Andrea Seppi

THE FLUX HOMOMORPHISM ON CLOSED HYPERBOLIC SURFACES AND ANTI-DE SITTER THREE-DIMENSIONAL GEOMETRY

Given a smooth spacelike surface Σ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation ρ : π1(S) → PSL2R × PSL2R where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to Σ a diffeomorphism φΣ of S. It turns out that φΣ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h ′ on S induced by the action of ρ on H 2 × H 2 . Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φΣ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h ′ ).

Introduction

Anti-de Sitter space is a real Lorentzian three-manifold of constant negative sectional curvature, which can be defined as the Lie group PSL 2 R endowed with the Lorentzian metric induced by the Killing form. In recent times, since the groundbreaking paper [START_REF] Mess | Lorentz spacetimes of constant curvature[END_REF] of Mess, its study has spread widely, mostly motivated by the relations between Anti-de Sitter space (which is denoted AdS 3 ) and Teichmüller theory of hyperbolic surfaces -and the present paper lies in this research direction as well.

Let us explain one of the instances of this relation. A remarkable construction permits to associate to a smooth spacelike surface Σ (topologically a disc) in AdS 3 a submanifold Λ Σ of H 2 × H 2 , where H 2 is the hyperbolic plane. This is essentially due to the fact that the space of timelike lines of AdS 3 is identified to H 2 × H 2 . Hence the submanifold Λ Σ is the set of timelike lines orthogonal to Σ -roughly speaking, the analogue of the Gauss map in this context.

It turns out, as observed in [START_REF] Barbot | Lorentzian Kleinian Groups. To appear in Handbook of Group Actions[END_REF] and [START_REF] Bonsante | Equivariant maps into Anti-de Sitter space and the symplectic geometry of H 2 × H 2[END_REF], that Λ Σ is always a Lagrangian submanifold, for the symplectic structure which makes H 2 × H2 a Kähler manifold, namely Ω = (π

* l Ω H 2 -π * r Ω H 2 ),
where Ω H 2 is the area form of H 2 , π l , π r are the projections to each factor of H 2 × H2 , and H2 denotes H 2 endowed with the opposite orientation. There are two classes of particular cases of this construction:

• If Σ has negative curvature, then Λ Σ is the graph of a symplectomorphism φ Σ of (H 2 , Ω H 2 ) -this is the prototype of Lagrangian submanifolds of H 2 × H 2 . This case was first considered in [START_REF] Krasnov | Minimal surfaces and particles in 3manifolds[END_REF]. • If Σ = Σ 0 has vanishing mean curvature (also called maximal, which implies negative curvature), then φ Σ 0 is minimal Lagrangian, that is, it is a symplectomorphism and its graph is a minimal surface in the Riemannian product H 2 × H 2 . See [START_REF] Bonsante | Maximal surfaces and the universal Teichmüller space[END_REF][START_REF] Torralbo | Minimal Lagrangian immersions in RH 2 × RH 2[END_REF][START_REF] Barbot | Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS3[END_REF][START_REF] Seppi | Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms[END_REF][START_REF] Seppi | Minimal discs in hyperbolic space bounded by a quasicircle at infinity[END_REF] for results in this direction.

From the results of [START_REF] Barbot | Lorentzian Kleinian Groups. To appear in Handbook of Group Actions[END_REF] and [START_REF] Bonsante | Area-preserving diffeomorphisms of hyperbolic plane and K-surfaces in Anti-de Sitter space[END_REF], it follows that being Lagrangian is essentially the only obstruction to inverting this construction, namely to realizing a submanifold of H 2 × H 2 as the image of the Gauss map of a spacelike surface in AdS 3 . However, the situation is extremely more interesting when we consider cocompact actions. More precisely, let S be a closed oriented surface of negative Euler characteristic. Suppose Σ is invariant by an action of the group π 1 (S), which preserves the orientation and the time-orientation of AdS 3 . This produces therefore a representation ρ of π 1 (S) in the isometry group Isom(AdS 3 ), which (from the definition of AdS 3 as PSL 2 R endowed with the bi-invariant metric) is naturally isomorphic to PSL 2 R × PSL 2 R. By the theory developed in [START_REF] Mess | Lorentz spacetimes of constant curvature[END_REF], it turns out that ρ = (ρ l , ρ r ) where ρ l and ρ r are Fuchsian representations -that is, ρ l (π 1 (S)) and ρ r (π 1 (S)) act freely and properly discontinuously on H 2 , with quotient a hyperbolic surface homeomorphic to S.

Let us identify the two quotient hyperbolic surfaces H 2 /ρ l (π 1 (S)) and H 2 /ρ r (π 1 (S)) by (S, h l ) and (S, h r ) respectively, where h l and h r are hyperbolic metrics on S. Then the Lagrangian submanifold Λ Σ of H 2 ×H 2 descends to a Lagrangian submanifold in the quotient (S × S, h l ⊕ h r ). In particular, if the cocompact surface Σ has negative curvature, then φ Σ induces a symplectomorphism, which we denote again by φ Σ : (S, Ω h l ) → (S, Ω hr ). On the other hand, again by the work of Mess, whenever one picks two Fuchsian representations ρ l and ρ r , there is abundance of embedded surfaces Σ ⊂ AdS 3 on which π 1 (S) acts freely and properly discontinuously by means of the representation ρ = (ρ l , ρ r ) :

π 1 (S) → PSL 2 R × PSL 2 R.
For instance, there always exists a (unique) invariant maximal surface Σ 0 (i.e. of vanishing mean curvature, see [START_REF] Barbot | Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS3[END_REF]), which induces the minimal Lagrangian diffeomorphism isotopic to the identity:

φ Σ 0 = φ ML : (S, h l ) → (S, h r ) .
It was already known from the results of [START_REF] Labourie | Surfaces convexes dans l'espace hyperbolique et CP 1structures[END_REF] and [START_REF] Schoen | The role of harmonic mappings in rigidity and deformation problems[END_REF] that such a minimal Lagrangian diffeomorphism φ ML exists and is unique, for any two closed hyperbolic surfaces (S, h l ) and (S, h r ).

Hence it is a natural question to characterize the symplectomorphisms

φ Σ =: (S, Ω h l ) → (S, Ω hr ) ,
which arise as the symplectomorphism associated to a cocompact surface Σ in AdS 3 . The main result of this paper is a new proof of the following:

Theorem. Let ρ l , ρ r : π 1 (S) → PSL 2 R be Fuchsian representations and let Σ ⊂ AdS 3 be a smooth, embedded spacelike surface invariant for the representation ρ = (ρ l , ρ r ) : π 1 (S) → Isom 0 (AdS 3 ), whose curvature is negative.

Then φ Σ = φ ML • ψ ,

where

• φ Σ : (S, Ω h l ) → (S, Ω hr ) is the diffeomorphism associated to Σ;

• φ ML : (S, h l ) → (S, h r ) is the unique minimal Lagrangian diffeomorphism isotopic to the identity; • ψ is a Hamiltonian symplectomorphism, for the area form Ω h l .

Given a symplectic manifold (M, Ω), a symplectomorphism ψ : (M, Ω) → (M, Ω) is Hamiltonian if there exists an isotopy ψ t with ψ 0 = id and ψ 1 = ψ and a smooth family of functions H t : M → R such that the generating vector field X t of the isotopy ψ t is the symplectic gradient of H t for every t. It turns out that Hamiltonian symplectomorphisms form a group, denoted Ham(M, Ω).

The above theorem is actually a consequence of the main result in [START_REF] Bonsante | Equivariant maps into Anti-de Sitter space and the symplectic geometry of H 2 × H 2[END_REF], where it was proved that any submanifold Λ Σ associated to an invariant surface Σ is in the same Ham(S × S, π * l Ω h l -π * r Ω hr )-orbit of the submanifold graph(φ ML ). This implies that, when Σ has negative curvature, and therefore Λ Σ is the graph of the symplectomorphism φ Σ : (S, Ω h l ) → (S, Ω hr ), then φ Σ and φ ML differ by a Hamiltonian symplectomorphism of (S, Ω h l ).

However, we provide here a new proof, when Σ has negative curvature. This relies on the construction of a map C h,h ′ , defined on Symp 0 (S, Ω h , Ω h ′ ) with values in the quotient of the de Rham cohomology group H 1 dR (S, R) ∼ = Hom(π 1 (S), R) by the discrete subgroup of 1-forms with periods integer multiples of 2π, which is identified to Hom(π 1 (S), 2πZ). This map is given by a rather algebraic construction, which involves differential-geometric invariants of the two hyperbolic surfaces h and h ′ . Hence a priori it depends on the choice of the two hyperbolic metrics. However, we then show that, when h = h ′ , this map coincides with the so-called flux homomorphism (see [START_REF] Calabi | On the group of automorphisms of a symplectic manifold[END_REF], [START_REF] Banyaga | Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique[END_REF] and [MS98, Chapter 6]):

Flux : Symp 0 (S, Ω) → H 1 dR (S, R) ,
post-composed with the projection to the quotient of H 1 dR (S, R) by the subspace Hom(π 1 (S), 2πZ). In particular, when h = h ′ , then the map C h,h turns out to be a homomorphism (this is proved directly, and is an important step of the proof), and only depends on the area form of h. The proof of the main theorem then follows from interpreting such map C h,h ′ in terms of Anti-de Sitter geometry, and showing essentially that it vanishes when the symplectomorphism φ Σ is associated to an invariant spacelike surface Σ. One then obtains our main result as a consequence of the classical fact that the group of Hamiltonian symplectomorphisms Ham(M, Ω) is exactly the kernel of the flux homomorphism Flux, applied to the symplectic manifold (M, Ω) = (S, Ω h l ).

The paper then terminates with some remarks about the converse statement of our main theorem. Namely, given φ of the form φ ML • ψ, for ψ ∈ Ham(S, Ω h l ), one can construct an invariant surface Σ in AdS 3 , which is the candidate to be a surface such that φ Σ = φ. This is indeed the case if Σ is embedded. However, such surface Σ will in general develop singularities. Although this generality is not considered here, in [START_REF] Bonsante | Equivariant maps into Anti-de Sitter space and the symplectic geometry of H 2 × H 2[END_REF] it was showed that one can always construct a smooth lift of Σ to the unit tangent bundle of the universal cover of AdS 3 , thus partially reversing the implication proved here.

Acknowledgements. I would like to thank Thierry Barbot, Francesco Bonsante and Alessandro Ghigi for many invaluable conversations.

Algebraic construction for the flux homomorphism

In this paper, S will denote a closed oriented surface of Euler characteristic χ(S) < 0. Moreover, we will usually fix two hyperbolic metrics (i.e. Riemannian metrics of constant curvature -1) h and h ′ on S.

In the first section, we will recall the classical definition of flux homomorphism, for a symplectic surface (S, Ω), from the point of view of symplectic geometry. Then we will give the definition of the alternative map C h,h ′ which a priori depends on the choice of hyperbolic metrics h and h ′ on S. We will study some of its properties, for instance the fact that it is a homomorphism in the case h = h ′ , and we will show that the two homomorphisms essentially coincide in this case.

The flux homomorphism. In the definitions below, we will only suppose that S is a closed oriented surface endowed with a symplectic form Ω. Later, we will consider Ω = Ω h as the area form induced by a hyperbolic metric h. Let us denote by Symp 0 (S, Ω) the group of symplectomorphisms ψ : (S, Ω) → (S, Ω) isotopic to the identity.

