ON THE PERIODS OF HARMONIC AND MEROMORPHIC FORMS ON KLEIN SURFACES

MEROMORPHIC FORMS ON KLEIN SURFACES

A. ALONSO GOMEZ and A. FERNANDEZ ARIAS

We extend to the wider setting of Klein surfaces some classical work of L.V.

Ahlfors on Riemann surfaces. We recall the notion of meromorphic form on a Klein surface and extend it to define complex differential forms on Klein surfaces, concept which differs from the standard definition of differential forms on a dif- ferentiable manifold. The spaces of forms on a Klein surface can be embedded into the corresponding spaces of forms of the complex double of the surface, which turns out to be a proper Riemann surface, what allows to introduce a Hilbert space structure on them. In this way we can introduce a metric in the spaces of forms on a Klein surface, so that the question of continuity of functionals on these spaces makes sense. Also making use of the complex double, it is possible to introduce the notion of intersection number of two curves. Finally, we relate some functionals associated with a curve, which might play the role of periods of the curve, with the intersection number when we apply these functionals to some specific forms.

AMS 2000 Subject Classification: 30F45.

Key words: Klein surface, harmonic and meromorphic form, period.

1. INTRODUCTION

In this paper we extend some results of L.V. Ahlfors on harmonic and meromorphic differentials on Riemann surfaces to the wider setting of Klein surfaces.

In Section 2 we recall some definitions and basic facts related to Klein surfaces and differential forms on Klein surfaces. In Section 3 we discuss briefly some classical definitions and results on harmonic and meromorphic differen- tials on Riemann surfaces. In Section 3 we also recall the natural way to endow the spaces of harmonic and meromorphic forms on a Riemann surface with an structure of Hilbert space and analyze this method for Klein surfaces. The main problem here is the absence of conjugation for a form in the Klein surface setting, on the other hand we can embed the space of forms on Klein surfaces into that of its complex double, which is, in fact, a classical Riemann surface,

REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 349–360

allowing in this way to introduce a metric in the spaces of forms on Klein surfaces.

In Section 4 we introduce some functionals on the spaces of forms on Klein surfaces, which will play the role of periods for the integrals of a form on a Klein surface. Finally, we extend the intersection theory to the non-orientable case, making use again of the complex double and find some expressions for the intersection number of two curves in terms of the above mentioned functionals applied to forms associated with the curves.

2. NOTATION AND BASIC FACTS

2.1.We recall that a Klein surface is a surfaceS endowed with a diana- lytic structure A, that is, there is defined on S an atlas of charts covering S such that the transition maps are either analytic or antianalytic. In the first case the transition maps will preserve the orientation while in the se- cond they change it. In the case that all these changes of parameters can be taken analytic, that is, orientation preserving, we obtain an orientable surface and, in fact, this is a classical Riemann surface, otherwise we obtain a proper nonorientable Klein surface.

With every proper Klein surface there is associated a Riemann surface Sc which is a double cover of S, with a canonical projection p :Sc → S and where is defined an antianalytic involution σ :S_c →S_c, without fixed points for which p◦σ=p.

2.2.A complex differential formω on a differentiable surface is a collec- tion of linear formsadx+bdywith differentiable complex valued functionsa, b satisfying the compatibility conditions

a₁ =a∂x

∂x1

+b ∂y

∂x1

, b₁=a∂x

∂y1

+b∂y

∂y1

,

when we change the local chart, that is, we pass from the coordinates (x, y) to (x₁, y₁). This can also be expressed in complex notation. An expression adx+bdy can be written in the formαdz+βdz, and with this notation the compatibility condition becomes

α1 =α∂z

∂z₁ +β ∂z

∂z₁, β1 =α ∂z

∂z₁ +β ∂z

∂z₁.

We might call ω harmonic when a, b were harmonic, equivalently when α, β were harmonic.

