Winding numbers and attaching Riemann surfaces

Winding numbers and attaching Riemann surfaces

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

Winding numbers and attaching Riemann surfaces

Georgios Dimitroglou Rizell Advisor: Tobias Ekholm

June 18, 2007

Abstract

Consider a smoothly embedded circle γ in C

and a closed arc

A ( γ. Suppose the projection of π

◦ γ to the first coordinate line is

in general position and has no selfintersection point on π

◦ 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 π

◦ 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.

Acknowledgements

I would like to express my deepest gratitude to Burglind Juhl-J¨oricke for suggesting the problem and for helping me along the way, as well as inspiring me and introducing me to interesting subjects in mathematics. In fact, Burglind was my mathematical advisor during this project but because of very 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 advisor

for this master thesis.

1 Prerequisites

1.1 Introduction

We call a Riemann surface with boundary a smooth compact manifold M with 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 compact set occurs to be crucial in various questions of several complex variables and symplectic geometry. Here we consider the following situation.

Let γ : S

−→ C

be a smoothly embedded connected closed curve in C

(S

denotes the circle). Suppose moreover that the projection γ

:= π

◦ γ onto the first coordinate is in general position, i.e. it is an immersion and the self-intersections are double points intersecting transversally. Let A ⊂ S

be a closed arc with nonempty interior. Shrinking A we may assume A is connected and that γ

(A) doesn’t contain any crossings of γ

. Let

E := γ

(A) × C ⊂ C

be the union of all complex lines parallell to the second coordinate axis passing through γ

(A). Let U be any neighbourhood of γ

(S

) ∪ E. We are interested in the existence of Riemann surfaces with boundary in U and with area bounded from below independently of the choice of U . Our result will be given in terms of the winding number of the curve γ

.

Definition 1.1. Let Γ be a compact real 1-dimensional manifold without boundary. The winding number at a ∈ C of a smooth map ζ : Γ → C\{a} is the integer W (ζ, a) :=

R

ζ 1

z−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 S

such that γ

satisfies the winding number condition.

(ii) For each neighbourhood U of γ(S

) ∪ 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 connected

boundary and a continuous mapping f : M −→ C

with f (∂M ) ⊂ U

and f holomorphic on the interior M \∂M of M , such that the area

of π

◦ f (M ) is bounded from below by some C > 0 depending only on

γ

.

The question emerged from the still open problem whether Herman ring cylinders exist for Henon mappings. Henon mappings are those polyno- mial automorphisms of C

which are interesting from the point of view of dynamics. Henon mappings have the form

H((z

, z

)) = (z

, az

+ p(z

)), (z

, z

) ∈ C

for a constant a ∈ C and a polynomial p. The Fatou set of the mapping consists of all points z ∈ C

which have a neighbourhood on which the iterates of H form a normal family. The Fatou set is open. Herman ring cylinders are Fatou components which are biholomorphic to A × C, where A is an annulus

A := {z ∈ C; r

< |z| < r

, r

, r

∈ R}

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

The existence of Herman rings was proved using a theorem of Arnold on linearization of real analytic diffeomorphisms of the circle and later using quasiconformal surgery. It is still not known whether Herman ring cylinders exist for Henon mappings. If they exist they are Runge domains in C

which are biholomorphic to A × C (see §§1.2 for more details). If such domains exist at all, they must be very exotic.

Note that a domain in C

is not a Runge domain if there is an attached Riemann surface which is not contained in the domain (see §§1.2). This leads to the following question.

Let U ⊂ C

be a domain which is biholomorphic to A × C. Consider a smooth closed curve in U which is not contractible. Can one deform it inside U to a curve which extends a Riemann surface? The question differs from the one treated in Theorem 1.1. In Theorem 1.1 we use parallel complex lines, 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 C through each point of a closed curve, but we do not know whether they are straightenable, i.e. can be mapped to a coordinate line by a biholomorphic mapping of C

.

