Simon Raulot

GREEN FUNCTIONS FOR THE DIRAC OPERATOR UNDER LOCAL BOUNDARY CONDITIONS AND APPLICATIONS

Keywords: 1991 Mathematics Subject Classification. 53A30, 53C27 (Primary), 58J50, 58C40 (Secondary) Manifolds with boundary, Conformally invariant operators, Dirac operator, Green function, Local boundary conditions. 1

In this paper, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the MIT bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in particular, solve the Yamabe problem on manifolds with boundary in some cases. Finally, using the MIT Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.

Introduction

A nice and powerful application in the study of the Green function of an elliptic differential operator appears in Schoen's approach [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] of the Yamabe problem. This famous problem of conformal Riemannian geometry can be stated as follow: given a connected compact Riemannian manifold (M n , g) with n ≥ 3, can one find a metric g in the conformal class [g] of g with constant scalar curvature? We first briefly recall some steps in the resolution of this problem. In [START_REF] Yamabe | On a deformation of Riemannian structures on compact manifolds[END_REF],

Yamabe claims to answer this question, however Trüdinger [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] points out a mistake in Yamabe's argument and can fix the proof in some cases. The remaining cases were treated by Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] and Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF]. In fact, solving the Yamabe problem is equivalent to find a smooth positive solution of a non linear elliptic equation involving a conformally invariant operator: the conformal Laplacian. This is a two order elliptic differential operator acting on functions defined by: L g := 4(n -1) n -2 ∆ g + R g and which relates the scalar curvature of two metrics in the same conformal class by the formula:

R g = f -n+2 n-2 L g f, if g = f 4
n-2 g ∈ [g] and where ∆ g and R g denote respectively the standard Laplacian and the scalar curvature of (M, g). Then finding a metric with constant scalar curvature in the conformal class of g is equivalent to find a smooth positive function f for the equation:

L g f = C f N -1 (1)
where C is a constant and N = 2n n-2 . The major difficulty in this problem comes from the fact that the Sobolev embedding H 2 1 (M) ֒→ L N (M) is not compact and then a standard variationnal approach cannot allow to conclude. However with a little work, one can show the existence of a non negative smooth function f solution of Equation (1) which is either positive or zero on M. So it remains to show that this function cannot vanish identically on M. In [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF], Aubin observed that if:

µ(M) < µ(S n ) (2)
then the function f has to be positive. Here µ(M) denotes the Yamabe invariant of M (which only depends of the conformal class of g) defined by: µ(M) := inf u∈C 1 (M),u =0 M uL g u dv(g)

M |u| N dv(g) 2 N . (3)
He also proved that if (M, g) is not conformally equivalent to the standard sphere (S n , g st ) then µ(M) ≤ µ(S n ) and moreover if n ≥ 6 and the manifold is not locally conformally flat then (2) holds. The last step in the resolution of the Yamabe problem was made by Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] which proved (2) in the remaining cases using the positive mass theorem. In 1987, Lee and Parker [START_REF] Lee | The Yamabe problem[END_REF] gave a proof of the Yamabe problem unifying Aubin's and Schoen's arguments.

The key point of their work uses Schoen's result and is strongly based on the study of the Green function of the conformal Laplacian and on the existence of an adaptated coordinates chart namely the conformal normal coordinates. In fact, using the development of the Green function near a point q ∈ M in these coordinates, they construct adapted test functions which include Aubin and Schoen's arguments. More precisely, in a neighbourhood of q, the conformal Laplacian's Green function can be decomposed into the sum of a singular part (which is nearly the Green function for the standard Laplacian in Euclidean space) and a regular part. The fact is that according to the manifold is or is not conformally flat, this regular part evaluated in q is either the mass of an asymptotically flat manifold (obtained by conformal change of metrics whose weight is exactly the Green function) either the squared norm of the Weyl tensor at q.

If now we suppose that the manifold M is spin, there exists another operator which has the same conformal properties than the conformal Laplacian operator: the Dirac operator. This operator is a first order elliptic differential operator acting on sections of the complex spinor bundle over (M, g). The relation between these two operators was made by Hijazi (see [START_REF] Hijazi | A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors[END_REF]) with an inequality (now called the Hijazi inequality) which relates their first eigenvalues and which allows to define a spin conformal invariant given by:

λ + min (M, [g], σ) := inf g∈[g]
λ + 1 (g)Vol(M, g)

1 n .
This invariant can be seen as an analogue of the Yamabe invariant in the spinorial setting. Moreover, using the Hijazi inequality, we can compare this invariant to the Yamabe invariant of (M, g) (see [START_REF]Première valeur propre de l'opérateur de Dirac et nombre de Yamabe[END_REF] and [START_REF] Bär | Lower eigenvalue estimate for Dirac operator[END_REF] for the case n = 2):

λ + min (M, [g], σ) 2 ≥ n 4(n -1) µ(M). (4)
This invariant has been (and is again) the main subject of many works, in particular in a serie of papers by Ammann, Humbert and Morel (see [START_REF]A spin-conformal lower bound of the first positive Dirac eigenvalue[END_REF], [Amm], [AHM] or [START_REF]Un problème de type Yamabe sur les variétés spinorielles compactes[END_REF]). In these papers, the authors show that this invariant shares a lot of the Yamabe's invariant properties. Indeed, in [START_REF]A spinorial analogue of Aubin's inequality[END_REF] it is shown that we can derive a spinorial analogue of Aubin's inequality, i.e. we have:

λ + min (M, [g], σ) ≤ λ min (S n , [g st ], σ st ) = n 2 ω 1 n n (5)
where ω n denotes the volume of the standard sphere. Moreover, if this inequality is strict, then with the help of the Hijazi inequality, it gives a spinorial proof of the Yamabe problem. On the other hand, Ammann [START_REF]A variational problem in conformal spin geometry[END_REF] also proves that the strict inequality in (5) gives the existence of a nonlinear partial differential equations for the Dirac operator involving a critical Sobolev embedding theorem. A natural question is then to ask if we can find sufficient conditions for which Inequality (5) is strict. A partial answer is given in [AHM] with the help of a detailed study of the Green function for the Dirac operator. Indeed, in this article, they considered the case of locally conformally flat manifolds where the behaviour of the Green function can be well understood. In fact, they show that the Dirac operator's Green function can be decomposed (around q) into the sum of a singular part (which is nearly the Green function for the Dirac operator in the Euclidean space) and a smooth spinor field. Then they define the mass operator which plays the same role as the constant term of the Green's function of the conformal Laplacian. An other application of the Green function of the Dirac operator can also be found in [START_REF] Ammann | Positive mass theorem for the Yamabe problem on spin manifolds[END_REF] where a simple proof of the positive mass theorem for the Yamabe problem is given.

The main idea of the proof is based on the fact that the Witten's spinor (see [START_REF] Lee | The Yamabe problem[END_REF] or [START_REF] Bartnik | The mass of an asymptotically flat manifold[END_REF]) is exactly obtained as the image of the Green function's of the Dirac operator under the conformal change of metric whose weight is given by the Green function of the conformal Laplacian.

In this paper, we define and study the Green's function of the Dirac operator on manifolds with boundary. More precisely, since the conformal aspect is underlying in this work, the choice of the boundary conditions is crucial for the following. It turns out that local boundary conditions seems to be appropriate for the study of the conformal aspect of the Dirac operator on manifolds with boundary.

In a first part, we will focus on the condition associated with a chirality operator (also called the chiral bag boundary condition, see [START_REF] Hijazi | Eigenvalue boundary problems for the Dirac operator[END_REF] for example). A direct application of this construction is motivated by previous results of the author (see [START_REF]The Hijazi inequality on manifolds with boundary[END_REF], [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF] or Section 2.2). Indeed, in [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF] we define an analogue of λ + min (M, [g], σ) on manifolds with boundary and we show that this invariant satisfies an inequality corresponding to (5). We note that if this inequality is strict then there exists a solution for the Yamabe problem on manifolds with boundary (see [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF]). Moreover, we show (in a forthcoming paper [START_REF]A variational problem in spin conformal geometry on manifolds with boundary[END_REF]) that it also implies the existence of a spinor field solution of a Yamabe type boundary problem for the Dirac operator under the chiral bag boundary condition. A natural question is then to ask if one can find sufficient conditions for which this inequality is strict. It appears that the study of the Green function of the Dirac operator under this boundary condition allows to give such a condition.

Secondly, we will study the Green function for the Dirac operator for an other local elliptic boundary condition: the MIT bag boundary condition. Unlike the chiral bag boundary condition, this condition exists on every spin manifold with boundary since it does not require some additional structures. As in the previous part, we will compute the development of this function around a boundary point where the geometry of the manifold is quite simple. Then as an application, we will give a proof of the positive mass theorem proved by Escobar in [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF] which appears in the resolution of the Yamabe problem on manifolds with boundary. Note that our result requires that the manifold is spin (which is a stronger assumption compared with Escobar's theorem) however we don't assume that the whole manifold is locally conformally flat with umbilic boundary but only that there exists a boundary point q ∈ ∂M with a locally conformally flat with umbilic boundary neighbourhood (see below for the definition of such a neighbourhood).

Convention: In this article, a point q ∈ ∂M in a Riemannian manifold (M n , g) has a locally conformally flat with umbilic boundary neighbourhood, if there exist a metric g ∈ [g] and a neighbourhood V ⊂ M (resp. U ⊂ R n + ) of q ∈ ∂M (resp. of 0 ∈ ∂R n + ) such that (U, ξ) and (V, g) are isometric. In a same way, a point q ∈ ∂M has a locally flat with umbilic boundary neighbourhood if g = g in the previous notation.

Acknowledgements: I would like to thank Oussama Hijazi and Emmanuel Humbert for their support. I am also very grateful to Marc Herzlich and Sebastián Montiel for their remarks and their suggestions.

2. Green function for the Dirac operator under the chiral bag boundary condition 2.1. Definitions and first properties. In this section, we give a rigorous definition of the Green function of the Dirac operator under the chiral bag boundary condition. This condition has been introduced by Gibbons, Hawking, Horowitz and Perry [START_REF] Gibbons | Positive mass theorems for black holes[END_REF] (see also [START_REF] Herzlich | The positive mass theorem for black holes revisited[END_REF]) on asymptotically flat manifolds with inner boundary in order to prove positive mass theorems for black holes. This condition defines an elliptic boundary condition for the Dirac operator and in the context of compact Riemannian spin manifolds with boundary, we can study the properties of its spectrum (see [START_REF] Farinelli | On the spectrum of the Dirac operator under boundary conditions[END_REF], [START_REF] Hijazi | Eigenvalues of the Dirac operator on manifolds with boundary[END_REF] or [START_REF]The Hijazi inequality on manifolds with boundary[END_REF] for example).

