Opening infinitely many nodes

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Opening infinitely many nodes

Martin Traizet

To cite this version:

Martin Traizet. Opening infinitely many nodes. Journal für die reine und angewandte Mathematik,

Walter de Gruyter, 2013, 684, pp.165–186. �hal-00785186�

Opening infinitely many nodes

Martin Traizet November 3, 2011

Abstract : we develop a theory of holomorphic differentials on a certain class of non-compact Riemann surfaces obtained by opening infinitely many nodes.

1 Introduction

This paper is about holomorphic 1-forms, on a certain class of non compact Riemann surfaces. The compact case is a classical subject. The space of holo- morphic 1-forms (also called holomorphic differentials) on a compact Riemann surface Σ has complex dimension equal to the genusg of Σ. Moreover, a holo- morphic differential ω can be defined by prescribing its integrals (also called periods) on a suitable set of cycles, namely the cyclesA1,· · · , Agof a canonical homology basis.

The non-compact case is mostly unexplored. In this case, the space of holo- morphic differentials on Σ is infinite dimensional. For any practical purpose, it is required to put a norm on this space. So what we are interested in are Banach spaces of holomorphic differentials on Σ. The question which we would like to answer is :

Can we define a holomorphic differential by prescribing its periods, and in which Banach space does it live ?

We will answer this question on a class of non-compact Riemann surfaces which are made of infinitely many Riemann spheres connected by small necks.

More precisely, consider an infinite graph Γ. Let V denote its set of vertices andE its set of edges. For each vertex v ∈V, consider a Riemann sphereSv. For any edgee∈E from the vertexv to the vertex v⁰, we connectSv and Sv⁰

by a small neck, see figure 1. We do this by opening nodes, a standard explicit construction.

We denote byγethe cycle around the neck corresponding to the edgee. Es- sentially we prove that we can define a holomorphic differentialωby prescribing its periodsαe on the cycles γe fore∈E, provided the prescribed periods sat- isfy the obvious homological obstruction obtained from Cauchy theorem in each Riemann sphere.

Figure 1: A small portion of an infinite graph and a picture of its associated Riemann surface.

One issue is to put norms on both the space of holomorphic differentials ω and the space of period vectorsα = (αe)_e∈E. Our choice was to leave the later norm as general as possible, and see what is the right norm on the space of holomorphic differentials so that the operator which maps a holomorphic differential ω to its period vector α is a Banach isomorphism. This result is Theorem 3.

We also prove that the holomorphic differentialωobtained by prescribing its periods depends smoothly (actually, analytically) on all parameters which enter the construction of Σ. Roughly speaking, these parameters can be thought of as the position and size of the necks.

Because our Riemann surface is not compact, one can ask what is the asymp- totic behavior of the holomorphic differentialω. This can be done using weighted norms and will be a recurrent question in this paper.

Finally, in the last section of the paper, we consider meromorphic differen- tials. In the compact case, it is well known that one can define a meromorphic differential by prescribing its poles, principal part at each pole and periods. We prove such a result in our non compact setting (a Mittag Leffler type result).

The theory of holomorphic differentials on non compact Riemann surfaces which we develop in this paper, besides being interesting in its own right, has applications to minimal surfaces theory, via the classical Weierstrass Represen- tation. It can be used to glue infinitely many minimal surfaces together. See [5] for an example.

I would like to thank the referee for valuable comments and suggestions.

2 Opening nodes

Consider a connected graph Γ. It may have multiple edges and edges from a vertex to itself (this is usually called a multi-graph). We denote byV its set of

vertices andE its set of edges. We are mostly interested in the case whereV is not finite. We assume that Γ is oriented, which means that each edge has an orientation. The orientation is arbitrary and will only be used to orient certain curves. Ife∈E is an oriented edge from vertexvto vertex v⁰, we say thatvis the starting point ofeandv⁰ is the endpoint.

For a vertexv ∈V, the set of edges havingv as starting point (resp. end- point) will be denotedE_v⁻ (resp. E_v⁺). The set of edges adjacent to the vertex v is denoted Ev =E_v⁻∪E_v⁺. The degree of v is the cardinal of Ev, which we assume is always finite. Note that together with the fact that Γ is connected, this implies thatV is countable.

To each vertexv∈V we associate a Riemann sphere C∪ {∞}denoted Sv. For each oriented edge e∈E from v to v⁰, we choose a pointp⁻_e in Sv and a pointp⁺_e in Sv⁰. We assume that for each vertexv, the points p⁻_e for e∈E_v⁻ andp⁺_e fore∈E_v⁺ are distinct.

We consider the disjoint union of allS_v forv∈V, and for each edgee∈E, we identify the pointsp⁻_e andp⁺_e. This creates a node which we callp_e. We call Σ₀ the resulting Riemann surface with nodes.

We open nodes in the standard way as follows, see for example [3]. We use the notation D(a, R) for the disk of center a and radius R in C. We consider, for each edge e ∈ E from v to v⁰, some local complex coordinates z⁻_e :D_e⁻⊂Sv

→∼ D(0, ρ) andz⁺_e :D⁺_e ⊂Sv⁰

→∼ D(0, ρ) in a neighborhood ofp⁻_e and p⁺_e respectively, such that z_e⁻(p⁻_e) = 0 and z⁺_e(p⁺_e) = 0. We assume that for each vertexv, the disks D_e⁻ fore∈E_v⁻ and D⁺_e fore∈E⁺_v are disjoint in Sv.

