Bernd Ammann

Pierre Jammes

THE SUPREMUM OF CONFORMALLY COVARIANT EIGENVALUES IN A CONFORMAL CLASS

Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metrics g conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with respect to g is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2.

Introduction

The goal of this article is to prove the following theorems.

Theorem 1.1. Let (M, g 0 , χ) be compact Riemannian spin manifold of dimension n ≥ 2. For any metric g in the conformal class [g 0 ], we denote the first positive eigenvalue of the Dirac operator on (M, g, χ) by λ + 1 (D g ). Then sup

g∈[g0]
λ + 1 (D g )Vol(M, g) 1/n = ∞.

Theorem 1.2. Let (M, g 0 , χ) be compact Riemannian manifold of dimension n ≥ 3. For any metric g in the conformal class [g 0 ], we denote the first positive eigenvalue of the conformal Laplacian L g := ∆ g + n-2 4(n-1) Scal g (also called Yamabe operator) on (M, g, χ) by λ + 1 (L g ). Then

sup

g∈[g0] λ + 1 (L g )Vol(M, g) 2/n = ∞.
The Dirac operator and the conformal Laplacian belong to a large family of operators, definded in details in subsection 2.3. These operators are called conformally covariant elliptic operators of order k and of bidegree ((n -k)/2, (n + k)/2), acting on manifolds (M, g) of dimension n > k. In particular, our definition includes formal self-adjointness.

The above theorems can be generalized to the following:

1
bernd.ammann at gmx.de, pierre.jammes at univ-avignon.fr 1 Theorem 1.3. Let P g be a conformally covariant elliptic operator of order k, of bidegree ((n -k)/2, (n + k)/2) acting on manifolds of dimension n > k. We also assume that P g is invertible on S n-1 × R (see Definition 2.4). Let (M, g 0 ) be compact Riemannian manifold. In the case that P g depends on the spin structure, we assume that M is oriented and is equipped with a spin structure. For any metric g in the conformal class [g 0 ], we denote the first positive eigenvalue of P g by λ + 1 (P g ). Then sup

g∈[g0] λ + 1 (P g )Vol(M, g) k/n = ∞.
The interest in this result is motivated by three questions. At first, we found that the infimum inf

g∈[g0]
λ + 1 (D g )Vol(M, g) 1/n has a rich geometrical structure [START_REF]A variational problem in conformal spin geometry[END_REF], [START_REF]The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions[END_REF], [START_REF] Ammann | Mass endomorphism and spinorial Yamabe type problems[END_REF], [START_REF] Ammann | The first conformal Dirac eigenvalue on 2-dimensional tori[END_REF]. In particular it is strictly positive [START_REF]A spin-conformal lower bound of the first positive Dirac eigenvalue[END_REF] and under some condition preventing the blowup of spheres it is attained [START_REF]A variational problem in conformal spin geometry[END_REF], [START_REF]The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions[END_REF].

The second motivation comes from comparing this result to results about other differential operators. Let us recall that for the Hodge Laplacian ∆ g p acting on p-forms, we have sup g∈[g0] λ 1 (∆ g p )Vol(M, g) 2/n = +∞ for n ≥ 4 and 2 ≤ p ≤ n-2 ( [START_REF] Colbois | Eigenvalues of the Laplacian acting on p-forms and metric conformal deformations[END_REF]). On the other hand, for the standard Laplacian ∆ g acting on functions, we have sup g∈[g0] λ k (∆ g )Vol(M, g) 2/n < +∞ (the case k = 1 is proven in [START_REF] Soufi | Immersions minimales, première valeur propre du laplacien et volume conforme[END_REF] and the general case in [START_REF] Korevaar | Upper bounds for eigenvalues of conformal metrics[END_REF]). See [START_REF] Jammes | Extrema de valeurs propres dans une classe conforme[END_REF] for a synthetic presentation of this subject.

The essential idea in the proof is to construct metrics with longer and longer cylindrical parts. We will call this an asymptotically cylindrical blowup. Such metrics are also called Pinocchio metrics in [START_REF] Ammann | Spin-Strukturen und das Spektrum des Dirac-Operators[END_REF][START_REF] Ammann | Dirac eigenvalues and total scalar curvature[END_REF]. In [START_REF] Ammann | Spin-Strukturen und das Spektrum des Dirac-Operators[END_REF][START_REF] Ammann | Dirac eigenvalues and total scalar curvature[END_REF] the behavior of Dirac eigenvalues on such metrics has already been studied partially, but the present article has much stronger results. This provides the third motivation.

Preliminaries

2.1. Notations. In this article B y (r) denotes the ball of radius r around y, S y (r) = ∂B y (r) its boundary. The standard sphere S 0 (1) ⊂ R n in R n is also denoted by S n-1 , its volume is ω n-1 . For the volume element of (M, g) we use the notation dv g .

For sections u of a vector bundle V → M over a Riemannian manifold (M, g) the Sobolev norms L 2 and H s , s ∈ N, are defined as

u 2 L 2 (M,g) := M |u| 2 dv g u 2 H s (M,g) := u 2 L 2 (M,g) + ∇u 2 L 2 (M,g) + . . . + ∇ s u 2 L 2 (M,g) .
The vector bundle V will be suppressed in the notation. If M and g are clear from the context, we write just L 2 H s . The completions of the compactly supported sections of V with respect to these norms are called L 2 (M, g) and H s (M, g).

Removal of singularities.

In the proof we will use the following removal of singularity lemma.

Lemma 2.1 (Removal of singularities lemma).

Let Ω be a bounded open subset of R n containing 0. Let P be an elliptic differential operator of order k on Ω, f ∈ C ∞ (Ω), and let u ∈ C ∞ (Ω \ {0}) be a solution of

P u = f (1) on Ω \ {0} with lim ε→0 B0(2ε)-B0(ε) |u|r -k = 0 and lim ε→0 B0(ε) |u| = 0 ( 2
)
where r is the distance to 0. Then u is a (strong) solution of (1) on Ω. The same result holds for sections of vector bundles over relatively compact open subset of Riemannian manifolds.