Let (M n , g, σ) a n-dimensionnal connected compact Riemannian spin manifold with non empty smooth boundary ∂M. We will denote by Σ g (M) the spinor bundle over (M, g), ∇ the Riemannian or spinorial Levi-Civita connections and "•" the Clifford multiplication. The Dirac operator is then the first order elliptic differential operator acting on Σ g (M) locally given by:

D g ϕ = n i=1 e i • ∇ e i ϕ,
for all ϕ ∈ Γ Σ g (M) and where {e 1 , ..., e n } is a local g-orthonormal frame of the tangent bundle. We now briefly recall the definition of the chiral bag boundary condition. From now on, we assume that there exists a chirality operator, i.e. a linear map

Γ : Σ g (M) -→ Σ g (M)
from the spinor bundle over (M, g) which satisfies the following properties:

Γ 2 = Id, Γϕ, Γψ = ϕ, ψ ∇ X (Γψ) = Γ(∇ X ψ), X • Γψ = -Γ(X • ψ), (6)
for all X ∈ Γ(TM) and for all spinor fields ψ, ϕ ∈ Γ Σ g (M) . If ν denotes the (inner) unit normal vector field to the boundary, an easy computation shows that the fiber preserving endomorphism:

v • Γ : S g -→ S g
from the restricted spinor bundle S g := Σ g (M) |∂M is an involution. So this spinor bundle splits into the direct sum S g = V + g ⊕ V - g where V ± g is the eigensubbundle associated with the eigenvalues ±1. Thus one can check that the orthogonal projection

B ± CHI : L 2 (S g ) -→ L 2 (V ± g ) ϕ -→ 1 2 (Id ± ν • Γ)ϕ, onto the eigensubbundle V ±
g defines an elliptic boundary condition for the Dirac operator. More precisely, the Dirac operator

D g : H ± g = {ϕ ∈ H 2 1 / B ± CHI (ϕ |∂M ) = 0} -→ L 2 Σ g (M) (7)
is a Fredholm operator and its spectrum (under this boundary condition) consists of entirely isolated real eigenvalues with finite multiplicity. In the following, we will denote by λ ± k (g) k∈Z this set of eigenvalues, i.e. we have:

   D g ϕ ± k = λ ± k (g)ϕ ± k on M B ± CHI (ϕ ± k |∂M ) = 0 along ∂M
where (ϕ ± k ) k∈Z can be chosen as being a spectral resolution. Let π 1 , π 2 : M × M → M be the projection on the first and the second component and let:

Σ g (M) ⊠ Σ g (M) * := π * 1 Σ g (M) ⊗ π * 2 Σ g (M) *
i.e. the fiber bundle whose fiber over (x, y) is given by

Σ g (M) ⊠ Σ g (M) * (x,y) := Hom Σ y (M), Σ x (M) ,
where Σ y (M) denotes the fiber of the spinor bundle over (M, g). We are now ready to give the definition of the Green function for the Dirac operator under the chiral bag boundary condition.

Definition 1. A Green function for the Dirac operator under the chiral bag boundary condition (or a chiral Green function) is given by a smooth section:

G ± CHI : M × M \ ∆ → Σ g (M) ⊠ Σ g (M) * (8)
locally integrable on M × M which satisfies, in a weak sense, the following boundary problem:

   D g G ± CHI (x, y) = δ y Id ΣyM B ± CHI G ± CHI (x, y) = 0, for x ∈ ∂M \ {y}, for all x = y ∈ M (where ∆ := {(x, y) ∈ M × M / x = y}).
In other words, we have:

M G ± CHI (x, y)ψ ± 0 , D g ϕ(x) dv(x) = ψ ± 0 , ϕ(y) for all y ∈ M, ψ ± 0 ∈ Σ y (M) satisfying ν •Γψ ± 0 = ∓ψ ± 0 and ϕ ∈ Γ Σ g (M) such that B ± CHI (ϕ |∂M ) = 0.
Remark 1. Here we recall that on the Euclidean space, we have a corresponding notion of Green function for the Dirac operator (see [AHM] for example). In fact, one can easily check that there exists a unique Green function given by:

G eucl : R n × R n \ ∆ -→ ΣR n ⊠ ΣR n * (x, y) -→ -1 ω n-1
x-y |x-y| n •, where ω n stands for the volume of the n-dimensional round sphere. This Green function can be seen (up to a conformal change of metric) as the inverse of the Dirac operator on the standard sphere S n .

We are now going to give the development of the chiral Green function near a boundary point q ∈ ∂M and when the manifold has a particular geometry near this point. From now on, we assume that the Dirac operator is invertible under the chiral bag boundary condition, i.e. Ker ± (D g ) = {0} where Ker ± (D g ) denotes the kernel of the Dirac operator with domain in H ± g .

In order to get this expansion, we have to identify (locally) the spinor bundle over (M, g) with the (trivial) spinor bundle over an open set of the half-space R n + endowed with the Euclidean metric. In fact, in a previous work (see [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF] or [START_REF]Aspect conforme de l'opérateur de Dirac sur une variété à bord[END_REF]), it is shown that if F q : U -→ V stands for the Fermi coordinate system around q ∈ V where U (resp. V) is an open set in R n + (resp. in M) then we have a canonical trivialization given by:

Σ ξ (U) -→ Σ g (V) ψ -→ ψ.
where ξ is the euclidean metric on R n + . This identification is closely related to those given in [START_REF] Bourguignon | Spineurs, opérateurs de Dirac et variations de métriques[END_REF] and [START_REF]A spinorial analogue of Aubin's inequality[END_REF].

Remark 2. In [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF] (see also [START_REF]Aspect conforme de l'opérateur de Dirac sur une variété à bord[END_REF]), we show that it is sufficient to choose a constant spinor field

ψ 0 ∈ Σ 0 (R n + ) such that B ± CHI ψ ± 0 (q) = 0 at one point q ∈ ∂M so that B ± CHI ψ ± 0 |V∩∂M = 0
In the following, we will note indifferently a spinor over U and its image on V under this trivialization. The expansion of the chiral Green function is then given by: Proposition 1. Assume that there exists a point q ∈ ∂M which has a locally flat with umbilic boundary neighbourhood. Then the chiral Green function exists and has, in the above trivialization, the following expansion (near q):

G ± CHI (x, q)ψ ± 0 = - 2 ω n-1 x -q |x -q| n • ψ ± 0 + m ± CHI (x, q)ψ ± 0 , (9)
where

ψ ± 0 ∈ Σ q (M) is a spinor field satisfying ν • Γψ ± 0 = ∓ψ ± 0 (in other words B ± CHI (ψ ± 0 ) = 0)
and where m ± CHI ( . , q)ψ ± 0 is a smooth spinor field such that D g m ± CHI ( . , q)ψ ± 0 = 0 around q. Moreover, along ∂M, we have:

B ± CHI m ± CHI (. , q)ψ ± 0|∂M = 0. ( 10
)
Proof: We only prove this result for the boundary condition B - CHI since the proof is similar for B + CHI . We trivialize the spinor bundle over V (around the boundary point q) using the preceding discussion. In fact, we can isometrically identify (V, g) with (U, ξ) and thus the spinor bundles over these spaces. Now we consider a smooth cut-off function η such that:

0 ≤ η ≤ 1, η ∈ C ∞ (M) η ≡ 1 on B + q (δ), supp(η) ⊂ B + q (2δ) ⊂ V where B +
q (2δ) is the half-ball centered in q and with radius 2δ > 0 contained in V. Let Ψ = D g ηG eucl ( . , q)ψ 0 with ψ 0 ∈ Σ q (M) such that ν • Γψ 0 = ψ 0 . Using Remark 2, we then have that this boundary condition is fulfilled along V ∩ ∂M for the spinor field ψ 0 . The spinor field Ψ is smooth on M \ {q} and an easy calculation leads to Ψ |B + q (δ)\{q} = 0. Thus we can extend this spinor all over M by Ψ(q) = 0. Since the Dirac operator is supposed to be invertible on H -, there exists a unique spinor field m - CHI ( . , q)ψ 0 ∈ H -such that D g m - CHI ( . , q)ψ 0 = -Ψ. Now for x ∈ M \ {q}, we let:

G - q (x)ψ 0 = 2η G eucl (x, q)ψ 0 + m - CHI (x, q)ψ 0 . We first check that if x ∈ ∂M \ {q} then this spinor field satisfies the chiral bag boundary condition. If x ∈ supp(η) c ∩ ∂M then B - CHI G - q (x)ψ 0 = B - CHI m - CHI (x, q)ψ 0 = 0 since by construction m ± CHI ( . , q)ψ 0 ∈ H -. Secondly, if x ∈ supp(η) ∩ ∂M \ {q} we get B - CHI G - q (x)ψ 0 = B - CHI G eucl (x, q)ψ 0 .
Using the expression of the Euclidean Green function given in Remark 1 and the properties (6) of the chirality operator Γ, we obtain

ν • Γ x -q |x -q| n • ψ 0 = x -q |x -q| n • ν • Γψ 0 = x -q |x -q| n • ψ 0 since ψ 0 is chosen such that ν • Γψ 0 = ψ 0 and thus B - CHI G - q (x)ψ 0 = 0.
An easy calculation shows that for all ϕ ∈ H -, we have

M G - q (x)ψ 0 , D(ϕ) dv(x) = ψ 0 , ϕ(q) ,
and so G - q ( . ) is a chiral Green function. Unicity follows from the hypothesis ker ± (D g ) = {0}. In fact, if we assume that there exists another chiral Green function G - q ( . ), we can show that the spinor field G - q ( . )ψ 0 -G - q ( . )ψ 0 is in the kernel of the Dirac operator (using the classical regularity theorems, see [START_REF] Schwartz | Hodge Decomposition-A method for solving boundary value problems[END_REF] for example). Since the Dirac operator (under the chiral bag boundary condition) is supposed to be invertible, this spinor vanishes identically and so unicity follows directly.

In the following, the chiral Green function will be indifferently denoted by G ± CHI ( . , q) or G ± q ( . ).

We now look at the behaviour of the chiral Green function under a conformal change of metric.

In fact, we prove the following result:

Proposition 2. Let g = f 2 g be a metric in the conformal class of g. If G ± CHI ( . , q) (resp. G ± CHI ) denote the chiral Green function for the Dirac operator D g (resp. D g ), then:

G ± CHI ( . , q) = f -n-1 2 ( . ) f -n-1 2 (q) G ± CHI ( . , q) (11)
for q ∈ ∂M.

Proof: First recall that given two conformal metrics g and g = f 2 g, there exists a canonical identification between the spinor bundle over (M, g) and the one over (M, g) (see [START_REF] Hitchin | Harmonic spinors[END_REF] or [START_REF] Hijazi | A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors[END_REF]). It is given by the bundle isomorphism (which is a fiberwise isometry)

F : Σ g (M) -→ Σ g (M) (12)
such that the Dirac operators satisfy the relation

F -1 D g (F(ψ)) = f -n+1 2 D g (f n-1 2 ψ), (13)
for all ψ ∈ Γ Σ g (M) . On the other hand, since G - q is the chiral Green function for the Dirac operator, we have:

M G - q (x)ψ 0 , D g ϕ dv(g) = ψ 0 , ϕ(q)
for all ψ 0 ∈ Σ q (M) such that B - CHI (ψ 0 ) = 0 and ϕ ∈ Γ Σ g (M) such that B - CHI (ϕ |∂M ) = 0. We can now prove that the section defined by (11) is a chiral Green function for the Dirac operator D g . For

ϕ ∈ Γ Σ g (M) such that B - CHI (ϕ |∂M ) = 0, we let Φ = f -n-1 2 F(ϕ) ∈ Γ Σ g (M) .
This spinor field satisfies B ± CHI (ϕ |∂M ) = 0 because of the conformal covariance of the chiral bag boundary condition (see for example [START_REF]Aspect conforme de l'opérateur de Dirac sur une variété à bord[END_REF]). We now write:

M G - q (x)F(ψ 0 ), D g Φ dv(g) = M f -n-1 2 (x) f -n-1 2 (q)f -n+1 2 (x) G - q (x)ψ 0 , D g ϕ f n dv(g) = f -n-1 2 (q) M G - q (x)ψ 0 , D g ϕ dv(g) = ψ 0 , f -n-1 2 (q)ϕ(q) that is: M G - q (x)F(ψ 0 ), D g (Φ) dv(g) = F(ψ 0 ), Φ(q) for all F(ψ ± 0 ) ∈ Σ q (M) such that B ± CHI F(ψ ± 0 ) = 0 and ϕ ∈ Γ Σ g (M) such that B ± CHI (ϕ |∂M ) = 0.
We have then checked that G q ( , ) is a chiral Green function. Unicity follows directly. Remark 3. Propositions 1 and 2 imply that the chiral Green function exists on locally conformally flat manifolds with umbilic boundary on which the Dirac operator is invertible under the chiral bag boundary condition.

