Winding numbers and attaching Riemannsurfaces

Georgios Dimitroglou Rizell

U.U.D.M. Project Report 2007:24

Examensarbete i matematik, 20 poäng

Handledare och examinator: Tobias Ekholm

Juni 2007

Department of Mathematics

Uppsala University

Winding numbers and attaching Riemann surfaces

Georgios Dimitroglou Rizell

Advisor: Tobias Ekholm

June 18, 2007

Abstract

Consider a smoothly embedded circle γ in C2 and a closed arc

A ( γ. Suppose the projection of π1 ◦ γ to the first coordinate line is

in general position and has no selfintersection point on π1 ◦A. Denote

by E the union of all complex lines parallel to the second coordinate

line and passing through a point of A. We give necessary and sufficient

conditions for each neighbourhood of γ∪E to contain the boundary of

a Riemann surface of area bounded from below. The conditon is given

in terms of winding numbers for the projection π1 ◦ γ parallel to the

first coordinate axis and uses classical results on extension of immer-

sions of curves to branched immersions of surfaces into the plane. The

question emerged from the still open problem of existence of Herman

ring cylinders for Henon mappings.

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Acknowledgements

I would like to express my deepest gratitude to Burglind Juhl-Joricke forsuggesting the problem and for helping me along the way, as well as inspiringme and introducing me to interesting subjects in mathematics. In fact,Burglind was my mathematical advisor during this project but because ofvery unusual circumstances it was not possible formally to list her as advisor.I am also very grateful to Tobias Ekholm for helping out as formal advisorfor this master thesis.

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1 Prerequisites

1.1 Introduction

We call a Riemann surface with boundary a smooth compact manifold Mwith boundary such that the open part M\∂M is a Riemann surface, i.e.an analytic 1-dimensional complex manifold.

The existence of Riemann surfaces with boundary in a given compactset occurs to be crucial in various questions of several complex variables andsymplectic geometry. Here we consider the following situation.

Let γ : S1 −→ C2 be a smoothly embedded connected closed curve in C2

(S1 denotes the circle). Suppose moreover that the projection γ1 := π1 ◦ γonto the first coordinate is in general position, i.e. it is an immersion and theself-intersections are double points intersecting transversally. Let A ⊂ S 1

be a closed arc with nonempty interior. Shrinking A we may assume A isconnected and that γ1(A) doesn’t contain any crossings of γ1. Let

E := γ1(A) ×C ⊂ C2

be the union of all complex lines parallell to the second coordinate axispassing through γ1(A). Let U be any neighbourhood of γ1(S1) ∪ E. Weare interested in the existence of Riemann surfaces with boundary in U andwith area bounded from below independently of the choice of U . Our resultwill be given in terms of the winding number of the curve γ1.

Definition 1.1. Let Γ be a compact real 1-dimensional manifold withoutboundary. The winding number at a ∈ C of a smooth map ζ : Γ → C\{a} isthe integer W (ζ, a) := 1

2πi

∫ζ

1z−a

dz. We will say that ζ satisfies the winding

number condition if W (ζ, a) ≥ 0 for all a ∈ C\ζ(Γ).

We prove the following theorem.

Theorem 1.1. Let γ, A and E be as above. Then the following are equiv-alent.

(i) There is an orientation of S1 such that γ1 satisfies the winding numbercondition.

(ii) For each neighbourhood U of γ(S1) ∪ E there exists a connected Rie-mann surface attached to U with area bounded from below. More pre-cisely, there exists a connected Riemann surface M with connectedboundary and a continuous mapping f : M −→ C2 with f(∂M) ⊂ Uand f holomorphic on the interior M\∂M of M , such that the areaof π1 ◦ f(M) is bounded from below by some C > 0 depending only onγ1.