For the proof of Theorem 1.1 we need some prerequisites from complex

analysis in several variables and a classical theorem on extension of immer-

sions of closed curves in C to branched immersions of Riemann surfaces into

C. We will explain these results below in §2 and §3 and prove the theorem

in section §3.

1.2 Some prerequisites in several complex variables

Definition 1.2 (H¨ormander [4], 2.5.1, 2.5.5). A domain U ⊂ C

is a do- main of holomorphy if there exists a holomorphic function f ∈ H(U ) which cannot be extended to a holomorphic function g ∈ H(V ) in a domain V ) U . Definition 1.3 (H¨ormander [4], 2.7.1). A domain U ⊂ C

is a Runge domain if all holomorphic functions on U can be approximated uniformly on compacts by polynomials and moreover U is a domain of holomorphy.

Definition 1.4. The polynomially convex hull of a compact set K ⊂ C

, denoted by ˆ K is the set

K := {z ∈ C ˆ

; |p(z)| ≤ max

|p(u)| for all polynomials p}.

Theorem 1.2. Let M be a compact Riemann surface with boundary and f : M → C

a continuous function such that f |

is analytic, then f (M ) is contained in the polynomially convex hull of f (∂M ).

Proof. Let p be any polynomial in C

. p ◦ f is an analytic function on M \∂M . Now the maximum modulus theorem for analytic functions on Rie- mann surfaces give that max

|p(z)| ≤ max

|p(z)|. Consequently f (M ) is a subset of the polynomially convex hull of f (∂M ).

Theorem 1.3. Let U ⊂ C

be a domain. Suppose there exists a connected Riemann surface M with boundary and a continuous mapping f : M −→ C

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

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

K ˆ

:= {z ∈ U ; |h(z)| ≤ max

|h| for all holomorphic functions h on U } is a compact subset of U . (See H¨ormander [4], Theorem 2.5.5).

By H¨ormander [4], Theorem 2.7.3, ˆ K

= ˆ 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 ⊂ f (∂M ) ˆ

. But f (M ) ∩ U is open in f (M ) since U is open. Therefore f (M ) ∩ U cannot be closed in M , otherwise it would coincide with f (M ) since f (M ) is connected. But

f (M ) ∩ U = f (M ) ∩ f (∂M ) ∩ U ˆ

must be compact since f (∂M ) ∩ U is compact. The contradiction proves the ˆ

theorem.

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 ∼ = R

to extend to a branched immersion of a Riemann surface. A discussion of the origin of the theorem is given in [5]. Here we will use the modern language of Soul´e [5] and present one of the proofs given in [5]. This proof differs from earlier ones.

Throughout this section we will let P denote a disjoint union of a finite set 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 an orientation preserving diffeomorphism except at a finite number of points in M \∂M , which we call critical points. Moreover at the critical points of f , the mapping is equivalent to the map z

, i.e. for any critical point z

there is a chart φ : U ⊂ M −→ V ⊂ C, z

∈ U , 0 ∈ V , such that f ◦ φ

(z) = z

, z ∈ V .

The images of the critical points under f are called branch points, and the number 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 branched immersion F : M −→ C such that

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

(ii) F |

= 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 call the bounded component of its complement the inside of the loop and the unbounded component of it’s complement the outisde. Note that this def- inition is independent of the orientation. We will call an embedded loop positive 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 the

complement C\ζ(P ). Also, if one of the regions has winding number n then

the other regions has winding number n ± 1. The region with the bigger

winding number is said to be on the left of the arc while the region with the

smaller winding number is said to be on the right.

(left) (right)

n n+1

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

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

Theorem 2.1. Let M be a compact oriented real 2-dimensional manifold with boundary ∂M oriented with the induced boundary orientation. Let F : M −→ R

, f = F |

be smooth mappings. Then for any regular value a ∈ R

\F (∂M ) of F for which DF |

is orientation preserving for all x ∈ F

(a), the number of preimage points |F

(a)| = W (f, a).