Alling and Greenleaf [4] define a meromorphic formωon a Klein surface as a collection of linear expressions αdz, where α is a meromorphic function associated with a local chart, in such a way that for analytic, that is, orientable

changes of coordinates, the compatibility condition is α₁(z₁) =α(z(z₁))dz

dz1

,

whereas for antianalytic changes of coordinates the compatibility condition is α1(z1) =α(z(z1))∂z

∂z1

,

so that if we write the given complex form as adx+bdy, the compatibility condition becomes in the case of an orientable change of coordinates the same as defined for general differentiable manifolds, that is,

a₁ =a∂x

∂x₁ +b ∂y

∂x₁, b₁=a∂x

∂y₁ +b∂y

∂y₁, whereas in the nonorientable case

a1 =a∂x

∂x1

+b ∂y

∂x1

, b1=a∂x

∂y1

+b∂y

∂y1

,

so that this change of coordinates differs from the general one and we cannot consider a form in this sense as a form in a differentiable manifold.

However, we shall take this definition for analytic or meromorphic forms and also for complex differentiable forms in general on a Klein surface.

2.3. We can conclude that, given a path γ : [0,1] → S such that γ^∗ = γ([0,1])⊂U∩U1, whereU, U1 are the domains of definition of the parameters z=x+yi,z1=x1+y1i, if the change of coordinateslz,z1 =z◦z⁻¹₁ is orientable then for C¹ complex formsω on a Klein surface we have

Z

γ

adx+bdy= Z

γ

a₁dx₁+b₁dy₁, whereas if the change of coordinates is nonorientable, then

Z

γ

adx+bdy= Z

γ

a1dx1+b1dy1 = Z

γ

a1dx1+b1dy1. So that, the number ReR

γadx+bdy remains unchanged, whereas Im

Z

γ

adx+bdy= Im Z

γ

a1dx1+b1dy1

if the change of coordinates is orientable, and Im

Z

γ

adx+bdy=−Im Z

γ

a1dx1+b1dy1, otherwise.

3. THE SPACES ΓRc(S), ΓCc(S) FOR A KLEIN SURFACE S LetS be a Klein surface andSc its complex double. Let ω=adx+bdy be a closed form complex form on S, where a, b are C¹ complex functions, which can also be written in the form ω=αdz+βdz,so that it satisfies the above described compatibility conditions.

For the case of real forms this condition coincides with the standard compatibility condition for differential forms on differentiable manifolds and, therefore, in this case and only in this case a complex differential form on a Klein surface can be considered a differential form in the sense of a differential form on a differentiable manifold.

Letω be a closed form satisfying the above compatibility condition. We can then define an associated form ωe in the complex double Sc. Let U = {U_i}_i∈

Nbe a covering ofSby simply connected domains, so thatωis expressed as ω =aidxi+bidyi in Ui with respect to the parameter zi =xi+ iyi inUi. GivenU_i∈ U, letU_ic¹,U_ic² be the corresponding open parametric sets inS_c over Ui. We shall set ωe=adxi+bdyi inU_ic¹ wherezi =xi+ iyi is the parameter in U_ic¹ and ωe = adx_i +bdy_i in U_ic² with parameter z_i = x_i −iy_i. It can be checked that ωe is a well-defined form onSc.

Whenωis a meromorphic form, that is,ωhas the formω=ϕ_i(z_i) dz_i in U_i, the above definition ofωe becomesωe =ϕ_i(z_i)dz_i inU_ic¹ and ωe=ϕ_i(z_i)dz_i in U_ic².Here, it can be checked that ωe is a well-defined meromorphic form on the Riemann surface Sc.

In the case of real closed forms, we have a well defined scalar product hλ₁, λ₂i=

Z Z

Sc

λ₁∧λ^∗₂

on S_c. In particular, we can use this scalar product in the case whenλ₁ =ωf₁, λ2 =fω2, whereω1, ω2 are closed differential forms inS. We shall say thatωf1, ωf₂ are symmetric forms inS_c.

In the complex case, we have an Hermitean product, namely, hλ₁, λ2i=

Z Z

Sc

λ1∧λ^∗₂,

where in both cases λ^∗ denotes the conjugate form ofλ, i.e., if λ=adx+bdy thenλ^∗=−bdx+ady with respect to the same coordinates. The fact that this is a well defined form follows from Cauchy-Riemann equations of the change of coordinates.

In our particular case of symmetric forms, we shall consider the norm kωke _c =

Z Z

Sc

ωe∧ωf^∗,

or, in the complex case,

kωke _c = Z Z

Sc

ωe∧ωf^∗,

and denote by Γ¹Rc(S_c), Γ¹Cc(S_c) the spaces of closed real and complex C¹ forms on S_c with finite norm, respectively. We also consider the spaces Γ¹Rh(Sc), Γ¹Ch(Sc) of real and complex harmonic forms on Sc and the space M(S_c) of meromorphic forms. We clearly have

Γ¹Rh(S_c)⊂Γ¹Ch(S_c)⊂Γ¹Cc(S_c), M(S_c)⊂Γ¹Ch(S_c).