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The question emerged from the still open problem whether Herman ringcylinders exist for Henon mappings. Henon mappings are those polyno-mial automorphisms of C2 which are interesting from the point of view ofdynamics. Henon mappings have the form

H((z1, z2)) = (z2, az1 + p(z2)), (z1, z2) ∈ C2

for a constant a ∈ C and a polynomial p. The Fatou set of the mappingconsists of all points z ∈ C2 which have a neighbourhood on which theiterates of H form a normal family. The Fatou set is open. Herman ringcylinders are Fatou components which are biholomorphic to A × C, whereA is an annulus

A := {z ∈ C; r1 < |z| < r2, r1, r2 ∈ R}

such that the Henon mapping on this component is conjugate to irrationalrotation of the annulus and contraction in the direction of the second co-ordinate axis. They are the counterparts of Herman rings which appear indynamics of rational mappings on the Riemann sphere. Herman rings areFatou components which are conformally equivalent to an annulus such thatthe rational mapping on it is irrational rotation.

The existence of Herman rings was proved using a theorem of Arnold onlinearization of real analytic diffeomorphisms of the circle and later usingquasiconformal surgery. It is still not known whether Herman ring cylindersexist for Henon mappings. If they exist they are Runge domains in C2 whichare biholomorphic to A × C (see §§1.2 for more details). If such domainsexist at all, they must be very exotic.

Note that a domain in Cn is not a Runge domain if there is an attachedRiemann surface which is not contained in the domain (see §§1.2). Thisleads to the following question.

Let U ⊂ C2 be a domain which is biholomorphic to A × C. Consider asmooth closed curve in U which is not contractible. Can one deform it insideU to a curve which extends a Riemann surface? The question differs fromthe one treated in Theorem 1.1. In Theorem 1.1 we use parallel complexlines, but only those that pass through points of a part of the closed curve.In the just mentioned question we consider injectively immersed copies of Cthrough each point of a closed curve, but we do not know whether they arestraightenable, i.e. can be mapped to a coordinate line by a biholomorphicmapping of C2.

For the proof of Theorem 1.1 we need some prerequisites from complexanalysis in several variables and a classical theorem on extension of immer-sions of closed curves in C to branched immersions of Riemann surfaces intoC. We will explain these results below in §2 and §3 and prove the theoremin section §3.

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1.2 Some prerequisites in several complex variables

Definition 1.2 (Hormander [4], 2.5.1, 2.5.5). A domain U ⊂ Cn is a do-main of holomorphy if there exists a holomorphic function f ∈ H(U) whichcannot be extended to a holomorphic function g ∈ H(V ) in a domain V ) U .

Definition 1.3 (Hormander [4], 2.7.1). A domain U ⊂ Cn is a Rungedomain if all holomorphic functions on U can be approximated uniformly oncompacts by polynomials and moreover U is a domain of holomorphy.

Definition 1.4. The polynomially convex hull of a compact set K ⊂ Cn,denoted by K is the set

K := {z ∈ Cn; |p(z)| ≤ maxu∈K |p(u)| for all polynomials p}.

Theorem 1.2. Let M be a compact Riemann surface with boundary andf : M → Cn a continuous function such that f |M\∂M is analytic, thenf(M) is contained in the polynomially convex hull of f(∂M).

Proof. Let p be any polynomial in Cn. p ◦ f is an analytic function onM\∂M . Now the maximum modulus theorem for analytic functions on Rie-mann surfaces give that maxz∈f(M)|p(z)| ≤ maxz∈f(∂M)|p(z)|. Consequentlyf(M) is a subset of the polynomially convex hull of f(∂M).

Theorem 1.3. Let U ⊂ Cn be a domain. Suppose there exists a connectedRiemann surface M with boundary and a continuous mapping f : M −→ Cn

with f(∂M) ⊂ U and f holomorphic on the interior of M . If f(M) is notcontained in U then U is not a Runge domain.

