Juan Pablo Borthagaray

Patrick Ciarlet
email: patrick.ciarlet@ensta-paristech.fr

Nonlocal models for interface problems between dielectrics and metamaterials

Consider two materials with permittivities/diffusivities of opposite sign, and separated by an interface with a corner. Then, when solving the classic (local) models derived from electromagnetics theory, strong singularities may appear. For instance the scalar problem may be ill-posed in H 1 . To address this difficulty, we study here a nonlocal model for scalar problems with sign-changing coefficients. Numerical results indicate that the proposed nonlocal model has some key advantages over the local one.

I. INTRODUCTION

In electromagnetism, one can model materials that exhibit (almost) real-valued and strictly negative electric permittivity and/or magnetic permeability, within given frequency ranges. These so-called metamaterials, or lefthanded materials, raise unusual questions. Among others, proving the existence of electromagnetic fields, and computing them, is a challenging issue for a problem set in a domain Ω ⊂ R n (n = 2, 3) divided into a classical dielectric material and a metamaterial, when the frequency is in the above mentioned frequency range (see for instance [START_REF] Fernandes | Well-posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials[END_REF][START_REF] Wallén | Surface modes of negative-parameter interfaces and the importance of rounding sharp corners[END_REF]). The main issue is that the problem is ill-posed on some situations (see § II.), and as a consequence its numerical solution is unstable. To adress this difficulty, we propose to reformulate/transform the problem by using a nonlocal framework ( § III.), and then to study the numerical approximation of the nonlocal problem ( § IV.). Another approach would be to add a small, fixed imaginary part to the sign-changing coefficient that appears in the principal part of the PDE, in regions where it takes negative values [START_REF] Chakrabarti | Finite checkerboards of dissipative negative refractive index[END_REF].

II. MODEL PROBLEM

Let us consider a problem in a two-dimensional domain, set in the time-harmonic regime with pulsation ω > 0. Then, the transmission problems in the transverse magnetic (TM) and transverse electric (TE) modes can be reduced to scalar problems like div(σ∇u)

+ ω 2 ςu = f in Ω,
with a source term f , and (σ, ς) equal to ( -1 , µ) or (µ -1 , ), where is the electric permeability, µ is the magnetic permeability, plus boundary conditions. The study of Maxwell's equation with sign-changing coefficients has been carried out in [START_REF] Bonnet-Ben Dhia | Two-dimensional Maxwell's equations with sign-changing coefficients[END_REF]-[4]- [START_REF] Brenner | Hodge decomposition for two-dimensional time-harmonic Maxwell's equations: impedance boundary condition[END_REF]. In particular, in [START_REF] Bonnet-Ben Dhia | T -coercivity for the Maxwell problem with sign-changing coefficients[END_REF], by resorting to the T -coercivity approach, it was shown that electric and magnetic Maxwell transmission problems are well-posed as soon as the associated three-dimensional scalar problems are well-posed. Thus, we focus on a model scalar problem. Consider a partition Ω = Ω 1 ∪ Ω 2 , with Ω 1 ∩ Ω 2 = ∅ and Σ = ∂Ω 1 ∩ ∂Ω 2 the interface. We study a problem in which there is a dielectric material in Ω 1 and an idealized metamaterial in

Ω 2 . Namely, let σ k ∈ L ∞ (Ω k ) (k = 1, 2) be real-valued functions satisfying σ 1 ≥ c 1 > 0 a.e. in Ω 1 and σ 2 ≤ c 2 < 0 a.e. in Ω 2 , where c k (k = 1, 2) are constant numbers. Define the sign-changing σ ∈ L ∞ (Ω) by σ Ω k = σ k , and consider ς ∈ L ∞ (Ω).
We complement the PDE with a homogeneous Dirichlet boundary condition and assume that the right hand side function f belongs to L 2 (Ω). Thus, we look for solutions in the function space H 1 0 (Ω), which is equipped with the usual energy-norm v H 1 0 (Ω) := ( Ω |∇v| 2 dx) 1/2 . Due to the compactness of the embedding H 1 0 (Ω) ⊂ L 2 (Ω), it is enough to consider the principal part of the PDE, Au = -div(σ∇u), and we are led to analyze mathematically the problem:

find u ∈ H 1 0 (Ω) such that -div(σ∇u) = f in Ω. (1)
Above, the differential operators, namely the divergence and gradient operators, are local. Hence (1) may be called a local problem [START_REF] Du | A Nonlocal Vector Calculus, nonlocal volume-constrained problems, and nonlocal balance laws[END_REF]. In variational form, it writes:

find u ∈ H 1 0 (Ω) s.t. ∀v ∈ H 1 0 (Ω), Ω σ(x)∇u(x) • ∇v(x) dx = Ω f v dx. ( 2
)
When σ 1 and σ 2 are constants, Costabel and Stephan [START_REF] Costabel | A direct boundary integral equation method for transmission problems[END_REF] proved that, if the interface Σ is of class C 2 , then the operator A is Fredholm if and only if the contrast κ σ = σ 2 /σ 1 is different from -1. In general situations and independently of the geometry of Σ, problem (1) is of Fredholm type if and only if the contrast lies outside some interval I c , called the critical interval, which always contains the value -1.

The range of the critical interval depends on the interface. For example, Bonnet-Ben Dhia, Dauge and Ramdani [START_REF] Bonnet-Ben Dhia | Analyse spectrale et singularités d'un problème de transmission non coercif[END_REF] analyzed the effect of corners: they proved that in a square minus square geometry it holds that I c = [-3, -1/3]. Furthermore, the solution exhibits strong singularities at these regions when the contrast approaches the critical interval. The influence of corners at the interface between the two materials for problem (1) set in a particular geometry (with one corner of particular aperture) has been mathematically clarified in [START_REF] Bonnet-Ben Dhia | Radiation condition for a non-smooth interface between a dielectric and a metamaterial[END_REF]. In such case, if the contrast lies inside the critical interval, Fredholm property is lost because of the existence of two strongly oscillating singularities at the corner, responsible for the ill-posedness in the classical framework. These singularities can be interpreted as waves propagating towards or outwards the corner. Then selecting the outgoing singularity by means of a limiting absorption principle allows to recover Fredholmness of the problem. A numerical method using Perfectly Matched Layers near the corner has then been derived, which gives satisfactory results [START_REF] Bonnet-Ben Dhia | On the use of perfectly matched layers at corners for scattering problems with sign-changing coefficients[END_REF].

III. NONLOCAL MODEL PROBLEM

We propose a method to shrink the critical interval for the contrast whenever the interface Σ has corners, without having to cope with the singularities there. Mathematically, the ill-posedness of problem (1) means that the usual energy-norm is not adequate to measure the behavior of the solution near the interface. To address this difficulty, we choose to utilize norms able to capture the phenomena near the interface by means of introducing a nonlocal interaction model for the materials. Namely, we use the function space H s (Ω) = {v ∈ H s (Ω) s.t. ṽ ∈ H s (R n )} where ṽ is the continuation of v by 0 to R n \ Ω, for some exponent 0 < s < 1, equipped with the norm

v H s (Ω) = R n ×R n |ṽ(x) -ṽ(y)| 2
|x -y| n+2s dydx.

In essence, these norms are nonlocal. The nonlocal counterpart of the variational formulation (2) writes

find u ∈ H s (Ω) s.t. ∀v ∈ H s (Ω), R n ×R n σ(x, y) (ũ(x) -ũ(y))(ṽ(x) -ṽ(y)) |x -y| n+2s dydx = Ω f v dx. (3)
Assuming that σ 1 and σ 2 are constants, we propose to use

σ(x, y) =      σ 1 if x ∈ Ω 1 , y ∈ Ω 1 ∪ Ω c , σ 2 if x ∈ Ω 2 , y ∈ Ω 2 ∪ Ω c , 1 2 (σ 1 + σ 2 ) if x ∈ Ω 1 , y ∈ Ω 2 . (4)
These nonlocal norms are weaker than the usual energy-norm involved in the analysis of (2). So when considering (3), we are expanding the domain of the operator associated to the problem, while reducing its range. Hence, in this nonlocal framework, we expect that the operator remains of Fredholm type for a wider range of contrasts. The exponent s accounts for the nonlocality of the problem: using an appropriate scaling, in the limit s → 1 one recovers the local problem (1) almost everywhere.