By the above considerations we can identify the subspaces of symmetric forms of these spaces with the corresponding spaces of forms on S, namely

Γ¹Rc(S), Γ¹Cc(S), Γ¹Rh(S), Γ¹Ch(S), M(S).

Making use of this fact, we can consider these spaces as normed spaces, where we setkωk=keωk_c for a C¹ form ω inS.

Following again Ahlfors [3], we can consider Γ¹Cc(S_c) as a subspace of the Hilbert space ΓCc(Sc) of the forms ω = adx+bdy, where now we only imposea, bto beL² functions. With the scalar product and the norm defined here in the same formal way as for Γ¹, the space Γ becomes a Hilbert space.

Thus, the spaces ΓRc(S_c), ΓCc(S_c) are defined following [3], so that they are Hilbert spaces and ΓRc(S), ΓCc(S) can be defined as the closures of Γ¹Rc(S), Γ¹Cc(S), respectively, and they also become Hilbert spaces.

4. THE FUNCTIONALS R, L, |I|

As a consequence of the considerations about the integral along a path γ with range into a parametric domain, we conclude that for a C¹ path γ the number

Rγ(ω) = Z

γ

Reω,

can be well defined for an arbitrary form ω ∈ Γ¹C(S) on a Klein surface S along a C¹ path inS.

We remark that we can also define the absolute value of the integral of the imaginary part

Z

γ

Imω ,

for a simple closed curve γ, that is, γ does not intersect itself. Starting from the initial point γ(0), we can choose a finite chain of charts U₁, . . . , U_n, such that the change of coordinates in Ui ∩Ui+1 is orientable except perhaps at U_n∩U₁, where γ(0) = γ(1) lies, that is γ(0) = γ(1)∈ U_n∩U₁. In fact, the only critical value might be precisely γ(0) =γ(1). We shall consider for small

values oftthe parameters in U1 while for values oftnear 1 the parameters of the corresponding γ(t) will be those ofU_n.Since the point γ(0) =γ(1) forms a negligible set, the integral of the imaginary part ofω is well defined and only depends on the initially chosen chart (U1, z1), in such a way that for different choices of this chart we can obtain either a particular value of R

γImω, or its opposite, so that the absolute value

R

γImω

remains unchanged.

For a non simple closed curve γ, we can decompose it in simple closed curves γ¹, . . . , γⁿ corresponding to the subarcs γⁱ = γ_|[ti−1,ti], where t₀ < t₁

< . . . < t_nare those points for which γ(ti−1) =γ(t_i).

Given a differential formω onS, we can define the number

|I|_γ(ω) =

n

X

i=1

Z

γⁱ

Imω .

For a single simple closed curveγ,it follows easily that this number is subad- ditive with respect to ω, that is,

|I|_γ(ω1+ω2)≤ |I|_γ(ω1) +|I|_γ(ω2) and also satisfies |I|_γ(αω) =|α| |I|_γ(ω) for a realα.

Finally, we shall introduce the functionalLγ,asLγ(ω) =Rγ(ω)+|I|_γ(ω).

Next, we consider the question of the continuity of Rγ,|I|_γ and Lγ.We shall consider these functionals defined on the spaces ΓRc(S), ΓCc(S) of closed forms. First of all, we shall deal withRγ.Proceeding as in the orientable case, see Ahlfors and Sario [3], we make use of the existence of partitions unity{e_i}, associated with open coverings on S.

We can cover γ^∗ by a finite open covering {V_i}_i=1,...,n such that the V_i are simply connected parametric domains and e1, . . . , en is a finite partition of unity subordinated to this covering. Then we can set

(4.1)

Z

γ

Reω=

n

X

i=1

Z

γ

Re (eiω) =

n

X

i=1

Re Z

(dei)ω,

where the integral in the right hand side is extended over the upper half plane.