Proof. Suppose U is Runge domain. Hence U is a domain of holomorphy orequivalently it is holomorphically convex, i.e. for each compact K ⊂ U itsholomorphically convex hull

KU := {z ∈ U ; |h(z)| ≤ maxK |h| for all holomorphic functions h on U}

is a compact subset of U . (See Hormander [4], Theorem 2.5.5).By Hormander [4], Theorem 2.7.3, KU = K ∩U for each compact subset

of U . By Theorem 1.2 f(M) is a subset of the polynomially convex hull of

f(∂M), consequently f(M)∩U ⊂ ˆf(∂M)U . But f(M)∩U is open in f(M)since U is open. Therefore f(M) ∩ U cannot be closed in M , otherwise itwould coincide with f(M) since f(M) is connected. But

f(M) ∩ U = f(M) ∩ ˆf(∂M) ∩ U

must be compact since ˆf(∂M)∩U is compact. The contradiction proves thetheorem.

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2 On extending immersions of curves to branched

immersions of Riemann surfaces

2.1 Introduction and formulation of theorem

We need a classical theorem which presents necessary and sufficient condi-tions for a set of closed curves in general position in C ∼= R2 to extend toa branched immersion of a Riemann surface. A discussion of the origin ofthe theorem is given in [5]. Here we will use the modern language of Soule[5] and present one of the proofs given in [5]. This proof differs from earlierones.

Throughout this section we will let P denote a disjoint union of a finiteset of real 1-dimensional compact oriented manifolds without boundary, i.e.a finite set of loops, mapped by ζ : P −→ C in general position.

Definition 2.1. Let M be be a Riemann surface with boundary. A contin-uous map f : M −→ C is called a branched immersion if it is locally anorientation preserving diffeomorphism except at a finite number of points inM\∂M , which we call critical points. Moreover at the critical points of f ,the mapping is equivalent to the map zn, i.e. for any critical point z0 thereis a chart φ : U ⊂ M −→ V ⊂ C, z0 ∈ U , 0 ∈ V , such that f ◦φ−1(z) = zn,z ∈ V .The images of the critical points under f are called branch points, and thenumber n is called the degree of the branch point.

Definition 2.2. Let P be a set of loops like above, f : P −→ C. An exten-sion of f is a compact Riemann surface M with boundary and a branchedimmersion F : M −→ C such that

(i) ∂M = P , and the induced orientation on ∂M from M agree with theorientation of P

(ii) F |∂M= f .

M is called an extension surface and F is called an extension map.

For an embedded loop in the plane, i.e. a Jordan curve, we will callthe bounded component of its complement the inside of the loop and theunbounded component of it’s complement the outisde. Note that this def-inition is independent of the orientation. We will call an embedded looppositive if the winding number in its inside is 1, and negative if it is -1.

Suppose some arc of ζ(P ) is a connected subset not containing any self-intersections. Note that such an arc has exactly two regions adjecent in thecomplement C\ζ(P ). Also, if one of the regions has winding number n thenthe other regions has winding number n ± 1. The region with the biggerwinding number is said to be on the left of the arc while the region with thesmaller winding number is said to be on the right.

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(left)(right)

n n+1

Figure 1: An arc of a loop in general position, n, n + 1 denote the windingnumbers in the respective regions

It turns out that the existence of an extension of ζ only depends on thewinding number of ζ. As a first step we present the following theorem.

Theorem 2.1. Let M be a compact oriented real 2-dimensional manifoldwith boundary ∂M oriented with the induced boundary orientation. Let F :M −→ R2, f = F |∂M be smooth mappings. Then for any regular valuea ∈ R2\F (∂M) of F for which DF |x is orientation preserving for all x ∈F−1(a), the number of preimage points |F−1(a)| = W (f, a).

Proof. A proof for a version of the theorem for nonoriented manifolds usingmod2 winding numbers is outlined in [2]. A similar approach may be usedfor oriented manifolds.

From this theorem we immediately get a necessary condition for a set ofloops to have an extension. We formulate this in a corollary.

Corollary 2.2. Suppose there is an extension F : M −→ C of ζ. Then ζsatisfies the winding number condition.