Proof. We show that u is a weak solution of (1), and then it follows from standard regularity theory, that it is also a strong solution. This means that we have to show that for any given compactly supported smooth test function ψ : Ω → R we have

Ω uP * ψ = Ω f ψ.
Let η : Ω → [0, 1] be a test function that is identically 1 on B 0 (ε), has support in B 0 (2ε), and with

|∇ m η| ≤ C m /ε m . It follows that sup |P * (ηψ)| ≤ C(P, ψ)ε -k , on B 0 (2ε) \ B 0 (ε) and sup |P * (ηψ)| ≤ C(P, ψ) on B 0 (ε) and hence Ω uP * (ηψ) ≤ Cε -k B0(2ε)\B0(ε) |u| + C B0(ε) |u| ≤ C B0(2ε)\B0(ε) |u|r -k + C B0(ε) |u| → 0. ( 3
)
We conclude

Ω uP * ψ = Ω uP * (ηψ) + Ω uP * ((1 -η)ψ) = Ω uP * (ηψ) →0 + Ω (P u)(1 -η)ψ → Ω f ψ (4)
for ε → 0. Hence the lemma follows.

Condition (2) is obviously satisfied if Ω |u|r -k < ∞. It is also satisfied if Ω |u| 2 r -k < ∞ and k ≤ n, (5)
as in this case

B0(2ε)\B0(ε) |u|r -k 2 ≤ Ω |u| 2 r -k B0(2ε)\B0(ε) r -k ≤C .
2.3. Conformally covariant elliptic operators. In this subsection we present a class of certain conformally covariant elliptic operators. Many important geometric operators are in this class, in particular the conformal Laplacian, the Paneitz operator, the Dirac operator, see also [START_REF] Fegan | Conformally invariant first order differential operators[END_REF][START_REF]Second order conformal covariants[END_REF][START_REF] Gover | Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus[END_REF] for more examples.

Such an operator is not just one single differential operator, but a procedure how to associate to an n-dimensional Riemannian manifold (M, g) (potentially with some additional structure) a differential operator P g of order k acting on a vector bundle. The important fact is that if g 2 = f 2 g 1 , then one claims

P g2 = f -n+k 2 P g1 f n-k 2 .
(6) One also expresses this by saying that P has bidegree ((n -k)/2, (n + k)/2).

The sense of this equation is apparent if P g is an operator from C ∞ (M ) to C ∞ (M ). If P g acts on a vector bundle or if some additional structure (as e.g. spin structure) is used for defining it, then a rigorous and careful definition needs more attention. The language of categories provides a good formal framework [START_REF] Lane | Categories for the working mathematician[END_REF]. The concept of conformally covariant elliptic operators is already used by many authors, but we do not know of a reference where a formal definition is carried out that fits to our context. (See [START_REF] Kolář | Natural operations in differential geometry[END_REF] for a similar categorial approach that includes some of the operators presented here.) Often an intuitive definition is used. The intuitive definition is obviously sufficient if one deals with operators acting on functions, such as the conformal Laplacian or the Paneitz operator. However to properly state Theorem 1.3 we need the following definition.

Let Riem n (resp. Riemspin n ) be the category n-dimensional Riemannian manifolds (resp. n-dimensional Riemannian manifolds with orientation and spin structure). Morphisms from (M 1 , g 1 ) to (M 2 , g 2 ) are conformal embeddings (M 1 , g 1 ) ֒→ (M 2 , g 2 ) (resp. conformal embeddings preserving orientation and spin structure).

Let Laplace n k (resp. Dirac n k ) be the category whose objects are {(M, g), V g , P g } where (M, g) in an object of Riem n (resp. Riemspin n ), where V g is a vector bundle with a scalar product on the fibers, where

P g : Γ(V g ) → Γ(V g ) is an elliptic formally selfadjoint differential operator of order k. A morphism (ι, κ) from {(M 1 , g 1 ), V g1 , P g1 } to {(M 2 , g 2 ), V g2 , P g2 } consists of a conformal embedding ι : (M 1 , g 1 ) ֒→ (M 2 , g 2 ) (
preserving orientation and spin structure in the case of Dirac n k ) together with a fiber isomorphism κ : ι * V g2 → V g1 perserving fiberwise length, such that P g1 and P g2 satisfy the conformal covariance property [START_REF] Ammann | The first conformal Dirac eigenvalue on 2-dimensional tori[END_REF]. For stating this property precisely, let f > 0 be defined by ι * g 2 = f 2 g 1 , and let

κ * : Γ(V g2 ) → Γ(V g1 ), κ * (ϕ) = κ • ϕ • ι.
Then the conformal covariance property is

κ * P g2 = f -n+k 2 P g1 f n-k 2 κ * . (7)
In the following the maps κ and ι will often be evident from the context and then will be omitted. The transformation formula [START_REF] Ammann | Mass endomorphism and spinorial Yamabe type problems[END_REF] then simplifies to (6).

Definition 2.2. A conformally covariant elliptic operator of order k and of bidegree ((n-k)/2, (n+k)/2) is a contravariant functor from Riem n (resp. Riemspin n ) to Laplace n k (resp. Dirac n k ), mapping (M, g) to (M, g, V g , P g ) in such a way that the coefficients are continuous in the C k -topology of metrics (see below). To shorten notation, we just write P g or P for this functor.

It remains to explain the C k -continuity of the coefficients.

For Riemannian metrics g, g 1 , g 2 defined on a compact set K ⊂ M we set

d g C k (K) (g 1 , g 2 ) := max t=0,...,k (∇ g ) t (g 1 -g 2 ) C 0 (K) .
For a fixed background metric g, the relation

d g C k (K) ( • , •
) defines a distance function on the space of metrics on K. The topology induced by d g is independent of this background metric and it is called the C k -topology of metrics on K. Definition 2.3. We say that the coefficients of P are continuous in the C k -topology of metrics if for any metric g on a manifold M , and for any compact subset K ⊂ M there is a neighborhood U of g| K in the C k -topology of metrics on K, such that for all metrics g, g| K ∈ U, there is an isomorphism of vector bundles κ :

V g | K → V g | K over the identity of K with induced map κ * : Γ(V g | K ) → Γ(V g | K )
with the property that the coefficients of the differential operator

P g -(κ * ) -1 P g κ *
depend continuously on g (with repsect to the C k -topology of metrics).

2.4. Invertibility on S n-1 × R. Let P be a conformally covariant elliptic operator of order k and of bidegree (

(n -k)/2, (n + k)/2). For (M, g) = S n-1 × R, the operator P g is a self-adjoint operator H k ⊂ L 2 → L 2 (see Lemma 3.

and the comments thereafter).

Definition 2.4. We say that P is invertible on S n-1 × R if P g is an invertible operator H k → L 2 where g is the standard product metric on S n-1 × R. In order words there is a constant σ > 0 such that the spectrum of

P g : Γ H k (V g ) → Γ L 2 (V g ) is contained in (-∞, -σ] ∪ [σ, ∞) for any g ∈ U .
In the following, the largest such σ will be called σ P .

We conjecture that any conformally covariant elliptic operator of order k and of bidegree ((n -k)/2, (n + k)/2) with k < n is invertible on S n-1 × R.