Equation (4.1), that is, the Stokes Theorem, is still valid locally in a parametric domain. This leads to

(4.2)

Z

γ

Re(eiω)

≤ Z

γ

eiω

=

Z Z

H⁺

(dei)ω and if we set

ω=a_idx+b_idy, de_i = ∂e_i

∂xdx+∂e_i

∂ydy

in Vi, from the classical H¨older inequality we get

Z Z

H⁺

(de_i)ω

=

Z Z

H⁺

∂e_i

∂xb_i−∂e_i

∂ya_i

dxdy (4.3)

≤ Z Z

H⁺

∂ei

∂xbi

dxdy+ Z Z

H⁺

∂ei

∂yai

dxdy

≤ Z Z

H⁺

∂e_i

∂x 2

dxdy

!¹₂ Z Z

H⁺

|b_i|²dxdy ¹₂

+ Z Z

H⁺

∂ei

∂y 2

dxdy

!¹

2Z Z

H⁺

|a_i|²dxdy ¹

2

≤2 Z Z

H⁺

∂e_i

∂x 2

+ ∂e_i

∂y 2!

dxdy

!¹₂ Z Z

H⁺

|a_i|²+|b_i|² dxdy

¹₂

≤C_ikωk.

From (4.1),(4.2) and (4.3) we conclude that

(4.4) |R_γ(ω)| ≤Ckωk

for some C >0.Since R_γ is a real linear functional, this implies continuity.

A similar argument can be applied to|I|_γ if we decompose γ into simple parts γ_i. Let us consider a particularγ_i and a finite covering of γ_i by simply connected domains V_i and in such a way that, as in the first part of the section, the change of parameters in Vi ∩Vi+1 is orientable. In this way, we have defined a particular choice for the value R

γiImω, then arguing as we did above for R

γiReω.

We conclude that

(4.5)

Z

γi

Imω

≤C_ikωk,

and, as a consequence, also that|I|_γ(ω)≤Ckωkfor some C >0.Though |I|_γ is not a linear functional, it can also be seen that (4.5) implies the continuity

of |I|_γ.In fact,

|I|_γ(ω1)− |I|_γ(ω2) =

n

X

i=1

Z

γi

Imω1

−

n

X

i=1

Z

γi

Imω2

(4.6)

≤

n

X

i=1

Z

γi

Imω₁

− Z

γi

Imω₂

≤

n

X

i=1

Z

γi

Imω₁− Z

γi

Imω₂

=

n

X

i=1

Z

γi

Im (ω₁−ω₂)

=|I|_γ(ω₁−ω₂)≤Ckω₁−ω₂k.

We remark that the termsR

γiImω₁,R

γiImω₂ are determined by the choice of Im(ω₁−ω₂), so that the inequalities hold without ambiguity. The Lipschitz condition (4.5) implies uniform continuity of |I|_γ. Both facts, the uniform continuity of R_γ, |I|_γ imply the uniform comtinuity of L_γ. Also, we get the stronger Lipschitz condition

|L_γ(ω1)−Lγ(ω2)| ≤Ckω₁−ω2k.

We remark that the functionals Rγ, |I|_γ, Lγ are only defined for C¹- forms, that is, in Γ¹(S). We have checked that they are uniformly continuous, in fact, satisfy a Lipschitz condition on Γ¹Cc(S), so that all of them can be extended by continuity to their completions.

5. REPRODUCING FORMULAE FOR THE FUNCTIONALS R_γ, |I|_γ, L_γ

Given a simple closed curve γ,that is, with no selfcrosssing points, and once fixed a covering ofγ^∗by simply connected parametric domainsU1, . . . , Un

with orientable change of parameters in U_i∩Ui−1 except perhaps in U_n∩U₁, we have a well-defined value R

γω, for a C¹-form ω, that is, for ω ∈ Γ¹C(S).

In this space we have a well-defined linear functional T_γ(ω) =R

γω and, as a consequence, on the isomorphic space Γ¹Cs(Sc).

If we restrict this functional to the subspace Γ¹Cc(S) of closed forms ω in Γ¹C(S) then, by the results of the previous section, the functional T_γ : Γ¹Cc(S)→Cis continuous on Γ¹Cc(S) and, as a consequence, on the isomor- phic space Γ¹Ccs(Sc) of closed symmetricC¹-forms onSc.Therefore, it can be extended to its closure Γ_csC(S_c) which is a Hilbert space, so that we obtain a representation of Tγ in the form Tγ(ω) =hω, ae γi, where aγ ∈ΓC(Sc).