Proof. Since F is a branched immersion it follows from the definition thatthe set of critical points, and therefore the set of branch points, is finite.Consequently all other points in C\ζ(P ) are regular values whose preimagepoints (if any) are points on M where the orientation is preserved. Afterapplying Theorem 2.1 we conclude that W (ζ, a) is non-negative for all a ∈C\ζ(P ) which is not a branch point. Consequently W (ζ, a) is non-negativefor all a ∈ C\ζ(P ) since the winding number is constant in each connectedcomponent and moreover each connected component is open and containsuncountably many points.

It turns out that the winding number condition is actually sufficient forthe existence of an extension surface when ζ is in general position. Thisleads to the main theorem of this section.

Theorem 2.3. Let P be a finite set of loops, and ζ : P −→ C a map ingeneral position. Then there is an extension F : M −→ C of ζ iff ζ satisfiesthe winding number condition.

7

Before we prove this theorem we need to establish some additional no-tation and notions.

2.2 Annotated diagrams and branch cuts

A branched covering of C can be constructed explicitly as follows. We willtake the disjoint union some numbered copies of C, call the sheets. Wethen make a finite number of branch cuts between pairs of sheets. A branchcut between two sheets, say Ci,Cj , i 6= j is performed along some smoothembedded arc α : [0, 1] → C and is denoted as in Figure 2.

(i,j)

Figure 2: A branch cut between sheet i and j

Let S denote the space created. It is a Riemann surface and the naturalprojection π : S −→ C is easily verified to be an analytic branched coveringof C, i.e. it is a surjective branched immersion. The branch points of themap π are the endpoints of the branch cut curves, they have degree 2. Aregion in a branched covering space S constructed as above will be calledunbounded if it is a neighbourhood of ∞ in at least one of the sheets.

The idea of the proof is as now as follows; for a given set of loops Pmapped by ζ : P −→ C in general position, construct a branched coveringof C as above, and then lift ζ to the branched covering space, i.e. find a ζsuch that π ◦ ζ = ζ. The goal is to find a lifting ζ(P ) such that

(i) it is an embedding

(ii) it is the boundary of a compact manifold M ⊂ S

(iii) the orientation of ∂M induced by the embedding ζ coincides with theorientation of ∂M induced by M\∂M (an open submanifold of S),which in turn has the standard orientation induced by S.

If this can be done, then we have created an extension π |M of π |∂M whichcan be seen as a reparametrization of ζ since π |∂M ◦ζ = ζ.

To describe a construction of a branched covering of C together with alifting of ζ we establish the notion of an annotated diagram.

Definition 2.3. An annotaded diagram is:

1 A set of loops P mapped by ζ in general position into the plane

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2 The number of sheets, m

3 A finite set of embedded closed arcs αi : [0, 1] −→ C in the plane whichare disjoint and transversal to ζ, we call these branch cut curves. Toeach branch cut curve we associate two numbers {i, j}, i 6= j, 1 ≤ i, j ≤m. These numbers represent the sheets which are glued together by thebranch cut.

4 For each arc of some loop in P which starts and ends at some branchcut curve(s) without meeting any branch cut curve in between, or foran entire loop if it doesn’t meet any branch cut curve, we associate anumber 1 ≤ i ≤ m. This represents the sheet in which the lifted arclies.

5 For each connected component A of C\(ζ(P ) ∪ {branch cuts}) andsheet number 1 ≤ i ≤ m we associate a label l(A, i) which is either inor out. The labels denote whether the region should be a part of theextension surface or not.

Each annotated diagram represents some branched covering of C by somebranched covering space S and some lifting of ζ which we call ζ. Howeverin general the lifting ζ defined by an annotated diagram may not even becontinuous.

Definition 2.4. An annotated diagram is called valid if it satisfies the fol-lowing conditions:

C1 Suppose a branch cut between sheets i and j crosses an arc of ζ(P ).Since they intersect transversally, at least locally the branch cut dividesthe arc in two peices. If the two pieces of the arc on each side of thecut is annotated in sheet k respectively l, then either k = l /∈ {i, j} or{i, j} = {k, l}.