Let us recall the definition of Hamiltonian symplectomorphisms (see [MS98, Chapter 6] as a reference): Definition 2.1. Let (S, Ω) be a closed symplectic manifold. Then a symplectomorphism ψ : (S, Ω) → (S, Ω) is Hamiltonian if there exists a smooth isotopy ψ • : [0, 1] → Symp 0 (S, Ω) , with ψ 0 = id and ψ 1 = ψ, and a smooth map

H • : [0, 1] × S → R ,
such that (if we denote H t (p) = H(t, p)):

Ω(X t , •) = dH t ,
where X t is the generating vector field of the isotopy ψ t , namely:

d dt ψ t = X t • ψ t .
It turns out that the space of Hamiltonian symplectomorphisms is a group, which we denote by Ham(S, Ω). Let us now define the flux homomorphism Flux : Symp 0 (S, Ω) → H 1 dR (S, R) . Definition 2.2. Given a symplectomorphism ψ : (S, Ω) → (S, Ω) isotopic to the identity, take an isotopy ψ t with ψ 0 = id and ψ 1 = ψ, and define

Flux(ψ) = 1 0 [Ω(X t , •)]dt ∈ H 1 dR (S, R) ,
where X t is the generating vector field of the isotopy ψ t , namely:

d dt ψ t = X t • ψ t .
It turns out that, if there exists a homotopy in Symp 0 (S, Ω) between the two paths ψ • , ψ ′ • : [0, 1] → Symp 0 (S, Ω) with fixed endpoints, then the value of Flux(ψ) does not change. Since the group Symp 0 (S, Ω) is simply connected, see [Pol01, Section 7.2], Flux is well-defined on Symp 0 (S, Ω). In fact, by the Moser isotopy argument (see [START_REF] Mcduff | A survey of the topological properties of symplectomorphism groups[END_REF]), the inclusion of Symp(S, Ω) into Diff(S) is a homotopy equivalence, and Diff 0 (S) is contractible by [START_REF] Earle | The diffeomorphism group of a compact Riemann surface[END_REF].

The flux homomorphism provides the following characterization of Hamiltonian symplectomorphisms:

Theorem 2.3. The sequence

1 / / Ham(S, Ω) i / / Symp 0 (S, Ω) Flux / / H 1 dR (S, R) / / 1
is a short exact sequence of groups. In particular, a symplectomorphisms ψ : (S, Ω) → (S, Ω) is Hamiltonian if and only if Flux(ψ) = 0.

An alternative homomorphism. Let us now fix two hyperbolic metrics h and h ′ on S, let Ω h and Ω h ′ be the area forms induced by h and h ′ , and let ψ : (S, Ω h ) → (S, Ω h ′ ) be a symplectomorphism. We denote by Isom(T S, ψ * h ′ , h) the subbundle of the bundle End(T S), whose fibers over x ∈ S are linear orientation-preserving automorphisms of T x S which are isometries between the metrics ψ * h ′ and h. Now, given a symplectomorphism ψ and a smooth section b of the bundle Isom(T S, ψ * h ′ , h), we will define a smooth 1-form R) . The 1-form η ψ,b will a priori depend on the hyperbolic metrics h, h ′ , even when h and h ′ coincide.

η ψ,b ∈ Z 1 dR (S,
We start by defining the 1-form locally. Let {v 1 , v 2 } be an oriented orthonormal frame for h, on an open subset U of S, and let ω be the associated connection form for the Levi-Civita connection ∇ of h, which is defined by:

∇ v v 1 = ω(v)v 2 . Analogously, since ψ * h ′ = h(b•, b•), let ω ′ be the connection form associated to the oriented ψ * h ′ -orthonormal frame {v ′ 1 , v ′ 2 } := {b -1 v 1 , b -1 v 2 } and to the Levi-Civita connection ∇ ′ of ψ * h ′ , that is: ∇ ′ v (b -1 v 1 ) = ω ′ (v)(b -1 v 2 ) . Then define η ψ,b = ω ′ -ω .
The following lemma shows that this definition does not depend on the choice of the orthonormal frame, and therefore this local definition provides a well-defined global 1-form on S.

Lemma 2.4. The definition of η ψ,b does not depend on the choice of the oriented orthonormal frame v 1 , v 2 .

Proof. Let { v 1 , v 2 } be another oriented orthonormal frame for h, on the open set U , which we can assume simply connected. Then there exists a smooth function θ : U → R (unique up to multiples of 2π) such that

v i = R θ (v i ) ,
where R θ ∈ Γ ∞ (U, Isom(T S, h)) is the section given, at every point x ∈ U , by counterclockwise rotation fixing x of angle θ for the metric h. By a direct computation,

∇ v v 1 = R θ ∇ v v 1 + dθ(v)J h R θ v 1 ,
where

J h = R π/2 is the almost-complex structure of h. Hence ω(v) = h(∇ v v 1 , v 2 ) = h(R θ ∇ v v 1 , R θ v 2 )+dθ(v)h(R θ v 2 , R θ v 2 ) = ω(v)+dθ(v) .
Observing that, since

ψ * h ′ = h(b•, b•), the rotation R ′ θ for the metric ψ * h ′ coincides with b -1 R θ b, the orthonormal frame { v ′ 1 , v ′ 2 } is {R ′ θ v ′ 1 , R ′ θ v ′ 2 }
, and thus by the same computation:

ω ′ (v) = ω ′ (v) + dθ(v) .
Therefore ω ′ -ω = ω ′ -ω on U , and this concludes the proof.

Hence we defined a 1-form η ψ,b associated to the pair (ψ, b). Let us show that this form is closed:

Lemma 2.5. The 1-form η ψ,b is closed.
Proof. By the well-known formula for the curvature form, one has dω = -Ω h , and analogously

dω ′ = -Ω ψ * h ′ = -Ω h , since ψ is a symplectomorphism. Therefore dη ψ,b = dω ′ -dω = 0.
Unfortunately, the cohomology class of η ψ,b defined in this way is not independent of the choice of b, once the symplectomorphism ψ isotopic to the identity is fixed. However, it can only differ in a controlled way, namely by cohomology classes with integer periods: Lemma 2.6. Let b and b be smooth sections of

Isom(T S, ψ * h ′ , h). Then [η ψ,b ] -[η ψ, b ] ∈ H 1 dR (S, 2πZ) .
Let us first explain the meaning of H 1 dR (S, 2πZ). Recall that there is a standard isomorphism e : H 1 dR (S, R) → Hom(π 1 (S), R), which is defined by

e([η]) : [γ] → γ η .
Then we denote H 1 dR (S, 2πZ) as the subgroup of H 1 dR (S, R) such that H 1 dR (S, 2πZ) = e -1 (Hom(π 1 (S), 2πZ)) . That is, H 1 dR (S, 2πZ) consists of cohomology classes of 1-forms with periods which are integer multiples of 2π.

Proof of Lemma 2.6. Given b and b, there exists a smooth function ϑ : S → S 1 such that b = R ϑ • b. Consider the composition of vector space homomorphisms

C ∞ (S, S 1 ) / / H 1 dR (S, R) e / / Hom(π 1 (S), R)
where the first arrow is

ϑ → [dϑ] ,
as dϑ is locally well-defined as a 1-form.

Let us observe that the composition

F : C ∞ (S, S 1 ) → Hom(π 1 (S), R)
can be expressed as

F (ϑ) = 2πϑ * : π 1 (S) → R ,
where ϑ * is the map induced by ϑ from π 1 (S) to π 1 (S 1 ) ∼ = Z < R. Hence F has image in Hom(π 1 (S), 2πZ). Now, pick an oriented orthonormal frame {v 1 , v 2 } for h. Then let ω ′ be the connection form of ψ * h ′ with respect to the frame

{ v ′ 1 , v ′ 2 } = { b -1 v 1 , b -1 v 2 } = {b -1 R -ϑ v 1 , b -1 R -ϑ v 2 } , which coincides with {R ′ -ϑ b -1 v 1 , R ′ -ϑ b -1 v 2 } = {R ′ -ϑ v ′ 1 , R ′ -ϑ v ′ 2 } . If ω ′ is the connection form of ψ * h ′ with respect to the frame {v ′ 1 , v ′ 2 }
, by the same computation as in Lemma 2.4, we have ω ′ = ω ′ -dϑ. Therefore η ψ,b and η ψ,b ′ differ by the 1-form dϑ, which has integer coefficients as remarked above.

Remark 2.7. From the construction in the above proof, it follows that, given b and b,

[η ψ,b ] = [η ψ, b ] in H 1 dR (S, R) if and only if b = R θ • b for θ : S → R.
That is, if and only if the function ϑ : S → S 1 above can be lifted to a R-valued map.

Lemma 2.6 enables us to give the following definition:

Definition 2.8. Given two hyperbolic metrics h and h ′ on S with area forms Ω h and Ω h ′ , we define

C h,h ′ : Symp 0 (S, Ω h , Ω h ′ ) → H 1 dR (S, R)/H 1 dR (S, 2πZ) by C h,h ′ (ψ) := [η ψ,b ]
, where b is any smooth section of Isom(T S, ψ * h ′ , h). Some properties of the 1-form η ψ,b . We will now derive an equivalent expression for the previously defined 1-form η ψ,b , which will be used later.

Lemma 2.9. Let h be a hyperbolic metric on S,

let b ∈ Γ ∞ (End(T S)) non- singular. If ∇ is the Levi-Civita connection of h and ∇ ′ is the Levi-Civita connection of h ′ = h(b•, b•), then: (1) ∇ ′ v w = b -1 ∇ v b(w) + α(v)J h ′ (w) ,
where J h ′ is the almost-complex structure induced by the metric h ′ and α is the 1-form defined by

α(v) = h(b -1 J h (⋆d ∇ b), J h v) ,
if J h is the almost-complex structure induced by h.

Recall that d ∇ b denotes the exterior derivative, which is the 2-form with values in T S defined by:

(2)

d ∇ b(v, w) = ∇ v (b(w)) -∇ w (b(v)) -b[v, w] ,
where v and w are any two vector fields. Furthermore, ⋆d ∇ b is the Hodge dual of the 2-form d ∇ b, which can be defined as

⋆d ∇ b = d ∇ b(v 1 , v 2 ) ,
where {v 1 , v 2 } is an oriented orthonormal frame (and this expression does not depend on the chosen oriented orthonormal frame).

Proof of Lemma 2.9. Observe that b -1 ∇b is a connection on S compatible with the metric h ′ -but not symmetric in general. Hence the difference between ∇ ′ and b -1 ∇b is a 1-form with values in the bundle of h ′ -skewsymmetric endomorphisms of T S. Hence we can write

∇ ′ v w = b -1 ∇ v b(w) + α(v)J h ′ (w)
, where α is a 1-form and

(3)

J h ′ = b -1 J h b
is the almost-complex structure induced by the metric h ′ . Then from the definitions, we have

d ∇ b(v, w) = ∇ v b(w) -∇ w b(v) -b[v, w] = (b∇ ′ v w -α(v)bJ h ′ (w)) -(b∇ ′ w v -α(w)bJ h ′ (v)) -b[v, w] = b(∇ ′ v w -∇ ′ w v -[v, w]) -α(v)J h b(w) + α(w)J h b(v) = -α(v)J h b(w) + α(w)J h b(v) ,
since we are assuming ∇ ′ is torsion-free.