Examples. Example 1: The Conformal Laplacian Let

L g := ∆ g + n -2 4(n -1)
Scal g , be the conformal Laplacian. It acts on functions on a Riemannian manifold (M, g), i.e. V g is the trivial real line bundle R. Let ι : (M 1 , g 1 ) ֒→ (M 2 , g 2 ) be a conformal embedding. Then we can choose κ := Id : ι * V g2 → V g1 and formula [START_REF] Ammann | Mass endomorphism and spinorial Yamabe type problems[END_REF] holds for k = 2 (see e.g.[13, Section 1.J]). All coefficients of L g depend continuously on g in the C 2 -topology. Hence L is a conformally covariant elliptic operator of order 2 and of bidegree ((n -2)/2, (n + 2)/2).

The scalar curvature of

S n-1 × R is (n -1)(n -2). Hence the spectrum of L g on S n-1 × R of L g coincides with the essential spectrum of L g and is [σ L , ∞) with σ L := (n -2) 2 /4. Hence L is invertible on S n-1 × R if (and only if) n > 2.
Example 2: The Paneitz operator Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 5. The Paneitz operator P g is given by

P g u = (∆ g ) 2 u -div g (A g du) + n -4 2 Q g u
where

A g := (n -2) 2 + 4 2(n -1)(n -2) Scal g g - 4 n -2 Ric g , Q g = 1 2(n -1) ∆ g Scal g + n 3 -4n 2 + 16n -16 8(n -1) 2 (n -2) 2 Scal 2 g - 2 (n -2) 2 |Ric g | 2 .
This operator was defined by Paneitz in the case n = 4, and it was generalized by Branson in [START_REF]Group representations arising from Lorentz conformal geometry[END_REF] to arbitrary dimensions ≥ 4. We also refer to Theorem 1.21 of the overview article [START_REF] Branson | Differential operators canonically associated to a conformal structure[END_REF]. The explicit formula presented above can be found e.g. in [START_REF] Hebey | Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients[END_REF]. The coefficients of P g depend continuous on g in the C 4 -topology As in the previous example we can choose for κ the identity, and then the Paneitz operator P g is a conformally covariant elliptic operator of order 4 and of bidegree ((n -4)/2, (n + 4)/2).

On S n-1 × R one calculates

A g := (n -4)n 2 Id + 4π R > 0
where π R is projection to vectors parallel to R.

Q g := (n -4)n 2 8 .
We conclude

σ P = Q = (n -4)n 2 8 and P is invertible on S n-1 × R if (and only if) n > 4.
Examples 3: The Dirac operator.

Let g = f 2 g. Let Σ g M resp. Σ gM be the spinor bundle of (M, g) resp. (M, g). Then there is a fiberwise isomorphism β g g : Σ g M → Σ g M , preserving the norm such that

D g • β g g (ϕ) = f -n+1 2 β g g • D g f n-1 2 ϕ ,
see [START_REF] Hitchin | Harmonic spinors[END_REF][START_REF] Baum | Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten[END_REF][START_REF] Hijazi | Spectral properties of the Dirac operator and geometrical structures[END_REF] for details. Furthermore, the cocycle conditions

β g g • β g g = Id and β ĝ g • β g ĝ • β g g = Id
hold for conformal metrics g, g and ĝ. We will hence use the map β g g to identify Σ g M with Σ gM . Hence we simply get

D g ϕ = f -n+1 2 • D g f n-1 2 ϕ . (8)
All coefficients of D g depend continuously on g in the C 1 -topology. Hence D is a conformally covariant elliptic operator of order 1 and of bidegree ((n -1)/2, (n + 1)/2).

The Dirac operator on S n-1 × R can be decomposed as D vert + D hor , where the first part is the sum over the derivations (and Clifford multiplication) along S n-1 and where D hor = ∂ t • ∇ ∂t , where ∂ t • is Clifford multiplication with ∂ t , t ∈ R. D vert and D hor anticommute. The spectrum of D vert is just the spectrum of the Dirac operator on S n-1 , and hence we see with [START_REF] Bär | The Dirac operator on space forms of positive curvature[END_REF] specD

vert = {± n -1 2 + k | k ∈ N 0 }.
The operator (D hor ) 2 is the ordinary Laplacian on R and hence has spectrum [0, ∞). Together this implies that the spectrum of the Dirac operator on

S n-1 × R is (-∞, -σ D ] ∪ [σ D , ∞) with σ D = n-1 2 . Hence D is invertible on S n-1 × R if (and only if) n > 1.
In the case n = 2 these statements are only correct if the circle S n-1 = S 1 carries the spin structure induced from the ball. In our article all circles S 1 carry this bounding spin structure due to the geometry of the asymptotically cylindrical blowups.

Example 4: The Rarita-Schwinger operator and many other Fegan type operators are conformally covariant elliptic operators of order 1 and of bidegree ((n -1)/2, (n + 1)/2). See [START_REF] Fegan | Conformally invariant first order differential operators[END_REF] and in the work of T. Branson for more information.

Example 5: Assume that (M, g) is a Riemannian spin manifold that carries a vector bundle W → M with metric and metric connection. Then there is a natural first order operator Γ(ΣM ⊗ W ) → Γ(ΣM ⊗ W ), the Dirac operator twisted by W . This operator has similar properties as conformally covariant elliptic operators of order 1 and of bidegree ((n-1)/2, (n+1)/2). The methods of our article can be easily adapted in order to show that Theorem 1.3 is also true for this twisted Dirac operator. However, twisted Dirac operators are not "conformally covariant elliptic operators" in the above sense. They could have been included in this class by replacing the category Riemspin n by a category of Riemannian spin manifolds with twisting bundles. In order not to overload the formalism we chose not to present these larger categories.

The same discussion applies to the spin c -Dirac operator of a spin c -manifold.

Asymptotically cylindrical blowups

3.1. Convention. From now on we suppose that P g is a conformally covariant elliptic operator of order k, of bidegree ((n -k)/2, (n + k)/2), acting on manifolds of dimension n and invertible on S n-1 × R.

3.2.

Definition of the metrics. Let g 0 be a Riemannian metric on a compact manifold M . We can suppose that the injectivity radius in a fixed point y ∈ M is larger than 1. The geodesic distance from y to x is denoted by d(x, y).

