Cross Ratios, Anosov Representations and the Energy Functional on Teichmüller Space

François LABOURIE ^{∗}

October 28, 2008

Résumé

Nous étudions deux classes de représentations linéaires d’un groupe de surface : les représentations de Hitchin et les représentations symplectiques maximales. En reliant ces représentations à des birapports, nous montrons qu’elles sont déplaçantes, c’est à dire que leurs longueurs de translation sont grossièrement contrôlées par celles du graphe de Cayley. Ceci nous permet de montrer que le groupe modulaire agit proprement sur l’espace de ces représentations et que la fonctionnelle énergie associée à une telle représentation est propre. Nous en déduisons alors l’existence de surfaces minimales dans les quotients d’espaces symétriques associés et en tirons deux conséquences : un résultat de rigidité pour les représentations symplectiques et un résultat partiel concernant la description de la composante de Hitchin en termes purement holomorphes.

Abstract

We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence of minimal surfaces in the quotient of the associated symmetric spaces, a fact which leads to two consequences: a rigidity result for maximal symplectic representations and a partial result concerning a purely holomorphic description of the Hichin component.

1 Introduction

Let S be a closed connected oriented surface of genus greater than one. Monodromies of hyperbolic structures on S define a distinguished class of homomorphisms from the fundamental group π_{1}(S) into PSL(2, ℝ). In this paper we study two generalisations of these surface group representations, one in which we replace PSL(2, ℝ) by PSL(n, ℝ) and one in which we generalise PSL(2, ℝ) = PSp(2, ℝ) to PSp(2n, ℝ).

∗Univ. Paris-Sud, Laboratoire de Mathématiques, Orsay F-91405 Cedex; CNRS, Orsay cedex, F-91405

The first generalisation uses the irreducible representation of PSL(2, ℝ) in PSL(n, ℝ). A Fuchsian representation from π_{1}(S) into PSL(n, ℝ) is a representation which decomposes as the product of a faithful cocompact representation from π_{1}(S) to PSL(2, ℝ) and the irreducible representation from PSL(2, ℝ) to PSL(n, ℝ) (see [30] and Section 4.1). The representations we study, called Hitchin representations, are those which may be deformed into a Fuchsian representation. In [26], Hitchin studies the moduli space of reductive (i.e. whose Zariski closure is a reductive group) Hitchin representations. For a combinatorial point of view on a related subject, see Fock and Goncharov in [16].

The second generalisation exploits the fact that the homogeneous space M associated to PSp(2n, ℝ) is Hermitian symmetric and thus carries an invariant symplectic form ω (see Section 4.2.1 for details). Given a representation ρ from π_{1}(S) to PSp(2n, ℝ), if f is any ρ-equivariant map from the universal cover of S to M, then f^{∗}ω is invariant under the action of π_{1}(S). The following number

τ(ρ) = \frac{n}{2π} ∫_{S} f^{∗}ω

is then an integer independent of the choice of f. This number, called the Toledo invariant of ρ, remains constant under continuous deformations of the representation and satisfies a generalised Milnor-Wood Inequality (see [43])

|τ(ρ)| ⩽ n|χ(S)|.

By definition, a maximal symplectic representation is one for which the Toledo invariant attains the upper bound in this inequality. The notion of maximality and a suitable version of the Milnor-Wood Inequality extend to all Hermitian symmetric spaces. W. Goldman shows in [18, 19] that maximal representations in PSL(2, ℝ) are precisely monodromies of hyperbolic structures. In the general case, these maximal representations have been extensively studied by Bradlow, García-Prada, Gothen, Mundet i Riera (as well as Xia in a specific example) ([3, 5, 4, 17, 22, 45]) using Higgs bundle techniques on one hand and Burger, Iozzi and Wienhard ([6, 9, 8, 44]) using bounded cohomology techniques on the other hand.

Both type of representations – Hitchin representations and maximal symplectic representations – can be thought of as generalisations of the PSL(2, ℝ)- representations which arise from monodromies of hyperbolic structures and hence as generalising Teichmüller-Thurston theory.