IV. NUMERICAL EXPERIMENTS

We use a conforming Finite Element Method to discretize the nonlocal problem (3), cf. [START_REF] Acosta | A short FEM implementation for a 2D homogeneous Dirichlet problem of a fractional Laplacian[END_REF]. The computations are carried out for two sample geometries: Symmetric geometry: Ω = (-1, 1) × (0, 1), Ω 1 = (-1, 0) × (0, 1), Ω 2 = (0, 1) 2 ; Square minus square geometry: Ω = (-1, 1) × (0, 2), Ω 1 = int(Ω \ Ω 2 ), Ω 2 = (0, 1) 2 .

In Ω 1 , we fix σ 1 = 1, while σ 2 belongs to (-∞, 0). The right-hand side f is equal to 1 over Ω. In principle, the nonlocality exponent s may span (0, 1). Figure 1 displays some of the outputs of these simulations. 3). First result: numerical solution for the symmetric geometry, with s = 0.999 and σ 2 = -0.999. Other results: numerical solutions for the square minus square geometry, with s = 0.7. From left to right, σ 2 is taken equal to {-0.5, -0.9, -0.99}.

In the symmetric geometry, we are considering s close to 1, so that the nonlocal problem approaches the local one, and for the latter the critical interval reduces to I c = {-1}. Regarding the problem set in the square minus square geometry, we checked that the solution is numerically stable for σ 2 = -0.5, namely that it does not depend significantly on the meshize. Observe that for the corresponding local problem, this value lies within the critical interval. So, it seems that the nonlocal model shrinks the critical interval for the contrast. Increasing the value of |σ 2 | leads to larger amplitudes, as can be seen in the rightmost two figures.

V. CONCLUSION

For interface problems between dielectrics and metamaterials, it appears that nonlocal models may help derive acceptable numerical solutions, even when their local counterparts fail. The next step will be the numerical analysis of the nonlocal method, based on the T -coercivity theory.

Fig. 1 :

 1 Fig.1: Computational results for the nonlocal problem (3). First result: numerical solution for the symmetric geometry, with s = 0.999 and σ 2 = -0.999. Other results: numerical solutions for the square minus square geometry, with s = 0.7. From left to right, σ 2 is taken equal to {-0.5, -0.9, -0.99}.


Masoud Hassani
email: masoud.hassani@alumni.univ-avignon.fr

ON THE IRREDUCIBLE ACTION OF PSL(2, R) ON THE 3-DIMENSIONAL EINSTEIN UNIVERSE

, and is one element of the classification of cohomogeneity one actions on Ein 1,2 ([5]).

1. Introduction 1.1. Einstein universe. Let R 2,n+1 denote a (n + 3)-dimensional real vector space equipped with a non-degenerate symmetric bilinear form q with signature (2, n + 1). The nullcone of R 2,n+1 is

N 2,n+1 = {v ∈ R 2,n+1 \ {0} : q(v) = 0}.
The (n + 1)-dimensional Einstein universe Ein 1,n is the image of the nullcone N 2,n+1 under the projectivization:

P : R 2,n+1 \ {0} -→ RP n+2 .
The degenerate metric on N 2,n+1 induces a O(2, n + 1)-invariant conformal Lorentzian structure on Einstein universe. The group of conformal transformations on Ein 1,n is O(2, n + 1) [START_REF] Frances | Géometrie et dynamique lorentziennes conformes[END_REF].

A lightlike geodesic in Einstein universe is a photon. A photon is the projectivisation of an isotropic 2-plane in R 2,n+1 . The set of photons through a point p ∈ Ein 1,n denoted by L(p) is the lightcone at p. The complement of a lightcone L(p) in Einstein universe is the Minkowski patch at p and we denote it by M ink(p). A Minkowski patch is conformally equivalent to the (n + 1)-dimensional Minkoski space E 1,n [START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF].

The complement of the Einstein universe in RP n+2 has two connected components: the (n + 2)dimensional Anti de-Sitter space AdS 1,n+1 and the generalized hyperbolic space H 2,n : the first (respectively the second) is the projection of the domain R 2,n+1 defined by {q < 0} (respectively {q > 0}).

An immersed submanifold S of AdS 1,n+1 or H 2,n is of signature (p, q, r) (respectively Ein 1,n ) if the restriction of the ambient pseudo-Riemmanian metric (respectively the conformal Lorentzian metric) is of signature (p, q, r), meaning that the radical has dimension r, and that maximal definite negative and positive subspaces have dimensions p and q, respectively. If S is nondegenerate, we forgot r and simply denote its signature by (p, q).