We choose a smooth function

F ∞ : M \ {y} → [1, ∞) such such that F ∞ (x) = 1 if d(x, y) ≥ 1, F ∞ (x) ≤ 2 if d(x, y) ≥ 1/2 and such that F ∞ (x) = d(x, y) -1 if d(x, y) ∈ (0, 1/2].
Then for L ≥ 1 we define F L to be a smooth positive function on M , depending only on d(x, y), such that

F L (x) = F ∞ (x) if d(x, y) ≥ e -L and F L (x) ≤ d(x, y) -1 = F ∞ (x) if d(x, y) ≤ e -L . For any L ≥ 1 or L = ∞ set g L := F 2 L g 0 .
The metric g ∞ is a complete metric on M ∞ . The family of metrics (g L ) is called an asymptotically cylindrical blowup, in the literature it is denoted as a family of Pinocchio metrics [START_REF] Ammann | Dirac eigenvalues and total scalar curvature[END_REF], see also Figure 1.

3.3.

Eigenvalues and basic properties on (M, g L ). For the P -operator associated to (M, g L ), L ∈ {0} ∪ [1, ∞) (or more exactly its selfadjoint extension) we simply write P L instead of P gL . As M is compact the spectrum of P L is discrete.

We will denote the spectrum of P L in the following way . . . ≤ λ - 1 (P gL ) < 0 = 0 . . . = 0 < λ + 1 (P gL ) ≤ λ + 2 (P gL ) ≤ . . . , where each eigenvalue appears with the multiplicity of the multiplicity of the eigenspace. The zeros might appear on this list or not, depending on whether P gL is invertible or not. The spectrum might be entirely positive (for example the conformal Laplacian Y g on the sphere) in which case λ - 1 (P gL ) is not defined. Similarly, λ + 1 (P gL ) is not defined if the spectrum of (P gL ) is negative.

3.4. The asymptotic analysis of (M ∞ , g ∞ ). The asymptotic analysis of non-compact manifolds as (M ∞ , g ∞ ) is more complicated than in the compact case. Nevertheless (M ∞ , g ∞ ) is an asymptotically cylindrical manifolf for there exists nowadays an extensive literature. We will need only very few of these properties that will be summarized in this subsection. Proofs will only be sketched.

Different approaches can be used for the proof. The following lemma shows that (M ∞ , g ∞ ) carries a b-structure in the sense of Melrose. "Manifolds with b-structures" form a subclass of "Manifolds with a Lie structure at infinity", also called "Lie manifolds" [START_REF] Ammann | On the geometry of Riemannian manifolds with a Lie structure at infinity[END_REF], [START_REF]Pseudodifferential operators on manifolds with a Lie structure at infinity[END_REF], [START_REF] Ammann | Sobolev spaces on Lie manifolds and regularity for polyhedral domains[END_REF]. We choose to use Melrose's b-calculus [START_REF] Melrose | The Atiyah-Patodi-Singer index theorem[END_REF] in this article as this calculus is more widely known, but similar statements hold in the larger category of "Manifolds with a Lie structure at infinity". Proof. Let S y M be the unit tangent bundle at y. For any X ∈ S y M let γ X be the geodesic with γ(0) = X.

Then for small ε > 0 the map Φ :

S y M × [0, ε) → M, (X, t) → γ X (t) is smooth and a diffeomorphism from S y M × (0, ε) to B y (ε) \ {0}. We define M := M ∞ ∪ : S y M × [0, ε)/ ∼ where ∼ indicates that (X, t) ∈ S y M × (0, ε) is glued together with Φ(X, t).
M is a manifold with boundary S y M and interior M ∞ . Expressing the metric in normal coordinates, one sees that d(x, y) 2 g extends to an exact b-metric on M .

From this observation may properties already follow with standard arguments similarly as in the compact case. The operator P g∞ has a self-adjoint extension, denoted by P ∞ . The essential spectrum of P ∞ coincides with the essential spectrum of the P -operator on the standard cylinder S n-1 × R which is contained in (-∞, -σ P ] ∪ [σ P , ∞). Hence the spectrum of P ∞ in the interval (-σ P , σ P ) is discrete as well. Eigenvalues of P ∞ in this interval will be called small eigenvalues of P ∞ . Similarly we use the notation λ ± j (P ∞ ) for the small eigenvalues of P ∞ . Proposition 3.2. Let P be a conformally covariant elliptic operator. Then on (M ∞ , g ∞ ) we have

(∇ ∞ ) s u L 2 (g∞) ≤ C( u L 2 (g∞) + P ∞ u L 2 (g∞) )
for all s ∈ {0, 1, . . . , k}.

Proof. Choose a λ ∈ R that is not in the spectrum of P . The continuity of the coefficients of P together with the fact that P g0 extends to M implies that P g∞ is an operator compatible with the b-structure. Hence we we can apply [START_REF] Melrose | The Atiyah-Patodi-Singer index theorem[END_REF]Proposition] for Q := P -λ. We see that P -λ is an isomorphism from H k to L 2 . Hence a constant C > 0 exists with

C (P -λ)u L 2 ≥ u H k .
using the triangle inequality we get

u H k ≤ Cλ u L 2 (g∞) + P ∞ u L 2 (g∞)
which is equivalent to the statement.

3.5. The kernel. Having recalled these previously known facts we will now study the kernel of the conformally covariant operators.

If g and g = f 2 are conformal metrics on a compact manifold M , then

ϕ → f - n -k 2 ϕ
obviously defines an isomorphism from ker P g to ker P g . It is less obvious that a similar statement holds if we compare g 0 and g ∞ defined before:

Proposition 3.3. The map

ker P 0 → ker P ∞ ϕ 0 → ϕ ∞ = F -n-k 2 ∞ ϕ 0
is an isomorphism of vector spaces.

Proof. Suppose ϕ 0 ∈ ker P 0 . Using standard regularity results it is clear that sup

|ϕ 0 | < ∞. Then M∞ |ϕ ∞ | 2 dv g∞ ≤ M\By (1/2) |ϕ ∞ | 2 dv g∞ + sup |ϕ 0 | 2 By(1/2) F -(n-k) ∞ dv g∞ ≤ 2 k M\By (1/2) |ϕ 0 | 2 dv g0 + sup |ϕ 0 | 2 ω n-1 1/2 0 r n-1 r k dr < ∞. (9)
Furthermore, formula (6) implies P ∞ ϕ ∞ = 0. Hence the map is well-defined. In order to show that it is an isomorphism we show that the obvious inverse ϕ ∞ → ϕ

0 := F n-k 2
∞ ϕ ∞ is well defined. To see this we start with an L 2 -section in the kernel of P ∞ .