Let us choose an oriented orthonormal frame {v 1 , v 2 } for h. Recall that the Hodge dual of the 2-form

d ∇ b equals ⋆d ∇ b = d ∇ b(v 1 , v 2 ). Then ⋆d ∇ b = -α(v 1 )J h bJ h (v 1 ) -α(v 2 )J h bJ h (v 2 ) = -(J h bJ h )(α(v 1 )v 1 + α(v 2 )v 2 ) .
Therefore, for every vector v,

α(v) = -h((J h bJ h ) -1 (⋆d ∇ b), v) = -h(J h b -1 J h (⋆d ∇ b), v) = h(b -1 J h (⋆d ∇ b), J h v) .
This concludes the proof.

Using Equation (1), we can now express the 1-form η ψ,b in the following way: if {v 1 , v 2 } is an oriented orthonormal frame, then:

η ψ,b (w) = ω ′ (w) -ω(w) = h ′ (∇ ′ w b -1 v 1 , b -1 v 2 ) -h(∇ w v 1 , v 2 ) = h(b∇ ′ w b -1 v 1 , v 2 ) -h(∇ w v 1 , v 2 ) = h(α(w)J h v 1 , v 2 ) = α(w) , since J h v 1 = v 2 ,
and we have used Equation (3). Therefore, using Lemma 2.9, we have the following:

Corollary 2.10. Given two hyperbolic metrics h and h ′ on S with area forms Ω h and Ω h ′ , a symplectomorphism ψ ∈ Symp 0 (S, Ω h , Ω ′ h ), and a smooth section b of Isom(T S, ψ * h ′ , h),

(4) η ψ,b = h(b -1 J h (⋆d ∇ b), J h •) ,
Observe that, from Equation (4), η ψ,b vanishes if and only if ⋆d ∇ b vanishes identically, which is equivalent to d ∇ b = 0. Hence from Corollary 2.10, we obtain:

Corollary 2.11. Given two hyperbolic metrics h and h ′ on S with area forms Ω h and Ω h ′ , a symplectomorphism ψ ∈ Symp 0 (S, Ω h , Ω ′ h ), and a smooth section b of Isom(T S, ψ * h ′ , h),

η ψ,b = 0 if and only if d ∇ b = 0 .
The group structure. Let us now study how the map C h,h ′ (defined in Definition 2.8) transforms under composition of symplectomorphisms. In particular, this will prove that C h,h : Symp 0 (S, Ω h ) → H 1 dR (S, R)/H 1 dR (S, 2πZ) is a group homomorphism.

Lemma 2.12. Let h and h ′ be hyperbolic metrics on S. Then for ψ ∈ Symp 0 (S, Ω h ) and ψ ∈ Symp 0 (S, Ω h , Ω h ′ ),

C h,h ′ ( ψ • ψ) = C h,h (ψ) + C h,h ′ ( ψ) .
Proof. Let b and b be smooth sections of End(T S) such that

ψ * h ′ = h( b•, b•) and ψ * h = h(b•, b•). Let us denote ψ * b = (dψ) -1 b(dψ). Then ψ * b ∈ Γ ∞ (Isom Σ (ψ * h, ψ * ψ * h ′ )) and ( ψψ) * h ′ = h((b • ψ * b)•, (b • ψ * b)•) .
To compute [η ψψ ], let us pick an orthonormal frame {v 1 , v 2 } for h, let ω be the associated connection form and ω ′′ the connection form of ( ψψ) * h ′ with respect to the frame

{(b • ψ * b) -1 v 1 , (b • ψ * b) -1 v 2 }. Let ω ′ be the connection form of ψ * h for the frame {b -1 v 1 , b -1 v 2 }. Then η ψψ,(b•ψ * b) = ω ′′ -ω = (ω ′′ -ω ′ ) + (ω ′ -ω) . Now clearly ω ′ -ω = η ψ,b .
On the other hand, ω ′ and ω ′′ are the pull-backs, via ψ, of the connection forms of h and ψ * h ′ , and thus ω ′′ -ω ′ = ψ * η ψ, b . This shows that:

η ψψ,(b•ψ * b) = η ψ,b + ψ * η ψ, b . Taking cohomology classes, [ψ * η ψ, b ] = [η ψ, b
] since ψ is isotopic to the identity, and therefore

C h,h ′ ( ψ • ψ) = C h,h (ψ) + C h,h ′ ( ψ) ,
which concludes the proof.

In particular, if we choose h = h ′ , then Lemma 2.12 shows that, if ψ, ψ ∈ Symp 0 (S, Ω h ),

C h,h ( ψ • ψ) = C h,h (ψ) + C h,h ′ ( ψ) . Hence C h,h : Symp 0 (S, Ω h ) → H 1 dR (S, R)/H 1 dR (S, 2πZ
) is a group homomorphism. Moreover, if ψ varies smoothly in Symp 0 (S, Ω h ), then the associated form [η ψ,b ] varies smoothly. Hence C h,h is smooth, for the structure of infinite-dimensional Lie group of Symp 0 (S, Ω h ). To summarize, we have: Corollary 2.13. Given a hyperbolic metric h on S, the map

C h,h : Symp 0 (S, Ω h ) → H 1 dR (S, R)/H 1 dR (S, 2πZ) is a Lie group homomorphism.
The coincidence in the quotient of Flux and C h,h . In this subsection we restrict to the case in which the two metrics h and h ′ coincide. In this case, the map C h,h is a group homomorphism, as already observed. The main result of this part is the fact that the map C h,h coincides with the flux homomorphism, composed with the projection from H 1 dR (S, R) to H 1 dR (S, R)/H 1 dR (S, 2πZ). Proposition 2.14. Given a hyperbolic metric h on a closed oriented surface S, let Ω h be the induced area form. Then the following diagram is commutative:

(5) Symp 0 (S, Ω h ) Flux / / C h,h ) ) ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ H 1 dR (S, R) π H 1 dR (S, R)/H 1 dR (S, 2πZ)
Surprisingly, Proposition 2.14 implies that the homomorphism C h,h (when the two hyperbolic metrics coincide) does not depend on the hyperbolic metric h, but only on its area form Ω h .

To prove Proposition 2.14, we will first prove that the two maps coincide at the infinitesimal level. Recall that the Lie algebra symp(S, Ω h ) consists of smooth vector fields satisfying divX = tr ∇X = 0. The exponential map exp : symp(S, Ω h ) → Symp 0 (S, Ω h ) coincides with the flow of the vector field X, namely exp(tX) = ψ t ∈ Symp 0 (S, Ω h ) where

d dt ψ t = X • ψ t .
Moreover, since H 1 dR (S, 2πZ) is discrete, the tangent space to the quotient H 1 dR (S, R)/H 1 dR (S, 2πZ) at the identity element is naturally identified with H 1 dR (S, R) itself. With this convention:

Lemma 2.15. Let X ∈ symp(S, Ω h ), and let ψ t ∈ Symp 0 (S, Ω h ) be the flow of X. Then

d dt t=0 C h,h (ψ t ) = [Ω h (X, •)] .
Proof. In order to compute C h,h (ψ t ), recall that C h,h is defined as the class of the 1-form η ψt,bt , where we can choose a smoothly varying family b t ∈ Isom(T S, ψ * t h, h). First of all, observe that, by differentiating the condition

h(dψ t •, dψ t •) = h(b t •, b t •) we obtain that h(∇ v X, w) + h(v, ∇ w X) = h( ḃ(v), w) + h(v, ḃ(w)) , where ḃ = d dt t=0 b t
and as usual ∇ is the Levi-Civita connection of h. In other words, the symmetric part of the operators ∇ X and ḃ coincide. Since ψ t preserves the area, the generating field X is divergence-free, that is tr ∇X = 0. On the other hand, since det b t = 1, we have tr ḃ = 0. Therefore there exists a function f : S → R such that

∇X = ḃ + f J h ,
where J h is the almost-complex structure determined by the metric h. Now, let us take exterior derivatives, applied to an othonormal frame {v 1 , v 2 }.

Using that the curvature of h is identically -1, one obtains

d ∇ (∇X)(v 1 , v 2 ) = J h X ,
while on the other hand, using that J h is parallel (namely ∇J h = 0), we have

d ∇ (f J h )(v 1 , v 2 ) = -df (v 1 )v 1 -df (v 2 )v 2 = -grad f .
Hence we have the following formula for the Hodge duals, which will be used later in the proof:

(6) ⋆ d ∇ ḃ = J h X -grad f .
Now from Corollary 2.10, we have:

d dt t=0 η ψt,bt = d dt t=0 h(b -1 t J h (⋆d ∇ b t ), J h •)
Recall that b 0 is the identity operator, hence

d ∇ b 0 = 0. Hence d dt t=0 η ψt,bt = h(J h (⋆d ∇ ḃ), J h •) = h(⋆d ∇ ḃ, •) = h(J h X -grad f, •) = Ω h (X, •) + df .
This shows that

d dt t=0 η ψt,bt = [Ω h (X, •)] ,
and thus concludes the proof.

The proof of Proposition 2.14 will be a consequence of the following lemma about (infinite-dimensional) Lie groups.

Lemma 2.16. Let G and H be Lie groups, possibly of infinite dimension. Let F 1 , F 2 : G → H be Lie group homomorphism and let (F 1 ) * , (F 2 ) * : g → h be the induced Lie algebra homomorphisms. If (F 1 ) * = (F 2 ) * , then

F 1 = F 2 .
Let us now conclude the proof of Proposition 2.14.

Proof of Proposition 2.14. Observe that both π • Flux and C h,h (the latter by Corollary 2.13) are Lie groups homomorphisms from Symp 0 (S, Ω) to H 1 dR (S, R)/H 1 dR (S, 2πZ). In Lemma 2.15, we showed that the differential of

C h,h is (C h,h ) * (X) = [Ω h (X, •)] ∈ H 1 dR (S, R)
, for any X ∈ symp(S, Ω h ). On the other hand, observe that ψ t = exp(tX), where X is the generating vector field of the 1-parameter subgroup ψ t , and exp is the Lie group exponential map of Symp 0 (S, Ω h ). Hence, by definition

Flux(ψ t ) = 1 0 [Ω h (tX, •)]ds = t[Ω h (X, •)] . Therefore (Flux) * (X) = d dt t=0 Flux(ψ t ) = [Ω h (X, •)] .
By appying Lemma 2.16, the proof follows.

The following is a direct consequence of Proposition 2.14 and the fact that H 1 dR (S, 2πZ) is discrete in H 1 dR (S, R): Corollary 2.17. Given a hyperbolic metric h on the closed oriented surface S, the connected component of the identity in Ker(C h,h ) coincides with Ham(S, Ω h ), where Ω h is the area form induced by h.

Application to Anti-de Sitter geometry

In this section, we will start by providing the necessary preliminary notions on Anti-de Sitter geometry, from the basic definitions concerning Antide Sitter space. Then we will show how a symplectomorphism of hyperbolic surfaces is associated to equivariant surfaces in Anti-de Sitter space, with the special case of minimal Lagrangian diffeomorphisms. We conclude with the proof of our main result and some remarks.