Consider again the disjoint union of all Riemann spheres S_v, v ∈ V. For each edge e∈ E from v to v⁰, choose some complex number t_e ∈D(0, ρ²). If te 6= 0, remove the disk |z_e⁻| ≤ ^|t_ρ^e| from D_e⁻ and |z_e⁺| ≤ ^|t_ρ^e| from D⁺_e. We identify each pointz∈D_e⁻ with the pointz⁰∈D⁺_e such thatz_e⁻(z)z_e⁺(z⁰) =t_e. This creates a neck connectingSv andSv⁰. Ifte = 0, then we identifyp⁻_e and p⁺_e as before to create a node.

Doing this for all edges defines a (possibly noded) Riemann surface which we call Σt. Heretdenotes the sequence (te)_e∈E. When allte are nonzero, Σt

is a genuine Riemann surface. It is compact when Γ is a finite graph.

3 Regular differentials

Regular differentials are the natural generalization of holomorphic 1-forms to Riemann surfaces with nodes.

Definition 1 (Bers [4]) A differentialω on a Riemann surface with nodes Σ is called regular if it is holomorphic away from the nodes and for each node p(obtained by identifying p⁻ and p⁺), it has simple poles at p⁻ andp⁺ with opposite residues.

For each edge e∈ E, the boundary ∂D_e⁺ is homologous in Σt to −∂D_e⁻. We define the cycleγe as the homology class of∂D⁺_e. We have

∀v∈V, X

e∈E⁻v

γe= X

e∈Ev⁺

γe. (1)

We want to define a regular differentialω on Σ_t by prescribing its periods on the cyclesγe:

∀e∈E, Z

γe

ω=αe, αe∈C By Cauchy theorem and equation (1), a necessary condition is

∀v∈V, X

e∈Ev⁻

αe= X

e∈Ev⁺

αe. (2)

If Γ is a finite graph, equation (2) is the only obstruction to define a regular differential on Σt :

Theorem 1 (Fay [2]) If Γ is a finite graph, then for t small enough, the op- eratorω7→(R

γ_eω)_e∈E is an isomorphism from the space of regular differentials onΣt to the space of vectors (αe)_e∈E which satisfy the compatibility condition (2)

Remark 1 Fay’s theorem is more general, as he does not require the parts Sv

to be spheres.

3.1 Admissible coordinates

In the case of an infinite graph, we need to make some assumptions on the coordinates used to open nodes. We denote by z the standard coordinate in each sphereSv, and writez_e^± to designate eitherz⁺_e orz_e⁻. For eache∈E, the ratio

z^±_e z−p^±e

extends continuously atp^±_e so is bounded from above and below in D^±_e by positive numbers. We require these numbers to be independent ofe∈E.

The following definition summarizes our requirements on the coordinates.

Definition 2 We say that the coordinates (z_e^±)e∈E are admissible if 1. all pointsp⁻_e andp⁺_e are different from∞,

2. for each e ∈ E, z_e^± : D^±_e →^∼ D(0, ρ) is a diffeomorphism such that z_e^±(p^±_e) = 0 (ρis independent ofe),

3. for eachv∈V, the disks D_e⁻ fore∈E_v⁻ andD⁺_e fore∈E_v⁺ are disjoint inSv,

4. there exists positive constants c1 andc2 such that

∀e∈E, ∀z∈D^±_e, c1|z_e^±(z)| ≤ |z−p^±_e| ≤c2|z_e^±(z)|.

Remark 2 If the coordinates are admissible, then for each edge e ∈ E, the round diskD(p^±_e, ρc1)is included in the topological diskD^±_e. In particular, point 3 implies that for each vertexv∈V, the round disksD(p⁺_e, ρc1)fore∈E_v⁺ and D(p⁻_e, ρc1) fore∈E_v⁻ are disjoint.

3.2 The `

case

Next, as Σt is non compact, the space of regular differentials is infinite di- mensional so we need to define a norm on this space. Fix some real number 0< ≤ρc1. For each vertex v∈V, let Ωv, be the Riemann sphereSv minus the disks D(p⁻_e, ) fore ∈ E_v⁻ and D(p⁺_e, ) for e ∈ E_v⁺. Let H¹_∞(Σt) be the space of regular differentials on Σ_t such that the norm

||ω||∞= sup

v∈V

sup

Ω_v,

ω dz

is finite. Let `^∞(E) be the space of bounded sequences of complex numbers (αe)_e∈E with the sup norm. Our first result, which generalizes Theorem 1 to the infinite case, is

Theorem 2 Assume that the degree of the vertices of Γ is bounded and that the coordinates are admissible. Then for t small enough (in `^∞ norm), the operator ω→(R

γ_eω)e∈E is an isomorphism of Banach spaces fromH¹_∞(Σt) to the subspace of sequences in`^∞(E)which satisfy (2).

This result is a corollary of Theorem 3 and proposition 1 below.

Remark 3 If follows from the theorem that the spaceH¹_∞(Σ_t)does not depend on the choice of (taking another will define an equivalent norm).

3.3 Examples of admissible coordinates

Example 1 Assume that for each v ∈ V, the points p⁺_e for e ∈ E_v⁺ and p⁻_e fore∈ E_v⁻ are at distance greater than 2ρ from each other. We can take the coordinatesz^±_e(z) =z−p^±_e onD_e^±=D(p^±_e, ρ). These coordinates are admissible withc₁=c₂= 1.