We calculate

M F k ∞ |ϕ 0 | 2 dv g0 = M∞ |ϕ ∞ | 2 dv g∞ .
Using again [START_REF] Ammann | The first conformal Dirac eigenvalue on 2-dimensional tori[END_REF] we see that this section satisfies P 0 ϕ 0 on M \ {y}. Hence condition ( 5) is satisfied, and together with the removal of singularity lemma (Lemma 2.1) one obtains that the inverse map is well-defined. The proposition follows.

4. Proof of the main theorem 4.1. Stronger version of the main theorem. We will now show the following theorem. Theorem 4.1. Let P be a conformally covariant elliptic operator of order k, of bidegree ((n -k)/2, (n + k)/2), on manifolds of dimension n > k. We assume that P is invertible on S n-1 × R.

If lim inf L→∞ |λ ± j (P L )| < σ P , then λ ± j (P L ) → λ ± j (P ∞ ) ∈ (-σ P , σ P ) for L → ∞.
In the case Spec(P g0 ) ⊂ (0, ∞) the theorem only makes a statement about λ + j , and conversely in the case that Spec(P g0 ) ⊂ (-∞, 0) it only makes a statement about λ - j . Obviously this theorem implies Theorem 1.3.

(λ + j (P L )) ≤ λ + j (P ∞ ). (10)
Let ϕ 1 , . . . , ϕ j be sequence of L 2 -orthonormal eigenvectors of P ∞ to eigenvalues λ + 1 (P ∞ ), . . . , λ + j (P ∞ ) ∈ [-λ, λ], λ < σ P . We choose a cut-off function χ : M → [0, 1] with χ(x) = 1 for -log(d(x, y)) ≤ T , χ(y) = 0 for -log(d(x, y)) ≥ 2T , and |(∇ ∞ ) s χ| g∞ ≤ C s /T s for all s ∈ {0, . . . , k}.

Let ϕ be a linear combination of the eigenvectors ϕ 1 , . . . , ϕ j . From Proposition 3.2 we see that

(∇ ∞ ) s ϕ L 2 (M∞,g∞) ≤ C ϕ L 2 (M∞,g∞)
where C only depends on (M ∞ , g ∞ ). Hence for sufficiently large T P ∞ (χϕ) -χP ∞ ϕ L 2 (M∞,g∞) ≤ kC/T ϕ L 2 (M∞,g∞) ≤ 2kC/T χϕ L 2 (M∞,g∞)

for sufficiently large T as χϕ L 2 (M∞,g∞) → ϕ L 2 (M∞,g∞) for T → ∞. The section χϕ can be interpreted as a section on (M, g L ) if L > 2T , and on the support of χϕ we have g L = g ∞ and P ∞ (χϕ) = P L (χϕ). Hence standard Rayleigh quotient arguments imply that if P ∞ has m eigenvalues (counted with mulitplicity) in the intervall [a, b] then P L has m eigenvalues in the intervall [a -2kC/T, b + 2kC/T ]. Taking the limit T → ∞ we obtain [START_REF]Pseudodifferential operators on manifolds with a Lie structure at infinity[END_REF].

By exchanging some obvious signs we obtain similarly lim sup L→∞ (-λ - j (P L )) ≤ -λ - j (P ∞ ).

We assume that we have a sequence L i → ∞, and that for each i we have a system of orthogonal eigenvectors ϕ i,1 , . . . , ϕ i,m of P Li , i.e. P Li ϕ i,ℓ = λ i,ℓ ϕ i,ℓ for ℓ ∈ {1, . . . , m}. Furthermore we suppose that λ i,ℓ → λℓ ∈ (-σ P , σ P ) for ℓ ∈ {1, . . . , m}.

Then

ψ i,ℓ := F Li F ∞ n-k 2 ϕ i,ℓ satisfies P ∞ ψ i,ℓ = h i,ℓ ψ i,ℓ with h i,ℓ := F Li F ∞ k λ i,ℓ .
Furthermore

ψ i,ℓ 2
L 2 (M∞,g∞) = M F Li F ∞ -k |ϕ i,ℓ | 2 dv gL i ≤ sup M |ϕ i,ℓ | 2 M F Li F ∞ -k dv gL i Because of M FL i F∞ -k
dv gL ≤ C r n-1-k dr < ∞ (for n > k) the norm ψ i,ℓ L 2 (M∞,g∞) is finite as well, and we can renormalize such that ψ i,ℓ L 2 (M∞,g∞) = 1. Lemma 4.2. For any δ > 0 and any ℓ ∈ {0, . . . , m} the sequence ψ i,ℓ C k+1 (M\By (δ),g∞) i is bounded.

Proof of the lemma. The sections ϕ i,1 , . . . , ϕ i,ℓ are orthogonal. For any fixed δ 0 (given by the previous lemma), it follows for sufficiently large i that

for l = ℓ, and hence in the limit ε → 0 (and δ 0 → 0) we get the orthogonality of ψ1 , . . . , ψm .

1

 1

Figure 1 .

 1 Figure 1. Asymptotically cylindrical metrics g L (alias Pinocchio metrics) with growing nose length L.

Lemma 3 . 1 .

 31 The manifold (M ∞ , g ∞ ) is an exact b-metric in the sense of[START_REF] Melrose | The Atiyah-Patodi-Singer index theorem[END_REF] Def. 2.8].

4. 2 .

 2 The supremum part of the proof of Theorem 4.1. At first we prove that lim sup L→∞

(11) 4 . 3 .

 43 The infimum part of the proof of Theorem 4.1. We now prove lim inf L→∞ (±λ ± j (P L )) ≥ ±λ ± j (P ∞ ).

2 ( 15 )

 215 M\By(δ0) ψ i,ℓ , ψ i, l dv g∞ = M\By(δ0) ϕ i,ℓ , ϕ i, l dv gL i = By(δ0) ϕ i,ℓ , ϕ i, l dv gL i ℓ , ψ i, l dv g∞ ≤ ε Because of strong L 2 convergence on M \ B y (δ 0 ) this implies M\By (δ0)ψℓ , ψl dv g∞ ≤ ε 2

Acknowledgements We thank B. Colbois, M. Dahl, E. Humbert and O. Hijazi for many related discussions. We thank R. Gover for some helpful comments on conformally covariant operators, and for several references. The first author wants to thank cordially the Einstein institute at Potsdam-Golm for its hospitality which enabled to write the article.

Proof of the lemma. After removing finitely many i, we can assume that λ i ≤ 2 λ and e -Li < δ/2. Hence F L = F ∞ and h i = λ i on M \ B y (δ/2). Because of

we obtain boundedness of ψ i in the Sobolev space H sk (M \ B y (3δ/4), g ∞ ), and hence, for sufficiently large s boudnedness in C k+1 (M \ B y (δ), g ∞ ). The lemma is proved.