Anti-de Sitter space. Let κ be the Killing form on the Lie group PSL 2 R, which is a bi-invariant bilinear form on the Lie algebra sl 2 R of signature (2, 1). The bilinear form κ induces a Lorentzian metric on PSL 2 R, which we will denote by g κ . Then Anti-de Sitter space of dimension 3 is defined as:

AdS 3 := PSL 2 R, 1 8 g κ .
Therefore AdS 3 is a Lorentzian manifold, homeomorphic to a solid torus, of constant sectional curvature. It turns out that, due to the normalization factor 1/8, its sectional curvature is -1. From the construction, it follows that the identity component of the isometry group of AdS 3 is:

Isom 0 (AdS 3 ) = PSL 2 R × PSL 2 R , where (α, β) ∈ PSL 2 R × PSL 2 R acts on AdS 3 by (α, β) • γ = α • γ • β -1 .
The group PSL 2 R is isomorphic to the group of orientation-preserving isometries of H 2 , the hyperbolic plane which we consider in the upper-half plane model:

H 2 := {z ∈ C | ℑ(z) > 0}, |dz| ℑ(z) .
Then, using the structure of the isometry group, one can show the following fact, which will be of fundamental importance in this paper. We recall that a differentiable curve γ : I → AdS 3 is timelike if g κ ( γ, γ) < 0 at every point γ(t), spacelike if g κ ( γ, γ) > 0, and lightlike if g κ ( γ, γ) = 0.

Fact 3.1. There is a 1-1 correspondence

{timelike geodesics in AdS 3 } ↔ H 2 × H 2 ,
which is defined by associating to (x, y) ∈ H 2 × H 2 the timelike geodesic

(7) L x,y := {γ ∈ PSL 2 R | γ(y) = x} .
This correspondence is equivariant with respect to the action of PSL 2 R × PSL 2 R on H 2 × H 2 by isometries of H 2 on each factor, and on the set of timelike geodesics induced by isometries of AdS 3 .

Indeed, L x 0 ,x 0 is a closed timelike geodesic for every x ∈ H 2 , since it is a maximal compact subgroup of the Lie group PSL 2 R, hence the restriction of the bilinear form is negative definite. It is a geodesic since it is a 1-parameter group, and the Riemannian exponential map coincides with the Lie group exponential map for bi-invariant metrics as g κ is. It also turns out that the length of L x 0 ,x 0 is π, and the arclength parameter is 1/2 the angle of rotation of elliptic elements of PSL 2 R fixing x 0 .

The equivariance of such 1-1 correspondence can be easily checked, since:

(α, β) • L x,y = L α(x),β(y) .
This also shows that every timelike geodesic of AdS 3 is of the form L x,y for some x, y, since it is the image of L x 0 ,x 0 under some isometry (α, β) of AdS 3 . There is a natural notion of boundary at infinity of Anti-de Sitter space, topologically a torus, which is defined in the following way:

∂ ∞ AdS 3 := RP 1 × RP 1 ,
where we declare that a sequence γ n ∈ PSL 2 R converges to a pair (p, q) ∈ RP 1 × RP 1 if and only if there exists a point x ∈ H 2 such that:

(8) γ n (x) → p γ -1 n (x) → q .
It is not difficult to check that, if Equation (8) holds for some point x ∈ H 2 , then it holds for every other point x ′ ∈ H 2 . It also turns out that every isometry of AdS 3 , of the form (α, β) ∈ PSL 2 R × PSL 2 R, extends to the boundary, with the obvious action of (α, β) on RP 1 × RP 1 .

From invariant spacelike surfaces to symplectomorphisms of hyperbolic surfaces. Recall that in this paper S denotes a closed oriented hyperbolic surface of Euler characteristic χ(S) < 0. A smoothly embedded surface Σ in AdS 3 is spacelike if its induced metric is a Riemannian metric. One of the main objects of this paper will be spacelike surfaces Σ which are preserved by a representation

ρ : π 1 (S) → PSL 2 R × PSL 2 R ,
which acts freely and properly discontinously on Σ. Following Mess, there is a precise description of these objects:

Fact 3.2 ([Mes07]
). Suppose Σ is a smooth spacelike surface in AdS 3 and

ρ = (ρ l , ρ r ) : π 1 (S) → PSL 2 R × PSL 2 R ,
is a representation which acts freely and properly discontinously on Σ. Then:

• The representations ρ l : π 1 (S) → PSL 2 R and ρ r : π 1 (S) → PSL 2 R
are Fuchsian, that is, each of them acts freely and properly discontinously on

H 2 . • The frontier ∂ ∞ Σ of Σ in ∂ ∞ AdS 3 ∼ = RP 1 × RP 1 is the graph of the unique ρ l -ρ r -equivariant homeomorphism of RP 1 .
In general, given two representations ρ 1 and ρ 2 , we say that a map

f is ρ 1 - ρ 2 -equivariant if, f • ρ 1 (τ ) = ρ 2 (τ )
• f for every τ . If ρ l and ρ r are Fuchsian, it is well-known that there exists a unique homeomorphism f : RP 1 → RP 1 which is ρ l -ρ r -equivariant, which is obtained as the value induced on RP 1 = ∂ ∞ H 2 by the lift to H 2 of any diffeomorphism between the quotient closed surfaces H 2 /ρ l (π 1 (S)) and H 2 /ρ r (π 1 (S)) isotopic to the identity.

Hence, let ρ l and ρ r be Fuchsian representations, so that the oriented topological surface S is homeomorphic to the hyperbolic surfaces H 2 /ρ l (π 1 (S)) and H 2 /ρ r (π 1 (S)). We will denote by h and h ′ the hyperbolic metrics induced on S in this way. We will now define the symplectomorphism

φ Σ : (S, Ω h l ) → (S, Ω hr ) ,
where Ω h l and Ω hr are the area forms induced by the metrics h l and h r and by the orientation of S. For this purpose, consider the map

(9) ι Σ : Σ → H 2 × H 2 ,
which maps γ ∈ Σ to the pair (x, y) which represents the unique timelike geodesic orthogonal to Σ at γ, using Fact 3.1. Let π l , π r : H 2 × H 2 → H 2 denote the projections on the first and second factor. Then we have: ]). Given two Fuchsian representations ρ l and ρ r and any smooth spacelike surface Σ in AdS 3 invariant by the representation ρ = (ρ l , ρ r ) :

Fact 3.3 ([KS07],[BS10],[ BS16
π 1 (S) → PSL 2 R × PSL 2 R , • For γ ∈ Σ, if the curvature of the induced metric on Σ is different from zero at γ, then π l • ι Σ and π r • ι Σ are local diffeomorphisms in a neighborhood of γ.
• If the curvature of the induced metric on Σ is everywhere different from zero, then π l • ι Σ and π r • ι Σ are global diffeomorphisms which extend to homeomorphisms from ∂ ∞ Σ to RP 1 . • In this case, the composition

φ Σ = (π r • ι Σ ) • (π l • ι Σ ) -1 induces a symplectomorphism φ Σ : (S, Ω h l ) → (S, Ω hr )
isotopic to the identity, where Ω h l and Ω hr are the area forms of h l and h r .

In the second point, we remark that, since Σ is invariant by ρ(π 1 (S)) with quotient homeomorphic to the closed surface S, with χ(S) < 0, by the Gauss-Bonnet formula the curvature of the induced metric is always different from zero if and only if it is negative everywhere.

For the last point, it follows from the definitions that π l • ι Σ is ρ-ρ lequivariant, and

π r • ι Σ is ρ-ρ r -equivariant, hence the composition (π r • ι Σ ) • (π l • ι Σ ) -1 induces a map in the quotient from (S, h l ) = H 2 /ρ l (π 1 (S)) to (S, h r ) = H 2 /ρ r (π 1 (S)).
In [START_REF] Bonsante | Maximal surfaces and the universal Teichmüller space[END_REF] and [START_REF] Bonsante | Area-preserving diffeomorphisms of hyperbolic plane and K-surfaces in Anti-de Sitter space[END_REF], some more precise properties of the map φ Σ were given. That is: We recall again that, for a connection ∇, d ∇ b is a 2-form with values in T S, defined in Equation (2).

Remark 3.5. Let us make some comments on these conditions. Condition (1) is equivalent to saying that b is a smooth section of the subbundle Isom(T S, φ * Σ h r , h l ), whose fiber over the point x ∈ S consists of the linear isometries b x : (T x S, φ * Σ h r ) → (T x S, h l ). First, such a section b is not uniquely determined. In fact, if b satisfies the conditions (1) and (2), then R • b still satisfies (1) and (2), where R ∈ Γ ∞ (Isom(T S, h l , h l )) -that is, the tensor b can be post-composed with an isometry of the metric h l at any point. Conditions (1) and (2) imply that φ Σ preserves the area forms of h l and h r , hence φ Σ : (S, Ω h l ) → (S, Ω hr ) is a symplectomorphism.

Further, if b satisfies (1), ( 2) and (3), then for any number θ 0 ∈ (0, 2π) also the section R θ 0 • b satisfies (1), (2) and (3), where

R θ 0 = (cos θ 0 )id + (sin θ 0 )J h l ∈ Γ ∞ (Isom(T S, h l , h l ))
is the section which at every point x ∈ (S, h l ) rotates T x S counterclockwise of an angle θ 0 , for the metric h l . (Here J h l is the almost-complex structure induced by the metric h l .) Finally, as condition ( 4) is an open condition, if b satisfies (1), (2), (3) and (4), then R θ 0 • b still satisfies (1), (2), (3) and (4) for small θ 0 .

Finally, we sketch an argument explaining why Fact 3.4 is true. On the metric universal cover H 2 of (S, h l ), a particular section b ∈ Γ ∞ (End(T H 2 )) satisfying the above properties can be constructed explicitly as

(10) b = (id + J Σ B Σ ) -1 • (id -J Σ B Σ ) ,
where B Σ is the shape operator, and J Σ is the almost-complex structure of the first fundamental form, both computed with respect to the embedding

σ := (π l • ι Σ ) -1 : H 2 → AdS 3 .
It can be checked that b is ρ l -invariant, and thus it defines a section b ∈ Γ ∞ (End(T S)), which always satisfies (1), (2), (3) and (4). See [BS16, Proposition 4.9] for more details.

Minimal Lagrangian diffeomorphisms. A particular example of the above correspondence occurs when Σ is a maximal surface, that is, its shape operator B Σ satisfies tr B Σ ≡ 0. In this case, the associated map φ Σ : (S, h l ) → (S, h r ) will be minimal Lagrangian, namely, it is a symplectomorphism for the induced area forms Ω h l and Ω hr , and the graph of φ Σ is a minimal surface in (S × S, h l ⊕ h r ). Hence this is a notion which depends on the hyperbolic metrics h l and h r , not only on the symplectic form. The term Lagrangian comes from the fact that, if φ Σ is a symplectomorphism, then the graph of φ Σ is a Lagrangian surface in the symplectic manifold (S × S, π * l Ω h l -π * r Ω hr ). Recall that an operator b ∈ Γ ∞ (End(T S)) is h-self-adjoint, for h a Riemannian metric on S, if

h(b•, •) = h(•, b•) .
The correspondence between maximal surfaces and minimal Lagrangian diffeomorphisms is consequence of the following characterization (see [START_REF] Labourie | Surfaces convexes dans l'espace hyperbolique et CP 1structures[END_REF]): Lemma 3.6. Let h l and h r be hyperbolic metrics on S. Then φ ML : (S, h l ) → (S, h r ) is minimal Lagrangian if and only if there exists a smooth h l -selfadjoint section b L ∈ Γ ∞ (End(T S)) such that:

(1)

φ * Σ h r = h l (b L •, b L •); (2) det b L = 1; (3) d ∇ l b L = 0,
where ∇ l is the Levi-Civita connection of the metric h l . In this case, the tensor b L (which is unique) is called Labourie operator.