C2 Suppose some arc of ζ(P ) doesn’t cross any branch cut and is an-notated in sheet i. If L, R are the adjecent regions of the arc inC\(ζ(P ) ∪ {branch cuts}) which is to the left respectively to the rightof the arc, then l(L, i) = in, l(R, i) = out while l(A, k) = l(B, k) forall j 6= k 6= i.

C3 If there is some branch cut between sheets i, j ending in a region Aof C\(ζ(P ) ∪ {branch cuts}) then l(A, i) = l(A, j).

C4 If two regions A, B of C\(ζ(P )∪ {branch cuts}) are separated by abranch cut between sheets i, j, then l(A, i) = l(B, j), l(A, j) = l(B, i)and l(A, k) = l(B, k) for all k 6= i.

C5 At each self-intersection of ζ, the two arcs that intersect are labelledin different sheets.

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C6 If a region A is an unbounded component of C\(ζ(P )∪{branch cuts}),then l(A, i) = out for all i.

j

i

(C1) (C2)

i

i

jk

k l(A,i)=in l(B,i)=out

A B

(C5)

(i,j)

Figure 3: Conditions for a valid diagram

Lemma 2.4. If P is a set of loops mapped by ζ in general position havinga valid annotated diagram, then there is an extension π |M : M −→ C of ζ.

Proof. Let S be the branched covering space constructed according to thediagram. Let ζ denote the lifting of ζ defined by lifting each arc to the sheetin which it is annotated.

(C1) assures that the lifting is continuous and smooth, (C5) assures thatit never intersects in the same sheet. Thus, ζ is an embedding.

(C6) implies that the regions labelled in are relatively compact in thebranched covering space of C.

(C3) and (C4) assures that the collection of regions labelled in are notbounded by any branch cut.

By (C2) it follows that the arcs in a sheet separates the regions labelledin from the regions labelled out and that the orientation of ζ(P ) agreeswith the boundary orientation induced by the standard orientation of S inthe region labelled in.

We construct M\ζ(P ) by taking the union of the regions in S\ζ(P ) whichare labelled in.

2.3 The proof of Theorem 2.3

Proof. (⇒): This direction was proved in Corollary 2.2.(⇐): This will be proved in two steps. First it will be proved when ζ is anembedding, and then in the general case.

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2.3.1 The proof in the case when P is embedded

Let ζ be an embedding of P which satisfies the winding number condition.Since ζ is already an embedding it will turn out that we don’t need any

branch cut curves to create a valid annotated diagram. We simply startwith an empty diagram which only consists of a number of sheets. Morespecifically, we take as many sheets as there are positive loops in ζ. Sincethere are no branch cuts in the diagram, we will be assigning sheet numbersto entire loops instead of just arcs.

Lemma 2.5. Suppose P is like above and that there is an annotated diagramwithout branch cuts where we denote the loops in sheet i by Si,1, . . . , Si,ni

.If for all sheets i, ζ restricted to Si,1, . . . , Si,ni

satisfies the winding numbercondition and moreover the maximum windning number for the regions inthis sheet is 1, then we can make the diagram valid.

Proof. Observe that all condition except (C2) and (C6) are automaticallysatisfied. All we have to do is to make sure that the labels are correct.

Take any region A ⊂ C that is a component of the complement ofSi,1, . . . , Si,ni

. The winding number of A restricted to these loops is either1 or 0. We’ll set l(A, i) = in if the winding number is 1 and l(A, i) = out ifthe winding number is 0. It is easy to check that the new diagram satisfiescondition (C2) and C(6).

We now construct a diagram of P and ζ in a finite number of steps:

Step 1: Let S1,1, . . . , S1,n1⊂ P denote the loops which bound the unbounded

component of C\ζ(P ). They must all be positive, since otherwise the wind-ing number would be −1 in an adjecent region to the right of them. We addthese loops to the diagram and give each loop sheet number 1.