Hence after passing to a subsequence ψ i,ℓ converges in C k,α (M \ B y (δ), g ∞ ) to a solution ψℓ of

By taking a diagonal sequence, one can obtain convergence in C k,α loc (M ∞ ) of ψ i,ℓ to ψℓ . It remains to prove that ψ1 ,. . . , ψm are linearly independent, in particular that any ψℓ = 0. For this we use the following lemma.

Lemma 4.3. For any ε > 0 there is δ 0 and i 0 such that

for all i ≥ i 0 and all ℓ ∈ {0, . . . , m}. In particular,

Proof of the lemma. Because of Proposition 3.2 and

≤ C for all s ∈ {0, . . . , k}. Let χ be a cut-off dunction as in Subsection 4.2 with T = -log δ. Hence

On the other hand (B y (δ) \ {y}, g ∞ ) converges for δ → 0 to S n-1 × (0, ∞) in the C ∞ -topology. Hence there is a function τ (δ) converging to 0 such that

Using the obvious relation

we obtain with ( 13) and ( 14)

The right hand side is smaller than ε for i sufficiently large and δ sufficiently small. The main statement of the lemma then follows for δ 0 := δ 2 . The Minkowski inequality yields.


Antoine Printz

The issue of inclusion in the new EU public policy framework

Over the past few years, we have gone through a major evolution in our political view of society and how it is organised. The social question is now seen through the prism of the inclusion/exclusion dichotomy. As we are confronted with new phenomena whereby isolated individuals are relegated and isolated, the concept of exclusion is being brought to the foreground and gradually taking over attempts to provide a sociological and political description of social realities. Starting in the 1980s, scholars have been calling for a new ontology of social problems. The division of society into classes is no more, and it has been replaced by a patchwork of individual positions, affiliations to various groups, and economic, social, professional categories. The social question can therefore no longer be understood in terms of class exploitation, but should rather be considered in terms of social exclusion, a pathological process that desocializes individuals in economic, civic, cultural and spatial terms 1 . Exclusion is seen as one facet of a more nuanced view of how to define an individual's place in society, beyond economic reductionism, which can contribute to developing a new policy agenda.

Over the course of twenty years, the fight against exclusionor against exclusions, as the plural is now meaningfulhas become a central part of public policies. Taking exclusion into account, results in a new approach of social risks, based on individual citizenhood and dignity 2 .

The requalificationwhether actual or perceived -of social risks, which are becoming 'life risks' as a result of their increasingly individual nature 3 , combined with the lower emphasis placed on exploitation in the public discourse, naturally results in the adoption of a new perspective in which the inability to create an integrated society stems from a 'subjective' failure of solidarity processes 4 .

Inclusion is defined in contrast to this concept of exclusion, as its pure semantic opposite. However, the concept of inclusion does not have an agreed-upon definition, with many scholars pointing out inconsistencies or vagueness in how the term is defined [START_REF] Bauer | Inclusion et planification : vers un territoire inclusif[END_REF] . So, what is the contribution of this perspective to actual policy-making? What are the socio-economic implications of this shift in the public policy framework? Works in cognitive sociology on public policy have shown how adopting a new framework as a strong reaction to putative social conditions offered specific cognitive and normative resources for policy-making [START_REF] Cefaï | La construction des problèmes publics. Définitions de situations dans des arènes publiques[END_REF] . How can the introduction of a conceptual dichotomy between inclusion and exclusion provide a framework for the interpretation of society? How does this framework restrict and guide policy-making? In this essay, we will take a brief look at how social action can be thematised through the prism of inclusion at the EU level, in order to identify symbolic and concrete frameworks that determine the form, content and implementation of social policies.

Inclusion according to the European Commission

The first place where inclusion is thematised at an institutional level is the EU, which has a structuring influence as one of the main sources of funding for inclusion policies. With an increasing integration at the EU level, characterised by an ideological convergence and concrete limitations [START_REF] Surel | L'intégration européenne vue par l'approche cognitive et normative des politiques publiques[END_REF] , we tend to consider this level as an essential one in the cognitive structuring of public policies even at a local scale, which choseor had to choose?the inclusion framework. The term's first appearance in EU texts was in the Lisbon strategy 8 , in 2000, and the topic has always been approached from an economic point of view. This first step was the beginning of a EU process intended to coordinate initiatives against poverty and exclusion, and the introduction into the language of EU social policy of a concept that would then become increasingly important [START_REF] Jaeger | L'inclusion : un changement de finalité pour le travail social ?[END_REF] . In 2010, the Commission establishes the term in its general work programme, defining the EU's post-crisis strategy for the following decade: economic growth must be green, smart, and inclusive [START_REF]Communication from the commission. Europe 2020: a strategy for smart, sustainable and inclusive growth (COM[END_REF] . Social inclusion is integrated into the policy agenda of the EU and, by extension, of each member state. Still, definitions of the term are rarely provided. One of the few extensive definitions, outside of indicator descriptions, can be found in COM (2003) 773: Social inclusion is a process which ensures that those at risk of poverty and social exclusion gain the opportunities and resources necessary to participate fully in economic, social and cultural life and to enjoy a standard of living and well-being that is considered normal in the society in which they live [START_REF]Rapport conjoint sur l'inclusion sociale[END_REF] .

Economicism and individualism

As these policies attempted to focus on social exclusion in order to develop a multidimensional and complex perspective of the processes involved in desocialization, it appears though that they have been unable to avoid being too reductive. The development of indicators is a good proof of this trend towards simplification: inclusion is essentially defined in terms of contribution to productive processes and of consumption capacity [START_REF] Atkinson | Indicators and targets for social inclusion in the European union[END_REF] .

Inclusion is defined as a process through which people overcome exclusion, and the indicator used to measure it is the rate of people at risk of poverty or social exclusion. This indicator is based on a combination of three sub-indicators, all of which are strongly linked to the economic aspects of social life. The first sub-indicator is the risk of poverty, with the poverty line defined as 60% of a country's median income. The second measures the percentage of households with low work intensity, i.e. where fewer than 20% of working-age household members have worked during the year. Finally, the third sub-indicator measures material deprivation and is based on nine items: a situation of severe material deprivation occurs when people have access to fewer than six of these items 13 . While the indicators used are not just economic in nature, they remain tied to material aspects of life and, as such, cannot be used to measure cultural participationexcept by measuring who owns a television set -, social participationexcept by measuring who has access to a telephoneor civic participation except by measuring employment.