Hence we have the following: Fact 3.7. Given two Fuchsian representations ρ l and ρ r , if Σ 0 is a maximal surface in AdS 3 invariant by the representation

ρ = (ρ l , ρ r ) : π 1 (S) → PSL 2 R × PSL 2 R , then the associated diffeomorphism φ Σ 0 : (S, h l ) → (S, h r ) is minimal La- grangian.
The proof of Fact 3.7 follows from checking that the tensor b defined in Equation ( 10) is h l -self-adjoint if Σ 0 is a maximal surface, that is, if tr B Σ 0 ≡ 0. See [START_REF] Bonsante | Maximal surfaces and the universal Teichmüller space[END_REF].

The following is a theorem of existence and uniqueness of minimal Lagrangian maps isotopic to the identity between any two compact oriented hyperbolic surfaces. It was proved in [START_REF] Labourie | Surfaces convexes dans l'espace hyperbolique et CP 1structures[END_REF] and [START_REF] Schoen | The role of harmonic mappings in rigidity and deformation problems[END_REF], and it can also be inferred from the existence and uniqueness of the invariant maximal surface Σ 0 , see [START_REF] Barbot | Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS3[END_REF] and [START_REF] Bonsante | Maximal surfaces and the universal Teichmüller space[END_REF].

Theorem 3.8. Given any two hyperbolic metrics h, h ′ on the compact oriented surface S, there exists a unique minimal Lagrangian diffeomorphism φ ML : (S, h) → (S, h ′ ) isotopic to the identity.

Proof of the main theorem. Let us now prove the main result of this paper, which was already stated in the introduction. As usual, given two Fuchsian representations ρ l , ρ r : π 1 (S) → PSL 2 R, we denote (S, h l ) = H 2 /ρ l (π 1 (S)) and similarly (S, h r ) = H 2 /ρ r (π 1 (S)), and Ω h l and Ω hr are the induced area forms. Theorem 3.9. Let ρ l , ρ r : π 1 (S) → PSL 2 R be Fuchsian representations and let Σ ⊂ AdS 3 be a smooth, embedded spacelike surface invariant for the representation ρ = (ρ l , ρ r ) : π 1 (S) → Isom 0 (AdS 3 ), whose curvature is negative. Then φ Σ = φ ML • ψ , where

• φ Σ : (S, Ω h l ) → (S, Ω hr ) is the diffeomorphism associated to Σ;

• φ ML : (S, h l ) → (S, h r ) is the unique minimal Lagrangian diffeomorphism isotopic to the identity; • ψ is a Hamiltonian symplectomorphism, for the area form Ω h l .

Proof. Let ψ = φ -1 ML • φ Σ . Let b the section of Isom(T S, φ * Σ h r , h l ) given by Fact 3.4, and b L the analogous section for φ ML , which was also introduced in Lemma 3.6. Using Lemma 2.12, we have

C h,h ′ (φ Σ ) = C h,h ′ (φ ML ) + C h,h (ψ) . Now, since d ∇ l b = d ∇ l b L =
0 by Fact 3.4 (and Lemma 3.6), using Corollary 2.11 we have η φ

Σ ,b = η φ ML ,b L = 0, hence C h,h ′ (φ Σ ) = C h,h ′ (φ ML ) = 0 and therefore C h,h (
ψ) = 0. From Definition 2.8 and Proposition 5, this implies that Flux(ψ) ∈ H 1 dR (S, 2πZ). On the other hand, consider the maximal surface Σ 0 invariant by the representation ρ (as in Fact 3.7). By the results in [START_REF] Mess | Lorentz spacetimes of constant curvature[END_REF], the surfaces Σ and Σ 0 induce closed embedded surfaces in a (so-called maximal globally hyperbolic) three-manifold M homeomorphic to S × R. Let us call Σ and Σ 0 be the surfaces induced in the quotient. Both Σ and Σ 0 are homotopic to S × {⋆}. Hence one can find a smooth isotopy f t : S → M which is a homotopy equivalence at every time t, such that f 0 (S) = Σ 0 , and f 1 (S) = Σ, and f t (S) is a smooth spacelike surface for every t. Lifting to AdS 3 , the surfaces Σ t which project to f t (S) induce maps φ Σt which vary smoothly. If we put φ Σt = φ ML •ψ t , the corresponding classes Flux(ψ t ) vary smoothly, and are all in H 1 dR (S, 2πZ) by the same argument as the first part of this proof.

Moreover, ψ 0 is the identity, and therefore Flux(ψ 0 ) = 0. Since H 1 dR (S, 2πZ) is discrete, it then follows that Flux(ψ 1 ) = Flux(ψ) = 0. Hence by Theorem 2.3, ψ is Hamiltonian. This concludes the proof.

Let us remark that the second part is exactly in the spirit of Corollary 2.17, by using the fact that there is a smooth interpolation between φ Σ and φ ML . By specializing to the case in which the two hyperbolic metrics coincide, we get the following: Corollary 3.10. Given a Fuchsian representation ρ 0 : π 1 (S) → PSL 2 R, let φ Σ : H 2 /ρ 0 (π 1 (S)) → H 2 /ρ 0 (π 1 (S)) be a symplectomorphism induced by a smooth embedded spacelike surface Σ ⊂ AdS 3 invariant by (ρ 0 , ρ 0 )(π 1 (S)) < PSL 2 R × PSL 2 R. Then φ Σ is Hamiltonian.

Reconstructing smooth spacelike surfaces. We conclude by a discussion about the question whether the condition that φ = φ ML • ψ for a symplectomorphism φ : (S, h l ) → (S, h r ), where φ ML is minimal Lagrangian and ψ is Hamiltonian, is also sufficient to obtain φ as the symplectomorphism associated to an invariant surface in AdS 3 . In fact, by Theorem 3.9 this is a necessary condition.

Using the tools from [START_REF] Bonsante | Area-preserving diffeomorphisms of hyperbolic plane and K-surfaces in Anti-de Sitter space[END_REF], given a symplectomorphism φ : (S, Ω l ) → (S, Ω r ) and any section b ∈ Γ(Isom(T S, φ * h r , h l )), a map σ φ,b : H 2 → AdS 3 can be reconstructed explicitly, such that σ φ,b is equivariant for the action of ρ l : π 1 (S) → PSL(2, R) on H 2 and of the pair ρ = (ρ l , ρ r ) on AdS However, the map σ φ,R θ b can be singular. Observe also that the function θ is not uniquely determined, but there are possible choices which differ by adding a constant. In general, σ φ,R θ b might be non-singular only for some of these choices of θ (i.e. up to adding a constant function).

In [START_REF] Bonsante | Equivariant maps into Anti-de Sitter space and the symplectic geometry of H 2 × H 2[END_REF], the authors showed that such a map σ φ,R θ b always lifts to an embedding into the future timelike unit tangent bundle of the universal cover AdS 3 . Finally, let us remark that if one picks a section b ∈ Γ(Isom(T S, φ * h r , h l )) such that [η φ, b ] ∈ H 1 dR (S, 2πZ), then one can still define a map σ φ, b as in Equation (11). However, σ φ,b and σ φ, b will lift to maps from H 2 → AdS 3 which are equivariant for different lifts to Isom( AdS 3 ) of the representation ρ : π 1 (S) → Isom(AdS 3 ).

Lemma 2.1 in [START_REF] Bonsante | Equivariant maps into Anti-de Sitter space and the symplectic geometry of H 2 × H 2[END_REF] shows that all the ρ l -ρ-equivariant embeddings into AdS 3 lift to embeddings into AdS 3 which are equivariant for the same lift of ρ. This implies that the map σ φ, b will never be an embedding if [η φ, b ] ∈ H 1 dR (S, 2πZ) but [η φ, b ] = 0.

Fact 3. 4 .

 4 Given two Fuchsian representations ρ l and ρ r and any smooth spacelike surface Σ in AdS 3 invariant by the representation ρ = (ρ l , ρ r ) : π 1 (S) → PSL 2 R × PSL 2 R , let φ Σ : (S, Ω h l ) → (S, Ω hr ) be the associated symplectomorphism. Then there exists a smooth section b ∈ Γ ∞ (End(T S)) such that:(1)φ * Σ h r = h l (b•, b•); (2) det b = 1;(3) d ∇ l b = 0, where ∇ l is the Levi-Civita connection of the metric h l ; (4) tr b = -2.

  3 . The map σ φ,b : H 2 → AdS 3 is defined by the condition that σ φ,b (x) is the unique isometry σ ∈ PSL(2, R) (recalling that AdS 3 is the Lie group PSL(2, R)) such that(11) σ(φ(x)) = x dσ φ(x) • dφ x = -b x .Moreover, such map is orthogonal to the foliation in timelike lines of the formL x,φ(x) provided b satisfies d ∇ l b = 0 ([BS16, Corollary 5.7]), but in general it is not an immersion. Therefore, given a section b ∈ Γ(Isom(T S, φ * h r , h l )), suppose [η φ,b ] = 0 ∈ H 1 dR (S, R).(This assumption is satisfied if φ = φ ML • ψ, for φ ML : (S, h l ) → (S, h r ) the minimal Lagrangian diffeomorphism and ψ : (S, Ω l ) → (S, Ω l ) Hamiltonian, by an argument similar to above.) Then by the same argument of the proof of Lemma 2.6, one can find a function θ : S → R such that η φ,R θ b = 0, which is equivalent to d ∇ l (R θ b) = 0 by Corollary 2.11. If σ φ,R θ b is an embedding, then the image Σ = σ φ,R θ b (H 2 ) is a smoothly embedded spacelike surface, invariant by the representation ρ = (ρ l , ρ r ), whose associated map is φ.


Thomas Pradeu
email: thomas.pradeu@u-bordeaux.fr

Lynn Chiu

Jean Gayon

Peter Godfrey-Smith

Matt Haber

Philippe Huneman

Maureen O'malley

Sahotra Sarkar

Keywords:

Which domains of biology do philosophers of biology primarily study? The fact that philosophy of biology has been dominated by an interest for evolutionary biology is widely admitted, but it has not been strictly demonstrated. Here I analyse the topics of all the papers published in Biology & Philosophy, just as the journal celebrates its thirtieth anniversary. I then compare the distribution of biological topics in Biology & Philosophy with that of the scientific journal Proceedings of the National Academy of Science of the USA, focusing on the recent period 2003-2015. This comparison reveals a significant mismatch between the distributions of these topics. I examine plausible explanations for that mismatch. Finally, I argue that many biological topics underrepresented in philosophy of biology raise important philosophical issues and should therefore play a more central role in future philosophy of biology.

Introduction

Even though the phrase "philosophy of biology" was coined by Whewell in 1840 [START_REF] Gayon | Philosophy of Biology: An Historico-critical Characterization[END_REF]) and despite the fact that interesting philosophical and conceptual explorations of biology appeared between the 1920s and 1960s1 , philosophy of biology only emerged as a well-identified domain in the Anglo-Saxon world during the 1970s, and has since then blossomed [START_REF] Hull | The History of the Philosophy of Biology[END_REF]. Today, philosophy of biology has its own international society, the International Society for the History, Philosophy, and Social Studies of Biology (ISHPSSB, founded in 19892 ), and it is disseminated in several academic journals, the most representative of which is certainly Biology & Philosophy, launched by Michael Ruse in 1986 (Ruse was the editor of the journal from 1986 to 2000; Kim Sterelny has been the editor from 2000 until January 2017).

