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LOWER BOUNDS FOR THE EIGENVALUES OF THE Spin c DIRAC OPERATOR ON SUBMANIFOLDS

Roger Nakad, Julien Roth

To cite this version:

Roger Nakad, Julien Roth. LOWER BOUNDS FOR THE EIGENVALUES OF THE Spin c DIRAC

OPERATOR ON SUBMANIFOLDS. Archiv der Mathematik, Springer Verlag, 2015, 104 (5), pp.451-

461. �hal-01112320�

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OPERATOR ON SUBMANIFOLDS

ROGER NAKAD AND JULIEN ROTH

ABSTRACT. We prove lower bounds for the eigenvalues of theSpincDirac operator on submanifolds. These estimates are expressed in terms of extrinsic and intrinsic quanti- ties. We also give estimates involving the Energy-Momentum tensor as well as conformal bounds. The limiting cases of these estimates give rise to particular spinor fields, called generalized twisted Killing spinors, which are also studied.

1. INTRODUCTION ANDPREMILINARIES

The limiting cases of estimates for eigenvalues of the Dirac operator on compact (with or without boundary) manifolds give rise to examples of special geometries. For instance, equality in the classical inequalitiy of Friedrich [3]

λ16 n 4(n−1)inf

M ScalM,

wherenis the dimanesion of the manifoldM andScalM its scalar curvature, forces the manifold to be Einstein with positive scalar curvature, due to the fact that the eigenspinor associated with the first eigenvalue is then a real Killing spinor.

This is also the case for the conformal inequality of Hijazi [9]

λ16 n 4(n−1)µ1,

whereµ1is the first eigenvalue of the Yamabe operator as well for the inequality involving the Energy-momentum tensor [10]

λ161 4inf

M(ScalM+|Qϕ|2),

whereQϕis the Energy-Momentum tensor associated with a first eigenspinorϕ.

On the other hand, in the recent years, many estimates have been proved for the eigenvalues of theSpincDirac operator. A great difference between both cases if that the equality cases forSpinclower bounds are less and therefore give larger classes of limiting manifolds (see [8, 13, 14] for instance).

In this note, we prove a new lower bound for the eigenvalues of theSpincDirac operator on submanifolds of Spinc manifolds (see Theorem 2.1). This generalizes for theSpinc case, previous estimates by Hijazi-Zhang [11, 12] and Ginoux-Morel [5]. The limiting case of this estimate is characterized by the existence of particular spinor fields called generalized twisted Killing spinors. We will study these particular spinor fields and show that under a natural assumption on the dimension and the codimension of the submanifold, they are in facttwisted Killing spinorswhich generalize naturally the usual Killing spinors for twisted spinor bundles.

Finally, we also prove conformal estimates (Theorem 2.3) as well as estimates involving the Energy-Momentum tensor associated with an eigenspinor (Theorem 2.2).

2010Mathematics Subject Classification. 53C27, 53C42.

Key words and phrases. Submanifolds, Dirac Operator, Eigenvalues,Spincspinors, Twisted Killing spinors.

1

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2 R. NAKAD AND J. ROTH

We recall briefly some basic facts about Spinc manifolds and their hypersurfaces (the reader can refer to [1, 2, 4, 5]). Let(fMm+n,eg)be a RiemannianSpincmanifold and Mm a submanifold isometrically immersed into Mf. Assume thatM is also Spinc and denote byN M the normal bundle of the immersion ofM intoMf. We denote byieΩ(resp.

iΩ) the curvature2-form of the corresponding auxiliary line bundle. Since the manifolds M andMfareSpinc, there exists aSpincstructure on the bundleN M. We denote byΣN theSpincbundle ofN M and let

Σ :=

ΣM⊗ΣN ifmornis even,

ΣM⊗ΣN⊕ΣM⊗ΣN otherwise

It is well known that there is a natural isomorphism betweenΣandΣMf|M. Moreover, we denote by∇the covariant derivative onΣdefined by

∇:=

ΣM ⊗Id + Id⊗∇ΣN ifmornis even,

ΣM ⊗Id + Id⊗∇ΣN ⊕ ∇ΣM⊗Id + Id⊗∇ΣN otherwise.