The way in which these indicators are designed strongly implies that a specific lifestyle is being promoted. Thus, there is a risk that policies intended to fight exclusion might have an unintended yet central normalising component. Inclusion simply means following this 'normal' lifestyle, which is essentially focused on consumption. Those who are seen as excluded, and who therefore should be included, are those who deviate from this standard where consumption and a focus on material goods are the standard 14 . In this sense, it is worth noting that the issue of social exclusion could be solved -by the Commission's definition, that is, and according to the goal of reducing the number of people in poverty or social exclusion by 20 millionsimply by providing a few million households with televisions or washing machines. This caricature is not meant as a genuine argument, but it does highlight the deeply restrictive nature of the EU's perspective on social exclusion and, therefore, inclusion.

It should be noted, however, that alongside this main indicator, the Commission has added a limited series of indicators related to education. In the more comprehensive list of thirteen inclusion indicators, three are related to illiteracy, school leaving, and poor educational performance. While these are not directly tied to economic participation, a relationship still exists: the ability to read is not seen as an obstacle to citizenhood as it is a major obstacle to being a productive worker. Again, the end goal is the same: what matters is inclusion in the economic sphere, based on production and consumption, which takes over the entire social question. As a result, most policies intended to reduce social exclusion are approached through the angle of job creation, which is especially visible in strategic documents published by the EU 15 . In this perspective, the fight against exclusion and poverty is always reduced to productive aspects [START_REF] Lebrun | Pour la Solidarité. 17 European Commission, Social investment[END_REF] . In theory, of course, the concept of inclusion covers more than just an economic perspectiverelevant texts also refer to cultural and social aspects -, but an analysis of the issue reveals the central role of economic participation in how inclusion is thematised at the EU level.

The emphasis placed on the concept of social investment confirms this tendency, and demonstrates the EU policies' focus on individual abilities. The Commission defines social investment as a series of measures seeking to 'strengthen people's current and future capacities, and improve their opportunities to participate in society and the labour market 17 '. Upon closer scrutiny, it seems that the term actually covers all operations aimed at empowering and enabling individuals so that they can join the productive sphere, with consequences on policies: '[s]ocial investment helps people to adapt to societal challenges [START_REF]Social investment: Commission urges Member States to focus on growth and social cohesionfrequently asked questions[END_REF] '. By looking at the European Social Fund (ESF), for instance, which is the EU's first structural fund and the one that is closest to social inclusion policies, we realise that two types of policy are considered: one provides direct assistance to people, and the other targets systems and structures [START_REF] Di Nardo | Le fonds social européen et l'inclusion sociale[END_REF] . A closer analysis of the details of the ESF's significant investments reveals that most policies deal with helping individuals in order to enable them and improve the employability of excluded people. Measures supported by the ESF, which are intended as responses to the specific needs of excluded people, consist in little more than coaching, training, or personal growth activities, always with an emphasis on entering the labour market, which is seen as the main vector for people's inclusion.

What does this mean for cities?

In 2016, under the Dutch presidency, during an informal meeting of EU ministers in charge of urban issues, the European Council made a commitment to adjust the cross-cutting objectives of the Europe 2020 strategy to urban policies. This adjustment was requested by the European Parliament, as this process is essential [START_REF] Van Lierop | EU Urban Agenda -State of play (PE 577.986[END_REF] . The meeting resulted in the 'Pact of Amsterdam [START_REF]Netherlands Presidency of the Council of the European Union[END_REF] ', providing guidelines for the EU's urban agenda. This document reaffirms the priorities defined in the European strategy, applying the three key words 'green, smart, inclusive' to urban policies. Based on a proposal by the European Parliament, who intends to make urban policy one of its central tools, a European urban agenda must be perfectly aligned with the EU's overall strategy and objectives, and in particular with the Europe 2020 strategy [START_REF] Westphal | Draft report on the urban dimension of EU policies[END_REF] .

In this context, once again, social inclusion is primarily considered from an economic perspective, the goal being to allow people living in poverty or exclusion to live with dignity and play an active role in society: urban development policies often use workers as a point of reference, rather than citizens or simply residents. Kerstin Westphal, explains the need for adequate urban equipment, in a rather striking way: 'lack of appropriate infrastructure can cause psychological pressure and stress on workers [START_REF] Westphal | [END_REF] '. So is urban planning mostly intended for workers? In any case, the EU's urban policy agenda does not look beyond an economic perspective.

The ERDF's interface: a territorialised European policy

The urban dimension of the EU's social inclusion policies will be implemented by several tools, including the European Regional Development Fund (ERDF). The ERDF provides funding for projects that contribute to the development of a territorial policy for economic, social and environmental cohesion. The European Regional Development Fund (ERDF) is the European Union's financial lever for successfully achieving its cohesion and regional development policy. In other words, the ERDF aims to reduce economic, social and territorial disparities between the 28 Member States of the European Union by co-financing projects designed to promote smart, sustainable and inclusive growth as defined in the Europe 2020 strategy [START_REF] Dussart | Les projets financés par le FEDER en Région de Bruxelles-Capitale. Programmation 2007-2013 et perspectives[END_REF] .

Along with the ESF and the EAFRD 25 , the ERDF is the third financial resource for inclusion policies. This institutional position is not insignificant, as the requirements for funding results in most ERDF funds going to projects that focus on topics that are not tackled by the ESF, which is the main fund used for inclusion policies. This means the ERDF relies on a perhaps broader thematisation of the concepts of inclusion and exclusion, moving beyond the reductionist view of exclusion as poverty as adopted by the ESF: political and cultural aspects are therefore more the remit of the ERDF.

For the second programming period, covering the 2014-2020 period, the Brussels-Capital region received 200 million euro for a call for proposals involving specific policy orientations, which are described and developed in the ERDF's operational programme (OP) for the Brussels-Capital Region [START_REF]Programme Opérationnel (PO) de la programmation FEDER[END_REF] . The terms of the funding involved a delegation of public intervention to the associative, parastatal, and private sectors; in this context, the authorities' role is limited to funding, i.e. selecting projects and assessing them once they have been implemented. Forty-six projects were selected based on the criteria of 'reinforcement of the region's economic, social and territorial cohesion 27 ', building on the EU's cohesion policy, the Europe 2020 strategy. The projects were divided into four categories: Social inclusion falls into the latter category, with eleven projects selected in the Brussels-Capital Region seeking to include people who find themselves excluded [START_REF]The eleven projects are: Altaïr nursery, Marchandise nursery, Gosselies nursery, Schaerbeek CPAS nursery, Ulens nursery, Charbonnage nursery, Abbaye de Forest, Masui 4ever, De Vaartkapoen, Actie zkt Burger[END_REF] . This category of spending received 15% of the total funds allocated to the Brussels-Capital Region, and its overall purpose was to reduce social, economic and environmental inequalities by improving living conditions for disadvantaged neighbourhoods and populations 29 . The projects selected covered three kinds of concrete initiatives: child care, increased cultural activities in disadvantaged neighbourhoods, and increased participation of residents to planning projects in their neighbourhood. These initiatives consist in making infrastructures, equipments and services available so as to reinforce individual abilities, provide social support for empowerment. This can involve making resources available to individuals, e.g. child care facilitieswhich are seen as a way to eliminate factors preventing women from working -, or launching cultural projects with potential to produce a 'leverage effect 30 '. As these projects are influenced by the EU's idea of inclusion, economics permeates the various approaches of social intervention and there is a constant underlying link between this type of urban development and the economic dimension. This strong presence of economics is also present in policy-making, as (one of) the main driver(s) of inclusion policies.