A special feature of philosophy of biology is that it is closely tied to biology as it is practised. Several founders, in particular [START_REF] Hull | What philosophy of biology is not[END_REF], insisted that philosophers of biology should be well informed by the most recent biological results, and, if possible, offer analyses that are interesting and useful for biologists themselves. Philosophers of biology often publish their papers in scientific journals, and some of them are widely cited by biologists. Reciprocally, a number of biologists (including Ernst Mayr, Richard Lewontin, and Francisco Ayala) have played a major role in the content and the institutional organization of philosophy of biology, and it is not rare for biologists to publish papers in a journal such as Biology & Philosophy (examples include Richard Dawkins, Richard Lewontin, David Haig, and Richard Michod, among many others).

But which domains of biology are philosophers of biology interested in? It has often been observed that philosophy of biology is dominated by evolutionary studies, but the extent and exact nature of this domination have not been carefully analysed. In addition, it seems useful to describe in detail the respective role other biological domains play in philosophy of biology. To address these issues, I analyse all the papers published in Biology & Philosophy since the journal was launched in 1986, exactly thirty years ago. I show that two topics are dominant: evolution and, to a much lesser extent, the psychological and cognitive sciences. I then compare this distribution of papers in Biology & Philosophy with the distribution of papers published by a major generalist scientific journal, the Proceedings of the National Academy of Science of the USA (hereafter PNAS), during the same period 3 . This analysis shows that there is a significant mismatch between the topics studied by biologists and those studied by philosophers of biology (as reflected in Biology & Philosophy). I examine explanations that have been or could be advanced to account for this mismatch, and I show that many of them are unconvincing. In particular, the idea that biological fields not focusing directly on evolution do not raise important philosophical problems is ruled out, on the basis of several examples. My conclusion is that philosophers of biology should widen their perspectives, and complement the work done about evolutionary biology by investigations about a variety of other biological fields, as this would lead to a richer and more integrative philosophy of biology.

What domains of biology do philosophers of biology study?

In this section, I analyse the distribution of biological domains reflected by all the papers published in Biology & Philosophy in the last thirty years, starting from 1986. For the period between 1986 and 2002, I reconstructed an analysis made by Jean [START_REF] Gayon | Philosophy of Biology: An Historico-critical Characterization[END_REF]. For the period between 2003 and 2015, I carried out my own analysis.

Biological domains in Biology & Philosophy from 1986 to 2002 [START_REF] Gayon | Philosophy of Biology: An Historico-critical Characterization[END_REF] analysed the themes of all the papers published in Biology & Philosophy from 1986 to 2002. Here I recall Gayon's methodology and his main results. From 1986 to 2002, Biology & Philosophy published a total of 421 papers, that is, on average 25 papers per year. Gayon classified all the papers published in Biology & Philosophy during this period into three broad categories: "Studies on particular biological theories, concepts or methods" (approximately 48%); "Philosophical questions of general interest regarding biology and the living world" (approximately 37%); "Articles with a clear historical dimension" (approximately 15%). The first category was then divided into six subcategories: Evolution; Taxonomy; Species; Ecology; Genetics; Other (which includes development, psychology, physiology, cognitive sciences, etc.). The second category was also divided into six categories: Evolutionary epistemology; Ethics and biology (especially evolutionary ethics); Nature/culture; Function, teleology, design; Reflections in biology in general (laws, autonomy of biology, etc.); Other (which includes philosophy of mind, emotions, religion, etc.). Gayon's main conclusions (2009: 208-209) were that:

• 72% of the thematic papers published in Biology & Philosophy from 1986 to 2002 (those belonging to the category "Studies on particular biological theories, concepts or methods") deal with evolution broadly speaking. • Physiology, biochemistry, and biophysics are strikingly absent.

• There is no significant change in the representation of the different biological domains over time. • This stability suggests that philosophers of biology are not particularly sensitive to scientific novelty in biology over time.

Because my main aim here was to offer a comprehensive view of the biological domains represented in the papers published in Biology & Philosophy, I reconstructed, with Gayon's help, the tables found in [START_REF] Gayon | Philosophy of Biology: An Historico-critical Characterization[END_REF] From 2003 to 2015, 499 papers were published in Biology & Philosophy, 38 papers per year on average. I start with a few explanations about the methodology that I adopted. The main difference with Gayon's analysis is that I did not use categories and subcategories. I used a total of 23 categories4 , reflecting directly those of the scientific journal PNAS, in order to enable the systematic comparison between the representation of biological domains in the two journals, which constitutes the main objective of Section 3. Therefore, the categories used here are: Agricultural sciences; Anthropology; Applied biological sciences; Biochemistry; Biophysics and computational biology; Cell biology; Developmental biology; Ecology; Environmental sciences; Evolution; Genetics; Immunology; Medical sciences; Microbiology; Neuroscience; Pharmacology; Physiology; Plant biology; Population biology; Psychological and cognitive sciences; Sustainability science; Systems biology; and Others5 .

Moreover, I followed Gayon in excluding book reviews, unless a whole issue or a whole section of the journal was devoted to a particular book, because I considered that in those cases the theme of the book had been deemed by the editor of Biology & Philosophy to be of broad interest for the community of philosophers of biology. Editorials of a very general nature have been excluded. However, editors' introductions to special issues were included, as they often offered arguments explaining the importance of the special issue theme.

What are the main results of this analysis? Table S2 (Supplementary material) gives a detailed account of the biological domains represented in Biology & Philosophy from 2003 to 2015. Figure 2 shows the main results, which can be summarized as follows:

• 62% of the papers published in Biology & Philosophy from 2003 to 2015 were about evolution. More precisely, 41% were about evolution in general, 6% about systematics and the species problem, and 15% about human evolution (including evolutionary psychology, evolutionary ethics, and cultural evolution). • The presence of some domains increased over time. This includes the psychological and cognitive sciences, as well as microbiology (even though in the latter case it is due almost exclusively to the single special issue mentioned above). Other domains were stable over time.

• 14% of the papers published in

The discrepancy between today's philosophy of biology and biology

To get a more precise idea about how philosophers of biology see biology, it is useful to compare the distribution of biological domains in Biology & Philosophy to that found in a scientific journal. PNAS is an appropriate choice, for several reasons. Launched in 1915 as the official scientific journal of the National Academy of Sciences of the USA and published weekly ever since, it is highly respected in the broad scientific community 7 . More importantly for the present study, PNAS uses a discipline-based system of classification for the papers it publishes (called "Sections" in PNAS), unlike many other scientific journals8 . As explained above, I used PNAS's 22 categories to classify biological papers to classify the papers of Biology & Philosophy as well.

How are different biological fields represented in PNAS? Table S3 (Supplementary material) shows this representation over the periods 1986-2002 and 2003-2015. Nonetheless, because my main purpose was comparative and because PNAS used systematically its classification system only starting from 1996, the present analysis focuses on the period 2003-2015 (that is, the same as the one used above about Biology & Philosophy). During this period, PNAS published a total of 51,896 papers, 42,934 of which were about biology (approximately 83%). Table S4 (Supplementary material) details, for each domain, the number of papers and the percentage with regard to the total number of biological papers. Both the overall representation of each biological domain in PNAS from 2003 to 2015 and the evolution of each domain through time can be of interest. Overall, the six dominant biological domains in PNAS (those that represent strictly more than 5% of the total biological papers) are, by order of percentage amounts: biochemistry (12%), neuroscience (12%), medical sciences (10%), and computational biology (9%), immunology and inflammation (7%), and microbiology (7%). Even if these disciplines dominate in the long run, their relative representation has sometimes changed through the years. Indeed, as shown in Figure S2 (Supplementary material), some biological domains became more represented while others became less so in PNAS from 2003 to 2015. For example, biochemistry represented 17% of the The most important question for our present purposes is how the distributions of biological domains in Biology & Philosophy (Figure 2) and PNAS (Figure 3) compare. The most immediate lessons that can be drawn from this comparison are: Thus, the comparison between Biology & Philosophy and PNAS suggests the existence of a very significant mismatch between the biological domains studied by philosophers of biology and those by biologists themselves. The present analysis gives strength to the remark made by [START_REF] Sarkar | A companion to the philosophy of biology[END_REF] that, given the diversity and richness of the different fields of current biology, a philosophy of biology focused exclusively or almost exclusively on evolution is "myopic".10 Although the mismatch described here has been anticipated by many actors in the field (e.g., [START_REF] Griffiths | Philosophy of Biology[END_REF], it seemed important to demonstrate and describe it more clearly. The crucial question now is how this mismatch might be explained.

Possible explanations for the mismatch between biology and philosophy of biology

This section examines explanations that have been or could be advanced to account for the mismatch between biology and philosophy of biology in terms of distribution of biological domains. I show that many of these explanations are unconvincing. In particular, using examples coming from different disciplines of current biology, I reject the idea that biological fields not focusing directly on evolution do not raise important philosophical problems.

A simple historical contingency

Perhaps the most straightforward explanation is one insisting on historical contingency: according to this explanation, it just happened that the founders, both philosophers and biologists, were interested in evolution. This includes, naturally, David Hull, Michael Ruse, and Marjorie Grene11 , as well as many others after them (e.g., [START_REF] Brandon | Adaptation and Evolutionary Theory[END_REF][START_REF] Wimsatt | Randomness and perceived-randomness in evolutionary biology[END_REF][START_REF] Sober | The nature of selection: evolutionary theory in philosophical focus[END_REF][START_REF] Lloyd | The structure and confirmation of evolutionary theory[END_REF][START_REF] Sober | Philosophy of Biology[END_REF]. Among biologists who contributed to the emergence of the field of philosophy of biology, evolutionists were clearly dominant (Ernst Mayr, Richard Lewontin, Francisco Ayala, Michael Ghiselin, etc.). Though this historical contingency is probably part of the explanation, it is clearly not sufficient. This kind of explanation presupposes in part what is in question: it says that philosophy of biology has focused on evolution because its founders were interested in evolution, without questioning who these "founders" are, why they are considered as founders, and why exactly they chose to focus on evolutionary issues. Looking carefully at the history of the philosophy of biology, the "founder-based" argument seems questionable. One of the first philosophers writing specifically about biology in the 1960s and 1970s was Ken Schaffner, who wrote mainly about molecular biology (broadly construed) and about the potential reduction of Mendelian genetics to molecular genetics [START_REF] Schaffner | Approaches to Reduction[END_REF][START_REF] Schaffner | The Watson-Crick Model And Reductionism[END_REF]). Schaffner was not alone in his interest for molecular biology. Reducibility to the molecular level became one of the most intensively discussed topics among the philosophers of biology of the 1970s (e.g., [START_REF] Ruse | Reduction, Replacement, and Molecular Biology[END_REF][START_REF] Hull | Philosophy of biological science[END_REF][START_REF] Wimsatt | Reductive Explanation: A Functional Account[END_REF][START_REF] Darden | Interfield Theories[END_REF]. Moreover, an interest for molecular biology has clearly been present in philosophy of biology from the 1970-1980s onward (e.g., [START_REF] Darden | Interfield Theories[END_REF][START_REF] Kitcher | Genes[END_REF][START_REF] Kitcher | 1953 and all That. A Tale of Two Sciences[END_REF][START_REF] Burian | On Conceptual Change in Biology: The Case of the Gene[END_REF][START_REF] Rosenberg | The structure of biological science[END_REF][START_REF] Waters | Why the Anti-Reductionist Consensus Won't Survive: The Case of Classical Mendelian Genetics[END_REF][START_REF] Sarkar | Reductionism and functional explanation in molecular biology[END_REF], so what requires an explanation is why this branch of philosophy of biology has not flourished as much as the evolutionary branch. To some extent, therefore, it seems that it is those who did philosophy of evolution who, in retrospect, have been considered as the founders of the field, at least as much as the other way around.12 Overall, therefore, historical contingency is probably part of the explanation for the domination of evolution in philosophy of biology, but it cannot be the whole story.