We now prove a self-adjointness result for the chiral bag boundary condition which will be very important for the following and more particulary for the application given in Section 2.2. Proposition 3. For (x, y) ∈ M × M \ ∆, we have:

G ± CHI * (x, y) = G ± CHI (y, x). (14)
In fact, if

ψ ± 0 ∈ Σ x (M) (resp. ϕ ± 0 ∈ Σ y (M)) such that ν • Γψ ± 0 = ∓ψ ± 0 for x ∈ ∂M (resp. ν • Γϕ ± 0 = ∓ϕ ± 0 for y ∈ ∂M), then: G ± CHI (x, y)ϕ ± 0 , ψ ± 0 Σx(M) = ϕ ± 0 , G ± CHI (y, x)ψ ± 0 Σy(M)
. Proof: We use the fact that there exists a spectral resolution of the space of L 2 -spinors, i.e. for all ψ ∈ Γ Σ g (M) ,

∃(A k ) k∈Z ⊂ C such that ψ = k∈Z A k ϕ k where ϕ k is a smooth spinor field satisfying D g ϕ k = λ k (g)ϕ k on M B - CHI (ϕ k |∂M ) = 0 along ∂M. First note that: M×M\∆ G - CHI (x, y)ϕ i , ϕ j dv(x)dv(y) = M M G - CHI (x, y)ϕ i , ϕ j dv(x) dv(y) = M M G - CHI (x, y)ϕ i , 1 λ j D g ϕ j dv(x) dv(y) = 1 λ j M ϕ i , ϕ j dv(y) = 1 λ j δ ij ,
since D is assumed to be invertible and thus λ j = 0 for all j. In a same way, we easily show that:

M×M\∆ ϕ i , G - CHI (y, x)ϕ j dv(x)dv(y) = 1 λ i δ ij ,
We thus have proved that for all (i, j)

∈ Z 2 : M×M\∆ G - CHI (x, y)ϕ i , ϕ j dv(x)dv(y) = M×M\∆ ϕ i , G - CHI (y, x)ϕ j dv(x)dv(y).
By linearity, we have:

M×M\∆ G - CHI (x, y)ϕ, ψ dv(x)dv(y) = M×M\∆ ϕ, G - CHI (y, x)ψ dv(x)dv(y), for all ψ, ϕ ∈ Γ Σ g (M) ) such that B - CHI (ψ |∂M ) = B - CHI (ϕ |∂M ) = 0. Now we consider x 0 ∈ M such that y → G - CHI (y, x 0 ) and let f ε : R n → R such that f ε → δ x 0 where δ x 0 is the Dirac's distribution at x 0 . So if ψ ∈ Γ Σ g (M) satisfies B - CHI (ψ |∂M ) = 0 and ψ(x 0 ) = ψ 0 , we obtain: M ϕ(y), G - CHI (y, x 0 )ψ 0 dv(y) = M G - CHI (x 0 , y)ϕ(y), ψ 0 dv(y).
Using the same method for the y-variable with y 0 = x 0 leads to:

ϕ 0 , G - CHI (y 0 , x 0 )ψ 0 Σy 0 (M) = G - CHI (x 0 , y 0 )ϕ 0 , ψ 0 Σx 0 (M)
, and this concludes the proof.

We now show that the chiral Green functions G - CHI ( . , q) and G + CHI ( . , q) are related by the action of the chirality operator. In fact, we prove the following result: Proposition 4. Let (M n , g) be a connected compact Riemannian spin manifold with a boundary point q ∈ ∂M which has a locally conformally flat with umbilic boundary neighbourhood. Then we get:

G - CHI ( . , q) = -ΓG + CHI ( . , q)Γ. (15)
Proof: Without loss of generality, we can assume that the metric g is such that there is a neighbourhood of q ∈ ∂M isometric to an open set of the Euclidean half-space. Then in this chart, the chiral Green function G - CHI ( . , q) admits the development (9). Now define the section G( . , q) = -ΓG + CHI ( . , q)Γ on M \ {q}. First note that we can easily check that G( . , q) ∈ H -. Then a direct computation shows that the spinor field given by G - CHI ( . , q)ψ 0 -G( . , q)ψ 0 for ψ 0 such that B - CHI (ψ 0 ) = 0, is harmonic. Hence since the Dirac operator is supposed to be invertible under the chiral bag boundary condition, we conclude that G( . , q)ψ 0 = G - CHI ( . , q)ψ 0 is the unique chiral Green function.

We are now ready to define the chiral mass operator (see [AHM] for the case of manifolds without boundary). The name of this operator comes from its tight relation with the concept of mass in General Relativity which also appears in the context of the Yamabe problem (see [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] or [START_REF] Lee | The Yamabe problem[END_REF] for the closed case and [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF] for the boundary case).

Definition 2. Let (M n , g, σ) be a connected compact Riemannian spin manifold with non empty smooth boundary ∂M. Suppose that there exits a boundary point q ∈ ∂M which has a locally flat with umbilic boundary neighbourhood. The chiral mass operator is then defined by:

m ± CHI (q) : V ± q -→ V ± q ψ ± 0 -→ m ± CHI (q, q)ψ ± 0
where m ± ( . , q)ψ ± 0 is given in Proposition 1 and

V ± q = 1 2 Id ± ν • Γ Σ q (M) .
Thanks to Proposition 14, we can deduce the following result which will be very useful for the next section.

Proposition 5. For a point q ∈ ∂M with a locally flat with umbilic boundary neighbourhood, the chiral mass operator m ± CHI (q) is linear and symmetric. Proof: An easy computation shows that the chiral Green function is linear. Using this fact, it clearly follows that the chiral mass operator is also linear. The symmetry of the chiral mass operator comes from the symmetry of the chiral Green function proved in Proposition 3.

Remark 4. In the following, we will refer to the "negative chiral mass operator" (resp. "positive chiral mass operator") for m - CHI (q) (resp. m + CHI (q)). As a direct consequence, we obtain: Corollary 6. For a point q ∈ ∂M with a locally flat with umbilic boundary neighbourhood, the (pointwise) spectrum of the chiral mass operator m ± CHI (q) is real. Moreover, if κ is an eigenvalue for the negative chiral mass operator, then -κ is an eigenvalue for the positive chiral mass operator.

Proof: Using Proposition 4, we can easily check that the positive and the negative chiral mass operators satisfy the relation:

Γm - CHI (q) = -m + CHI (q)Γ.
So consider an eigenspinor ψ 0 ∈ V - q for the negative chiral mass operator associated with the eigenvalue κ, i.e. m - CHI (q)ψ 0 = κψ 0 and ν • Γψ 0 = ψ 0 . Using the preceding formula and the fact that ν • Γ Γψ 0 = -Γψ 0 , we observe that m + CHI (q)(Γψ 0 ) = -κΓψ 0 and thus -κ is an eigenvalue for the positive chiral mass operator.

In the next section, we give a direct application of the construction of the chiral Green function and the chiral mass operator.

Application:

The chiral bag invariant. This part is devoted to a direct application of the preceding construction of the chiral Green function for the Dirac operator. This application concerns a spin conformal invariant on manifolds with boundary introduced in [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF]. We first begin with a brief introduction on this invariant. We have seen in Section 3.1 (see [START_REF] Hijazi | Eigenvalue boundary problems for the Dirac operator[END_REF] for more details) that the spectrum of the Dirac operator under the chiral bag boundary condition consists of entirely isolated real eigenvalues with finite multiplicity. If we denote by λ ± 1 (g) the first eigenvalue of the Dirac operator D g under the boundary condition B ± CHI , then the chiral bag invariant is defined by:

λ ± min (M, ∂M) := inf g∈[g] |λ ± 1 (g)|Vol(M, g) 1 n , (16)
where [g] denotes the conformal class of g and Vol(M, g) is the volume of the manifold M equipped with the Riemannian metric g ∈ [g]. A very useful formula for the following is given by the variational characterization of the chiral bag invariant. In fact, it is shown in [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF] (see also [START_REF]A spin-conformal lower bound of the first positive Dirac eigenvalue[END_REF]) that:

λ ± min (M, ∂M) = inf ϕ∈C ± g M |D g ϕ| 2n n+1 dv(g) n+1 n M Re D g ϕ, ϕ dv(g) , (17)
where C ± g is the L 2 -orthogonal of Ker ± (D g ) in H ± g and D g is the Dirac operator in the metric g.

Remark 5.

(1) The above definition seems to depend on the boundary condition chosen B + g or B - g , however it doesn't (see [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF]). Taking in account this fact, we will denote by λ min (M, ∂M) the chiral bag invariant in what follows and we will use the B - g condition.

(2) Using the Hijazi inequality proved in [START_REF]The Hijazi inequality on manifolds with boundary[END_REF], we can compare the chiral bag invariant with the Yamabe invariant µ(M, ∂M) of the manifold (see [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF]). In fact, if n ≥ 3, we have:

λ min (M, ∂M) 2 ≥ n 4(n -1) µ(M, ∂M). (18)
(3) In [Rau06c], we have shown a spinorial analogous of Aubin's (or Escobar's) inequality

[Aub76] ([Esc92]
, for the non empty boundary case). More precisely, we proved that if n ≥ 2:

λ min (M, ∂M) ≤ λ min (S n + , ∂S n + ) = n 2 ω n 2 1 n . ( 19
)
This inequality is the analogue of the one obtained by Ammann, Humbert and Morel [START_REF]A spinorial analogue of Aubin's inequality[END_REF] in the boundaryless case.

A natural question is to find a sufficient condition under which Inequality (19) is strict. Keeping in mind the work of Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF], Escobar [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF] and Ammann, Humbert and Morel [AHM], we are going to see that the construction of the chiral mass operator gives an answer for a certain class of manifolds. More precisely, we prove the following result:

Theorem 7. Let (M n , g, σ) be a n-dimensional (n ≥ 2) connected compact Riemannian spin manifold with non empty smooth boundary ∂M. Suppose that there exists a point q ∈ ∂M which has a locally conformally flat with umbilic boundary neighbourhood. Moreover, we assume that the Dirac operator D g is invertible on H -(or on H + ) and that the negative chiral mass operator m - CHI (q) (or the positive chiral mass operator m + CHI (q)) is not identically zero. Then we get: λ min (M, ∂M) < λ min (S n + , ∂S n + ). The proof of this theorem is in the same spirit as the one of Inequality (19). Indeed, the idea is to construct a test-spinor to estimate in the variational characterization (17) of λ min (M, ∂M). In [START_REF]On a spin conformal invariant on manifolds with boundary[END_REF], the test-spinor was constructed from a Killing spinor on the hemisphere satisfying the chiral bag boundary condition and with support contained in an open set of a trivialization around a boundary point. In order to prove Theorem 7, it is not enough to extend by zero the test-spinor away from the open set of trivialization. In fact, the good extention is given by the chiral Green function. More precisely, we show: Theorem 8. Let (M, g, σ) be a n-dimensional connected compact Riemannian spin manifold (n ≥ 2) with non empty boundary ∂M. Suppose that there exists a point q ∈ ∂M which has a locally conformally flat with umbilic boundary neighbourhood. Assume also that there exits on M \ {q} a spinor field ψ ± such that:

   D g ψ ± = 0 on M \ {q} B ± CHI (ψ ± |∂M ) = 0 along ∂M (20)
which admits the following development around q:

ψ ± = x r n • ψ ± 0 + ψ ± 1 + θ ± , ( 21
)
where ψ ± 0 , ψ ± 1 ∈ Σ q M are spinors satisfying ν • Γψ ± 0 = ∓ψ ± 0 , ν • Γψ ± 1 = ∓ψ ± 1 and such that: Re ψ ± 0 , ψ ± 1 < 0 and Re x • ψ ± 0 , ψ ± 1 = 0. ( 22
)
We also assume that θ ± is a smooth spinor field all over M satisfying θ ± = O(r), B ± CHI (θ ± |∂M ) = 0 and which is harmonic around q. Under these hypothesis, we get:

λ min (M, ∂M) < λ min (S n + , ∂S n + ) = n 2 ω n 2 1 n .
Proof: For ε > 0, we let ρ := ε

1 n+1 and ε 0 := ρ n ε f x ε n 2 with f (r) = 1
1+r 2 and we consider the spinor field defined by:

ψ ± ε =            f x ε 1 -x ε • ψ ± 0 -ε 0 ψ ± 1 if r ≤ ρ -ε 0 ψ ± -ηθ ± + ηf ρ ε n 2 ψ ± 0 if ρ ≤ r ≤ 2ρ ε 0 ψ ± if 2ρ ≤ r (23)
where r = d(x, q) and η is a cut-off function equal to zero on M \ B + q (2ρ), 1 on B + q (ρ) and satisfying |∇η| ≤ 2 ρ . We first check that B ± CHI (ψ ± ε|∂M ) = 0. For r ≤ ρ, we have:

B ± CHI (ψ ± ε|∂M ) = B ± CHI f x ε 1 - x ε • ψ ± 0 -ε 0 ψ ± 1 |∂M = f x ε 1 - x ε • B ± CHI (ψ ± 0 |∂M ) -ε 0 B ± CHI (ψ ± 1 |∂M ) = 0 since B ± CHI (ψ ± 0 |∂M ) = B ± CHI (ψ ± 1 |∂M ) = 0.
In the same way, since ψ ± and θ ± satisfy

B ± CHI (ψ ± |∂M ) = B ±
CHI (θ ± |∂M ) = 0, we easily check that for ρ ≤ r ≤ 2ρ or r ≥ 2ρ, we also have B ± CHI (ψ ± ε|∂M ) = 0. Without loss of generality, we can assume that |ψ ± 0 | 2 = 1. Since ψ ± and θ ± are harmonic around q, we compute that:

D g ψ ± ε =              n ε f r ε n 2 +1 1 -x ε • ψ ± 0 if r ≤ ρ ε 0 ∇η • θ ± + f r ε n 2 ∇η • ψ ± 0 if ρ ≤ r ≤ 2ρ 0 if 2ρ ≤ r.
In order to obtain an estimate of the numerator of the variational characterization (17) of λ min (M, ∂M), one can check that:

|D g ψ ± ε | 2n n+1 =              n 2n n+1 ε -2n n+1 f r ε n if r ≤ ρ |ε 0 ∇η • θ ± + f r ε n 2 ∇η • ψ ± 0 | 2n n+1 if ρ ≤ r ≤ 2ρ 0 if 2ρ ≤ r.
Hence we have:

B + q (ρ) |D g ψ ± ε | 2n n+1 dx = ε n-2n n+1 n 2n n+1 B + q ( ρ ε ) f n dx ≤ R n + f n dx
and:

B + q (2ρ)\B + q (ρ) |D g ψ ± ε | 2n n+1 dx ≤ Cε n(2n-1) n+1 + Cε n(3n-1 n+1 ≤ Cε n(2n-1)
n+1 .

These estimates lead to:

M |D g ψ ± ε | 2n n+1 dv(g) n+1 n ≤ ε n-1 n 2 I 1+ 1 n 1 + Cε n 2 n+1 = ε n-1 n 2 I 1+ 1 n 1 + o(ε n-1 )
where

I = R n + f n dx.
For the next, we let κ = Re ψ ± 0 , ψ ± 1 and we focus on an estimate of the denominator of the variational characterization (17) of λ min (M, ∂M). We write:

Re D g ψ ± ε , ψ ± ε |B + q (ρ) = Re n ε f r ε n 2 +1 1 - x ε • ψ ± 0 , f r ε n 2 1 - x ε • ψ ± 0 -ε 0 ψ ± 1 = n ε f r ε n - n ε ε 0 κ f r ε n 2 +1 + n ε ε 0 f r ε n 2 +1 Re x ε • ψ ± 0 , ψ ± 1 .
Integrating on B + q (ρ) gives:

B + q (ρ) Re D g ψ ± ε , ψ ± ε dx = n ε B + q (ρ) f r ε n dx - n ε ε 0 κ B + q (ρ) f r ε n 2 +1 dx + n ε ε 0 B + q (ρ) f r ε n 2 +1 Re x ε • ψ ± 0 , ψ ± 1 dx = nε n-1 B + q ( ρ ε ) f (r) n dx -ε 0 κ B + q ( ρ ε ) f (r) n 2 +1 dx + A ε
where:

A ε = nε 0 ε -n B + q (ρ) f r ε n 2 +1 Re x ε • ψ ± 0 , ψ ± 1 dx.
However A ε = 0 since by hypothesis, we assumed that:

Re x ε • ψ ± 0 , ψ ± 1 = 0.
Moreover an easy computation leads to:

B + q ( ρ ε ) f (r) n dx = I + O(ε n 2 n+1 )
and since ε 0 ∼ ε n-1 when ε → 0, we find:

B + q (ρ) Re D g ψ ± ε , ψ ± ε dx ≥ nε n-1 I -C 0 κε n-1 + o(ε n-1 )
where

C 0 = R n + f (r) n 2 +1 dx.
On the other hand, we compute:

Re D g ψ ± ε , ψ ± ε |B + q (2ρ)\B + q (ρ) = -Re ε 0 ∇η • θ ± , ε 0 (ψ ± -ηθ ± ) + ηf ρ ε n 2 ψ ± 0 +Re f ρ ε n 2 ∇η • ψ ± 0 , ε 0 (ψ ± -ηθ ± ) + ηf ρ ε n 2 ψ ± 0 = -Re ε 0 ∇η • θ ± + f ρ ε n 2 ∇η • ψ ± 0 , ε 0 ψ ± since Re ∇η • θ ± , θ ± = 0, Re ∇η • ψ ± 0 , ψ ± 0 = 0 and Re ∇η • ψ ± 0 , θ ± = -Re ∇η • θ ± , ψ ± 0 .
This leads to the following estimation:

Re D g ψ ± ε , ψ ± ε |B + q (2ρ)\B + q (ρ) ≥ Cε 2n-2 ρ 1-n
and thus we have:

B + q (2ρ)\B + q (ρ) Re D g ψ ± ε , ψ ± ε dx = o(ε 2(n-1) ).
Now using the fact that Re D g ψ ± ε , ψ ± ε |M\B + q (2ρ) = 0, we get:

M Re D g ψ ± ε , ψ ± ε ≥ nε n-1 I 1 -C 0 κε n-1 + +o(ε n-1 ) .
The variational characterization (17) of λ min (M, ∂M) gives:

λ min (M, ∂M) ≤ nI 1 n 1 + o(ε n-1 ) 1 -C 0 κε n-1 + o(ε n-1 ) = λ min (S n + , ∂S n + ) + C 0 κε n-1 + o(ε n-1 )
where C 0 is a positive constant. However, we assumed that κ := Re ψ ± 0 , ψ ± 1 < 0 so we can finally conclude that:

λ min (M, ∂M) < λ min (S n + , ∂S n + ).
The proof of Theorem 7 is then reduced to prove the existence of a spinor field satisfying the hypothesis of Theorem 8.

Proof of Theorem 7: We show that under the hypothesis of Theorem 7, there exists a spinor field on M \ {q} satisfying (20), ( 21) and ( 22). Without loss of generality, we can assume (using the conformal covariance of λ min (M, ∂M)) that for the metric g, there exists a boundary point q ∈ ∂M with locally flat and umbilic boundary neighbourhood. On the other hand, since the Dirac operator is supposed to be invertible on H ± , Proposition 1 allows to write that (around a point q ∈ ∂M), the chiral Green function admits the following development:

G ± CHI (x, q)ψ ± 0 = - 2 ω n-1 x -q |x -q| n • ψ ± 0 + m ± CHI (x, q)ψ ± 0 ,
where ψ ± 0 is a spinor such that ν • Γψ ± 0 = ∓ψ ± 0 and B ± CHI m ± CHI ( . , q)ψ ± 0|∂M = 0. Since the chiral mass operator is supposed to be non identically zero, we can choose a spinor ψ ± 0 which is an eigenspinor for the chiral mass operator m ± CHI (q) associated with the eigenvalue ±κ with κ ∈ R. Moreover, Corollary 6 insures that the eigenvalues ±κ are of opposite signs and so one of the two is positive and the other is negative. However the point (1) of Remark 2 tells us that the chiral bag invariant λ min (M, ∂M) does not depend of the choice of the boundary condition and we can then choose ψ - 0 (for example) as being an eigenspinor for the negative chiral mass operator m - CHI (q) associated with the eigenvalue κ with κ > 0. Then the chiral Green function G - CHI ( . , q)ψ - 0 is given by:

G - CHI (x, q)ψ - 0 = - 2 ω n-1 x -q |x -q| n • ψ - 0 + κψ - 0 .
Using the notations of Theorem 8, we have ψ - 1 = -κψ - 0 and then we obtain:

Re x • ψ - 0 , ψ - 1 = κ Re x • ψ - 0 , ψ - 0 = 0. Thus the spinor field -ω n-1
2 G - CHI ( . , q)ψ - 0 satisfies the properties (20), ( 21) and ( 22) and so Theorem 8 allows to conclude.

As an application of this result, we obtain a spinorial proof of the Yambe problem on manifolds with boundary. In fact, we have: Corollary 9. Suppose that µ(M, ∂M) > 0 and that the manifold (M n , g, σ) (n ≥ 3) has a locally conformally flat with umbilic boundary point q ∈ ∂M. Assume moreover that the chiral mass operator is non identically zero, then there exists a metric g ∈ [g] such that the scalar curvature R g is a positive constant and the mean curvature H g is zero.

Proof: A sufficient condition for the existence of a solution of the Yamabe problem on manifolds with boundary is (see [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF]):

µ(M, ∂M) < µ(S n + , ∂S n + ), (24)
where µ(M, ∂M) is the Yamabe invariant of M endowed with the conformal class of g. The hypothesis µ(M, ∂M) > 0 implies that the Dirac operator is invertible, under the chiral bag boundary condition. Since the manifold has a locally conformally flat with umbilic boundary point q ∈ ∂M and since the chiral mass operator is supposed to be non zero, then we can apply Theorem 7 and we finally get:

λ min (M, ∂M) < λ min (S n + , ∂S n + ). However, using the Hijazi inequality (18) and the fact that

λ min (S n + , ∂S n + ) = n 2 ω n 2 1 n ,
we obtain (24) and thus the result follows direcly.

This spinorial proof of the Yamabe problem problem on manifolds with boundary is obviously more difficult than the one given by Escobar. However, it seems that the inequality proved in Theorem (7) gives existence of solutions to a Yamabe type problem for the Dirac operator under the chiral bag boundary condition involving critical Sobolev embedding theorems. A similar equation has been treated by B. Ammann for the boundaryless case (see [START_REF]A variational problem in conformal spin geometry[END_REF] or [Amm]).

The MIT Green function

In this section, we construct the Green function for the Dirac operator under the MIT bag boundary condition. As an application of this construction, we give a simple proof of a positive mass theorem proved by Escobar [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF] in the context of the Yamabe problem on manifolds with boundary.