Proof. A proof for a version of the theorem for nonoriented manifolds using mod2 winding numbers is outlined in [2]. A similar approach may be used for oriented manifolds.

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

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

Proof. Since F is a branched immersion it follows from the definition that the 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 preimage points (if any) are points on M where the orientation is preserved. After applying 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-negative for all a ∈ C\ζ(P ) since the winding number is constant in each connected component and moreover each connected component is open and contains uncountably many points.

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

Theorem 2.3. Let P be a finite set of loops, and ζ : P −→ C a map in

general position. Then there is an extension F : M −→ C of ζ iff ζ satisfies

the winding number condition.

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 will take the disjoint union some numbered copies of C, call the sheets. We then make a finite number of branch cuts between pairs of sheets. A branch cut between two sheets, say C

, C

, i 6= j is performed along some smooth embedded 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 natural projection π : S −→ C is easily verified to be an analytic branched covering of C, i.e. it is a surjective branched immersion. The branch points of the map π are the endpoints of the branch cut curves, they have degree 2. A region in a branched covering space S constructed as above will be called unbounded 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 P mapped by ζ : P −→ C in general position, construct a branched covering of C as above, and then lift ζ to the branched covering space, i.e. find a ˜ ζ such 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 the orientation 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 |

of π |

which can be seen as a reparametrization of ζ since π |

◦˜ ζ = ζ.

To describe a construction of a branched covering of C together with a lifting 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

2 The number of sheets, m

3 A finite set of embedded closed arcs α

: [0, 1] −→ C in the plane which are disjoint and transversal to ζ, we call these branch cut curves. To each 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 the branch cut.

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

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

Each annotated diagram represents some branched covering of C by some branched covering space S and some lifting of ζ which we call ˜ ζ. However in general the lifting ˜ ζ defined by an annotated diagram may not even be continuous.

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 divides the arc in two peices. If the two pieces of the arc on each side of the cut 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 in C\(ζ(P ) ∪ {branch cuts}) which is to the left respectively to the right of the arc, then l(L, i) = in, l(R, i) = out while l(A, k) = l(B, k) for all j 6= k 6= i.

C3 If there is some branch cut between sheets i, j ending in a region A of 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 a branch 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 labelled

in different sheets.

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

k j

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 having a valid annotated diagram, then there is an extension π |

: M −→ C of ζ.

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

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

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

(C3) and (C4) assures that the collection of regions labelled in are not bounded by any branch cut.

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

We construct M \ ˜ ζ(P ) by taking the union of the regions in S\ ˜ ζ(P ) which are 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 an

embedding, and then in the general case.

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 start with an empty diagram which only consists of a number of sheets. More specifically, we take as many sheets as there are positive loops in ζ. Since there are no branch cuts in the diagram, we will be assigning sheet numbers to entire loops instead of just arcs.

Lemma 2.5. Suppose P is like above and that there is an annotated diagram without branch cuts where we denote the loops in sheet i by S

, . . . , S

. If for all sheets i, ζ restricted to S

, . . . , S

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

Proof. Observe that all condition except (C2) and (C6) are automatically satisfied. 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 of S

, . . . , S

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

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

Step 1: Let S

, . . . , S

⊂ 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 add these loops to the diagram and give each loop sheet number 1.

.. .

Step m: Let S

, . . . , S

denote the loops that bound the unbounded component of C\ζ(P \ S

i=1

(S

∪ . . . ∪ S

)). Let α = S

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 positive loops.