We have the following identification between the Clifford multiplication X·M ϕ= (X·ω·ψ)|M,

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whereϕ=ψ|M,ψ∈Γ(ΣMf),ω :=ωn ifnis even andω =−iωn ifnis odd, with ωn =i[n+121· · ·νnthe complex volume element of the normal bundle. We also recall the SpincGauss formula

∇eXϕ=∇Xϕ+1 2

m

X

j=1

ej·B(X, ej)·ϕ, (2)

where∇e is the spinorial connection onMfandBis the second fundamental form ofM in Mf. We will denote byHthe mean curvature.

Now, let us define the following Dirac operators D=

m

X

j=1

ej· ∇ej, De:=

m

X

j=1

ej·∇eej,

and

DH= (−1)nω·De = (−1)nω·D+1

2H·ω·.

Clearly, from the spinorial Gauss formula (2), we haveDe =D−12H·. Moreover (see [1, Lemma 2.1]),DandDHare formally self-adjoint andDH2 =DeDewhereDeis the formal adjoint ofDewith respect to theL2-scalar productR

M(·,·)dvg.

We finish this section of preliminaries by the two following lemmas. The first one gen- eralizes the classical Lichnerowicz formula in the context of twistedSpinc spinor bun- dles. Before stating the lemma, we need to introduce the following function associ- ated to a spinor field ϕ ∈ Γ(Σ), RNϕ := 2Pn

i,j=1

D

ei·ej·Id⊗ RNei,ejϕ,|ϕ|ϕ2

E , on Mϕ ={x∈M |ϕ(x)6= 0}. Here,RNei,ej stands for the spinorial curvature of the normal bundleN M.

Proposition 1.1(Twisted Lichnerowicz formula). For any spinor fieldϕ ∈Γ(Σ), point- wise onMϕ, we have

D2ϕ, ϕ

=h∇∇ϕ, ϕi+1

4(ScalM +RNϕ)|ϕ|2+ i

2hΩ·ϕ, ϕi. (3)

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Proof:We give the proof in the case wheremornis even. The other case is similar. We compute the square of the Dirac operatorDacting onϕ=α⊗σ. We have

(4) D2ϕ=∇∇ϕ+1

2

m

X

i,j=1

ei·ej· Rei,ejϕ,

where∇is the formal adjoint of∇andRis the spinorial curvature associated with the connection∇. From the definition of∇, we see easily that

Rei,ejϕ= (RMe

i,ejσ)⊗α+σ⊗(RNe

i,ejα).

Then, a classical computation on each factor gives the desired formula.

We have this second elementary lemma

Proposition 1.2. For any spinor fieldϕ∈Γ(Σ), we have hiΩ·ϕ, ϕi ≥ −cm

2 |Ω||ϕ|2. (5)

Proof:Ifmornis even, thenΣM⊗ΣN and a spinorϕ∈Σcan be writtenϕ=σ⊗α.

Hence, we have

hiΩ·ϕ, ϕi = hiΩ·(σ⊗α), σ⊗αi=hi(Ω ·

Mσ)⊗α, σ⊗αi

= hi(Ω ·

Mσ), σi|α|2≥ −cm

2 |Ω||σ|2|α|2≥ −cm 2 |Ω||ϕ|2.

Note that we use the fact that the scalar product onΣis the product of the scalar products onΣM andΣN. We also use the classical estimate onM, that ishiΩ·σ, σi ≥ −c2m|Ω||σ|2 (see [8]). Ifmandnare odd, thenΣ = ΣM⊗ΣN⊕ΣM⊗ΣN and a spinorϕ∈Σis of the formϕ= (α⊗σ, σ0⊗α0). Thus, we have

hiΩ·ϕ, ϕi = hiΩ·(σ⊗α), σ⊗αi+hiΩ·(σ0⊗α0), σ0⊗α0i

≥ −cm

2 |Ω| |σ|2|α|2+|σ0|20|2

≥ −cm

2 |Ω||ϕ|2.

This concludes the proof.