1.2. The irreducible representation of PSL(2, R). A subgroup of O(2, n + 1) is irreducible if it preserves no proper subspace of R 2,n+1 . By [3, Theorem 1], up to conjugacy, SO • (1, 2) PSL(2, R) is the only irreducible connected Lie subgroup of O(2, 3).
On the other hand, for every integer n, it is well known that, up to isomorphism, there is only one n-dimensional irreducible representation of PSL(2, R). For n = 5, this representation is the natural action of PSL(2, R) on the vector space V = R 4 [X, Y ] of homogeneous polynomials of degree 4 in two variables X and Y . This action preserves the following quadratic form

q(a 4 X 4 + a 3 X 3 Y + a 2 X 2 Y 2 + a 1 XY 3 + a 0 Y 4 ) = 2a 4 a 0 - 1 2 a 1 a 3 + 1 6 a 2 2 .
The quadratic form q is nondegenerate and has signature [START_REF] Collier | The geometry of maximal representations of surface groups into SO[END_REF][START_REF] Di Scala | Connected subgroups of SO(2, n) acting irreducibly on R 2,n[END_REF]. This induces an irreducible representation

PSL(2, R) → O(2, 3) [2].
Theorem 1.1. The irreducible action of PSL(2, R) on the 3-dimensional Einstein universe Ein 1,2 admits three orbits:

• An 1-dimensional lightlike orbit, i.e. of signature (0, 0, 1)

• A 2-dimensional orbit of signature (0, 1, 1),
• An open orbit (hence of signature (1, 2)) on which the action is free.

The 1-dimensional orbit is lightlike, homeomorphic to RP 1 , but not a photon. The union of the 1dimensional orbit and the 2-dimensional orbit is an algebraic surface, whose singular locus is precisely the 1-dimensional orbit. It is the union of all projective lines tangent to the 1-dimensional orbit. Figure 1 describes a part of the 1 and 2-dimensional orbits in the Minkowski patch M ink(Y 4 ). We will also describe the actions on Anti de-Sitter space and the generalized hyperbolic space H 2,2 : Theorem 1.2. The orbits of PSL(2, R) in the Anti de-sitter component AdS 1,3 are Lorentzian, i.e. of signature [START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF][START_REF] Collier | The geometry of maximal representations of surface groups into SO[END_REF]. They are the leaves of a codimension 1 foliation. In addition, PSL(2, R) induces three types of orbits in H 2,2 : a 2-dimensional spacelike orbit (of signature (2, 0)) homeomorphic to the hyperbolic plane H 2 , a 2-dimensional Lorentzian orbit (i.e., of signature (1, 1)) homeomorphic to the de-Sitter space dS 1,1 , and four kinds of 3-dimensional orbits where the action is free:

• one-parameter family of orbits of signature (2, 1) consisting of elements with four distinct non-real roots,

• one-parameter family of Lorentzian (i.e. of signature (1, 2)) orbits consisting of elements with four distinct real roots, • two orbits of signature (1, 1, 1),

• one-parameter family of Lorentzian (i.e. of signature (1, 2)) orbits consisting of elements with two distinct real roots, and a complex root z in H 2 making an angle θ smaller than 5π/6 with the two real roots.

Remark 1.3. F. Fillastre indicated to us an alternative description for the last case stated in Theorem 1.2: these orbits correspond to polynomials whose roots in CP 1 are ideal vertexes of regular ideal tetraedra in H 3 .

Proofs of the Theorems

Let f be an element in V. We consider it as a polynomial function from C 2 into C. Actually, by specifying Y = 1, we consider f as a polynomial of degree at most 4. Such a polynomial is determined, up to a scalar, by its roots z 1 , z 2 , z 3 , z 4 in CP 1 (some of these roots can be ∞ if f can be divided by Y ).

It provides a natural identification between P(V) and the set CP 1 4 made of 4-tuples (up to permutation) (z 1 , z 2 , z 3 , z 4 ) of CP 1 such that if some z i is not in RP 1 , then its conjugate zi is one of the z j 's. This identification is PSL(2, R)-equivariant, where the action of PSL(2, R) on CP 1 4 is simply the one induced by the diagonal action on (CP 1 ) 4 .