...

Step m: Let Sm,1, . . . , Sm,nmdenote the loops that bound the unbounded

component of C\ζ(P\⋃m−1

i=1 (Si,1∪ . . .∪Si,ni)). Let α = Sm,j be such a loop.

Case 1: If α is positive we give it a sheet number that hasn’t already been used.This is possible since there are as many sheets as there are positiveloops.

Case 2: If α is a negative loop then from the winding number condition itfollows that α is contained in the inside of n > 0 positive loops, sinceotherwise the winding number in the adjecent region to the right ofα would be negative. For the same reason α is contained in at mostn − 1 negative loops. We can thus find some sheet i containing apositive loop added in a previous step such that the loop contains α

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and moreover no other loop in that sheet contains α. We add α tothis sheet.

By induction we will now show that if the diagram satisfies the conditionsof Lemma 2.5 after step m−1, then it also satisfies the conditions after stepm. After step 1 it obviously satisfies the conditions. Suppose the diagramsatisfies the conditions after step m−1. Any positive loop we add at step mis added to an unused sheet, so this preserves the conditions. Any negativeloop is added to a sheet such that there is only one loop in that sheet thatcontains it, and moreover this loop is positive. It is easy to see that thewinding number of the loops in this sheet still is either 0 or 1.

Using Lemma 2.5 we thus get a valid annotated diagram of P . Lemma2.4 gives us an extension.

1 1

2

1

Figure 4: An example of the algorithm applied to three embedded loops

2.3.2 The proof of the general case

Suppose ζ : P −→ C is in general position satisfying the winding numbercondition. We will use small changes in a diagram to reduce it to the casewhen the loops are embedded. We will use the term valid move for such achange that preserves the validity of an annotated diagram, and moreoverthe inverse of the move has the same property.

� � � � � �� � � � � �

C

A

B

D

A

C

i i

jjj

i(i,j)B

Figure 5: The Crossing Cutting move

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The Crossing Cutting move showed in Figure 5 is a valid move. To seethis, first let’s assume that the diagram to the left in Figure 5 is a partof a valid annotated diagram. Because of condition (C2) and (C3) we getthat l(A, i) = l(A, j) = in and l(B, i) = l(B, j) = out. We also get thatl(C, i) = l(D, i) = in and l(C, j) = l(D, j) = out from condition (C2). It isimmediate that we still get a valid diagram to the right in Figure 5 with thesame labels of A, B and C. All the conditions are fulfilled locally, and thisis all that we need to check. For the same reasons, the move from right toleft in in Figure 5 is also a valid move.

n−1n−1

n

n−2

n−1

n−2

n

Figure 6: The unnanotated Crossing Cutting move

Proof. If we perform the unannotated Crossing Cutting move to P breakingall the crossings, the new loops still satsify the winding number condition asseen in Figure 6. Since these loops never intersect and contain no crossingsthey are embedded. By §§§2.3.1 we can find a valid annotated diagram forthese loops. If we now apply the inverse of the annotated Crossing Cutting,we will get a valid annotated diagram for P . By Lemma 2.4 we have anextension of ζ.

1 1

2

1

2

(1,2)

(1,2)

1

(1,2)

12

2

1

Figure 7: An example of the algorithm applied to a loop in general position

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3 Proof of Theorem 1.1

3.1 The proof of (ii) ⇒ (i)

We will prove the following; let γ = (γ1, γ2) : S1 −→ C2 be a smoothembedding and A ⊂ S1 a closed connected arc with nonempty interior suchthat γ1 |A is an immersion and γ1(A) doesn’t contain any self-intersectionsof γ1. Suppose that for none of the two orientations of S1, γ1 satisfies thewinding number condition, or equivalently that the complement of γ1 hascomponents both of negative and of positive winding number. Then for anyC > 0 there is a neighbourhood U of γ(S1) ∪ E, where E := γ1(A) × C,such that there is no Riemann surface attached to U such that the area ofthe image of the attaching map projected to the first coordinate is greaterthan C.