2. EIGENVALUE ESTIMATES FOR SUBMANIFOLDS

Now, we have all the ingredients to state the eigenvalue estimates. We begin by the follow- ing basic estimates involving intrinsic terms (scalar curvature, curvature of the line bundle overM) and extrinsic terms (mean curvature and spinorial normal curvature). This result generalizes in theSpinc setting the estimate of Hijazi-Zhang [11] (extending to any codi- mension by Ginoux-Morel [5]).

Theorem 2.1. Let(Mm, g)be a compact RiemannianSpinc manifold isometrically im- mersed into a RiemannianSpincmanifold(Mfm+n,eg). Consider a non-trivial eigenspinor fieldϕ ∈ Γ(Σ)for the submanifold Dirac operatorDH, i.e.DHϕ = λϕ. Assume that m≥2and

ScalM+REϕ−cm|ΩM|> m−1

m ||H||2>0 onMϕ, then, we have

λ2≥ 1 4inf

Mϕ

r m

m−1(ScalM +RNϕ −cm|Ω|)− ||H||

2 . Moreover, if equality holds, thenϕis a twisted generalized Killing spinor.

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4 R. NAKAD AND J. ROTH

Proof: Let λ be an eigenvalue of the submanifold Dirac operator DH andq a smooth function, nowhere equal tom1. We consider the following modified connection∇λ,qdefined by

λ,qX ψ=∇Xψ+ 1−q

2(1−mq)X·H·ψ+qλX·ω·ψ,

for any spinor fieldψ∈Γ(Σ). Letϕbe an eigenspinor forDHassociated with the eigen- valueλ. Using the Twisted Lichnerowicz formula (3), we can easily compute

Z

M

|∇λ,qϕ|2vg = Z

M

(1 +mq2−2q)h λ2−1

4

ScalM +RNϕ

1 +mq2−2q−(m−1)kHk2 (1−mq)2

|ϕ|2

− i

2(1 +mq2−2q) <Ω·ϕ, ϕ >i vg

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Then, using Inequality (5) and by assumingm(ScalM+RNϕ−cm|Ω|)>(m−1)kHk2>0, we can chooseqso that

(1−mq)2= (m−1)kHk q m

m−1(ScalM +RNϕ −cm|Ω|)− kHk , (7)

onMϕ. Inserting (7) in (6), and since the complement ofMψ inM is of measure0, we conclude because the left member of (6) is nonnegative.

If equality occurs,∇λ,qX ϕ= 0, which implies that|ϕ|is constant onM and Dϕ= m(1−q)

2(1−mq)H·ϕ+mqλω·ϕ.

On the other hand, sinceϕis an eigenspinor forDHand the link betweenDHandD, we get

0 = λω·ϕ+H

2 ·ϕ− m(1−q)

2(1−mq)H·ϕ−mqλω·ϕ

= (1−mq)2λω·ϕ−(m−1)H 2 ·ϕ.

Moreover, equality also implies2|λ|=q

m

m−1(ScalM+RNϕ −cm|Ω|)− ||H||. From the expression ofqand the above relation, we haveω·ϕ= sgn(λ)||H||H ·ϕand thusϕsatisfies

Xϕ = −mfX ·ω·ϕ, withf = sgn(λ)2 q m

m−1(ScalM +RNϕ −cm|Ω|). That is,ψis a generalized twisted Killing spinor. Note that heref isa prioria function. We will see in Section 3 that under some assumptions on the dimensionsmandn, the functionf is

constant.

Now, we define the Energy-Momentum tensor associated with a spinor fieldψ∈Γ(Σ)on Mψby

Qψij =1

2(ei·ω· ∇ej +ej·ω· ∇ei, ψ

|ψ|2) Note that

Qψij= 1

2(ei·Mej +ej·Mei, ψ

|ψ|2)

so it is the intrinsic energy momentum tensor and it is the one appearing in the Einstein Dirac equation. We have the following estimate involving the Energy-Momentum tensor.

Theorem 2.2. Let(Mm, g)be a compact RiemannianSpinc manifold isometrically im- mersed into a RiemannianSpincmanifold(Mfm+n,eg). Consider a non-trivial eigenspinor fieldϕ ∈ Γ(Σ)for the submanifold Dirac operator DH, i.e. DHϕ = λϕ. Assume that m≥2and

ScalM +RNϕ + 4|Qϕ|2−cm|ΩM|>||H||2>0

(6)

onMϕ, then, we have λ2≥ 1

4inf

Mϕ ScalM+RNϕ + 4|Qϕ|2−cm|Ω|)− ||H||2 .