Actually, the complement of RP 1 in CP 1 is the union of the upper half-plane model H 2 of the hyperbolic plane, and the lower half-plane. We can represent every element of CP 1 4 by a 4-tuple (up to permutation) (z 1 , z 2 , z 3 , z 4 ) such that:

-either every z i lies in RP 1 , -or z 1 , z 2 lies in RP 1 , z 3 lies in H 2 and z 4 = z3 , -or z 1 , z 2 lies in H 2 and z 3 = z1 , z 4 = z2 . Theorems 1.1 and 1.2 will follow from the following proposition:

Proposition 2.1. Let [f ] be an element of P(V). Then:

• it lies in Ein 1,2 if and only if it has a root of multiplicity at least 3, or two distinct real roots z 1 , z 2 , and two complex roots z 3 , z 4 = z3 , with z 3 in H 2 and such that the angle at z 3 between the hyperbolic geodesic rays [z 3 , z 1 ) and [z 3 , z 2 ) is 5π/6, • it lies in AdS 1,3 if and only it has two distinct real roots z 1 , z 2 , and two complex roots z 3 , z 4 = z3 , with z 3 in H 2 and such that the angle at z 3 between the hyperbolic geodesic rays [z 3 , z 1 ) and 2 if and only if it has no real roots, or four distinct real roots, or a root of multiplicity exactly 2, or it has two distinct real roots z 1 , z 2 , and two complex roots z 3 , z 4 = z3 , with z 3 in H 2 and such that the angle at z 3 between the hyperbolic geodesic rays

[z 3 , z 2 ) is > 5π/6, • it lies in H 2,
[z 3 , z 1 ) and [z 3 , z 2 ) is < 5π/6.
Proof of Proposition 2.1. Assume that f has no real root. Hence we are in the situation where z 1 , z 2 lie in H 2 and z 3 = z1 , z 4 = z2 . By applying a suitable element of PSL(2, R), we can assume z 1 = i, and z 2 = ri for some r > 0. In other words,

f is in the PSL(2, R)-orbit of (X 2 + Y 2 )(X 2 + r 2 Y 2 ). The value of q on this polynomial is 2 × 1 × r 2 + 1 6 (1 + r 2 ) 2 > 0, hence [f ] lies in H 2,2
. Hence we can assume that f admits at least one root in RP 1 , and by applying a suitable element of PSL(2, R), one can assume that this root is ∞, i.e. that f is a multiple of Y .

We first consider the case where this real root has multiplicity at least 2:

f = Y 2 (aX 2 + bXY + cY 2 )
Then, q(f ) = 1 6 a 2 : it follows that if f has a root of multiplicity at least 3, it lies in Ein 1,2 , and if it has a real root of mulitplicity 2, it lies in H 2,2 .

We assume from now that the real roots of f have multiplicity 1. Assume that all roots are real. Up to PSL(2, R), one can assume that these roots are 0, 1, r and ∞ with 0 < r < 1.

f (X, Y ) = XY (X -Y )(X -rY ) = X 3 Y -(r + 1)X 2 Y 2 + rXY 3
Then, q(f ) = -1 2 r + 1 6 (r + 1) 2 = 1 6 (r 2 -r + 1) > 0. Therefore f lies in H 2,2 once more. The only remaining case is the case where f has two distinct real roots, and two complex conjugate roots z, z with z ∈ H 2 . Up to PSL(2, R), one can assume that the real roots are 0, ∞, hence:

f (X, Y ) = XY (X -zY )(X -zY ) = XY (X 2 -2|z| cos θXY + |z| 2 Y 2 )
where z = |z|e iθ . Then:

q(f ) = 2|z| 2 3 (cos 2 θ - 3 4 )
Hence f lies in Ein 1,2 if and only if θ = π/6 or 5π/6. The proposition follows easily.

Remark 2.2. In order to determine the signature of the orbits induced by PSL(2, R) in P(V), we consider the tangent vectors induced by the action of 1-parameter subgroups of PSL(2, R). We denote by E, P and H, the 1-parameter elliptic, parabolic and hyperbolic subgroups stabilizing i, ∞ and {0, ∞}, respectively.