Therefore, we can express the corresponding operatorsRγ,|I|_γ as Rγ(ω) =

Z

γ

Reω= Re Z

γ

ω= Reheω, aγi,

|I|_γ(ω) = Z

γ

Imω

=|Imhω, ae _γi|

and, as a consequence, also as Lγ(ω) = Reheω, aγi+|Imhω, ae γi|.

In general, for a closed Jordan curve γ in S which is decomposed into simple Jordan curves γ1, . . . , γn, that is, γ = γ1 ∼ γ2 ∼ . . . ∼ γn, there will exist corresponding elements a_γ1,...,a_γn in Γ_csC(S_c) such that

R_γ(ω) = Rehω, ae _γ1i+· · ·+ Reheω, a_γni= Rehω, ae _γ1 +· · ·+a_γni,

|I|_γ(ω) =|Imheω, a_γ1i|+· · ·+|Imhω, ae _γni|, and

(5.1) Lγ(ω) = Rehω, ae γ1 +· · ·+aγni+|Imhω, ae γ1i|+· · ·+|Imhω, ae γni|.

6. INTERSECTION THEORY ON KLEIN SURFACES In the case of a Riemann surface, that is, in the orientable case, the intersection number of two cycles is well defined. The orientability seems to be essential in this concept, so that we shall make use of the complex double associated with a Klein surface in order to extend, as far as it can be done, this notion to the non-orientable case.

Letγ1,γ2 be two closed curves inS.Three cases can arise here:

i) both ofγ₁,γ₂, preserve the orientation at their endpoints;

ii) one of them, say γ2, changes the orientation at its endpoint γ2(0) = γ₂(1),whereas γ₁ preserves the orientation at its endpoint γ₁(0) =γ₁(1);

iii) both ofγ1,γ2 change the orientation at their endpoints.

We shall analyze successively the different cases.

Let us first consider case i). In this case we liftγ₁, γ₂ toS_c, respectively γe1, γe2, such that p(γe1) = γ1, p(γe2) = γ2, where each of γe1,γe2 will be formed by two closed curves, say fγ₁¹,fγ₁²,fγ₂¹,fγ₂², where σ(fγ₁¹) =fγ₁²,σ(fγ₂¹) =fγ₂².

We can considerγe₁,γe₂ as 1-chains, see Ahlfors and Sario [3], so that the intersection numberγe1 ×γe2is defined. However, we observe that this number should be zero since given an intersection point p of γe₁,γe₂ then, at the point σ(p), there should be another intersection with opposite sign since σ changes orientation. However, if we fix one of the curves, say fγ₁¹ of γe1, we can then consider the number fγ₁¹ × γe2, so that for fγ₁² we should obtain the opposite

number, that is, fγ¹₁×γe₂ =−γe₂×fγ₁².Thus, the numberγ₁×γ₂ defined as γ₁×γ₂=

fγ₁¹×γe₂ =

fγ₁²×γe₂

is well defined and we shall adopt it as the intersection number of γ1and γ2. Next, consider case ii). Here γe₂ is not formed by two closed curves but by two open arcs fγ¹₂,fγ²₂, the final point fγ₂¹(1) of fγ₂¹, being the same point as the initial point fγ₂²(0) of fγ₂² and conversely. So that, they form together a closed curve fγ₂¹∼fγ₂².Here, we can again consider

γ₁×γ₂ =

fγ₁¹× γe₂=fγ₂¹ ∼fγ₂² =

−fγ₁²× γe₂ =fγ¹₂ ∼fγ₂² as intersection number.

Finally, in case iii),γe₁ and γe₂ will be formed both of them by two open arcs fγ₁¹,fγ₁² andfγ₂¹,fγ₂² respectively, so thatγe1 =fγ₁¹∼fγ₁² andγe2=fγ₂¹ ∼fγ₂² are closed Jordan curves and we shall define

γ₁×γ₂ =

fγ₁¹×γe₂ =

−fγ₁²×γe₂ .

It can be easily verified that in all cases we have the symmetry property γ₁×γ₂=γ₂×γ₁.