This is probably the most natural choice of coordinates. The proof of The- orem 3 would be significantly simplified if we restricted ourselves to these co- ordinates, so let me give other examples to motivate the use of more general coordinates.

Example 2 Assume that for eachv∈V, the pointsp⁺_e fore∈E_v⁺ andp⁻_e for e ∈ E_v⁻ are on the unit circle and are at distance greater than 4ρ from each other. We would like the inversionσ(z) = 1/z to be well defined on Σt. In this case we take the coordinates

z^±_e(z) =i z−p^±_e z+p^±e

.

Then z_e^±◦σ = z^±e, so provided all te are real numbers, σ is well defined on Σt. Let D^±_e be the inverse image ofD(0, ρ) byz_e^±. Computations shows that these disks are disjoint, and the coordinates are admissible withc1 = _1+ρ² and c2= _1−ρ² .

Example 3 Assume that the graph Γ is bipartite, meaning that the set of vertices has a partitionV =V⁺∪V⁻, so that all edges go from one vertex inV⁻ to a vertex inV⁺. Consider on each sphereS_v a meromorphic functionf_v with simple zeros atp⁻_e fore∈E_v⁻ andp⁺_e fore∈E_v⁺. For eache∈E fromv tov⁰, we takez⁻_e =fvandz_e⁺=fv⁰. Some hypothesis must be made on the functions (fv)_v∈V so that these coordinates are admissible. Consider a small complex number t and take te =t² for all e∈ E. Then we can define a meromorphic function f on Σt by f(z) = fv(z)

t ifv ∈ V⁺ and f(z) = t

fv(z) ifv ∈ V⁻. Indeed, for each edgee∈E from v tov⁰, if z∈D_e⁻ is identified withz⁰ ∈D⁺_e, we havefv(z)fv⁰(z⁰) =t² sof(z) =f(z⁰) andf is well defined.

The reason this example is interesting is that together with a Riemann sur- face, we construct a meromorphic function. It is in general not easy to define a meromorphic function on a given Riemann surface (think of Abel’s theorem).

This is particularly fruitful in the case of minimal surfaces, where the function f will be the Gauss map. See [5], [6] for an illustration of this idea.

3.4 Admissible norms

The`<sup>∞</sup> norm is maybe the most natural one, but for certain applications it is desirable to allow other norms. For example, in section 3.7 we use weighted`^∞ norms to study the decay of regular differentials. In the following definition, we try to consider norms as general as possible.

LetC^EandC^V denote the space of sequences of complex numbers (αe)e∈E

and (uv)v∈V indexed by edges and vertices, respectively. We assume that we are given two norms|| · ||E and|| · ||V defined on a subspaceBE ofC^E andBV

ofC^V.

Definition 3 We say that the norms || · ||E and || · ||V are admissible if the following conditions hold :

1. The spaces BE andBV are Banach spaces.

2. The norms|| · ||E and|| · ||V are monotonic, in the sense that if|αe| ≤ |α⁰_e| for alle∈E then||α||E≤ ||α⁰||E, and a similar definition for|| · ||V. 3. There exists a constantc3>0such that for all α∈ BE,

P

e∈Ev|α_e|

v∈V

_V ≤c₃||α||_E. 4. There exists a constantc4>0such that for all u∈ BV,

(|u_v|+|u_v⁰|)e∈E e=vv0

_E≤c₄||u||_V.

5. The space of sequences(λe,n)_e∈E,n∈Nfor which the norm

(sup_n∈N|λe,n|)_e∈E E

is defined is a Banach space.

3.5 Main result

We defineH¹(Σt) as the space of regular differentials on Σt such that the norm

||ω||Ω=

sup_{Ωv, dz}^ω

v∈V

_V

is defined. The following theorem, which is the main result of this paper, gen- eralizes Theorem 2 to this more general setup.

Theorem 3 Assume that the coordinates and the norms are admissible. Then fort small enough (in `^∞ norm), the operatorω 7→ (R

γeω)_e∈E is an isomor- phism fromH¹(Σ_t)to the subspace of sequences inBE which satisfy (2)

3.6 Examples of admissible norms

We define weighted `^p spaces for 1 ≤p ≤ ∞as follows. Consider a function σ:V →(0,∞). Given a sequence u= (uv)_v∈V, let

||u||_p,σ= X

v∈V

σ(v)^p|u_v|^p

!^1/p

ifp <∞,

||u||_∞,σ= sup

v∈V

σ(v)|uv|.

We call `<sup>p,σ</sup>(V) the space of sequences u ∈ C<sup>V</sup> for which the above norm is finite. Whenσ≡1, these are the usual`^p spaces.

We define a weight function on edges by σ(e) = ¹₂(σ(v) +σ(v⁰)) if e is an edge from v to v⁰. The space `^p,σ(E) of sequencesα ∈ C^E which have finite

`^p,σ norm is defined in the obvious way.

Proposition 1 Assume that there exists a positive numberc such that the de- gree of the vertices of Γ is bounded by c, and that for any adjacent vertices v andv⁰,σ(v)≤cσ(v⁰)andσ(v⁰)≤cσ(v). Then the norms || · ||E =|| · ||p,σ and

|| · ||V =|| · ||p,σ are admissible.

Proof. The first point is standard, the second is clear. Regarding the third

point, in the casep <∞we write X

v∈V

σ(v)^p X

e∈Ev

|αe|

!^p

≤ X

v∈V

σ(v)^pdeg(v)^p−1 X

e∈Ev

|αe|^p by Jensen inequality

≤ c^p−1X

v∈V

X

e∈E_v

σ(v)^p|αe|^p

= c^p−1X

e∈E

(σ(v)^p+σ(v⁰)^p)|αe|^p

≤ 2^pc^p−1X

e∈E

σ(e)^p|αe|^p

The casep=∞is straightforward.