The maximal symplectic representations and the Hitchin representations are known to share several fundamental properties, including:

• They are Anosov as defined in [30]. For Hitchin representations this is proved in [30], for maximal representations this is shown in [7] by Burger, Iozzi, Wienhard and the author.

• The Zariski closure of the images are reductive. For Hitchin representations, see Proposition 4.1.5. For maximal representations, this is proved by Burger, Iozzi and Wienhard in [8].

• They are discrete: see [30] for Hitchin representations and the proof by Burger, Iozzi and Wienhard in [8] for maximal representations.

The results presented in this paper extend this list of common features by showing that both types of representation have the property that we call displacing. More precisely, let Γ be a finitely generated subgroup of the isometry group of a metric space X: we say that Γ is displacing if, given a finite generating set G of Γ, there exist positive constants A and B such that for all elements γ of Γ

inf_{x∈X} d(x, γ(x)) ⩾ A inf_{η∈Γ} ∥ηγη^{−1}∥_{G} − B.

where ∥γ∥_{G} is the word length of γ with respect to G. It is easy to check that this definition is independent of the generating set G. Note that cocompact groups are always displacing, as are convex-cocompact groups whenever X is Hadamard (i.e. complete, nonpositively curved and simply connected). If ρ is a representation from a finitely generated group Γ with values in a connected semi-simple real Lie group G without compact factor and with trivial centre, then ρ is displacing if the group ρ(Γ) is displacing as a group of isometries of the associated symmetric space.

We now briefly summarise results of Delzant, Guichard, Mozes and the author in [13] which compare this notion to the fact that orbit maps are quasiisometries. While the two notions turn out to be equivalent for surface groups and more generally hyperbolic groups, they are not equivalent for every group: there are known examples which have displacing representations whose orbit maps are not quasiisometries and also nondisplacing representations for which the orbit maps are quasiisometries.

The starting point of this article is the following result.

Theorem 1.0.1 Hitchin and maximal symplectic representations are displacing.

It has already been observed by Burger, Iozzi, Wienhard and the author in [7] that the orbit maps are quasiisometries for maximal symplectic representations. Here we prove the theorem by relating Hitchin representations and maximal symplectic representations to cross ratios (cf Theorems 4.1.6 and 4.2.4).

The two main applications of this result are that

• the mapping class group acts properly on certain moduli spaces, and

• the energy functional on Teichmüller space is proper.

Let us be more specific. Let Hom_{H}(π_{1}(S), PSL(n, ℝ)) be the space of Hitchin homomorphisms and

Rep_{H}(π_{1}(S), PSL(n, ℝ)) = Hom_{H}(π_{1}(S), PSL(n, ℝ))/PSL(n, ℝ),

where the action of PSL(n, ℝ) is by conjugation.

Similarly, let Hom_{T} (π_{1}(S), PSp(2n, ℝ)) be the space of maximal symplectic homomorphisms and

Rep_{T} (π_{1}(S), PSp(2n, ℝ)) = Hom_{T} (π_{1}(S), PSp(2n, ℝ))/PSp(2n, ℝ).

Both spaces Hom_{H} and Hom_{T} are unions of connected components, each of which is a component of the corresponding space of reductive homomorphisms. Moreover, since they consist of reductive representations, the quotient spaces Rep_{H} and Rep_{T} are Hausdorff (hence locally compact). This last fact follows from the identification due to Hitchin [25] of reductive representations with polystable Higgs bundle, although more direct proofs could be obtained. The mapping class group M(S) – that is the group of outer automorphisms of π_{1}(S) which may be represented by orientation preserving diffeomorphisms – acts by precomposition on these spaces. The following result will be an immediate consequence of Theorem 1.0.1.

Theorem 1.0.2 The mapping class group M(S) acts properly by precomposition on the spaces Rep_{H}(π_{1}(S), PSL(n, ℝ)) and Rep_{T} (π_{1}(S), PSp(2n, ℝ)).

In the case of PSL(3, ℝ), W. Goldman proved in [20] that the mapping class group acts properly on the moduli space of convex ℝℙ^{2} structures. Moreover, together with Choi in [10], he identified this moduli space with the Hitchin component.