Case 2: If α is a negative loop then from the winding number condition it

follows that α is contained in the inside of n > 0 positive loops, since

otherwise the winding number in the adjecent region to the right of

α would be negative. For the same reason α is contained in at most

n − 1 negative loops. We can thus find some sheet i containing a

positive loop added in a previous step such that the loop contains α

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

By induction we will now show that if the diagram satisfies the conditions of Lemma 2.5 after step m − 1, then it also satisfies the conditions after step m. After step 1 it obviously satisfies the conditions. Suppose the diagram satisfies the conditions after step m − 1. Any positive loop we add at step m is added to an unused sheet, so this preserves the conditions. Any negative loop is added to a sheet such that there is only one loop in that sheet that contains it, and moreover this loop is positive. It is easy to see that the winding 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 . Lemma 2.4 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 number condition. We will use small changes in a diagram to reduce it to the case when the loops are embedded. We will use the term valid move for such a change that preserves the validity of an annotated diagram, and moreover the inverse of the move has the same property.

C

A

B D

A C

i i

j j

j

(i,j) i B

Figure 5: The Crossing Cutting move

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

n−1 n−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 breaking all the crossings, the new loops still satsify the winding number condition as seen in Figure 6. Since these loops never intersect and contain no crossings they are embedded. By §§§2.3.1 we can find a valid annotated diagram for these 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 an extension of ζ.

1 1

2 1

2

(1,2) (1,2)

1

(1,2)

1 2

2

1

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

3 Proof of Theorem 1.1

3.1 The proof of (ii) ⇒ (i)

We will prove the following; let γ = (γ

, γ

) : S

−→ C

be a smooth embedding and A ⊂ S

a closed connected arc with nonempty interior such that γ

|

is an immersion and γ

(A) doesn’t contain any self-intersections of γ

. Suppose that for none of the two orientations of S

, γ

satisfies the winding number condition, or equivalently that the complement of γ

has components both of negative and of positive winding number. Then for any C > 0 there is a neighbourhood U of γ(S

) ∪ E, where E := γ

(A) × C, such that there is no Riemann surface attached to U such that the area of the image of the attaching map projected to the first coordinate is greater than C.

Proof. Take any C > 0. Let γ(S

)

be a tubular neighbourhood of γ(S

) on the form

γ(S

)

= {z ∈ γ(S

); min

|z − u| < }

for some sufficiently small  > 0, i.e. choose  such that γ(S

) is a deforma- tion retract of γ(S

)

. This is possible by [3]. After choosing a smaller  if necessary we may assume that the area of π

(γ(S

)

) is less than C and less than the area of any bounded connected component of C\γ

(S

) (note that there are finitely many such components). Let

E

:= {(z

, z

) ∈ C

; min

|z

− u| < }

be a neigbourhood of E. Take 0 < δ <  so small that E

intersects γ(S

)

along a tubular neighbourhood of γ(A) in E

. This is possible since γ

(A) doesn’t contain any self-intersections of γ

and since γ

|

is an immersion.

Consider the open neighbourhood

U := γ(S

)

∪ E

of γ(S

) ∪ E. Observe that γ(S

) is a deformation retract of Cl(γ(S

)

) ∩ U which in turn is a deformation retract of U . Consequently the fundamental group of U is isomorphic to the fundamental group of γ(S

) ∼ = S

, hence isomorphic to Z.

Suppose M is a connected Riemann surface with connected boundary attached to U , i.e. f : M −→ C

, f (∂M ) ⊂ U for a continuouns map f on M which is holomorphic on the interior of M . Let ζ : ∂M ∼ = S

−→

U be the restriction of f to the boundary of M . Composing ζ with the

deformation retraction from U to γ(S

) and taking into account the fact

that the fundamental group of U is isomorphic to Z we conclude that ζ is

homotopic in U to a curve ˜ ζ : S

−→ γ(S

) which winds n ∈ Z times around

γ(S

). Hence W (π

◦ ζ, z) = W (π

◦ ˜ ζ, z) = nW (γ

, z) for all z / ∈ π

(U ).