Proof of Theorem 1.1. It follows from Proposition 2.1 that there are precisely three PSL(2, R)-orbits in Ein 1,2 :

-one orbit N comprising polynomials with a root of multiplicity 4, i.e. of the form [(sY -tX) 4 ] with s, t ∈ R. It is clearly 1-dimensional, and equivariantly homeomorphic to RP 1 with the usual projective action of PSL(2, R). Since d dt | t=0 (Y -tX) 4 = -4XY 3 is a q-null vector, this orbit is lightlike (but cannot be a photon since the action is irreducible), -one orbit L comprising polynomials with a real root of multiplicity 3, and another real root. These are the polynomials of the form [(sY -tX) 3 (s Y -t X)] with s, t, s , t ∈ R. It is 2-dimensional, and it is easy to see that it is the union of the projective lines tangent to N . The vectors tangent to L induced by the 1-parameter subgroups P and E at

[XY 3 ] ∈ L are v P = -Y 4 and v E = 3X 2 Y 2 + Y 4 . Obviously, v P is orthogonal to v E and v E + v P is spacelike. Hence L is of signature (0, 1, 1).
-one open orbit comprising polynomials admitting two distinct real roots and a root z in H 2 making an angle 5π/6 with the two real roots in ∂H 2 . The stabilizers of points in this orbit are trivial since an isometry of H 2 preserving a point in H 2 and one point in ∂H 2 is necessarily the identity.

Proof of Theorem 1.2. According to Proposition 2.1, the polynomials in AdS 1,3 have two distinct real roots, and a complex root z in H 2 making an angle θ greater than 5π/6 with the two real roots. It follows that the action in AdS 1,3 is free, and that the orbits are the level sets of the function θ. Suppose that M is a PSL(2, R)-orbit in AdS 1,3 . There exists

r ∈ R such that [f ] = [Y (X 2 + Y 2 )(X -rY )] ∈ M . The orbit induced by the 1-parameter elliptic subgroup E at [f ] is γ(t) = (X 2 + Y 2 ) (sin t cos t -r sin 2 t)X 2 -(sin t cos t + r cos 2 t)Y 2 + (cos 2 t -sin 2 t + 2r sin t cos t)XY .
Then q( dγ dt | t=0 ) = -2 -2r 2 < 0. This implies, as for any submanifold of a Lorentzian manifold admitting a timelike vector, that M is Lorentzian, i.e., of signature [START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF][START_REF] Collier | The geometry of maximal representations of surface groups into SO[END_REF].

The case of H 2,2 is the richest one. According to Proposition 2.1 there are four cases to consider:

• No real roots. Then f has two complex roots z 1 , z 2 in H 2 (and their conjugates). It corresponds to two orbits: one orbit corresponding to the case z 1 = z 2 : it is spacelike and has dimension 2. It is the only maximal PSL(2, R)-invariant surface in H 2,2 described in [2, Section 5.3]. The case z 1 = z 2 provides a one-parameter family of 3-dimensional orbits on which the action is free (the parameter being the hyperbolic distance between z 1 and z 2 ). One may assume that z 1 = i and z 2 = ri for some r > 0. Denote by M the orbit induced by PSL(2, R)

at [f ] = [(X 2 + Y 2 )(X 2 + r 2 Y 2 )
]. The vectors tangent to M at [f ] induced by the 1-parameter subgroups H, P and E are:

v H = -4X 4 +4r 2 Y 4 , v P = -4X 3 Y -2(r 2 + 1)XY 3 , v E = 2(r 2 -1)X 3 Y + 2(r 2 -1)XY 3 ,
respectively. The timelike vector v H is orthogonal to both v P and v E . It is easy to see that the 2-plane generated by {v P , v E } is of signature [START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF][START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF]. Therefore, the tangent space T [f ] M is of signature (2, 1).

• Four distinct real roots. This case provides a one-parameter family of 3-dimensional orbits on which the action is free -the parameter being the cross-ratio between the roots in RP 1 . Denote by M the PSL(2, R)-orbit at [f ] = [XY (X -Y )(X -rY )] (here as explained in the proof of Proposition 2.1, we can restrict ourselves to the case 0 < r < 1). The vectors tangent to M at [f ] induced by the 1-parameter subgroups H, P , and E are: Consider q(x) = 0 as a quadratic polynomial F in b. Since 0 < r < 1, the discriminant of F is non-negative and it is positive when c = 0. Thus, the intersection of the orthogonal complement of the spacelike vector v P with the tangent space T [f ] M is a 2-plane of signature [START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF][START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF]. This implies that M is Lorentzian, i.e., of signature (1, 2). • A root of multiplicity 2. Observe that if there is a non-real root of multiplicity 2, when we are in the first "no real root" case. Hence we consider here only the case where the root of multiplicity 2 lies in RP 1 . Then, we have three subcases to consider: -two distinct real roots of multiplicity 2: The orbit induced at