Evolution is more theoretical

The argument that evolution is more theoretical and therefore philosophically more interesting seems to have been important for some philosophers of biology, particularly Hull. In many statements, Hull says that only evolution offers a genuine scientific theory, in contrast with other biological domains such as physiology and anatomy (e.g., [START_REF] Hull | Philosophy of biological science[END_REF])) This view raises many difficulties.

i) Is it true that non-evolutionary domains are not theoretical?

Non-evolutionary domains can be theoretical. To take just a few examples, systems biology, immunology, and ecology have very strong theoretical components, but their theories have not aroused the interest of many philosophers of biology. Even domains that have often been perceived as non-theoretical, for example developmental biology, can in fact offer theories [START_REF] Minelli | Towards a Theory of Development[END_REF], but, here again, few philosophers have studied those theories. Therefore, it seems unlikely that the main reason why philosophers of biology have been so much attracted by evolution is the theoretical nature of this domain as such.

ii) Should evolution be seen exclusively as a theoretical field?

Clearly evolution offers one unifying framework for all biology, and some aspects of evolution are highly theoretical, but this should not hide the fact that evolution has also a key experimental component. Recent major advances in evolution have come from studies in "experimental evolution", such as those of Richard Lenski, Michael Travisano, and several others [START_REF] Lenski | Long-Term Experimental Evolution in Escherichia coli. I. Adaptation and Divergence During 2,000 Generations[END_REF][START_REF] Lenski | Dynamics of adaptation and diversification: a 10,000-generation experiment with bacterial populations[END_REF][START_REF] Sniegowski | Evolution of high mutation rates in experimental populations of E. coli[END_REF]). An exclusive focus on the theoretical dimension of evolution might lead philosophers of biology to miss the importance of these experimental approaches to evolution.

iii) Is the theoretical dimension essential to philosophy of science?

The focus of some philosophers of biology on theories is a testament of the priority given to theories in philosophy of science more generally (a clear example of this are the debates over the "syntactic" or "semantic" nature of evolutionary theory: e.g., [START_REF] Ruse | The philosophy of biology[END_REF][START_REF] Lloyd | The structure and confirmation of evolutionary theory[END_REF]). Starting from the 1980s, several discussions in philosophy of science (including [START_REF] Hacking | Representing and intervening : introductory topics in the philosophy of natural science[END_REF]), have questioned this focus on theories, emphasizing that many practical and experimental aspects of the sciences could be philosophically stimulating. The attention to experimental and practical aspects of science has also manifested in philosophy of biology, for example in the work of Marcel [START_REF] Weber | Philosophy of experimental biology[END_REF], Ken [START_REF] Waters | How Practical Know--How Contextualizes Theoretical Knowledge: Exporting Causal Knowledge from Laboratory to Nature[END_REF]Ingo Brigandt (2013a), among others. Many philosophers of science today consider that there is much more to philosophy of science than just theories.

Evolution is philosophically more interesting

If asked why evolution dominates their field, many philosophers of biology might say that evolution is philosophically very interesting, and probably more interesting than other biological fields (e.g., Ruse 1988: 80). Perhaps some fields -such as biochemistry, biophysics, or cell biology -are biologically important, but of little philosophical interest.

The philosophical dimension of evolutionary biology is naturally beyond doubt. Evolutionary theory is rich and "substrate-neutral", that is, it can be applied to many different levels of the biological world -a feature that has played an important role in the major debate over "units of selection" [START_REF] Lewontin | The Units of Selection[END_REF]) and evolutionary individuality [START_REF] Hull | Individuality and Selection[END_REF]. Over the last four decades or so, philosophers of biology have succeeded in showing that analysis of evolutionary biology can make key contributions to both general philosophy of science and general philosophy. Evolution can shed light on causality and explanation (for example the status of historical explanations) (e.g., [START_REF] Sober | The nature of selection: evolutionary theory in philosophical focus[END_REF], and the definition of scientific theories (e.g., [START_REF] Lloyd | The structure and confirmation of evolutionary theory[END_REF], among other issues. In addition, a careful examination of evolution can have major consequences for some of the most fundamental and traditional debates of philosophy and metaphysics, including the possibility of defining human nature [START_REF] Hull | On Human Nature[END_REF][START_REF] Lewens | Human Nature: The Very Idea[END_REF], the definition of natural kinds (e.g., [START_REF] Dupré | The disorder of things : metaphysical foundations of the disunity of science[END_REF], the emergence of cognition [START_REF] Sterelny | Thought in a hostile world: the evolution of human cognition[END_REF] and morality [START_REF] Joyce | The evolution of morality[END_REF], or the definition of individuality [START_REF] Hull | A Matter of Individuality[END_REF][START_REF] Hull | Individuality and Selection[END_REF][START_REF] Gayon | Darwinian Individuals[END_REF].

Can other biological domains be as philosophically interesting as evolution? Some examples show that biological domains that do not focus on evolutionary questions can raise major philosophical problems, and in some cases already have done so (though not, or not massively, in Biology & Philosophy). One obvious example is the philosophy of neuroscience13 . As we have seen, the neurosciences represent 12% of the biological papers published in PNAS but only 3% of the papers published in Biology & Philosophy. The neurosciences are of high philosophical interest, from at least two points of view [START_REF] Gold | Philosophy of Neuroscience[END_REF][START_REF] Bickle | Introduction[END_REF]. First, the neurosciences can contribute to a variety of traditional philosophical problems, as suggested by "neurophilosophy" (Patricia Churchland 1986). This includes our understanding of perception (e.g., [START_REF] Keeley | Making Sense of the Senses: Individuating Modalities in Humans and Other Animals[END_REF], representation [START_REF] Bechtel | Representations: From Neural Systems to Cognitive Systems[END_REF][START_REF] Bechtel | Investigating neural representations: the tale of place cells[END_REF], emotion (Hardcastle 1999a;[START_REF] Prinz | Gut reactions: a perceptual theory of emotion[END_REF], or pain (Hardcastle 1999b). Perhaps classic problems of philosophy of science (e.g., the structure of scientific theories) could be impacted as well, as suggested by Paul [START_REF] Churchland | A neurocomputational perspective: the nature of mind and the structure of science[END_REF]. More generally, many philosophers have suggested that the neurosciences can affect major questions of traditional philosophy and ethics, such as the nature of desire [START_REF] Schroeder | Three faces of desire[END_REF], moral cognition [START_REF] Prinz | The emotional construction of morals[END_REF]), but also free-will, self-control, and personal identity [START_REF] Roskies | Neuroethics for the New Millenium[END_REF]. Second, some questions raised by the neurosciences themselves can be worth exploring from a philosophical point of view, including the neural coding of information and the meaning of the word "information" in that context (e.g., [START_REF] Garson | The Introduction of Information into Neurobiology[END_REF], the problem of perceptual binding (i.e., how our perceptions become united into wholes even though all individual features that comprise them are distributed about the brain) [START_REF] Hardcastle | Consciousness and the neurobiology of perceptual binding[END_REF], or the issue of the neural correlates of consciousness [START_REF] Frith Null | The neural correlates of conscious experience: an experimental framework[END_REF][START_REF] Rees | Neural correlates of consciousness in humans[END_REF][START_REF] Block | Two neural correlates of consciousness[END_REF]. Though several of these issues have been examined by philosophers in journals specialized in ethics or philosophy of mind, I suspect that they would be treated differently (and in ways that would usefully complement those pre-existing explorations) if they were to become the focus of papers published in Biology and Philosophy.

Many other biological domains are philosophically interesting. Cell biology (8% in PNAS vs. 0% in Biology & Philosophy) raises some deep historical-philosophical questions (e.g., [START_REF] Bechtel | Discovering cell mechanisms: the creation of modern cell biology[END_REF], and additional philosophical topics that could be developed further. For example, [START_REF] Bechtel | The cell: locus or object of inquiry?[END_REF] has analysed the difference (and complementarity) between two trends in biological research about the cell, one that sees the cell as an explanandum, an entity to be decomposed into relevant intracellular mechanisms but that does not figure itself in biological explanations, and the other as an explanans, a functional and explanatorily relevant unit conceived as an integrated and organized system. More generally, cell biology raises crucial questions about what counts as a fundamental unit of life [START_REF] Canguilhem | La théorie cellulaire[END_REF]. Microbiology (7% in PNAS vs. 3% in Biology & Philosophy) has generated increasing amounts of philosophical literature in recent years (e.g., Franklin 2007;[START_REF] O'malley | Towards a philosophy of microbiology[END_REF][START_REF] Ereshefsky | Microbiology and the species problem[END_REF]O'Malley 2014). However, much philosophy of microbiology has focused on exploring "traditional" evolutionary topics (such as the species concept). Further philosophical work could be done on other aspects of microbiology, such as physiological, developmental and even clinical microbiology. Immunology (7% in PNAS vs. 0% in Biology & Philosophy) has much to say about several philosophical topics, including the definition of biological identity and individuality, and could be used to complement dominant evolutionary approaches to that topic [START_REF] Tauber | The Immune Self: Theory or Metaphor?[END_REF][START_REF] Pradeu | The Limits of the Self: Immunology and Biological Identity[END_REF][START_REF] Pradeu | Organisms or biological individuals? Combining physiological and evolutionary individuality[END_REF]. Systems biology 14 , a new and fastgrowing field, has already generated a rich philosophical literature, for example about part-whole relations, what constitutes a biological system (and which biological units qualify as "systems"), multilevel causation and the emergence of new functions at the system level [START_REF] O'malley | Fundamental issues in systems biology[END_REF], the possibility of formulating laws or at least design principles in biology [START_REF] Green | Revisiting generality in biology: systems biology and the quest for design principles[END_REF], the integration of mechanistic explanation and mathematical explanation (Brigandt 2013b), or the role in silico models can play in comparison with traditional experimental data (MacLeod and Nersessian 2013). Stem cell biology is also philosophically fascinating, for issues as diverse as modelling and interdisciplinarity [START_REF] Fagan | Philosophy of stem cell biology: knowledge in flesh and blood[END_REF], or the causality of tumour formation [START_REF] Laplane | Cancer stem cells: philosophy and therapies[END_REF].

Conclusion: Biologies and philosophy

This paper describes the significant mismatch in the biological domains studied by philosophers of biology and biologists, respectively, and asks how it can be explained. Obviously, future work will have to address these issues in greater detail. As suggested by two anonymous referees, a careful analysis of the talks given at ISHPSSB meetings and of the philosophy of biology papers published in generalist philosophy of science journals (such as Philosophy of Science, the British Journal for Philosophy of Science, etc.) would be very useful to offer a more balanced and precise picture. I hope I or others will be able to conduct this work in the near future.

Meanwhile, I will try to suggest a more personal conclusion. I have claimed that there is no compelling justification for the domination of evolutionary studies in philosophy of biology. In my closing remarks, I would like to argue that philosophy of biology should now adopt a much more open-minded and encompassing view, by exploring all the domains of today's biology. As the examples examined above show, many areas of the life sciences neglected by philosophers can be of considerable philosophical interest. But what exactly would be the benefits of this more diversified and integrative approach in philosophy of biology? I mention here four benefits.

