−1 with endpoints σ−1j−1(a), σ−1

j−1(b) in ∂Nj−1, otherwise disjoint from Nj−1,transversally intersecting ∂Nj−1, and such that Nj−1 ∪ γ is admissible. Also takean isotopic homeomorphism ς : Nj−1 ∪ γ → Mj−1 ∪ α so that ς |Nj−1 = σj−1 andς(γ ) = α.

On the other hand, consider in C2 a smooth regular Jordan arc λ agreeing withZ1(γ ∩ R1) near the endpoints Z1(σ

−1j−1(a)) and Z1(σ

−1j−1(b)), and such that

(b.1) (λ − Z1(γ ∩ R1)) ∩ Z1(Nj−1) = ∅, and(b.2) λ ⊂ C

2 − B(s0 + j − 2).

This choice of λ is possible thanks to property (a.3). Consider the admissible em-bedding Z1 : Nj−1 ∪ γ → C

2 given by Z1|Nj−1 = Z1 and Z1(γ ) = λ. Mergelyan’sTheorem provides a Runge compact region R2 and a holomorphic embedding Z2 :R2 → C

2 satisfying that

(c.1) R2 is a compact tubular neighborhood of Nj−1 ∪ γ ,(c.2) ‖Z2 − Z1‖1 < εj/2 on Nj−1 ∪ γ , and(c.3) Z2(R2 − N◦

j−1) ⊂ C2 − B(s0 + j − 2).

Since Z2(R2 −N◦j−1) is compact, (c.3) implies the existence of ε ∈ ]0, εj /2[ small

enough so that

Z2(R2 − N◦

j−1

) ⊂ C2 − B(s0 + j − 2 + ε). (4.4)

Set �j := (Nj = M,σj , Yj = X, εj ), where M and X are the data arising fromLemma 3.2 applied to the data

M = R2, X = Z2, r = s0 + j − 2 + ε, ξ = ε, and r = s0 + j − 1,

where σj : Nj → Mj is any homeomorphism with σj |Nj−1∪γ = ς . Observe that thelemma can be applied thanks to (c.3). Property (Aj ) follows from (a.1), (c.1), andLemma 3.2-(L.1). Property (Cj ) is implied by (a.2), (c.2), and Lemma 3.2-(L.2).Item (L.3) in Lemma 3.2 gives (Dj ). Finally, to check (Ej ) consider a point p ∈Nj − N◦

j−1 and let us distinguish cases. If p ∈ Nj − R◦2 then Lemma 3.2-(L.4) gives

Yj (p) ∈ C2 − B(s0 + j − 2) and we are done. Otherwise, p ∈ R2 − N◦

j−1, and in this

case Lemma 3.2-(L.2) and (4.4) guarantee that Yj (p) ∈ C2 − B(s0 + j − 2) as well.

This concludes the construction of the sequence {�j }j∈N satisfying the desiredproperties. �

Proper Holomorphic Embeddings of Riemann Surfaces

Set M := ∪j∈NNj and σ : M → N , σ |Nj= σj . Since {Mj }j∈N is an exhaustion

of N by Runge compact regions and σj is an isotopic homeomorphism for all j , thenσ is an isotopic homeomorphism as well, and statement (T.1) holds.

Properties (Bj ) and (Cj ), j ∈ N, imply that the sequence of holomorphic maps{Yj }j∈N uniformly converges on compact subsets of M to a holomorphic map X :M → C

2 satisfying

‖X − Y‖1 < ε0 on N. (4.5)

(Recall that Y1 = Y and N1 = N .) This implies (T.2) (see (4.3)).Let us check (T.3). Take p ∈ M −N◦. Then, there exists j ≥ 2 such that p ∈ Nj −

N◦j−1 and, by properties (Cj ) and (Ej ), X (p) ∈ C

2 −B(s0 +j −2−ε0) ⊂ C2 −B(s);

see (4.3).To check that X is injective, we have to work a little further. Take p,q ∈ M,

p �= q , and let us prove that X (p) �= X (q). Indeed, consider a large enough j0 ∈ N sothat {p,q} ⊂ Nj and d(p, q) > 1/j , ∀j ≥ j0. Then, for any j > j0, from properties(Bj ) and (Cj ), one has

∥∥Yj−1(p) − Yj−1(q)

∥∥ ≤ ∥

∥Yj−1(p) − Yj (p)∥∥ + ∥

∥Yj (p) − Yj (q)∥∥

+ ∥∥Yj (q) − Yj−1(q)

∥∥

< 2εj + ∥∥Yj (p) − Yj (q)

∥∥

≤ 1

j2

∥∥Yj−1(p) − Yj−1(q)

∥∥ + ∥

∥Yj (p) − Yj (q)∥∥;

see Definition 4.4. Therefore, ‖Yj (p) − Yj (q)‖ ≥ (1 − 1/j2)‖Yj−1(p) − Yj−1(q)‖,∀j > j0, and so

∥∥Yj0+i (p) − Yj0+i (q)

∥∥ ≥ ∥

∥Yj0(p) − Yj0(q)∥∥ ·

j0+i∏

j=j0+1

(

1 − 1

j2

)

, ∀i ∈ N.

Taking limits in the above inequality as i → ∞, we obtain that ‖X (p) − X (q)‖ ≥12‖Yj0(p) − Yj0(q)‖ > 0 (recall that Yj0 is an embedding), and we are done.

Let us check that X : M → C2 is proper. Consider a compact subset K ⊂ C

2.It suffices to prove that X −1(K) is compact in M. Take j0 ∈ N large enough sothat K ⊂ B(s0 + j0 − 2 − ε0). On the other hand, properties (Bj ) and (Ej ) give thatX (Nj − N◦

j−1) ⊂ C2 − B(s0 + j0 − 2 − ε0) for any j ≥ j0. Hence X −1(K) ⊂ Nj0−1

which is compact in M, and we are done.Finally, let us check that X is an immersion, hence an embedding. Take p ∈ M

and j0 ∈ N such that p ∈ Nj ∀j ≥ j0. Then

‖dX /ω‖(p) ≥ ‖dYj0/ω‖(p) −∑

j>j0

‖Yj − Yj−1‖1 ≥ ‖dYj0/ω‖(p) −∑

j>j0

εj

≥ ‖dYj0/ω‖(p) −∑

j>j0

�j−1 ≥ ‖dYj0/ω‖(p)

(

1 −∑

j>j0

1

2j

)

≥ 1

2‖dYj0/ω‖(p) > 0,

A. Alarcón, F.J. López

where we have used (Bj ), j > j0. The proof of Theorem 4.5 is done. �

The Main Theorem in the Introduction easily follows from Theorem 4.5. Indeed,let N be an open Riemann surface, let N be a conformal compact disc on N , and letY : N → C

2 be a holomorphic embedding with Y(∂N) ⊂ C2 − B(1). Then Theorem

4.5 provides an open domain M ⊂ N homeomorphic to N and a proper holomor-phic embedding X : M → C

2. Furthermore, if we substitute N for any hyperbolicisotopic subdomain of N , the arising domain M is hyperbolic as well.

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