Regarding the fourth point, we have for any edge e ∈ E from v to v⁰, σ(e)≤ ^c+1₂ σ(v) andσ(e)≤^c+1₂ σ(v⁰), so

X

e∈E e=vv0

σ(e)^p(|uv|+|uv⁰|)^p ≤ X

e∈E

σ(e)^p2^p−1(|uv|^p+|uv⁰|^p)

≤ (1 +c)^p 2

X

e∈E

σ(v)^p|uv|^p+σ(v⁰)^p|uv⁰|^p

= (1 +c)^p 2

X

v∈V

deg(v)σ(v)^p|u_v|^p

≤ c(1 +c)^p 2

X

v∈V

σ(v)^p|uv|^p

Finally, point 5 is true by the standard fact that `<sup>∞</sup>(N) is a Banach space and an infinite product of Banach spaces is a Banach space for the`^p norm of

the norms (cf exercice T page 243 in [1]).

3.7 Decay at infinity of normalized differentials

To illustrate the use of weights, let us consider the following example.

Example 4 Let Γ be an infinite graph which contains at least one cycle γ.

Give the cycleγ an orientation and orient the rest of Γ arbitrarily. Define the sequence (αe)e∈E as αe = 1 if e belongs to γ and αe = 0 otherwise. Then (α_e)_e∈E satisfies (2), so defines a regular differential ω ∈ H¹_∞(Σ_t). We call ω the normalized differential associated to the cycleγ.

Whent= 0,ω is supported on the spheresSv such that the vertexvbelongs to the cycleC, so has compact support. It is natural to ask what is the decay of ω at infinity whent6= 0. For this purpose, we pick an arbitrary vertexv0∈V and consider the weightσ(v) =r^d(v,v0), wherer >1 is a fixed real number, and d(v, v0) denotes the graphical distance fromv0tovon Γ. Thenα∈`^∞,σ(E), so by Theorem 3,ω ∈ H¹_∞,σ(Σt) fort small enough, so ω has exponential decay.

More precisely sup_{Ωv, dz}^ω

is bounded by some constant timesr^−d(v,v0). The rate of decayris arbitrary, but of course the largerr, the smaller tmust be.

4 Proof of Theorem 3

Convention 1 By a uniform constant, we mean a number which only depends on the numbersρ,c₁ andc₂ in definition 2 andc₃,c₄ in definition 3. We use the letter C to denote uniform constants. The same letter C can be used to denote various uniform constants.

Let L be the operator ω 7→ (R

γ_eω)e∈E from H¹(Σt) to the subspace of sequences inBE which satisfy (2). We prove that

1. Lis bounded, 2. Lis onto, 3. Lis injective.

1) Let e be an edge from v to v⁰. Let r =ρc1 ≥. By remark 2, we can replaceγeby the circleC(p⁺_e, r), so

Z

γe

ω

= Z

C(p⁺e,r)

ω

≤2πrsup

Ω_v0,

|ω

dz| ≤2πr sup

Ω_v,

|ω

dz|+ sup

Ω_v0,

|ω dz|

! .

By point 4 of the definition of admissible norms, we get||L(ω)||E≤C||ω||Ωfor some uniform constantC.

2) Let us prove next thatLis onto. Given (α_e)_e∈E∈ BE, satisfying (2), we are asked to construct a regular differentialω on Σ_t such that L(ω) = α. In each sphereS_v, we defineω=ω₁+ω₂with

ω1=ω⁺₁ +ω⁻₁ = 1 2πi

X

e∈E⁺v

αe

dz z−p⁺e

− 1 2πi

X

e∈Ev⁻

αe

dz z−p⁻e

(3)

ω2=ω₂⁺+ω₂⁻= X

e∈Ev⁺

∞

X

n=2

λ⁺_e,n

4

n

dz

(z−p⁺e)ⁿ + X

e∈Ev⁻

∞

X

n=2

λ⁻_e,n

4

n

dz

(z−p⁻e)ⁿ. (4) (Fori= 1,2,ω_i⁺ and ω_i⁻ denote the first and second sum of ω_i, respectively.) The meromorphic differentialω₁has the required periods and is entirely deter- mined by the data we are given. Condition (2) ensures that the residue ofω₁ at infinity vanishes, soω1 is holomorphic at infinity. The meromorphic differ- ential ω2 is a corrective term with no periods. The coefficients λ^±_e,n are to be determined.

Let us first assume that the series definingω2 converge and see what is the condition so that ω is a well defined differential on Σt. Then we solve theses equations, and finally prove that the series converge.

For any edgee∈E, define ϕ⁺_e = (z_e⁺)⁻¹◦

te

z⁻e

, ϕ⁻_e = (z⁻_e)⁻¹◦ te

ze⁺

= (ϕ⁺_e)⁻¹

Ifte6= 0,ϕ⁺_e is bi-holomorphic from the annulus ^|t_ρ^e| ≤ |z⁻_e| ≤ρto the annulus

|te|

ρ ≤ |z⁺_e| ≤ρ, and the pointzis identified with the pointϕ⁺_e(z) when defining Σ_t. In other words, these are the change of charts for the coordinates z_e⁺ and z⁻_e on Σ_t, soω is well defined on Σ_t if (ϕ⁺_e)^∗ω=ω for all edgese∈E.