Our second main application concerns the energy functional. We first recall briefly the general framework and refer to Paragraph 5 for precise definitions. Let ρ be a representation from π_{1}(S) to a connected semi-simple real Lie group G without compact factor and with trivial centre. Let M be the symmetric space associated to G and let M_{ρ} be the flat M-bundle over S defined by the representation ρ. Let Γ(S, M_{ρ}) be the space of smooth sections of M_{ρ}. If J is a complex structure on S and f an element of Γ(S, M_{ρ}), we define

Energy_{J}(f) = ∫_{S}⟨df ∧ df ∘J⟩.

Then, the energy functional e_{ρ}, associated to the representation ρ, is the map from the space of all complex structures on S to the real numbers defined by

e_{ρ}(J) = inf{Energy_{J}(f) | f ∈ Γ(S, M_{ρ})}.

The value of this function depends only on the isotopy class of the complex structure J, and hence defines a function on Teichmüller space. Denoted by e_{ρ} and also called the energy functional, this function is smooth on Teichmüller space (cf Paragraph 5.2). We shall prove the following result

Theorem 1.0.3 If ρ is a Hitchin representation or a maximal symplectic representation, then the energy functional e_{ρ} is a proper function on Teichmüller space.

It is classical that critical points of the energy functional are related to minimal surfaces. Indeed, using Gulliver’s definition of a branched immersion ([24]), we obtain the following consequence

Corollary 1.0.4 Let ρ be a Hitchin or maximal symplectic representation. Then there exists a minimal branched immersion from S into M/ρ(π_{1}(S)) which represents ρ at the level of homotopy groups.

This corollary will lead to two applications that we shall explain in the next section.

I wish to thank L. Lemaire for his help on minimal surfaces and harmonic mappings, F. Paulin for a very helpful comment on irreducible representations and many useful comments on the writing up, A. Iozzi for clarifications on Toledo invariant, M. Burger, O. García-Prada, O. Guichard and A. Wienhard for numerous conversations and useful remarks on inconsistencies of the first draft, as well as W. Goldman for his interest and his help on the writing up. I am also extremely grateful to S. Bradlow for his final comments on this article.

2 Application of the main results and outline of the paper

2.1 The minimal area and the Toledo invariant

We restrict ourselves in this paragraph to representations of surface groups with values in PSp(2n, ℝ).

Definition 2.1.1 [Minimal area] The minimal area of a representation ρ from π_{1}(S) to PSp(2n, ℝ) is

MinArea(ρ) = inf{e_{ρ}(J) | J ∈ T (S)}.

Definition 2.1.2 [Diagonal representation] A homomorphism ρ from π_{1}(S) with values in PSp(2n, ℝ) is diagonal if it factors as ρ = φ ∘ δ ∘ σ where σ is a cocompact homomorphism of π_{1}(S) into PSL(2, ℝ), δ is the diagonal mapping from PSL(2, ℝ) into

∏_{i=1}^{i=n} PSL(2, ℝ)

and φ is an embedding of this product in PSp(2n, ℝ) corresponding to a decomposition of ℝ^{2n} in a direct sum of 2-dimensional orthogonal symplectic vector spaces.

Note that the set of diagonal homomorphisms is invariant under conjugation.

Theorem 2.1.3 For every representation ρ, we have

\frac{n}{2π} MinArea(ρ) ⩾ |τ(ρ)|,

where τ(ρ) is the Toledo invariant of ρ. If furthermore ρ is maximal and

\frac{n}{2π} MinArea(ρ) = τ(ρ),

then ρ is a diagonal representation.

As a corollary of the proof, we have the following result

Corollary 2.1.4 Let ρ be a maximal representation. Assume that there exists a holomorphic equivariant map f from S to the associated symmetric space of PSp(2n, ℝ). Then ρ is diagonal and f is totally geodesic.