v H = -rY 4 + 2(r + 1)XY 3 -3X 2 Y 2 , v P = -2X 3 Y + 2rXY 3 , v E = X 4 -rY 4 + 3(r -1)X 2 Y 2 + 2(r + 1)XY 3 -2(r + 1)X 3 Y,
X 2 Y 2 is the image of the PSL(2, R)-equivariant map dS 1,1 ⊂ P(R 2 [X, Y ]) -→ H 2,2 , [L] → [L 2 ],
where R 2 [X, Y ] is the vector space of homogeneous polynomials of degree

v H = -2XY 3 , v P = Y 4 -2XY 3 , v E = Y 4 -X 4 -2X 2 Y 2 + X 3 Y -XY
v H = -4rY 4 -2X 3 Y + 2XY 3 , v P = -3X 2 Y 2 + 2rXY 3 -Y 4 , v E = X 4 -Y 4 -2rX 3 Y -2rXY 3 ,
respectively. The following set of vectors is an orthogonal basis for T [f ] M where the first vector is timelike and the two others are spacelike.

{(7r + 3r 3 )v H + (6 -2r 2 )v P + (5 + r 2 )v E , 4v P + v E , v H }.

Therefore, M is Lorentzian, i.e., of signature [START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF][START_REF] Collier | The geometry of maximal representations of surface groups into SO[END_REF].

Figure 1 .

 1 Figure 1. Two partial views of the intersection of the 1 and 2-dimensional orbits in Einstein universe with Mink(Y 4 ). Red: Part of the 1-dimensional orbit in Minkowski patch. Green: Part of the 2-dimensional orbit in Minkowski patch.

2 + 1 )

 21 respectively. A vector x = av H + bv P + cv E is orthogonal to v P if and only if 2ra + b(r + 1) + c(r + 1) 2 = 0. Set a = b(r + 1) + c(r + 1) 2 / -2r in q(x) = 2ra 2 -r c 2 + 2(r + 1)ab + 2(r + 1) 2 + ac(2r 2 -r + 5).

  L is the projective class of a Lorentzian bilinear form on R 2 ). The vectors tangent to the orbit at X 2 Y 2 induced by the 1-parameter subgroups P and E are v P = -2XY3 and v E = 2X 3 Y -2XY 3 , respectively. It is easy to see that the 2-plane generated by {v p , v E } is of signature (1, 1). Hence, the orbit induced at X 2 Y 2 is Lorentzian.-three distinct real roots, one of them being of multiplicity 2: Denote by M the orbit induced by PSL(2, R) at [f ] = [XY 2 (X -Y )]. The vectors tangent to M at [f ] induced by the 1-parameter subgroups H, P and E are:

	2 in two variables
	X and Y , endowed with discriminant as a PSL(2, R)-invariant bilinear form of signature
	(1, 2) [2, Section 5.3]. (Here,

  3 , respectively. Obviously, the lightlike vector v H + v P is orthogonal to T [f ] M . Therefore, the restriction of the metric on T [f ] M is degenerate. It is easy to see that the quotient of T [f ] M by the action of the isotropic line R(v H + v P ) is of signature[START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF][START_REF] Barbot | A primer on the (2+1) Einstein universe. Recent developments in pseudo-Riemannian geometry[END_REF]. Thus, M is of signature (1, 1, 1).-one real root of multiplicity 2, and one root inH 2 : Denote by M the orbit induced by PSL(2, R) at [f ] = [Y 2 (X 2 + Y 2 )]. The vectors tangent to M at [f ] induced by the 1parameter subgroups H, P and E arev H = 4Y 4 , v P = -2XY 3 , v E = 2X 3 Y + 2XY3 , respectively. Obviously, the lightlike vector v H is orthogonal T [f ] M . Therefore, the restriction of the metric on T [f ] M is degenerate. It is easy to see that the quotient of T [f ] M by the action of the isotropic line R(v H ) is of signature (1, 1). Thus M is of signature (1, 1, 1). • Two distinct real roots, and a complex root z in H 2 making an angle θ smaller than 5π/6 with the two real roots. Denote by M the orbit induced by PSL(2, R) at [f ] = [Y (X 2 + Y 2 )(X -rY )]. The vectors tangent to M at [f ] induced by the 1-parameter subgroups H, P and E are:

This work is funded by the French Ministry of Foreign Affairs, through Campus France. Grant number 878286E.