Claim 1 ω is well defined on Σt if and only if for all edges e ∈ E such that te6= 0

∀n≥0, Z

∂D⁻_e

(z−p⁻_e)ⁿ((ϕ⁺_e)^∗ω−ω) = 0,

∀n≥0, Z

∂De⁺

(z−p⁺_e)ⁿ((ϕ⁻_e)^∗ω−ω) = 0.

Proof. By the theorem on Laurent series, the holomorphic differential (ϕ⁺_e)^∗ω− ωis zero on the annulus ^|t_ρ^e|≤ |z_e⁻| ≤ρif and only if for alln∈Z,

Z

∂D⁻_e

(z_e⁻)ⁿ((ϕ⁺_e)^∗ω−ω) = 0. (5) Equation (5) for alln≥0 means that (ϕ⁺_e)^∗ω−ω extends holomorphically to the disk |z⁻_e| ≤ρ. Using z−p⁻_e as a coordinate, this is equivalent to the first condition of the claim. Ifn≤0, then by a change of variable

Z

∂D⁻_e

(z_e⁻)ⁿ((ϕ⁺_e)^∗ω−ω) = − Z

∂D⁺_e

(ϕ⁻_e)^∗

(z_e⁻)ⁿ((ϕ⁺_e)^∗ω−ω)

= −

Z

∂D⁺_e

te

ze⁺

ⁿ

(ω−(ϕ⁻_e)^∗ω)

So equation (5) for alln≤0 means that ω−(ϕ⁻_e)^∗ω extends holomorphically to the disk|z_e⁺| ≤ ρ. Using z−p⁺_e as a coordinate, this is equivalent to the

second condition of the claim.

Claim 2 ωis a well defined regular differential onΣ_tif and only if for alle∈E and alln≥2,

λ⁻_e,n= −1 2πi

4

ⁿZ

∂D⁺_e

(ϕ⁻_e −p⁻_e)ⁿ⁻¹ω, (6) λ⁺_e,n= −1

2πi 4

ⁿZ

∂D⁻e

(ϕ⁺_e −p⁺_e)ⁿ⁻¹ω. (7) Proof. Ifte= 0, thenϕ⁺_e−p⁺_e andϕ⁻_e−p⁻_e are identically zero. Hence equations (6) and (7) for alln≥2 mean that ω2 is holomorphic atp⁻_e and p⁺_e, soω has at most simple poles as required for a regular differential.

Ifte6= 0, we use the previous claim. The definition ofω=ω1+ω2gives Z

∂D⁻_e

(z−p⁻_e)ⁿω=

−αeifn= 0 2πiλ⁻_{e,n+1 4}n+1

ifn≥1 By a change of variable,

Z

∂D⁻_e

(z−p⁻_e)ⁿ(ϕ⁺_e)^∗ω = − Z

∂D⁺_e

(ϕ⁻_e)^∗

(z−p⁻_e)ⁿ(ϕ⁺_e)^∗ω

= −

Z

∂D⁺e

(ϕ⁻_e −p⁻_e)ⁿω

We have similar statements exchanging the roles of the + and − signs. The

claim follows.

Given a holomorphic differentialwon Ω=S

v∈V Ωv,, define fore∈Eand n≥2

F_e,n^± (w) = −1 2πi

4

nZ

∂D^∓e

(ϕ^±_e −p^±_e)ⁿ⁻¹w. (8) LetF^±(w) = (F_e,n^± (w))_e∈E,n≥2 andF(w) = (F⁺(w), F⁻(w)). By claim 2,ω is well defined on Σ_t if and only ifλ=F(ω). We want to find λas a fixed point of the mapλ7→F(ω₁+ω₂(λ)). To this effect, letB_L be the space of sequences λ= (λ⁺_e,n, λ⁻_e,n)_e∈E,n≥2 such that the following norm is defined :

||λ||L=

sup_n≥2max{|λ⁺_e,n|,|λ⁻_e,n|}

e∈E

_E. This is a Banach space by point 5 of definition 3.

Lemma 1 There exists a uniform constantC such that if||t||∞≤ ^c_4c^1ρ2 2

, 1. ||ω1||Ω≤ ^C||α||E,

2. ||ω2||Ω≤C||λ||L,

3. ||F(w)||L≤ ^C2||t||_∞||w||Ω.

Proof of point 1 : we have the straightforward estimate sup

Ω_v,

ω1

dz ≤ 1

2π X

e∈Ev

|αe|.

The conclusion follows by point 3 of definition 3.

Proof of point 2 :

sup

Ω_v,

ω₂⁺ dz

≤ X

e∈E⁺_v

∞

X

n=2

|λ⁺_e,n| ⁿ

4

ⁿ

≤

∞

X

n=2

1 4

ⁿ!

 X

e∈E⁺_v

sup

n≥2

|λ⁺_e,n|



.

We estimateω⁻₂ in the same way. The conclusion follows by point 3 of definition 3.

Proof of point 3 : Let e be an edge from v to v⁰. By definition of admissible coordinates, we have

|ϕ⁺_e(z)−p⁺_e| ≤c2|z_e⁺(ϕ⁺_e(z))|= c2|te|

|ze⁻(z)| ≤ c²₂|te|

|z−p⁻e|.