Proof. Take any C > 0. Let γ(S1)ε be a tubular neighbourhood of γ(S1) onthe form

γ(S1)ε = {z ∈ γ(S1); minu∈γ(S1)|z − u| < ε}

for some sufficiently small ε > 0, i.e. choose ε such that γ(S1) is a deforma-tion retract of γ(S1)ε. This is possible by [3]. After choosing a smaller ε ifnecessary we may assume that the area of π1(γ(S1)ε) is less than C and lessthan the area of any bounded connected component of C\γ1(S1) (note thatthere are finitely many such components). Let

Eε := {(z1, z2) ∈ C2; minu∈γ(A)|z1 − u| < ε}

be a neigbourhood of E. Take 0 < δ < ε so small that Eδ intersects γ(S1)δ

along a tubular neighbourhood of γ(A) in Eδ . This is possible since γ1(A)doesn’t contain any self-intersections of γ1 and since γ1 |A is an immersion.Consider the open neighbourhood

U := γ(S1)δ ∪ Eδ

of γ(S1)∪E. Observe that γ(S1) is a deformation retract of Cl(γ(S1)δ)∩Uwhich in turn is a deformation retract of U . Consequently the fundamentalgroup of U is isomorphic to the fundamental group of γ(S1) ∼= S1, henceisomorphic to Z.

Suppose M is a connected Riemann surface with connected boundaryattached to U , i.e. f : M −→ C2, f(∂M) ⊂ U for a continuouns map fon M which is holomorphic on the interior of M . Let ζ : ∂M ∼= S1 −→U be the restriction of f to the boundary of M . Composing ζ with thedeformation retraction from U to γ(S1) and taking into account the factthat the fundamental group of U is isomorphic to Z we conclude that ζ ishomotopic in U to a curve ζ : S1 −→ γ(S1) which winds n ∈ Z times aroundγ(S1). Hence W (π1 ◦ ζ, z) = W (π1 ◦ ζ, z) = nW (γ1, z) for all z /∈ π1(U).

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Case 1: If |n| = 0, ζ is homotopic to a constant map and hence W (π1◦ζ, z) = 0for all z /∈ π1(U). Note that π1 ◦ f extends π1 ◦ ζ and almost everypoint of C is a regular value of π1 ◦ f . Since π1 ◦ f is holomorphic itis orientation preserving at such preimage points. Thus Theorem 2.1applies and we can conclude that the area of π1(f(M)) is less than thearea of π1(U) = π1(γ(S1)δ) and hence less than C.

Case 2: Let |n| > 0. Since W (π1 ◦ ζ, z) = nW (γ1, z) for all z /∈ π1(U) andW (γ1, z) assumes both positive and negative values for such z, thesame holds for W (π1 ◦ ζ, z). Applying Theorem 2.1 we arrive at acontradiction.

We found a neighbourhood U with the required property.

3.2 The proof of (i) ⇒ (ii)

We will use the algorithm in §2 to construct an extension g : M −→ C ofγ1 := π1 ◦ γ1. This will be the first coordinate of the attaching map ofM . For the second coordinate we will use an approximation theorem forcontinuous functions on embedded arcs, and use the fact that the functionmay take any value on the arc A (recall that the set E := γ(A) × C iscontained in U).

Definition 3.1 (Hormander [4], 5.1.3). A complex analytic manifold M ofdimension n which is countable at infinity is a Stein manifold if

(i) M is holomorphically convex, i.e.

K := {z ∈ M ; |f(z)| ≤ supK |f | for every f ∈ H(M)}

is compact for every compact set K ⊂ M .

(ii) If z1, z2 ∈ M and z1 6= z2 then f(z1) 6= f(z2) for some f ∈ H(M).