3.1. Definitions and first properties. The MIT bag boundary condition has been introduced by physicists of the Massachusetts Institute of Technology in the seventies (see [CJJ + 74], [START_REF] Chodos | Baryon structure in the bag theory[END_REF] or [START_REF] Johnson | T bag model[END_REF]). The spectrum of the Dirac operator under this boundary condition has then been studied on compact Riemannian spin manifolds with boundary (see [START_REF] Hijazi | Eigenvalue boundary problems for the Dirac operator[END_REF] and [START_REF] Raulot | Optimal eigenvalue estimates for the Dirac operator on domains with boundary[END_REF]). Another very nice application of this boundary condition can be found in [START_REF]Conformal lower bounds for the Dirac operator on embedded hypersurfaces[END_REF] where the authors use its conformal covariance to give some estimates of the boundary Dirac operator involving a conformal invariant (which can be seen as an extrinsic version of the Hijazi inequality [START_REF] Hijazi | A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors[END_REF]).

We now briefly recall the definition of the MIT bag boundary condition. Consider the linear map

iν• : S g -→ S g that is B ∓ MIT (Φ 0|V∩∂M ) = 0.
Proof: One considers the function defined on V by f (p) = |iν • Φ 0 -Φ 0 | 2 (p) and we then show that f vanishes on V ∩ ∂M. Indeed, for 1 ≤ i ≤ n -1, we have:

e i (f ) = e i (|iν • Φ 0 -Φ 0 | 2 ) = 2 e i |Φ 0 | 2 -i ν • Φ 0 , Φ 0 = 2 e i |Φ 0 | 2 + 2 Im ν • Φ 0 , Φ 0 ,
where Im (z) is the imaginary part of a complex number z ∈ C. However, since the spinor field Φ 0 is parallel, we can assume that |Φ 0 | 2 = 1 and since the trivialization of the spinor bundle is an isometry, we get |Φ 0 | 2 = 1 and so e i |Φ 0 | 2 = 0. Using the compatibilty of the spinorial Levi-Civita connection with the Hermitian metric leads to:

e i Im ν • Φ 0 , Φ 0 = Im ∇ e i ν • Φ 0 , Φ 0 + 2 Im ν • ∇ e i Φ 0 , Φ 0 .
The local expression of the spinorial Levi-Civita connection ∇ acting on Σ g (M) gives:

Im ν • ∇ e i Φ 0 , Φ 0 = 1 4 1≤j =k≤n-1 Γ k ij Im ν • e j • e k • Φ 0 , Φ 0 - 1 2 Im ∇ e i ν • Φ 0 , Φ 0 .
Note that:

ν • e j • e k • Φ 0 , Φ 0 = Φ 0 , ν • e j • e k • Φ 0 ,
and so we have e i (f ) = 0 for all 1 ≤ i ≤ n -1. Since we assumed that f (q) = 0, we obtain immediately the result of this lemma.

We can now state the analogous result of Proposition 1 for the MIT Green function. Indeed, we have:

Proposition 11. Assume that there exists a point q ∈ ∂M which has a locally flat with umbilic boundary neighbourhood. Then the MIT Green function exists and admits the following development near q:

G ± MIT (x, q)ψ ± 0 = - 2 ω n-1 x -q |x -q| n • ψ ± 0 + m ± MIT (x, q)ψ ± 0 , (26)
where ψ ± 0 ∈ Σ q (M) is a spinor field satisfying i ν • ψ ± 0 = ±ψ ± 0 (in other words B ∓ MIT (ψ ± 0 ) = 0) and where m ± MIT ( . , q)ψ ± 0 is a smooth spinor field such that D g m ± MIT ( . , q)ψ ± 0 = 0 around q. Moreover, along ∂M, we have:

B ∓ MIT m ± MIT (. , q)ψ ± 0|∂M = 0. ( 27
)
Proof: The proof of this proposition follows the proof of Proposition 1 using Lemma 10 and the fact that the Dirac operator:

D g : H ± g := {ϕ ∈ H 2 1 (ΣM) / B ± MIT (ϕ |∂M ) = 0} -→ L 2 Σ g (M)
, defines an invertible operator.

Application:

The positive mass theorem for the Yamabe problem on manifolds with boundary. First we recall briefly the notion of mass for compact manifolds with boundary. For more details, we refer to [START_REF] Lee | The Yamabe problem[END_REF] for the boundaryless case and [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF] for the non empty boundary case. Let (M n , g) be a connected compact Riemannian manifold which is locally conformally flat with umbilic boundary. If we assume that the Yamabe invariant µ(M, ∂M) of the manifold M is positive and if q ∈ ∂M then there exists a unique Green function G q for the conformal Laplacian

L g := 4 n -1 n -2 ∆ g + R g
under the boundary condition

B g := - 2 n -2 ∂ ∂ν + H g ,
that is, a smooth function G q defined on M \ {q} which satisfies, in a weak sense, the boundary problem:

   L g G q = δ q on M B g G q|∂M = δ q along ∂M (28)
One can then check that if there exists a point q ∈ ∂M with a locally flat and umbilic boundary neighbourhood in the metric g ∈ [g], then the Green function G q admits the following expansion (near q):

G q (x) = 1 (n -2)ω n-1 r n-2 + A + α q (x) (29)
where A ∈ R and α q is a harmonic function (near q) which satisfies α q (q) = 0 and ∂αq ∂ν = 0. The positive mass theorem proved by Escobar can then be stated as follows:

The constant A satisfies A ≥ 0. Moreover, A = 0 if and only if M is conformally isometric to the round hemisphere S n + . We give here a proof of this result for spin manifolds. However, in our proof, it is not necessary to impose that the manifold is locally conformally flat with umbilic boundary. Indeed, it is sufficient to assume that the manifold has only one point q ∈ ∂M which has a locally conformally flat with umbilic boundary neighbourhood. The proof of this positive mass theorem is inspired of the work of Ammann and Humbert [START_REF] Ammann | Positive mass theorem for the Yamabe problem on spin manifolds[END_REF] and is based on the construction of the Green function of the Dirac operator. The chiral Green function used in Section 2 seems to be a good candidate for our purpose however it needs the existence of a chirality operator which is not the case in any dimension. That is why we will use another conformal covariant boundary condition which exists without additional assumptions, the MIT bag boundary condition. More precisely, we prove: Theorem 12. Let (M n , g, σ) be a connected compact Riemannian spin manifold with non empty smooth boundary. Assume that the Yamabe invariant is positive and that there exists a point q ∈ ∂M which has a locally conformally flat with umbilic boundary neighbourhood. Then the mass A of the manifold M satisfies A ≥ 0. Moreover, equality holds if and only if M is conformally isometric to the round hemisphere S n + . Proof: First note that we can assume that for the metric g, the point q ∈ ∂M has a locally flat with umbilic boundary neighbourhood. Proposition 11 allows to deduce that there exists a unique MIT Green function G - MIT ( . , q) for the Dirac operator which admits the following development near q ∈ ∂M:

G - MIT (x, q)ψ 0 = G eucl (x, q)ψ 0 + m - MIT (x, q)ψ 0 , (30)
where ψ 0 ∈ Σ q M is such that iν • ψ 0 = -ψ 0 and m - MIT ( . , q)ψ 0 ∈ Γ Σ g (M) is a harmonic spinor near q which satisfies:

B -

MIT m - MIT (. , q)ψ 0|∂M = 0. Without loss of generality, we can assume that |ψ 0 | 2 = 1. On the other hand, since µ(M, ∂M) > 0, there exists a unique Green function G q for the conformal Laplacian smooth on M \ {q} such that G q > 0 (see [START_REF] Escobar | The Yamabe problem on manifolds with boundary[END_REF]). Moreover, since the point q is supposed to have a locally flat with umbilic boundary neighbourhood in the metric g, the Green function G q can be written

G q (x) = 1 (n -2)ω n-1 r n-2 + A + α q (x)
near q ∈ ∂M and where A ∈ R and α q is a smooth function on M. Now consider the conformal change of the metric on M \ {q} given by:

g = (n -2)ω n-1 G q 4 n-2 g = G 4 n-2 q g.
Since the Green function satisfies the boundary problem (28) and using the fact that the scalar and mean curvatures in g and g are related by: R = G

-n+2 n-2 q L g G q and H = G -n n-2 q B g G q ,
we obtain that the scalar curvature of (M\{q}, g) is R = 0 and the mean curvature of (∂M\{q}, g) in M is H = 0. We can then identify the spinor bundle (M \ {q}, g) with the one over (M \ {q}, g) thanks to the bundle ismorphism F used in the proof of Proposition 2. Now consider the spinor field defined by: G q ( . )ψ 0 = G -n+1 n-2 q F G - MIT ( . , q)ψ 0 ∈ Γ Σ g (M \ {q}) . Using the conformal covariance (13) of the Dirac operator, we have:

D g G q ( . )ψ 0 = G -n-1 n-2 q F D g G - MIT ( . , q)ψ 0 = 0 (31)
on M \ {q}. Moreover since the MIT condition is also conformally invariant, we obtain:

B - MIT G q ( . )ψ 0|∂M = 0,
where B -MIT = 1 2 Id -iν • is the projection of the MIT bag boundary condition in the metric g. Now let ε > 0 and denote by B + q (ε) the half-ball centered at q with radius ε. Using the formula (31), the Schrödinger-Lichnerowicz formula and the fact that the scalar curvature of M \ {q} is zero, we get:

0 = M\B + q (ε) D 2 g G q (x)ψ 0 , G q (x)ψ 0 dv(g) = M\B + q (ε) ∇ * ∇(G q (x)ψ 0 ), G q (x)ψ 0 dv(g).
An integration by parts leads to:

∂B + q (ε) ∇ νε (G q (x)ψ 0 ), G q (x)ψ 0 ds(g) = ∂Mε ∇ ν (G q (x)ψ 0 ), G q (x)ψ 0 ds(g) + M\B + q (ε) |∇(G q (x)ψ 0 )| 2 dv(g) (32)
where ∂M ε = ∂M \ ∂M ∩ ∂B + q (ε) and ν (resp. ν ε ) is the inner unit vector field (resp. outer) normal to ∂M ε (resp. ∂B + q (ε)) in the metric g. An easy calculation gives:

∇ ν (G q (x)ψ 0 ) = n -1 2 H G q (x)ψ 0 -ν •D g G q (x)ψ 0 -D ∂M g G q (x)ψ 0 where D ∂M g := n-1 i=1 e i • ν • ∇ S e i
is the boundary Dirac operator acting on the restricted spinor bundle over ∂M endowed with the metric g. However, on ∂M ε we have H = 0 and D g G q (x)ψ 0 = 0, so (32) gives:

∂B + q (ε)
∇ νε (G q (x)ψ 0 ), G q (x)ψ 0 ds(g) = -∂Mε D ∂M g G q (x)ψ 0 , G q (x)ψ 0 ds(g)

+ M\B + q (ε)
|∇(G q (x)ψ 0 )| 2 dv(g).

On the other hand, using the conformal covariance of the MIT bag boundary condition, we have: iν •G q (x)ψ 0 = G q (x)ψ 0 , for all x ∈ ∂M ε and so:

D ∂M g G q (x)ψ 0 , G q (x)ψ 0 = iν •D ∂M g G q (x)ψ 0 , iν •G q (x)ψ 0 = -D ∂M g G q (x)ψ 0 , G q (x)ψ 0 .
Hence Inequality (32) gives:

0 ≤ M\B + q (ε)
|∇(G q (x)ψ 0 )| 2 dv(g) = 1 2 ∂B + q (ε)

ν ε |G q (x)ψ 0 | 2 ds(g). (33)
The vector field ν ε is the inner normal field to ∂B + q (ε) for the metric g and g is conformal to g (which is flat around q), then the vector field ν ε is colinear to ∂ ∂r , that is there exists a constant c > 0 such that ν ε = -c ∂ ∂r .