However, another perspective of inclusion appears in the OP, coveringthough with a lower budgetincreased participation of residents to the urban initiatives and projects in their neighbourhood 31 . Despite the lower priority given to such measures, their mere presence is extremely significant, as it reveals the appearance of an alternative conception of social inclusion: it is not only a means to an end, and it takes into account principles that build upon a different idea of social issues, involving a collaborative dimension. Still, despite these encouraging principles, none of the projects selected were based on this idea of inclusion: this means the funding body's intention to promote collaborative initiatives was not followed.

What public policies in favour of inclusion?

We can offer three areas of reflection following out analysis: the quantitative and rational approach that emerge from this thematisation of inclusion; the reduction of social issues to mere economic terms and the disappearance of political considerations to the profit of pragmatic initiatives; and the development of a functional model of social inclusion.

Measuring inclusion with numbers

The approach of inclusion seems to necessarily be very quantitative: '[w]hen measuring social inclusion, studies tend to rely on objective measures [START_REF] Cobigo | Shifting our Conceptualization of Social Inclusion[END_REF] '. This is typical of the processes involved in developing indicators used to assess ERDF projects; the Fund has a very strong tendency to reduce factors to relatively superficial metrics. For instance, projects involving cultural improvement of neighbourhoods are assessed in the most quantifiable way possible, but also in a way that is very removed from the residents' actual daily experiences: simply by counting the number of additional cultural institutions installed in the areas covered by the project. A finer analysis might involve the surface in square meters of additional cultural spaces 33 . This is a striking illustration of current public policies, which are characterised by a quantitative abstraction that is all the more concerning that the perspective of exclusion/inclusion was intended to move beyond economics when analysing poverty, by integrating it into a broader experiential and qualitative view of social marginalisation. Obviously, it is difficult to assess results using factors that are not objectively measurable, but it is nevertheless surprising that policies that are meant to promote social life are evaluated with no regard for people's qualitative experiences.

As we can see, the view of inclusion demonstrates a holistic rationality. Social life is seen as a binary issue with each individual being either 'in' or 'out'. There is no room for medium-term approaches, or for semi-inclusion. This perspective is what leads to numbersbased measures and objectives. Additionally, mathematical rationality results in a technical approach where those who fulfil the criteria to be considered 'in' are full members of society. The kind of interventions developed based on this view simply seek to help people enter the spheres from which they are excluded: once this is achievedmeaning inclusion is a matter of access policy -, the people are included and a social goal has been reached. As a result, the only social policies that are promoted are purely technical ones, aiming to facilitate access, streamline mobility and limit obstacles.

Apoliticism and reduction

In terms of public policies, the opposite of technicity is politics; and the development of strictly technical interventions could end up obliterating any room for political orientations. Rather than political decisions, the approaches we have seen promote technical measures. Social belonging and participation are seen as problems in the mechanisms of society, which can be solved through local measures focused on specific problematic issues. Yet exclusion is a highly political topic, calling for more than a purely pragmatic response 34 . Realistic responses to inclusion problems only tackle the effects of exclusion. Once these are solved, the problem of social exclusion appears to be over. In the current fight against exclusion, we are witnessing the emergence of public policies that only deal with situations that have already deteriorated. Focusing on exclusion means resigning oneself to trying to repair tears in the social fabric without taking into account the factors that cause the tears [START_REF] Castel | La montée des incertitudes : travail, protections, statut de l'individu[END_REF] .

The objective defined by the Commission is that 'people experiencing poverty and social exclusion [should be] enabled to live in dignity and take an active part in society [START_REF]Communication from the commission. Europe 2020: a strategy for smart, sustainable and inclusive growth (COM[END_REF] '. This is a concerning approach, as it seems to consider the issue of social exclusion to be a result of the obstacles it creates. The problematic factor is the consequences of exclusion and poverty on social participation, which should be shared taking into account the unequal distribution of material, territorial, and symbolic resources, so that people who are experiencing poverty can play an active and dignified part in society instead of just no longer experiencing poverty.

According to the Commission's objectives, the dignity that poorer people should have access to can be reduced to a handful of consumption and leisure practices: getting 20 million people out of social exclusion is simply a matter of money, employment and access to consumer goods. Our goal here is not to diminish the considerable importance of measures intended to provide excluded people access to jobs and consumption. Still, we believe that this reductive view of exclusion fails to take into account a series of aspects, and that it prevents the implementation of a genuine poverty reduction policy. Officially, poor people can remain poor provided they are active and have dignity.

The functional model of inclusion

As we can see, inclusion policies at the EU level are built around a specific view of inclusion. The end of marginalisation is no longer sought based on a causal approach of the social experience, as was the case for instance in the providentialist philosophy, but is rather seen as a by-product of economic performance 37 . When the Commission is required to justify the cost of social investment policies in its communication, it mentions a number of benefits for society: 'higher productivity, higher employment, better health and social inclusion, more prosperity and a better life for all 38 '.

The ideal social experience refers to societal performance in an individualised and vertical view. This model of social inclusion calls upon a highly individual approach of social life, which is no longer just about interpersonal relations, but about the inclusion of each individual in certain social spheres. The only goal of empowering individuals is to help them integrate into a system that already functions based on rules, regardless of individual contributions. Society exists outside of the individuals that inhabit it, and who are simply included into society following an adaptative rather than a contributive approach [START_REF] Kihlstrom | Luhmann's system theory in social work: Criticism and reflections[END_REF] . They can only adjust to existing conditions, and have no potential for participation: there is no room for a horizontal approach of social issues that might offer a genuine alternative to the functional solitude of people 40 .

1 .

 1 Promoting research and innovation 2. Promoting entrepreneurship and creating SMEs in high-growth industry 3. Promoting circular economy and resource efficiency 4. Improving the living conditions of disadvantaged neighbourhoods and populations