Moreover, if equality holds, thenϕis a twisted (symmetric) EM-spinor.

Proof:For any real functionqthat never vanishes, consider the modified covariant deriva- tive defined onΓ(Σ)by

Qeiψ=∇eiψ− 1

2mqei·H·ψ+ (−1)n+1qλei·ω·ψ+X

j

Qψijej·ω·ψ

Again, we can compute, for an eigenspinorϕ Z

M

|∇Qϕ|2vg = Z

M

(1 +mq)2h λ2−1

4

ScalM +RNϕ + 4|Qϕ|2

(1 +mq2) −kHk2 mq2

i|ϕ|2vg

−1 4 Z

M

(1 +mq2)h 2 mq(1 +mq2)

kHk2−< H·ϕ, ω·ϕ >2

|ϕ|4

i|ψ|2vg

−i 2

Z

M

<Ω·ϕ, ϕ >vg

(8)

Now, we use again (5) and if moreover,ScalM+RNϕ −cM|Ω|+ 4|Qϕ|2>kHk2>0, we take

q= v u u t

kHk m(q

ScalM +RNϕ −cM|Ω|+ 4|Qϕ|2− kHk) ,

and then by the Cauchy Schwarz inequality, we have kHk2−< H·ϕ, ω·ϕ >2

|ϕ|4 ≥0.

If equality holds, then ∇Qϕ = 0and equality occurs in the Cauchy-Schwarz inequality, that is,kHk2−< H·ϕ, ω·ϕ >2

|ϕ|4 = 0. Thus, proceeding as in the proof of Theorem 2.1, we deduce that ∇Xϕ = −Q(X) · ω · ϕ, that is ψ is a twisted (symmetric)

Energy-Momentum spinor (EM-spinor).

Note that, by a completely similar computation, we can obtain a lower bound in- volving both symmetric and skew-symmetric Energy-Momentum tensors as in [7]. We do not write it in this note.

Finally, following the idea of Hijazi [9], we consider a conformal change of the metrc

¯

g =e2ueg. LetΣ−→Σ,¯ ψ−→ψ¯be the corresponding isometry between the two spinor bundles. Recall that for 2 spinorsψandϕonΣand for any vector fieldXonMf, we have

(ϕ, ψ) = ( ¯ϕ,ψ)¯ ¯g and X¯¯·ψ¯=X·ψ Note that we have

D(e−(m−1)2 uψ) =e−(m+1)2 u

whereDdenotes the Dirac operator w.r.t the metricg. Moreover, the corresponding mean curvature is given by

H =e−2u(H−mgradNu)

Now, assume thatgradNu= 0, thenDHis also conformally covariant and we have DH(e−(m−1)2 uψ) =e−(m+1)2 uDHψ

From now on, we will consider regular conformal change of the metricg, i.e.,gradNu= 0.

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6 R. NAKAD AND J. ROTH

Theorem 2.3. Let(Mm, g)be a compact RiemannianSpinc manifold isometrically im- mersed into a RiemannianSpincmanifold(Mfm+n,eg). Consider a non-trivial eigenspinor fieldϕ∈Γ(Σ)for the submanifold Dirac operatorDH, i.e.DHψ=λψ. For any regular conformal change of metric¯g=e2ueg, assume thatm≥3and

ScalM e2u+RNϕ + 4|Qψ|2−cm|ΩM|>||H||2>0 onMϕ, then, we have

λ2≥1 4 inf

Mψ

q

ScalM e2u+RNψ + 4|Qψ|2−cm|Ω|)− ||H||

2 .