However:

1 = g(ν ε , ν ε ) = c 2 G 4 n-2 q g( ∂ ∂r , ∂ ∂r ) = c 2 (n -2) 4 n-2 ω 4 n-1 n-1 G 4 n-2
q and since the half-ball B + q (ε) is contained in the open (flat) set of the trivialization near q, the Green function for the conformal Laplacian admits the expansion (29), that is:

c 2 (n -2) 4 n-2 ω 4 n-1 n-1 1 (n -2)ω n-1 r n-2 + A + α q (x) 4 n-2 = 1.
An easy calculation then shows that c = ε 2 + o(ε 2 ) and finally we have ν ε = -(ε 2 + o(ε 2 )) ∂ ∂r .

We now give an estimate of ν ε |G q (x)ψ 0 | 2 on ∂B + q (ε); for this, we write:

|G q (x)ψ 0 | 2 = G -2 n-1 n-2 q |G - MIT (x, q)ψ 0 | 2 = (n -2) -2 n-1 n-2 ω -2 n-1 n-2 n-1 G -2 n-1 n-2 q
|G - MIT (x, q)ψ 0 | 2 = 1 r n-2 + (n -2)ω n-1 A + α q (x) -2 n-1 n-2 |G eucl (x, q)ψ 0 + m - MIT (x, q)ψ 0 | 2 where α q (x) = (n -2) ω n-1 α q (x). Using the expansion (25) of the MIT Green function, we get:

|G q (x)ψ 0 | 2 = 1 + (n -2)ω n-1 Ar n-2 + α q (x)r n-2 -2 n-1 n-2

× 1 + |m - MIT (x, q)ψ 0 | 2 + 2Re G eucl (x, q)ψ 0 , m - MIT (x, q)ψ 0 .

Now note that, with some calculations, we obtain:

∂ ∂r |G q (x)ψ 0 | 2 = -2(n -1)(n -2) ω n-1 A ε n-3 + o(ε n-3 ).
On the other hand, we have:

ds(g) = det(g)dx = G 2 n-1 n-2 q dx = ε -2(n-1) 1 + o(1) dx
where dx is the standard volume form of R n-1 . We then have finally shown that:

0 ≤ M\B + q (ε)
|∇(G q (x)ψ 0 )| 2 dv(g) = 1 2 ∂B + q (ε)

ν ε |G q (x)ψ 0 | 2 ds(g) = 1 2 vol ∂B + q (ε) -ε 2 + o(ε 2 ) ε -2(n-1) + o(ε -2(n-1) ) × -2(n -1)(n -2)ω n-1 Aε n-3 + o(ε n-3 ) = (n -1)(n -2) ω 2
n-1 A + o(1), hence A ≥ 0. Now assume that A = 0; using the preceding identity, we have that the spinor field G q ( . )ψ 0 is parallel on M \ {q}, that is ∇ X (G q ( . )ψ 0 ) = 0 for all X ∈ Γ T(M \ {q}) . Since the choice of ψ 0 is arbitrary, we easily construct a basis of parallel spinor fields over (M \ {q}, g). On the other hand, the spinor G q ( . )ψ 0 satisfies the MIT bag boundary condition, i.e.: iν •G q ( . )ψ 0|∂M = G q ( . )ψ 0|∂M .

So if we derive (along ∂M) this equality, we get: ∇ X iν •G q ( . )ψ 0 = -iA(X)•G q ( . )ψ 0 = 0 since the spinor field G q ( . )ψ 0 is parallel. Using the fact the spinor field G q ( . )ψ 0 has no zero (since it is a constant section), we deduce that the boundary ∂M is totally geodesic in M. The restriction to the boundary of a parallel spinor field from a basis of the spinor bundle over (M, g) gives a parallel spinor field on the restricted spinor bundle along the boundary. This is an easy consequence of the spinorial Gauss formula and of the fact that the boundary is totally geodesic. We can thus easily conclude that the boundary is isometric to the Euclidean space R n-1 , hence the manifold (M \ {q}, g) is isometric to the Euclidean half-space (R n + , ξ). Now consider I : (M \ {q}, g) → (R n + , ξ) an isometry and let f (x) = 1 + ξ f (x) /4 where ξ f (x) := ξ f (x), f (x) . Then M \ {q} endowed with the metric f -2 g = f -2 G 4 n-2 q g is isometric to the standard hemisphere S n + . The function f -2 G 4 n-2 q is smooth on M \ {q} and extends to a positive function all over M. Thus we have shown that M is conformally equivalent to the standard hemisphere (S n + , g st ).

which is an involution, so we can define the pointwise orthogonal projection

g is the eigensubbundle associated with the eigenvalues ±1 of the endomorphism iν. We can then check (see [START_REF] Hijazi | Eigenvalue boundary problems for the Dirac operator[END_REF]) that these projections define elliptic boundary conditions for the Dirac operator. We can now define the Green function for the Dirac operator under the MIT bag boundary condition. Definition 3. A Green function for the Dirac operator under the MIT bag boundary condition (or a MIT Green function) is given by a smooth section:

locally integrable on M × M which satisfies, in a weak sense, the following boundary problem:

In other words, we have:

, we can easily check (using the Green formula) that (D ± g ) * = D ∓ g where (D ± g ) * is the formal adjoint of the Dirac operator under the boundary condition B ± MIT . Thus the domain dom (D ± g ) * of the Dirac operator's adjoint is given by H ∓ g . The preceding definition of the MIT Green fonction is then consistent with the definition of weak solution of an equation. Indeed, the section G ± MIT (. , q)ψ ± 0 ∈ Γ Σ g (M) satisfies on H ± g and in a week sense the equation:

for all ϕ ∈ dom (D ± g ) * = H ∓ g . This definition is exactly the one given in Definition 3. In order to give a development of the MIT Green function in a neighbourhood of q ∈ ∂M, it is very useful to study the behaviour of the MIT condition under the trivialization of the spinor bundle induced by Fermi coordinates and given in Section 2. More precisely, we prove the following lemma: Lemma 10. Let U and V be the open sets of the trivialization given in Section 2 and let Φ 0 ∈ Γ Σ ξ (R n + ) be a parallel spinor such that iν • Φ 0 (q) = ±Φ 0 (q) at one point q ∈ V ∩ ∂M. Then we get:


Aldomar Rückert
email: aldomar.ruckert@gmail.com

Camilo Carneiro Filho

Roberto Georg Uebel
email: roberto.uebel@ufrgs.br

Aldomar Arnaldo

Rückert Posgea

/ Propur

Camilo Pereira

PPGEEI Carneiro Filho

Roberto Rodolfo Georg

Uebel Ppgeei

CIST2016 proceedings En quête de territoire(s) ? Looking for territories? Transborderings in South America Some Examples of Recent Researches AUTEURS

Keywords: South America, Transborder regions, Transbordering processes, International migrations régions transfrontalières, processus transfrontaliers, migrations internationales

The article deals with transbordering scenarios in South America, seen as one geopolitical region. Transborder regions and transbordering processes can be understood in various aspects such as reordering of economic flows, people migrations, new connections of infrastructure and information that tend to produce new territorial interactions and territorial networks. It presents multinational South American cartographies in great detail of data. It is proposed the regional delimitation of Iguaçu Transborder Region where there are intense cross-border flows of different kinds, legal and illegal. Finally it adds to the analysis of transbordering processes recent international migration transcontinental flows to Brazil.

INTRODUCTION

The article examines transbordering scenarii in South America, seen as a geopolitical region. The continent is under various transformation scenarii with both political and economic integration, including defense and security, where processes of bordering, debordering and rebordering are very clear. Border regions and transbordering processes can be understood in the geopolitical region of South America [START_REF] Costa | Le Brésil et l'Amérique du Sud : scénarios géopolitiques et défis de l'intégration[END_REF] in many aspects such as reordering economic flows of legal and illegal merchandises and flows of people and information, promoting changes of infrastructures that tend to produce new territorial interactions and territorial networks.

Analytical attention is given in this article to the necessity of constructing South American multinational cartographies in great detail. Besides, we provide a regional delimitation of the Iguaçu Transborder Region -as an international region in the macro transborder region of the La Plata River Basin-where there are intense cross-border flows of different kinds, whether legal or illegal. Finally, we deal with the analysis of transbordering processes in South America, through very recent international transcontinental immigration flows into Brazil.

TRANSBORDER REGIONS AND TRANSBORDERING PROCESSES IN SOUTH AMERICA

Transborder regions and transbordering processes are related to the creation of a new South American regional order -a geopolitical multinational region. The recent creation of UNASUL (Union of South American Nations, in 2008) drives regional territorial cohesion policies along with the creation of the Council of South American Defense. The reordering of economic flows and people relates, in turn, to the new territorial interactions consolidating a new territorial order in which transnational territorial networks are one of the defining conditions of globalisation and regional integration [START_REF] Roseira | Nova ordem sul-americana: reorganização geopolítica do espaço mundial e projeção internacional do Brasil, Tese de Doutorado[END_REF]. Crossborder flows in this new context give new dynamics to territorial networks in this geopolitical region where it is obvious that the growing process of territories becomes more internationalised.

Multinational cartographies

An important aspect in examining transborderings is concerned with the technical deficiencies of the multinational character mapping. A test using European NUTS classifications was held for MERCOSUL (Southern Common Market) + Bolivia and Chile (without Venezuela in this representation) with a view to organize multinational cartography without representing boundary lines between States [START_REF] Rückert | Total population of Mercosur plus Bolivia and Chile, by countries, levels II and III[END_REF], on the basis of table 1 data provided by national institutes of statistics.

Table 1. Levels of spatial divisions and number of statistical units by countries This multinational cartography (map 1) underlines the evident concentration of population in each country, and especially in the micro-regions on the coasts of Brazil, in the Plata River Basin and the Andes line, as well as the multinational region called MERCOSUL Geoeconomic Core located in the macro-region of the Plata Basin, e.g. the diagonal area that goes from Belo Horizonte-Rio de Janeiro (Brazil) to Buenos Aires, Cordoba (Argentina) and Santiago do Chile. Scheibe (2013) and [START_REF] Fonseca | Agenda de projetos prioritários de integração -IIRSA/COSIPLAN[END_REF] examined the relationship between demographic concentrations in level 3 statistical units in South America and the corresponding location of priority projects of IIRSA (Initiative for South American Regional Infrastructure Integration Initiative)/COSIPLAN (South American Council of Infrastructure and Planning) of UNASUL (Union of South American Nations). It was demonstrated that the Agenda of Priority Projects of Integration suits geographic micro-regions or departments where there is greater population concentration, although jobs are not always located in those regions. This finding is even more evident in three axis: MERCOSUL-Chile, Paraguay-Paraná and Capricorn (see map 2). Some regions benefit while others are excluded from this competitive integration in an internationalisation process of transnational territorial networks and by the re-concentration of industrial activities in the already most competitive regions, such as those located in the geo-economic core of MERCOSUL. Transborderings intensify the internationalisation process of just a few parts of the territories, mainly those regarded as competitive.