Arguing in the same way on the other side we obtain the following useful esti- mate

c²₁|te|

|z−p⁻e| ≤ |ϕ⁺_e(z)−p⁺_e| ≤ c²₂|te|

|z−p⁻e| (9)

We replace the circle∂D_e⁻ in the definition ofF_e,n⁺ by the circle C(p⁻_e, r) with r=c1ρ. Then we have fore∈E andn≥2

|F_e,n⁺ (w)| ≤ 1 2π2πr4

4

c²₂|t_e|

r ⁿ⁻¹

sup

Ω_v,

w dz

(10)

≤ 4r

4c²₂|te| r

sup

Ω_v,

w dz

≤ 16c²₂

² ||t||_∞sup

Ω_v,

w dz .

(The term in the middle parenthesis of the first line is less than one by our hypothesis ont). We estimateF_e,n⁻ (w) in the same way. The conclusion follows

by point 4 of definition 3.

It follows from the lemma that if ||t||_∞ is small enough (depending on), the mapλ7→F(ω1+ω2(λ)) is contracting fromBL to itself. By the contraction mapping theorem, there exists a uniqueλ∈ BL such thatF(ω1+ω2(λ)) =λ.

It remains to consider the convergence of the series defining ω2. First of all, we have actually seen in the proof of point 2 of the lemma that this series converges on Ωv,, soωis already defined on each Ωv,. It remains to prove that for each edgee∈E,ωis defined on the corresponding neck, namely the annular region in Σt bounded by the circlesC(p⁻_e, ) andC(p⁺_e, ).

Since ||ω||Ω is finite, sup_Ω

v,|ω|is finite for all v ∈V (although maybe not uniformly bounded). Sinceλ=F(ω), we have by equation (10)

4

ⁿ

|λ^±_e,n| ≤r c²₂|te|

r ⁿ⁻¹

sup

Ω_v,

ω dz

.

From this we conclude that if te = 0, then λ^±_e,n = 0 for all n ≥2 so there is no convergence issue. Ifte6= 0 and|te|is small enough, then|z^±_e(z)| ≥ ¹₂|te|^1/2 implies that |z−p^±_e| > 2^c22|t_r^e|, which implies by the above estimate that the series

∞

X

n=2

λ^±_e,n

4

ⁿ (z−p^±e)ⁿ

converges. Hence ω is defined on the two domains |z_e⁺| ≥ ¹₂|te|^1/2 and |z_e⁻| ≥

1

2|te|^1/2, which cover the neck. So we have proven thatL is onto.

Remark 4 It follows from the lemma that for ||t||∞ small enough (depending on),||ω||Ω≤ ^C||α||E.

3) Finally we prove thatLis injective for ||t||∞ small enough. Letωbe in the kernel ofL. For anye∈E,ωis holomorphic in the annulusc2|te|

ρ ≤ |z−p^±_e| ≤ c₁ρ (which is included in the annulus ^|t_ρ^e| ≤ |z_e^±| ≤ ρ), so we can write its Laurent series in this annulus as

ω=

∞

X

n=−∞

c^±_e,n dz (z−p^±e)ⁿ.

The coefficientsc^±_e,1 are all zero becauseL(ω) = 0. The sum forn≤0 extends holomorphically to the diskD(p^±_e, r). The sum for n≥2 extends holomorphi- cally to the outside of this disk. Therefore, the difference

ω− X

e∈Ev⁺

∞

X

n=2

c⁺_e,n dz

(z−p⁺e)ⁿ − X

e∈E⁻v

∞

X

n=2

c⁻_e,n dz (z−p⁻e)ⁿ

extends holomorphically to all of Sv. Since there are no holomorphic 1-form on the sphere, it is zero. Letλ^±_e,n= (⁴)ⁿc^±_e,n, thenω =ω2(λ). Sinceω is well defined on Σ_t, we haveλ=F(ω). By lemma 1,||λ||_L≤^C₂||t||_∞||λ||_L, soλ= 0 iftis small enough. Henceω= 0. This concludes the proof of Theorem 3.

Remark 5 We have proven in point 2 that any regular differentialω∈ H¹(Σt) can be writtenω=ω1(α) +ω2(λ), whereω1 andω2 are defined by (3)and (4).

Moreover,λ=F(ω), whereF(ω)is defined by (8).

5 Smooth dependance on parameters

By Theorem 3, for any α ∈ B_E satisfying (2), there exists a unique regular differentialωon Σt whose periods are prescribed byα. In this section, we prove thatω depends smoothly on the parameters in the construction of Σt, namely p= (p⁺_e, p⁻_e)_e∈E and t= (te)_e∈E. Before we can formulate this result, several points need to be addressed.

We assume that the parameterp= (p⁺_e, p⁻_e)_e∈Eis in a neighborhood of some central valuep= (p⁺_e, p⁻_e)e∈E in`^∞norm. In this context, we define Ωv,to be the sphereSvminus the disksD(p⁻

e, ) fore∈E⁻_v andD(p⁺

e, ) fore∈E_v⁺. We define Ωas the disjoint union of all Ωv,forv∈V. The norm || · ||Ωis defined as in section 3.4. The point here is that the domain Ω is fixed, whereas our former domain Ω depends on the parameter p. Thanks to remark 3, the new norm|| · ||ΩonH1(Σ_t) is equivalent to the former one.