Bring new biologists on board 14 Systems biology represents 1% of the biological papers published in PNAS and 0% of the papers published in Biology & Philosophy. The raw numbers indicate that, from 2009 (the year when the category "Systems biology" was created in PNAS) to 2015, PNAS published 240 papers about systems biology, with a tendency to increase over time, while Biology & Philosophy published 2.

Many philosophers of biology have interacted with evolutionary biologists, and sometimes have even had an influence on their debates. It is likely that many biologists working in other fields would be willing to interact with philosophers, but they would like the latter to share their enthusiasm for their current research topics, which are often quite distant from the issues that are exciting to philosophers. As an example, if one takes the three most recent biological "breakthroughs of the year" decided annually by the leading journal Science (HIV treatment as prevention in 2011; cancer immunotherapy in 2013; the CRISPR system, identified as a prokaryotic immune system and then recognized as genome editing system, in 2015)15 , none of them has been extensively discussed by philosophers of biology, even though they could all be very stimulating from a philosophical point of view. In fact, recent philosophical work done about the domains examined above (neuroscience, microbiology, immunology, systems biology) have already led to many papers cowritten by biologists and philosophers, which proves that bringing new biologists to philosophy of biology is perfectly possible.

Enrich traditional philosophical questions and/or responses to these questions In the same way as philosophers of biology demonstrated that raising issues of general philosophy of science (e.g., causality, explanation, the nature of models and theories) exclusively through the lens of philosophy of physics was problematic, it is equally unsuitable to raise issues of philosophy of biology only through the lens of philosophy of evolution. For example, the question of whether or not biology offers laws has been examined mainly from the point of view of evolution (e.g., [START_REF] Beatty | The evolutionary contingency thesis[END_REF], and perhaps raising this same question from the points of view of biophysics or systems biology would lead to novel answers, or perhaps to the reframing of the question itself. At the very least, it seems that integrating and articulating the answers given by these different domains would be useful.

Raise new philosophical issues

Philosophy of evolutionary biology has given rise to interesting philosophical questions, but those should not be seen as the only philosophically interesting questions. Neuroscience, for example, addresses issues about cognition and the "mind-body" problem (or its dissolution) that echo traditional and crucial questions of philosophy [START_REF] Churchland | Neurophilosophy: toward a unified science of the mind-brain[END_REF][START_REF] Churchland | The impact of neuroscience on philosophy[END_REF] but are not raised directly by philosophy of evolutionary biology. Similarly, longstanding philosophical issues about part-whole relations, boundaries, identity, etc., could be informed, or even sometimes significantly transformed, by a better knowledge of cell biology, ecology, immunology, or systems biology.

Enrich evolution itself

Even for philosophers of biology who would want to maintain that evolution should remain central to philosophy of biology, integrating lessons from a variety of fields would still be useful, as it would enrich our view of evolution. Some of the most exciting challenges in current evolutionary biology lie at the interface between evolution and development (e.g., [START_REF] Minelli | Evolving pathways: key themes in evolutionary developmental biology[END_REF][START_REF] Wagner | Homology, genes, and evolutionary innovation[END_REF], mechanisms of extended inheritance (e.g., Danchin et al. 2011), systems biology andcomputational biology (e.g., Wagner 2005;[START_REF] Wagner | Neutralism and selectionism: a network-based reconciliation[END_REF][START_REF] Koonin | The logic of chance: the nature and origin of biological evolution[END_REF], among many other fields (see [START_REF] Laland | Cause and effect in biology revisited: is Mayr's proximate-ultimate dichotomy still useful?[END_REF], and it is clear than many major philosophical issues come forth at these interfaces.

Overall, then, the extension of biological topics advocated here holds promises for philosophy at all levels, from general philosophy to philosophy of science and to philosophy of biology. I can only hope that the new generation of philosophers of biology will take up these topics, and that, thirty years from now, this paper will be considered pointless.

Figure 1 :

 1 Figure 1: Biological domains represented in Biology & Philosophy from 1986 to 2002 (based on (Gayon 2009)).

  Biology & Philosophy from 2003 to 2015 were about psychological and cognitive sciences. • Many biological domains were absent or virtually absent, including biophysics and computational biology, cell biology, immunology, physiology, and systems biology. • Some biological domains were represented only or mainly because a special issue had been devoted to them. Two clear examples concerned developmental biology (7 of a total of 19 papers about development from 2003 to 2015 belong to a single special issue on 'evo-devo', edited by Sahotra Sarkar and Jason Robert in March 2003, which could effectively have been counted as about evolution), and microbiology (12 of a total of 13 papers about microbiology from 2003 to 2015 belong to a single special issue on "Philosophy and the Microbe" edited by Maureen O'Malley in March 2013).

Figure 2 :

 2 Figure 2: Biological domains represented in Biology & Philosophy from 2003 to 2015.

  Figure 3 shows visually the distribution of biological domains in PNAS from 2003 to 2015.

Fig. 3 .

 3 Fig. 3. Biological domains represented in PNAS from 2003 to 2015. This figure shows the relative weight of each biological domain with regard to all the biological papers (not all the scientific papers more generally) published in PNAS.

Table S1 (see Supplementary material). The main results appear in Figure 1, and

  can be summarized as follows:• 24.5% of all the papers published in Biology & Philosophy from 1986 to 2002 were about evolution strictly speaking. • 55% of the papers published in Biology & Philosophy from 1986 to 2002 were about evolution broadly construed, which includes the species problem, taxonomy, evolutionary epistemology, and evolutionary ethics. • Domains such as physiology, biochemistry, and biophysics are absent.

	Genetics; Evolutionary epistemology; Evolutionary ethics; Function, teleology,
	design; Reflections on biology in general; Others (which, here, includes papers
	dealing with some underrepresented areas of biology, e.g., development or
	physiology, strictly historical papers, and papers that are of a very general
	philosophical nature). Second, I re-examined carefully all the papers published from
	1986 to 2002, and, with Gayon's help, I made some important corrections to his
	original data.
	The details of these modified data appear in
	. First, I did not
	use the three broad categories mentioned above; instead, I re-classified all the papers
	into 10 categories, to better reflect all the diverse biological areas found in Biology &
	Philosophy. The 10 categories are: Evolution; Taxonomy; Species; Ecology;

  papers published in PNAS in 2003, but only 9% in 2015, while environmental sciences moved from 1% in 2005 to 4% in 2015.

	PNAS%2003)2015%	Physiology* Plant*Biology* 4%* Popula;on*biology* 0%*	3%* sciences* Psychological* and*cogni;ve* Sustainability*science* 1%*	Systems* Biology** 1%*	Agricultural*sciences* 1%* Anthropology* 2%* Applied*biological*sciences* 2%*
	Pharmacology*	2%*
	1%*					Biochemistry*
						12%*
	Neuroscience*
	12%*
						Biophysics*&*Computa;onal*Biology*
						9%*
	Microbiology*
	7%*					Cell*biology*
						8%*
	Medical*sciences*
	10%*					Ecology*
						3%*
			Immunology*&*inflamma;on*		Evolu;on*
				7%*	Gene;cs*	5%*
					5%*	Developmental*biology*
						3%*
						Environmental*sciences*
						2%*

biological

•

  The representations of biological domains in PNAS and Biology & Philosophy from 2003 to 2015 are extremely different. • With regard to evolution, the comparison reveals what could be called the "5-60% rule": evolution represents only 5% of the papers published in PNAS, but approximately 60% of the papers published in Biology & Philosophy. • Several of the most widely represented biological domains in PNAS are absent or almost absent from Biology & Philosophy. This includes biochemistry (12% vs. 2%), biophysics and computational biology (9% vs. 0%), cell biology (8% vs. 0%), immunology (7% vs. 0%), and the neurosciences (12% vs. 3%) 9 . • The representation of biological domains in Biology & Philosophy through time remains unaffected by the changes in the representation of biological domains in PNAS: for example, environmental sciences were not more represented in Biology & Philosophy in 2015 than in 2003, even though this domain increased significantly its representation in PNAS between these two dates.

This includes the work ofWoodger (1929), Beckner (1959), and Goudge (1961), among others. Importantly,[START_REF] Byron | Whence Philosophy of Biology?[END_REF] has shown that, contrary to common thought, biological questions have not been entirely neglected by philosophers of the logical positivist period.

The first meeting of the group that eventually became the ISHPSSB was held at Virginia Tech in 1983, but the ISHPSSB was officially named in 1989 and its governance was formalized in 1990. I am grateful to Dick Burian for his detailed account about this history.

Naturally, the methodology used in this paper is extremely simple, and it would not be approved by statisticians. Moreover, I analysed the data of only one journal in philosophy of biology, Biology & Philosophy (based on the argument that it is the only journal devoted exclusively to philosophy of biology), but it is clear that philosophy of biology is often published in other journals, so my conclusions do not necessarily apply to philosophy of biology as a whole.

In my counting, my guiding question was always: What domain of biological sciences (if any) is used (or mainly used)? In most cases, this question was not problematic, as the papers were indeed exploring a specific biological domain. In a very limited number of cases, it was a bit trickier, in particular when the main theme of the paper was a general question, using biological domains in a rather distant way (as, for example, when a paper would raise the general issue of whether or not there are biological laws, without using a particular field to address this question). But, again, this situation was rare.

The category "Others" includes papers that analyse problems of general philosophy of science applied to biology (e.g., causality, information, reductionism, emergence, theories-models, laws, mechanisms, functions). If a paper had a general scope but was based on an extensive analysis of one given biological domain, it was classified in the corresponding entry (e.g., a paper on reductionism in developmental biology appears in the "Developmental biology" entry).

Interestingly, this domination of evolution in philosophy of biology seems to go against the intentions of the two successive editors of Biology & Philosophy, Michael Ruse and Kim Sterelny. In his first editorial, Sterelny mentions his desire to continue Ruse's project to not focus exclusively on evolution.

PNAS is particularly respected by evolutionists, and this journal publishes many papers about evolution, which was important to avoid any bias.

The classification of PNAS papers into precise sections was not frequent in the 1980s and the 1990s (many papers were attributed to the very broad category "research article"), and became systematic only with the issue ofNovember 12, 1996 (vol.

no. 23). As a consequence, all the figures given here (in the Supplementary material) concerning the pre-1996 period must be taken with caution.

The example of the medical sciences (10% vs. 2%) is interesting too, but slightly different because philosophers of medicine often publish their papers in their own journals.

"Traditionally, evolution has been the focus of most philosophical attention. While it surely remains true that 'nothing in biology makes sense except in light of evolution' (Dobzhansky, 1973), this tradition within the philosophy of biology is myopic insofar as it ignores much -if not most -of the work in contemporary biology."Sarkar and Plutynski (2008, p. xviii)

Grene was extremely interested in evolution (e.g.,[START_REF] Grene | Two evolutionary theories[END_REF]) and this explains, at least in part, why[START_REF] Hull | The History of the Philosophy of Biology[END_REF] described her as the "grandmother" of philosophy of biology, despite their many disagreements (e.g.,[START_REF] Hull | What philosophy of biology is not[END_REF].

The "invention of predecessors" is a classic phenomenon in science and elsewhere: see, e.g.,[START_REF] Canguilhem | The Object of the History of Sciences[END_REF].

One should keep in mind that philosophers of neuroscience often publish their research in cognitive science journals and in general philosophy of science journals. My point here is simply that very few of them publish their research in Biology & Philosophy.

In 2012 and 2014, the "breakthroughs of the year" decided by Science concerned physics, not biology.

Dick Burian and Jean Gayon also provided very useful information about the history of ISHPSSB. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme -grant agreement n° 637647 -IDEM.