INTENSE TRANSBORDERINGS IN THE TRIPLE BORDER BRAZIL-ARGENTINA-PARAGUAY

The triple border Brazil-Argentina-Paraguay, located in the macro-region of the La Plata River Basin (map 4) is a good example of an international region with intense transborderings [START_REF] Schweitzer | Intégration régionale et aménagement du territoire dans le MERCOSUL : frontières, réseaux et dynamiques transfrontalières, thesis directed[END_REF][START_REF] Rückert | Integração regional, a região transfronteiriça da bacia do Rio da Prata e os projetos de infraestruturas de conexão[END_REF]. This macro-region covers the south-central South American continent (a probable cross-border macro-region), almost matching the geo-economic core of MERCOSUL. The area formed by the triple-twin cities of Foz do Iguaçu/BRA, Ciudad del Este/PAR and Puerto Iguazú/ARG, along with the adjacent municipalities from the Iguaçu Transborder Region [START_REF] Filho | Processos de transfronteirização na Bacia do Prata: a Tríplice Fronteira Brasil-Argentina-Paraguai, Tese de Doutorado[END_REF]. This, in turn, tends to concentrate multinational nodosities, centrality flows, and to strongly interact through border trade, international tourism, power generation as well as the crossborder coexistence of various ethnic groups.

The Iguaçu Transborder Region

The Iguaçu Transborder Region (map 4) we provide is composed of the Iguaçu microgeographic region in Brazil, the departments of Iguazú in Argentina and Alto Paraná in Paraguay. This territorial delimitation explains why these political-administrative units have economic, cultural and social ties, and their main economic activities in tourism, power generation and trade. The linkages between the three sides of the border are intense and the economies of these cities are often more interconnected than with their respective countries.

TRANSCONTINENTAL INTERNATIONAL FLOWS: RECENT IMMIGRATION FOR CENTRAL SOUTH BRAZIL

In recent research, Uebel (2015) sought to identify not only the migratory routes after the three immigration booms experienced by Brazil (at the beginning of the century, in 2010 and in 2013-2014), but also to check networks and transbordering processes that carry out these flows, especially of Haitians and Senegaleses, the more noticeable by society, media and researchers.

In total computation of the immigration stock in Brazil, the data show that there were about 1,900,000 immigrants from 2007 to 2014 -overcoming the total estimated by the government and the press in approximately 600,000 immigrant citizens.

1 Brasiguaio is a combination of the Portuguese words Brasi(leiro) and (Para)guaio for Brazilian small farmers and peasants that have been going to Paraguay since the 1970s looking for new lands for agriculture.

The immigration of Haitians and their routes

According to the data being consulted, there were 2,503 immigrants from Haiti in the state of Rio Grande do Sul (Southern Brazil) in October 2014. They have mostly arrived after a long process of migration through domestic routes in Brazil, presented in map 5, following reports obtained from immigrants themselves during field research conducted in July 2014 in the states of Acre and Rio Grande do Sul.

FINAL CONSIDERATIONS

Analysing the South America geopolitical region with focus on transborder regions and transbordering processes electing various types of cross-border flows as possible indicators of these dynamics it was aimed to point out numerous research possibilities in the South American scale. Looking to avoid the excesses of local and micro-regional researches it was sought to articulate some research results at national and international levels, without neglecting the local and regional, although in multinational contexts.

In the specific case of changing borders nowadays it is recognised that scenarii are much more complex than those of the post-Cold War, because they are marked by constant openings and closings, borderings and reborderings movements that mark currently processes of transbordering. In constant motions challenging paradigms, South America points to several competing directions with respect to territorial transformations. The infrastructure connections, the construction of roads, and the various flows that connect the several territorial networks, including the above-mentioned transcontinental migrations, are some indicators of new directions being given by the political uses of the territory in this continental region.

Map 3 .

 3 The macro-region of the Plata River Basin Map 4. The Iguaçu Transborder Region Transborder flows: the legal and illegal trade in the Brazil-Paraguay border In the Iguaçu Transborder Region, the most intense cross-border flows occur along the dyad that separates Brazil and Paraguay. The largest contingents are trading workers from Ciudad del Este living on the Brazilian side; individuals who develop smuggling and embezzlement activities; Brazilian students in Paraguayan education institutions (medical schools, above all); brasiguaio 1 migrants; Paraguayan students studying in Foz do Iguaçu universities (UNILA-Latin America Integration University) and Paraguayans who seek jobs in Brazil.

Map 5 .

 5 Routes of Haitian immigrants into BrazilThe immigration of Senegaleses and their routes On the contrary, it was observed that the concentration of flows and characteristics of Senegalese immigrants were distinct (see map 6). Although they have earned a remarkable prominence in the media, academic and governmental debates from 2013 on, the Senegalese immigration group (in stock) only ranks 22nd in immigration levels in Rio Grande do Sul (Southern Brazil) from 2007 to 2014.New aspects of transbordering processes have been emerging: illegal immigration has been strong in border regions, mainly in the macro transborder region of the La Plata River Basin. Illegal flows between Uruguai, Argentina, Paraguai and Bolivia have been providing illegal handwork to industrial regions, mainly São Paulo. Recent transcontinenal flows of Haitians, and from the West coast of Africa more generally underline another kind of porosity to the Amazon region, through which these immigrants have come into Brazil. En quête de territoire(s) ? Looking for territories? Map 6. Routes of Senegalese immigrants into Brazil


Nicholas Ayache

James Duncan

published or not. The documents may come   L'archive ouverte pluridisciplinaire

20th anniversary of the medical image analysis journal (MedIA) This special issue of Medical Image Analysis gathers 32 articles written by prominent members of the current editorial board of the journal. Each author had the opportunity to discuss a personal vision of the past and future of the field. The result is a set of outstanding visionary articles that will count as a historical landmark for our readership.

It was shortly after the success of the first CVRMed conference 1 held in Nice in 1995 that we took the decision to launch a new journal entirely dedicated to the computational analysis of medical images. The founding editorial board members 2 of MedIA were among the authors of the articles of the first volume 3 of the journal published by Oxford University Press in 1996.

Twenty years later, the journal (now published by Elsevier Science since 2000) has become a reference for a vibrant community of researchers working in academics, clinics and industry. It has also become the premier journal of the MICCAI Society, which organizes the Medical Image Computing and Computer Assisted Intervention conference each year on a different continent.

Medical Image Analysis is now recognized as a flourishing research field at the intersection of Informatics, Computational Sciences and Medicine. Its progress contributes to the development of innovative computational tools to assist medical imaging professionals in clinical analysis and intervention.

It has been our pleasure to serve this journal and this research community from its origin; we look forward to continue serving both in the future.

  Nice, France) & James Duncan (New Haven, USA) Co-Editors-in-Chief, Medical Image Analysis


Arturo Ortiz

Hans Henrik Støleum

STUDIES OF ENTROPY MEASURES CONCERNING THE GAPS OF PRIME NUMBERS

The Shannon entropy is used as a basis for applying different lemmas and conjectures concerning the set of gaps between prime numbers Gp, thus estimating several measures of it. The same procedures are applied to artificially created number sets, to compare the size of their entropy against Gp.

1

Introduction

Shannon's (information) entropy is based on probabilities. If a distribution of probabilities is known, it can be estimated using the formula for discrete distributions or for continuous distributions [START_REF] Fouque | Close to uniform prime number generation with fewer random bits[END_REF].

Why entropy bounds may not be good enough? Previous works on the generation of prime numbers, such as [START_REF] Brandt | On generation of probable primes by incremental search[END_REF][START_REF] Joye | Fast generation of prime numbers on portable devices: An update[END_REF][START_REF] Maurer | Fast generation of prime numbers and secure public-key cryptographic parameters[END_REF], provide a proof (based on rather strong assumptions) that the output distribution of their algorithm has an entropy not much smaller than the entropy of the uniform distribution. This is a reasonable measure of the inability of an adversary to guess which particular prime was output by the algorithm, but it doesn't rule out the possibility of gaining some information about the generated primes. In particular, it doesn't rule out the existence of an efficient distinguisher between the output distribution and the uniform one. For example [START_REF] Joye | Fast generation of prime numbers on portable devices: An update[END_REF], let H max ... nbit prime. n ≥ 256 (1.1) H max -H < 1 -γ log 2 = 0.609949 γ is the Euler-Mascheroni constant. The entropy loss with respect to a perfectly uniform generation is less than 0.61 bit for any prime bit length.

Therefore the central part of calculating Shannon's entropy rest upon how does one obtain the probabilities. For example [START_REF] Minculeta | The entropy of a natural number[END_REF] uses

(1.2) H(n) = log Ω(n) - 1 Ω(n) r i=1 a i log a i
where Ω(n) is the sum of the total number of prime factors of the natural number n and a i is the multiplicity of each one of those prime factors. In particular for n = 2 • 3 2 • 5 3 the entropy is estimated as

(1.3) H(n) = log 6 - 1 6 (2 log 2 + 3 log 3) ≈ 1.011
In this work it usage will be made of several measures like Eq.1.3 to prove that the entropy of a given amount of gaps between prime numbers is less than the entropy of a uniform distribution of a similar quantity of natural or real numbers.

Entropy of Real Numbers

Shannon's entropy is defined as [START_REF] Shannon | A mathematical theory of communication[END_REF] (2.1)

H(x) = - n i=1 P (x i ) log b P (x i )
where P (x i ) is the probability mass funcion of the discrete random variable X with possible values {x 1 , • • • , x n } and log b is the logarithm in base b used.

For the continuous case we have [START_REF] Park | Maximum entropy autoregressive conditional heteroskedasticity model[END_REF] (2.2)

h(X) = - X f (x) log f (x)dx
where f is a probability density function whose support is a set X For the purposes of this work, suppose it is desired to measure the entropy of a set with a uniform distribution with a given amount of real numbers (a set of random reals R real ), and in order to make it congruent to any measure of entropy with prime numbers, the support set X = [2, N p ], where N p is a given number of primes up to a given bound. Using Eq.2.2, the entropy of real numbers can be defined as

(2.3) H(R real ) = - N p 2 1 N p -2 log 1 N p -2 dR real = log(N p -2)
By the Prime Number Theorem [START_REF] Hardy | Contributions to the theory of the riemann zetafunction and the theory of the distribution of primes[END_REF], the amount of prime numbers N p up to a given bound x is (2.4)

N p ∼ x ln x ∼ π(x)
π(x) the number of primes less than or equal to x, for any Real number x. Assume that for a given, concrete, measure, we have x = x max , and using Eq.2.2

(2.5)

H(Real) = - xmax ln xmax 2 1 xmax ln xmax -2 ln 1 xmax ln xmax -2 dx = ln x max ln x max - 2
for a chosen maximal x. Goldston, Pintz, and Yildirim [START_REF] Goldston | Primes in tuples i[END_REF] under appropiate unproved conjectures, showed that there are infinitely many primes P n , P n+1 such that: P n+1 -P n < 16. Therefore, applying Eq. (2.5)

(3.2) G(x) ∼ log 2 (x) so H(G(x)) = - xmax 1 log 2 (x max ) log 1 log 2 (x max -1) dx = -log 2 (x max ) log 1 log 2 (x max -1) (x max -1) = log -2 (x max ) log 1 log 2 (x max -1) (x max -1) = log 1 log 2 (x max -1) (x max -1)/ log 2
For comparison, we generate randomly uniform distributions of gaps from 2 to G(x max ) so

H(Random gaps) = - G(xmax) 2 1 G(x max ) -2 log 1 G(x max ) -2 dx = - 1 G(x max ) -2 log 1 G(x max ) -2 (G(x max ) -2) = -log 1 G(x max ) -2 = log (G(x max ) -2)
Thus resulting the inequality to be questioned

(3.3) log x max log x max -2 > log (G(x max ) -2) ∀x > x min ?
using Eq. 3.1 we obtain

(3.4) log x max log x max -2 > log (log(x max ) (log(x max ) -2 log log(x max ) + c) -2)
exponentiating both sides and scratching out the outermost "2"s we get

(3.5) x max log x max > log(x max ) (log(x max ) -2 log log(x max ) + c) (3.6) x max > log 2 (x max ) (log(x max ) -2 log log(x max ) + c) solving for x max (3.7) x max > 9.17162
Meaning that for x max > 9.17162, the entropy of the largest gap from those generated with a uniform distribution, is greater than the entropy of the largest gap between primes, below a given bound x, according to Cramér's conjecture. Moreover, for a uniformly distribution of gaps from 2 to max gaps = G(x max ), we will have always

(3.8) log (G(x max ) -2)
whatever G(x max ) is; taking now G(x max ) ∼ log 2 (x max ) (Cramer's conjecture) for x >> 1 (large x) we have the inequality

(3.9) log x max log x max -2 > log log 2 (x max ) -2
and solving for x max , we obtain Using a conjecture in [START_REF] Heath-Brown | Gaps between primes, and the pair correlation of zeros of the zetafunction[END_REF] (assuming the validity of the Riemann Hypothesis), we have that log (G(x max ) -2) again, we get (4.1) log

x max log x max -2 > log (log(x max ) (log(x max ) + log log log(x max )) -2)
which is always true for x > 120.027 (or for 5.69781 < x < 8.43901).