Case 1: If |n| = 0, ζ is homotopic to a constant map and hence W (π

◦ζ, z) = 0 for all z / ∈ π

(U ). Note that π

◦ f extends π

◦ ζ and almost every point of C is a regular value of π

◦ f . Since π

◦ f is holomorphic it is orientation preserving at such preimage points. Thus Theorem 2.1 applies and we can conclude that the area of π

(f (M )) is less than the area of π

(U ) = π

(γ(S

)

) and hence less than C.

Case 2: Let |n| > 0. Since W (π

◦ ζ, z) = nW (γ

, z) for all z / ∈ π

(U ) and W (γ

, z) assumes both positive and negative values for such z, the same holds for W (π

◦ ζ, z). Applying Theorem 2.1 we arrive at a contradiction.

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 of γ

:= π

◦ γ

. This will be the first coordinate of the attaching map of M . For the second coordinate we will use an approximation theorem for continuous functions on embedded arcs, and use the fact that the function may take any value on the arc A (recall that the set E := γ(A) × C is contained in U ).

Definition 3.1 (H¨ormander [4], 5.1.3). A complex analytic manifold M of dimension n which is countable at infinity is a Stein manifold if

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

K := {z ∈ M ; |f (z)| ≤ sup ˆ

|f | for every f ∈ H(M )}

is compact for every compact set K ⊂ M .

(ii) If z

, z

∈ M and z

6= z

then f (z

) 6= f (z

) for some f ∈ H(M ).

(iii) For every z ∈ M , there exists n functions f

, . . . , f

∈ H(M ) which form 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 (H¨ormander [4], 5.3.9). If N is a Stein manifold of dimension n, then there is an analytic embedding f : N → C

.

Theorem 3.3 (Stolzenberg [6]). If A = α([0, 1]) ⊂ C

is a smooth embedded arc, then any continuous function on A can be approximated uniformly by polynomials.

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

Theorem 1.1.

Proof. Let γ = (γ

, γ

) be an embedded circle in C

. By assumption we may choose an orientation of S

so that γ

satisfies the winding number condition.

Using the method in §2 we may then construct a Riemann surface M with boundary and an extension map g : M −→ C of a reparametrization of γ

such that g is analytic on M \∂M . We may assume that M is connected since ∂M ∼ = S

is connected. Using the arguments in §§2.2 we will identify γ

with g |

. We will also reparametrize γ

so that it is defined on ∂M and identify the arc A ⊂ S

with the corresponding arc in ∂M .

Let U be any neighbourhood of γ(∂M )∪E, where E := γ

(A)×C. Since γ(∂M ) is compact there is a an  > 0 such that the open set

γ(∂M )

:= {z ∈ C

; min

|z − u| < }

is contained in U . If we can find a continuous function h : M −→ C which is analytic on M \∂M such that max

|h − γ

| < , then (g(t), h(t)) ∈ U for 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 particular S is non-compact and thus an open Riemann surface. By Theorem 3.1 S is a Stein manifold and hence there is an analytic embedding φ : S −→ C

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

. We observe that

γ

◦ φ

|

: I −→ C

is a smooth function since φ is an embedding. By Theorem 3.3 it follows that we can approximate this function uniformly by polynomials in C

. Take a polynomial p in C

such that max

|p − γ

◦ φ

| < . It follows that

p ◦ φ |

: M −→ C

has the desired properties, so we may take f := (g, p ◦ φ |

) as an attaching map of M to U .

All that now remains is to show that for every such f , π

(f (M )) = g(M ) has area bounded from below by some constant C > 0. But this follows immediately since g |

= γ

is in general position and satisfies the winding number condition. Since the winding number in the unbounded component of C\γ

(S

) is 0 the winding number must be 1 in all components which are separated from the unbounded component by an arc (and there is at least one such component). Since such a component is a non-empty open set it has area greater than some C > 0, and by Theorem 2.1 it will therefore be in the image of g.

References

[1] J. Guenot and R. Narasimhan. Introduction a la theorie des surfaces de

Riemann, Monographie N 23, page 301. Geneva, 1976.

[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. H¨ormander. 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.