The coordinates z^±_e must depend in some ways on the parameters, else the requirement z^±_e(p^±_e) = 0 fixes the point p^±_e. We assume that for each edge e ∈ E, the coordinates z^±_e have the form z_e^±(z) = ζ_e^±(z−p^±_e, ξ_e^±), where ξ_e^± is a vector of complex parameters andζ_e^± is a holomorphic function in all its variables. For example 2, we would takeξ_e^±=p^±_e andζ_e^±(z, p^±_e) =i ^z

z+2p^±e. We assume that each parameterξ_e^±is in a finite dimensional complex vector space, possibly depending on the edge, but of uniformly bounded dimension.

We writeξ= (ξ_e⁺, ξ_e⁻)_e∈E and we assume that the coordinates z_e^± are defined and admissible for all values ofξ in a neighborhood of some central valueξ = (ξ⁺

e, ξ⁻

e)_e∈E in `^∞norm.

Theorem 4 The regular differential ω, restricted toΩ, depends smoothly on the parameters(p,t, ξ) in a neighborhood of(p,0, ξ)in`^∞ norm.

The norm on the domain space is the`^∞norm. The norm on the target space is the norm|| · ||Ω.

Remark 6 By a theorem of Graves-Taylor-Hille-Zorn, a map f from an open set of a complexBanach space E to a complex Banach space F, which is dif- ferentiable in the usual (Frechet) sense, is holomorphic, hence smooth. So the word “smoothly” in the theorem can be replaced by the word “analytically”. See the book [7] for the definition of a holomorphic map between complex Banach spaces and Theorem 14.3 for the statement and the proof. Also, Hartog’s theo- rem holds in the complex Banach space setup : ifE₁ andE₂ are Banach spaces andf :E₁×E₂ →F is separately holomorphic with respect to each of its two variables, then it is holomorphic (Theorem 14.27).

Proof of Theorem 4 (continued from the proof of Theorem 3) : recall that we foundλas a fixed point ofλ7→F(ω1) +F(ω2(λ)). By the following lemma and the fixed point theorem with parameters,λdepends smoothly on all parameters, soω=ω1+ω2(λ) also depends smoothly on all parameters.

LetH1(Ω) be the space of holomorphic differentialswon Ωsuch that||w||Ω

is defined.

Lemma 2 For (p,t, ξ) in a neighborhood of (p,0, ξ) in `^∞ norm, α ∈ BE, λ∈ BL andw∈ H1(Ω),

1. the map (p, α)7→ω₁(p, α)is smooth, 2. the map (p, λ)7→ω₂(p, λ)is smooth,

3. the map (p,t, ξ, w)7→F(p,t, ξ, w)is smooth.

To prove this lemma, we first consider an abstract result. The setup is the following : we have an infinite family of holomorphic functionsfn(xn) forn∈ N. The variable xn is in the poly-disk D(0, R) in C^dn. We assume that the dimensiondn is uniformly bounded. The functionfn takes value in a Banach spaceFn. LetE=Q

n∈NC^dn andF =Q

n∈NFn, both with the sup norm. Let x= (xn)_n∈Nandf(x) = (fn(xn))_n∈N

Lemma 3 If each ||fn|| is bounded on the poly-disk D(0, R) by a constant C independent ofn, then f :B(0, R)→F is smooth, whereB(0, R)denotes the ball of radiusRin E.

Proof. Consider some r < R. Since the function fn is holomorphic in the poly-diskD(0, R) and bounded byC, all its partial derivatives of orderkin the poly-diskD(0, r) have norm bounded byCk! (R−r)^−k by Cauchy’s estimates, which holds true for Banach valued holomorphic functions. The point is that this bound does not depend onn. It is then straightforward to check thatf is differentiable in the ballB(0, r), with differentialdf(x)(h) = (dfn(xn)(hn))n∈N. Sinceris arbitrary,f is differentiable in the open ballB(0, R). The fact thatfis smooth follows by induction (or using the theorem of Graves-Taylor-Hille-Zorn.)

Proof of point 1 of lemma 2. We deal with the termsω⁻₁ andω₁⁺separately (see equation (3) for the definition of these differentials).

For each edgee∈Efromvtov⁰, letEebe the space of bounded holomorphic functions on Ωv,with the sup norm. This is a Banach algebra for the pointwise product. If|p⁻_e −p⁻_e|<₂ then for anyz∈Ωv,, we have|z−p⁻_e|> ₂. Letfebe the holomorphic functionz7→ ¹

z−p⁻e

on Ω_v,. The mapp⁻_e 7→f_e is holomorphic and bounded by ² from the disk D(p⁻_e,₂) toEe. Let E be the product of the spacesEefore∈Ewith the sup norm. Letf = (fe)e∈E. By lemma 3, the map p7→f is smooth from the ballB(p,₂) toE.

We compose this map with the operator which maps (α, f)∈ BE× E to the differential

X

e∈E_v⁻

α_ef_edz in Ω_v,, forv∈V .

It follows from the definition of the norms that this operator is bilinear bounded from BE× E to H1(Ω). We deal with ω⁺₁ in the same way. This proves the first point of lemma 2.

The proof of point 2 of lemma 2 is very much similar. We first deal with the termω⁻₂. Letf_e,n∈ E_ebe the function

f_e,n(z) = 2

ⁿ 1 (z−p⁻e)ⁿ. If |pe−p⁻

e| < ₂, then ||fe,n||∞ ≤ 1. Define f = (f_e,n)_e∈E,n∈N. By lemma 3, the mapp 7→f is smooth from the ball B(p,₂) to the space E⁰ equal to the product of the spacesE_e fore∈E andn∈N, with the sup norm.