Granville's formula

A. Granville's formula [START_REF] Granville | Harald cramér and the distribution of prime numbers[END_REF], which claims to be true for infinitely many pairs of primes P n , P n+1 for which (5.1) where the right-hand-side of Eq.(6.2) is the "size" of P n and n is the index, or the ordered position in the sequence of primes, for P n . Therefore we can take this as x max . Go back to the entropy of Reals, taking n log n as the largest number of primes, but now in terms of the index n.

P n , P n+1 = G(P n ) > 2e -γ log 2 (P n ) =
H(Real) = - # of primes 2 1 #P -2 log 1 #P -2 dReal (6.3) = - 1 #P -2 log 1 #P -2 (#P -2) = log (#P -2)
we recall that PNT also says that the number of primes is approximately (6.4) #primes ∼ x log x but now we are taking x as x max and in turn this maximal number making it equal to the size of the largest prime, that is x max = n log n, so first (6.5) log

x max log x max -2
and substituting the value of x max and by Eq.( 6.2)

(6.6) log n log n log(n log n) -2 = log P n log P n - 2
and now this new formula for the size of entropy in terms of the index n can be used for those formulas of the gaps that are in terms of P n , being now P n ∼ n log n. We go back to Granville's formula given by Wolf, in terms of the Euler-Mascheroni constant. (6.7) which is true for n ≥ 9.

G(P n ) > 2e -γ log 2 (P n ) =

Upper bound of Jaroma 2005

Instead of Eq.( 8.2) the following can also be used [START_REF] Jaroma | An upper bound on the nth prime[END_REF] (9.1)

P n < (1.2) k < P < (1.2) k+1
Therefore P k+1 < (1.2) k+1 , which we substitute in Eq.(8.2)

(9.2) log (1.2) n log(1.2) n -2 > log log 2 (1.2) n -2 log(1.2) n+1 -2
Solving for n, we obtain n ∈ +Z ∧ n ≥ 16.

Estimations based on Tschebychef function, Robin 1983

We have been using (10.1) log

P n log P n -2 > log (G(P n ))
suppose now the following result [START_REF] Robin | Estimation de la fonction de tchebychef θ sur le k-ième nombre premier et grandes valeurs de la fonction ω (n) nombre de diviseurs premiers de n[END_REF] (10.2)

P n ≤ n log n + n (log log n -0.9385)
for n ≥ 7022 substituing the right hand side of Eq.( 10.2) on the left hand side of Eq.(8.2) (10.3)

log n log n + n (log log n -0.9385) log n log n + n (log log n -0.9385) -2 > log log 2 P n -2 log P n+1 -2
Even if we don't have a result like Eq.(10.2) for P n+1 , let us argue like Jaroma [START_REF] Jaroma | An upper bound on the nth prime[END_REF] and suppose that (10.4) P n+1 ≤ (n + 1) log(n + 1) + (n + 1) (log log(n + 1) -0.9385) substituting Eq.(10.2) and Eq.(10.4) in Eq.( 10.1) we obtain log n log n + n (log log n -0.9385) log n log n + n (log log n -0.9385) -2 > (10.5) log log (n log n + n (log log n -0.9385))

2 -2 log(n + 1) log(n + 1) + (n + 1) (log log(n + 1) -0.9385) -2

But this is true only for 16 ≤ n ≤ 32.

11. Upper and lower bounds for gaps between primes 11.1. kontorovich-Zhang 2013. Alex Kontorovich [START_REF] Kontorovich | Levels of distribution and the affine sieve[END_REF] explains the basis of Zhang's theorem: In April 2013 Zhang [START_REF] Zhang | Bounded gaps between primes[END_REF] proved that the Bounded Gaps Conjecture is true. In particular, (11.1) lim inf

n→∞ (P n+1 -P n ) < 7 × 10 7
The average gap P n+1 -P n is of size about log P n (by PNT):

(11.2) ∀x, x >> 1, ∃ (Pn+1, Pn) (P n+1 -P n ) < 7 × 10 7
in the range [x, 2x] [START_REF] Zhang | Bounded gaps between primes[END_REF]. This will be taken as an upper limit for calculating the entropy of gaps between prime numbers. (11.9)

P |k, P >2 P -1 P -2 = 3 -1 3 -2 = 2
8 is divided by 8, 4, 2 but 2 is the only prime, and cannot be used. We should have zero, but by definition the function resorts to 1. We have, then that (11.10)

f (k) = P |k, P >2 P -1 P -2 = O(log log k)
The right-hand-side of this last expression is our smooth "envelope", with k meaning gaps of any length. By PNT and Merten's theorem, G(x) > log(x), and using the bounds 2 and 7 × 10 7 , an estimate of the entropy of gaps between prime numbers is:

(11.11) H(G(P )) = - Again, by PNT, G(x) > log(x). The x in this last equation is the natural number, which has to be so large, we can accomodate the size of the maximum possible entropy of gaps. We can go back to the formula for entropy of reals.

(11.12)

H(Real) = log x max log x max - 2
for a chosen maximal x. We know that maxgap = 7 × 10 7 = G(x) > log(x); solving for x, we obtain the exact number x = e 7×10 7 , meaning that, provided x ≥ exp[7 × 10 7 ], the entropy of the gaps will remain forever lower than any given real number. For a uniform distribution of gaps, the entropy is

(11.13) log (G(x max ) -2)
we know from [START_REF] Zhang | Bounded gaps between primes[END_REF] that G(x max ) = 3 × 10 7 , and that G(x) > log(x), so:

(11.14) log ((x max ) -2)
is the minimum entropy attainable by a random gap generator, which doesn't "know" about the maximal gap. The entropy is therefore: (11.15) x = e 2+e 3×10 7

and from there onwards the entropy of random gaps must be larger, and for H(Real) for H(Real) to be always greater than the entropy of the gaps between prime numbers.

(
12. H(P n+1 -P (n) = min{H(x k+1 -x k )}, k ∈ R Theorem 12.1. The entropy of the gaps between prime numbers is smaller than any similar distribution made with random gaps, and of real numbers.

Proof. Using the results of all sections, it is shown that at least above a certain measure, it can be certain that the entropy of gaps between prime numbers will remain smaller than any random distribution of gaps of similar size, or of real numbers of comparable size.

( 3 .

 3 10) x max > 93.3545 Now, let (3.11) G(x max ) = log(x max ) (log(x max ) + log log log(x max )) 4. Heath-Brown conjecture

  (k)) dk (assuming smoothness) Solving we obtain 2.57231 × 10 7 .

  1.12292... log 2 (P n )Compare with the other formula given above (G(x max ) ∼ log 2 (x max ) -Eq.3.2-); this would lead to rethink a real number, the size of P n , i.e., x max = P n ; what is the probability of P n ? As a consequence of the Prime Number Theorem (PNT), one gets an asymptotic expression for the n th prime number, denoted by P

	where γ = 0.577216... is the Euler-Mascheroni constant. The problem with Granville's, and other similar results, is that the formula (5.2) log(x max / log x max ) -2 cannot be used for comparison of the entropy with the reals, because Eq.(5.2) is Assuming the Riemann Hypothesis, Cramér proved that [20] not in terms of P 6. Using a Generalization of Cramér's conjecture (6.1) P n+1 -P n = O P 0.5 n log P n

n . Said in another way, Eq.(5.2) is not a formula saying something about how to find directly P n , i.e., how large x max must be, so that P n is found. However there might a way to interpret this P n using a generalization of Cramér's conjecture (see next section). n

[START_REF] Ribenboim | The little book of bigger primes[END_REF]

: (6.2)

P n ∼ n log n

  Solving numerically for P n ; P n ∈ R gives P n > 128.703 and the next prime to this real number is 131. The best current unconditional result for P n+1 -P n is O P 0.535 P n < log 2 P n -log P n+1 the right hand side of the inequality will be taken as G(P n ).

	We use log(G(P n ) -2) so we are saying that either
	(6.8)	log	P n log P n	-2 > log 2e -γ log 2 (P n ) -2
	or that (by 6.2)
	(6.9)	log	n log n log(n log n)	-2 > log 2e -γ log 2 (n log n) -2
	n	due to R. Baker and G. Harman, 1996, so:
	(6.10)		log	P n log P n	-2 > log P 0.535 n	-2
	which renders P n > 3.6532 or P n > 5 or, in terms of the index n
	(6.11)	log	n log n log(n log n)	-2 > log (n log n) 0.535 -2
				7. Cramér's conjecture
	(7.1)			P n+1 -P n = O P 0.5 n log P n
	assuming the Riemann Hypothesis. so the entropy inequality will be written as
	(7.2)		log	P n log P n	-2 > log P 0.5 n log P n -2
	so P n > 5503.66 or P n > 5507.
		8. Stronger Form of Firoozbakt's Conjecture
	Sinha [20] deduces a stronger form of Firoozbakt's conjecture, from which it is
	deduced that
	(8.1) P n+1 -Now we are saying that [17]
	(8.2)	log	P n log P n	-2 > log log 2 P n -2 log P n+1 -2
	which is true for P n ≥ 17 ∧ P n+1 ≥ 19, or
	(8.3) log	n log n log(n log n)	-2 > log log 2 (n log n) -2 log((n + 1) log(n + 1)) -2

1.12292... log 2 (P n )

  P n strictly bounds from below the size of the gaps. Eqns. 11.4 and 11.6 tell us that on average one can expect that the lower bound of gaps between prime numbers is as low as possible. Since prime numbers are odd numbers, the smallest possible gap is 2 and using 7 × 10 7 as an upper bound, we now need to estimate a smooth envelope for prime gaps.

	11.2. Kontoyiannis 2008. Ioannis Kontoyiannis [11], cites a paper of Chebyshev,
	1852 where
	(11.3)	C(n)	P ≤n	log P P	∼ log n
	where the sum is over all primes not exceeding n, and furthermore proves that
	(11.4)	lim inf n→∞	C(n) log n	= 1,
	using as one of the arguments that the entropy contained in the the number of
	primes up to certain value, is inversely proportional to the number of primes within
	that range:
	(11.5)	H(N ) = log	1 P (N )
	11.3. Perepelyuk 2013. In [16] it is mentioned that
	(11.6)	lim inf n→∞		P n+1 -P n log P n	= 0
	meaning that log 11.4. Smooth envelope. Using Merten's theorem [13] and PNT , it can be shown
	that
	(11.7)	f (k) = O(log log k)
	where
	(11.8)	f (k) =	P |k, P >2	P -1 P -2
	(k even) and compare for different k. So f (6) = 2 > 1 = f (8); so gaps of length 6
	are asymptotically twice as common as gaps of length 8.
	So, using Eq.(11.8), for 6, we only have 3 as prime:

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