(iii) For every z ∈ M , there exists n functions f1, . . . , fn ∈ H(M) whichform a coordinate system at z.

Theorem 3.1 (Guenot and Narasimhan [1]). If N is an open (i.e. non-compact) Riemann surface then it is a Stein manifold.

Theorem 3.2 (Hormander [4], 5.3.9). If N is a Stein manifold of dimensionn, then there is an analytic embedding f : N → C2n+1.

Theorem 3.3 (Stolzenberg [6]). If A = α([0, 1]) ⊂ Cn is a smooth embeddedarc, then any continuous function on A can be approximated uniformly bypolynomials.

We are now ready to present the proof of the implication (i) ⇒ (ii) ofTheorem 1.1.

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Proof. Let γ = (γ1, γ2) be an embedded circle in C2. By assumption we maychoose an orientation of S1 so that γ1 satisfies the winding number condition.Using the method in §2 we may then construct a Riemann surface M withboundary and an extension map g : M −→ C of a reparametrization of γ1

such that g is analytic on M\∂M . We may assume that M is connectedsince ∂M ∼= S1 is connected. Using the arguments in §§2.2 we will identifyγ1 with g |∂M . We will also reparametrize γ2 so that it is defined on ∂Mand identify the arc A ⊂ S1 with the corresponding arc in ∂M .

Let U be any neighbourhood of γ(∂M)∪E, where E := γ1(A)×C. Sinceγ(∂M) is compact there is a an ε > 0 such that the open set

γ(∂M)ε := {z ∈ C2; minu∈γ(∂M)|z − u| < ε}

is contained in U . If we can find a continuous function h : M −→ C which isanalytic on M\∂M such that max∂M\IntA|h − γ2| < ε, then (g(t), h(t)) ∈ Ufor t ∈ ∂M and we are done. We construct this function as follows.

Since M is constructed according to the method described in §2 we ac-tually have M ⊂ S where S is a branched covering space of C. In particularS is non-compact and thus an open Riemann surface. By Theorem 3.1 Sis a Stein manifold and hence there is an analytic embedding φ : S −→ C3

by Theorem 3.2. Since ∂M\IntA is a closed and connected proper arc,I := φ(∂M\IntA) is an embedded closed interval in C3. We observe that

γ2 ◦ φ−1 |I : I −→ C

is a smooth function since φ is an embedding. By Theorem 3.3 it followsthat we can approximate this function uniformly by polynomials in C3. Takea polynomial p in C3 such that maxI |p − γ2 ◦ φ−1| < ε. It follows that

p ◦ φ |M : M −→ C

has the desired properties, so we may take f := (g, p ◦φ |M ) as an attachingmap of M to U .

All that now remains is to show that for every such f , π1(f(M)) = g(M)has area bounded from below by some constant C > 0. But this followsimmediately since g |∂M= γ1 is in general position and satisfies the windingnumber condition. Since the winding number in the unbounded componentof C\γ1(S1) is 0 the winding number must be 1 in all components which areseparated from the unbounded component by an arc (and there is at leastone such component). Since such a component is a non-empty open set ithas area greater than some C > 0, and by Theorem 2.1 it will therefore bein the image of g.

References

[1] J. Guenot and R. Narasimhan. Introduction a la theorie des surfaces deRiemann, Monographie N 23, page 301. Geneva, 1976.

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[2] V. Guillemin and A. Pollack. Differential Topology, page 87. Prenice-Hall, New Jersey, 1974.

[3] V. Guillemin and A. Pollack. Differential Topology, page 69. Prenice-Hall, New Jersey, 1974.

[4] L. Hormander. An Introduction to Complex Analysis in Several Vari-ables. North Holland Publishing Company, 1973.

[5] S. T. Soule. Branched extensions of codimension one maps. PhD thesis,Brown University, 2002.

[6] G. Stolzenberg. Uniform approximation on smooth curves. Acta Math-ematica, 1966.

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