Proof:Forψ∈Γ(Σ)an eigenspinor ofDHwith eigenvalueλ, letϕ:=e−(n−1)u2 ψ. then, we haveDHϕ=λe−uϕ. Recall that

eiψ=∇eiψ−1

2ei·du·ψ−1 2ei(u)ψ,

and ei = e−uei. Now, it is straightforward to get Qϕi,j = e−uQψi,j, hence |Qϕ|2 = e−2u|Qψ|2. Equation (8) is also true on(M , g). If, we apply it tof ϕ, we get

Z

M

|∇Qϕ|2vg = Z

M

(1 +mq)2h

(λe−u)2−1 4

ScalM+RNϕ + 4|Qϕ|2

(1 +mq2) −kH˜k2g mq2

i|ϕ|2gvg

−1 4

Z

M

(1 +mq2)h 2 mq(1 +mq2)

kHk˜ 2g−<H·ϕ, ω˜ ·ϕ >2g

|ϕ|4g

i|ϕ|2gvg

−i 2

Z

M

<Ω·ϕ, ϕ >gvg

(9)

sinceH˜ =e−uHandRNϕ =e−2uRNψ, we have

Z

M

|∇Qϕ|2vg = Z

M

(1 +mq)2e−2uh

(λ)2−1 4

ScalMe2u+RNψ + 4|Qψ|2

(1 +mq2) −kHk2 mq2

i|ϕ|2vg

−1 4

Z

M

(1 +mq2)e−2uh 2 mq(1 +mq2)

kHk2−< H·ψ, ω·ψ >2

|ψ|4

i|ϕ|2vg

−i 2

Z

M

<Ω·ϕ, ϕ >gvg

(10)

Now, we use again (5) and if moreover,ScalM e2u+RNψ −cM|Ω|+ 4|Qψ|2>kHk2>0, we take

q= v u u t

kHk m(q

ScalMe2u+RNψ −cM|Ω|+ 4|Qψ|2− kHk) ,

and then we use the Cauchy Schwarz inequality

kHk2−< H·ψ, ω·ψ >2

|ψ|4 ≥0,

to get the desired result.

By taking u as first eigenfunction of the Yamabe operator on M, we get the follow- ing corollary, whereµ1is the first eigenvalue of the Yamabe operator.

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Corollary 2.4. Under the assunptions of Theorem 2.3 and if µ1+RNψ + 4|Qψ|2−cm|ΩM|>||H||2>0 onMϕ, then, we have

λ2≥1 4 inf

Mψ

q

µ1+RNψ + 4|Qψ|2−cm|Ω|)− ||H||2 .

3. GENERALIZED TWISTEDKILLING SPINORS

We have seen in Theorem 2.1 that if equality occurs, then, the eigenspinorϕis in fact a generalized twisted Killing spinor, that is satisfies the equation

(11) ∇Xϕ=f X·ω·ϕ=f X ·

Mϕ, wherefis a real function. We prove the following

Proposition 3.1. Let (Mm, g) be a compact Riemannian Spinc manifold isometrically immersed into a RiemannianSpincmanifold(fMm+n,eg). Letϕ∈Γ(Σ)be a generalized twisted Killing spinor with real-valued functionf. Ifm > n+ 4, thenf is constant.

Proof:We define the following forms forp∈ {1,· · ·, m}, ωp(X1,· · ·, Xp) =D

(X1∧X2∧ · · · ∧Xp) ·

Mϕ, ϕE ,

We have the following easy facts (see [6, 8] for instance). For anyk>0, the formsω4k+1

andω4k+2are imaginary-valued whereas the formsω4k+3andω4kare real-valued. More- over, we have for anyp>0

p= ((−1)pf−f)ωp+1. In particular, we have for anyk>1

(12) df∧ω2k = 0.

Assume thatf is not constant and letx∈Msuch thatdf 6= 0, on a neighborhoodV ofx.

Hencedfis of dimensionm−1and we consider{e1,· · ·, em−1}an orthonormal frame ofdf. From this, we have

ω2k(ei1,· · ·, ei2k) = 0,

for any subset{i1,· · · , i2k} of{1,· · ·, m−1}. Thus, forl = m−1

2

, we deduce that the spinor fieldsϕ,ei1 ·

M ei2 ·

Mϕ,· · · andei1 ·

Mei2· · ·ei2l ·

M ϕare orthonormal onV. Consequently, the space spanned by these spinors is a vector subspace ofΣxof complex dimension

1 +

m−1 2

+

m−1 4

+· · ·+

m−1 2l

= 2m−2.