We compose this map with the operator which maps (λ, f) to the differential X

e∈E⁻v

X

n≥2

λ⁻_e,n2⁻ⁿfe,ndz.

It is straightforward to check that this operator is bilinear bounded fromBL×E⁰ toH1(Ω) (the series converges thanks to the term 2⁻ⁿ). We deal with ω⁺₂ in the same way. This proves point 2 of lemma 2.

To prove point 3 of the lemma, we replace the integration circle in the definition ofF_e,n⁺ by the fixed circleC(p⁻_e, r) withr=^ρc₂¹. For each edgee∈E, letA⁻_e be the fixed annulus ^r₂ <|z−p⁻

e|<^3r₂. LetE_e⁰⁰be the spaces of bounded holomorphic functions on A⁻_e. Assume that ||p−p||∞ < ^r₄. If z ∈ A⁻_e, then

r

4 < |z−p⁻_e| < 2r. Assume that |te| < ^c_8c^1ρ2

2 . Then for z ∈ A⁻_e, we have

c₂|t_e|

ρ <|z−p⁻_e| < c1ρ. Using the definition of admissible coordinates, we get

|te|

ρ <|z⁻_e(z)|< ρ, so ϕ⁺_e(z) is defined. Letfe,n∈ E_e⁰⁰ be the function fe,n(z) =

4

ⁿ

(ϕ⁺_e(z)−p⁺_e)ⁿ, z∈A⁻_e. By equation (9), we have

|ϕ⁺_e −p⁺_e| ≤ c²₂|t_e|

|z−p⁻e| ≤ 8c²₂|t_e| ρc1

If|te| ≤ ^ρc_32c¹2 2

, then fe,n is bounded by 1 onA⁻_e. LetE⁰⁰ be the product of the spacesE_e⁰⁰fore∈E andn∈N, with the sup norm. Letf = (fe,n)_e∈E,n∈N∈ E⁰⁰. By lemma 3, the map (p,t, ξ)7→f is smooth from a neighborhood of (p,0, ξ) toE⁰⁰.

We compose this map with the operator (f, w)7→

Z

C(p⁻

e,r)

f_e,n−1w

!

e∈E,n≥2

.

This operator is bilinear bounded fromE⁰⁰× H1(Ω) toBL. This follows readily from the definition of the norms. This proves that F⁺ depends smoothly on parameters. We deal withF⁻ in the same way. This concludes the proof of

lemma 2.

Remark 7 We have used the following fact : iff(z, w) is a bounded holomor- phic function ofz∈Ω⊂C andw∈B(0, R)⊂C^d, then the mapw7→f(·, w) is holomorphic from the open ballB(0, R) to the space of holomorphic functions on Ω with the sup norm. As in the proof of lemma 3, this follows from the fact that we have uniform estimates for the derivatives off with respect tow.

Remark 8 The partial differential of ω with respect to t at t = 0 can be computed as follows : let us fix all variables buttand use the notation _∂t^∂ for the differential with respect totat t= 0. We have

ω=ω1+ω2(F(t, ω)).

Take the differential with respect tot:

∂ω

∂t ·h=ω2(∂F

∂t(0, ω)·h) +ω2(F(0,∂ω

∂t ·h)).

The second term is zero becauseF= 0 when t= 0. To compute the first term we write

∂ϕ⁺_e(z)

∂t_e = 1

(ze⁺)⁰(p⁺e)z⁻e(z). From this we obtain

∂ω

∂t ·h= X

e∈E⁺v

hea⁺_e dz

(z−p⁺e)²+ X

e∈E⁻v

hea⁻_e dz (z−p⁻e)² with

a⁺_e = −1 2πi(ze⁺)⁰(p⁺e)

Z

∂D⁻_e

ω ze⁻

and a similar definition fora⁻_e. In particular, when the coordinates are given as in example 1 byz^±_e =z−p^±_e, we have

a⁺_e = −1 2πi

Z

∂D⁻_e

ω z−p⁻e

.

6 Estimate of ω on the necks

Whent = 0, ω2 = 0 so ω =ω1 is explicitly given by formula 3. Theorem 4 tells us that the restriction of ω to the domain Ω depends smoothly on the parametert, so gives us good control ofω on Ω for small values oftbut says nothing outside of Ω, i.e. on the necks.

In a neighborhood ofp⁺_e, we expectω'α_e ^dz

z−p⁺e

. The following proposition gives a precise statement.

Proposition 2 Assume that the degree of vertices is bounded. There exists a uniform constant C such that for tsmall enough and for any edge e∈E one has

ω

dz − αe

z−p⁺e

≤C||α||_∞ in the annulus|te|^1/2≤ |z_e⁺| ≤ρ,

ω

dz + α_e z−p⁻e

≤C||α||∞ in the annulus|te|^1/2≤ |z_e⁻| ≤ρ.

Proof : we estimateω1 andω2 separately. Considerz in the annulus |te|^1/2 ≤

|z⁺_e(z)| ≤ρ. Then fore⁰∈E_v⁺, e⁰6=e, one has |z−p⁺_e0| ≥ρc1 and fore⁰ ∈E_v⁻, one has|z−p⁻_e0| ≥ρc1. This readily gives the estimate

ω1

dz − αe

z−p⁺e

≤C||α||_∞.

By remark 4 with=ρc1, we have||ω||_∞≤C||α||_∞. Using thatλ=F(ω) and estimate (10), we have

4

n

|λ⁺_e,n| ≤Cr c²₂|te|

r ⁿ⁻¹

||α||_∞.