7. INTERSECTION NUMBER OF TWO CURVES AND PERIODS OF ASSOCIATED DIFFERENTIALS Letp1, p2be two points in the same parametric domain, mapped onto the unit disc U ={z| |z|<1} by the corresponding parameter. Letζ₁ =z(p₁), ζ2 = z(p2) such that |ζ₁|,|ζ₂|< r1 < r2 < 1, and consider the function in U defined as

v= logz−ζ₂ z−ζ₁.

The multivalued function v admits a singlevalued branch in r₁ <|z|<1.We fix a branch and set

Θ =





 1

z−ζ2 −_z−ζ¹

1

dz for |z| ≤r1, d(ev) forr1<|z|<1,

0 outsideU,

where e(z) is an auxiliary function of class C² on U which is identically one for |z|< r1 and identically zero for |z|> r2. It is easy to check that Θ∈C¹ and dΘ = 0 except at ζ₁, ζ₂.We remark that the definition of Θ is associated with a previous choice of a chart (U, z).

In the orientable case, the periods of Θ along a curveγ can be expressed in terms of the intersection number of γ and a simple arc joining ζ₁ and ζ₂. Proceeding in a similar way, we can obtain analogous results for the numbers Rγ(Θ),|I|_γ(Θ), Lγ(Θ) in the case of Klein surfaces.

First of all, we observe that any simple closed curveγ can be written in the formγ0∼γ¹whereγ0also is a simple closed curve which lies inU whereas γ¹ ⊂ S\U. Clearly, R_γ(Θ) = R_γ0(Θ), |I|_γ(Θ) = |I|_γ0(Θ), L_γ(Θ) = L_γ0(Θ), and it can easily be checked thatRγ0(Θ) = 0 and, as a consequence,Rγ(Θ) = 0.

Let nowcbe a simple arc joiningζ₁andζ₂insideU. Then the intersection number c×γ₀ is well-defined and it can easily be checked that

|I|_γ(Θ) =|I|_γ0(Θ) =c×γ0 = 2π|n(γ, ζ₂)−n(γ, ζ1)|.

If γ is not a simple curve and γ =γ1 ∼ . . .∼γn, with γi, i= 1, . . . , n, simple closed curves, then we get

|I|_γ(Θ) =

n

X

i=1

|I|_γi(Θ) =

n

X

i=1

|I|_γ0i(Θ) =

n

X

i=1

c×γ_0i, where γ0i is the curve inside U corresponding toγi.

Finally, if c = c₁ ∼c₂ ∼ . . . ∼c_m with c_j simple closed arcs contained in parametric domains, we define Θ(c) = Θ(c1) +· · · + Θ(cm) and obtain Re Θ(c) = 0 and

Lγ(Θ(c)) =|I|_γ(Θ(c)) =

m

X

j=1 n

X

i=1

cj×γi.

Now, using (4.1), we conclude that Lγ(Θ(c)) =

m

X

j=1 n

X

i=1

cj×γi=

m

X

j=1 n

X

i=1

|hΘ(c_j), aγii|.

Acknowledgements. The second author was partially supported by the MEC Project MTM-090960.

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Sci. U.S.A.42(1956), 758–760.

[2] L.V. Ahlfors,Abel’s theorem for open Riemann surfaces. Seminars on analytic functions II, pp. 7–19. Institute for Advance Study (Princeton), New York, 1958.

[3] L.V. Ahlfors and L. Sario, Riemann Surfaces. Princeton Univ. Press, Princeton, NJ, 1960.

[4] N.L. Alling and N. Greenleaf, Foundations of the Theory of Klein Surfaces. Lectures Notes in Math.219. Springer-Verlag, Berlin, 1971.

[5] I. Bˆarz˘a,Integration on nonorientable Riemann surfaces. In: K. Sekigawa and S. Dixmier (Eds.),Proc. Internat. Workshop on Almost Complex Structures(Sofia, 1992), pp. 63–

97. World Sci. Publ., River Edge, NJ, 1994.

[6] M. Schiffer and D. Spencer, Functionals of Finite Riemann Surfaces. Princeton Univ.

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Received 1 October 2008 Universidad Polit´ecnica de Madrid Dpto de Matem´aticas

E.U.I.T. Agricolas, Ciudad Universitaria [email protected]

and

Facultad de Ciencias, UNED Dpto de Matem´aticas Fundamentales

Ciudad Universitaria 28040 Madrid, Spain [email protected]