Since the complex dimension ofΣis d(m, n) =

(

2m+n2 ifm+nis even 2m+n−12 ifm+nis odd

we conclude that2m−2 6 d(m, n). Therefore,f is constant if 2m−2 > d(m, n)which corresponds to the condition expressed in the statement of the Proposition. Indeed, on one hand, if m+nis even, then2m−2 > d(m, n)is equivalent tom−2 > n+m2 , that is m > n+ 4. On the other hand, if fm+nis odd, then22m−2> d(m, n)is equivalent to m−2 > n+m−12 , that ism > n+ 3. But sincem+nis odd, thenmcannot be equal to

n+ 4. hence we also havem > n+ 4.

Hence, we deduce the following

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8 R. NAKAD AND J. ROTH

Corollary 3.2. Let(Mm, g)be a compact RiemannianSpinc manifold isometrically im- mersed into a RiemannianSpincmanifold(Mfm+n,eg). ifm > n+ 4and equality occurs in Theorem 2.1, thenM admits a twisted Killing spinor.

Acknowledgment.The first author is indebted to the Center for Advanced Mathemati- cal Sciences (CAMS, Lebanon) and the University of Paris-Est, Marne-la-Vall´ee for their hospitality and support.

REFERENCES

[1] C. B¨ar,Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Glob. Anal. Geom. 16 (1998) 573-596.

[2] J. P. Bourguignon, O. Hijazi, J. L. Milhorat, A. Moroianu & S. Moroianu,A spinorial approach to Riemann- ian and conformal geometry, Monograph in Mathematics, EMS.

[3] T. Friedrich,Der erste Eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkr¨ummung,Math. Nach. 97 (1980), 117-146.

[4] T. Friedrich,Dirac operators in Riemannian Geometry, Graduate studies in mathematics, Volume 25, Amer- icain Mathematical Society.

[5] N. Ginoux & B. Morel,On eigenvalue estimates for the submanifold Dirac operator, Int. J. Math. 13 (2002), No. 5, 533-548.

[6] N. Grosse & R. Nakad, Complex generalized Killing spinors on Riemannian Spinc manifolds, arxiv:1311.0969 (to appear in Results in Mathematics).

[7] G. Habib,Energy-Momentum tensor on foliations, J. Geom. Phys. 57 (2007), 2234-2248.

[8] M. Herzlich et & Moroianu,Generalized Killing spinors and conformal eigenvalue estimates for Spinc manifold, Ann. Glob. Anal. Geom. 17 (1999), 341-370.

[9] O. Hijazi,A conformal Lower Bound for the Smallest Eigenvalue of the Dirac Operator and Killing Spinors, Commun. Math. Phys. 104, (1986) 151–162.

[10] O. Hijazi,Lower bounds for the eigenvalues of the Dirac operator, J. Geom. Phys., 16 (1995) 27-38.

[11] O. Hijazi & X. Zhang,Lower bounds for the Eigenvalues of the Dirac Operator, Part I. The Hypersurface Dirac Operator, Ann. Global Anal. Geom.19, (2001) 355-376.

[12] O. Hijazi & X. Zhang,Lower bounds for the Eigenvalues of the Dirac Operator, Part II. The Submanifold Dirac Operator, Ann. Global Anal. Geom.19, (2001) 163-181.

[13] R. Nakad,Lower bounds for the eigenvalues of the Dirac operator onSpincmanifolds, J. Geom. Phys. 60 (2010), 1634-1642.

[14] R. Nakad & J. Roth,TheSpincDirac operator on hypersurfaces and applications, Diff. Geom. Appl.,31 (1), (2013), pp 93-103

(J. Roth) LAMA, UNIVERSITE´PARIS-ESTMARNE-LA-VALLEE´ , CITE´DESCARTES, CHAMPS SURMARNE, 77454 MARNE-LA-VALLEE CEDEX´ 2, FRANCE

E-mail address:julien.roth@u-pem.fr

(R. Nakad) NOTREDAMEUNIVERSITY-LOUAIZE, FACULTY OFNATURAL ANDAPPLIED SCIENCES, DE- PARTMENT OFMATHEMATICS ANDSTATISTICS, P. O. BOX72, ZOUKMIKAEL, LEBANON

E-mail address:rnakad@ndu.edu.lb

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