2.2 The Hitchin map is surjective

In his article [26], N. Hitchin gives explicit parametrisations of the Hitchin components Rep_{H}(π_{1}(S), PSL(n, ℝ)). Namely, given a choice of a complex structure J over S, he produces a homeomorphism

H_{J} : Q(2, J) ⊕ … ⊕ Q(n, J) → Rep_{H}(π_{1}(S), PSL(n, ℝ)),

where Q(p, J) denotes the space of holomorphic p-differentials on the Riemann surface (S, J). The first step in the construction of this map uses results from K. Corlette’s seminal paper [11] – see also [15, 27] – to identify conjugacy classes of representations with harmonic mappings to symmetric spaces. The second step is to associate holomorphic differentials to a harmonic mapping with values in a symmetric space. This is accomplished by means of a construction very similar to the construction of characteristic classes in Chern-Weil theory – see Paragraph 8.1.3.

However, in this construction, the homeomorphism H_{J} depends on the choice of the complex structure J. In particular, this choice breaks the mapping class group symmetry. The construction thus does not give any information on the topological nature of the quotient of Rep_{H}(π_{1}(S), PSL(n, ℝ)) by the mapping class group.

We now explain a construction which is equivariant with respect to the action of the mapping class group and which conjecturally leads to a complex analytic description of the quotient. Let E^{(n)} be the vector bundle over Teichmüller space whose fibre above the (isotopy class of the) complex structure J is

E_{J}^{(n)} = Q(3, J) ⊕ … ⊕ Q(n, J).

The dimension of the total space of E^{(n)} is the same as that of the Hitchin component

Rep_{H}(π_{1}(S), PSL(n, ℝ))

since the dimension of the "missing" quadratic differentials in E_{J}^{(n)} accounts for the dimension of Teichmüller space. The Hitchin map ^{1} is then the map from E^{(n)} to Rep_{H}(π_{1}(S), PSL(n, ℝ)) defined by

1We are aware that this terminology is awkward since this Hitchin map is some kind of an inverse of what is usually called the Hitchin fibration.

(J, ω) → H_{J}(0, ω).

It follows from Hitchin’s construction that this map is equivariant with respect to the mapping class group action. We prove

Theorem 2.2.1 The Hitchin map is surjective.

Our strategy is to identify E^{(n)} with the moduli space of equivariant minimal surfaces in the associated symmetric space and then, by tracking a critical point of the energy, to prove that there exists an equivariant minimal surface for every representation.

We conjectured in [30] that the Hitchin map is a homeomorphism. This would be a consequence of the following:

Conjecture 2.2.2 If ρ is a Hitchin representation, then there exists a non degenerate minimum of e_{ρ}.

This conjecture is well known to be true for n = 2. For n = 3, it is proved in [31] by relating real projective structures, affine spheres and Blaschke metrics (as in [28, 31] or in [34]). By our previous discussion, Conjecture 2.2.2 would imply the following consequence which sheds light on the action of the mapping class group M(S) on the Hitchin components:

Conjecture 2.2.3 The quotient Rep_{H}(π_{1}(S), PSL(n, ℝ))/M(S) is homeomorphic to the total space of the vector bundle E – in the orbifold sense – over the Riemann moduli space, whose fibre at a point J is

E_{J} = Q(3, J) ⊕ … ⊕ Q(n, J).

Again, by the previous discussion this result is true for n = 2 and n = 3.

2.3 Outline of the paper

• 3. Cross ratios and the boundary at infinity. We recall the basic definitions (cross ratios, periods), explain how cross ratios are related to flows and finally show how this relation helps to control the growth of the periods (Proposition 3.3.1).

• 4. Representations and cross ratios. We explain that Hitchin and maximal symplectic representations are reductive, how they generate curves in Grassmannian spaces and how they relate to cross ratios.

• 5. The energy functional and the minimal area. We recall basic results about existence of equivariant harmonic mappings in symmetric spaces as well as classical definitions and results concerning minimal surfaces, energy and Teichmüller space.

• 6. Displacing representations and the energy functional. We introduce the notion of displacing representations. Using the relation with cross ratios, we show that the Hitchin and maximal symplectic representations are displacing. We then prove the main properties of displacing representations and obtain Theorems 1.0.2 and 1.0.3. As a consequence, we deduce the existence of equivariant minimal surfaces for our two main examples.

• 7. The Toledo invariant and the minimal area. We prove Theorem 2.1.3.

• 8. The Hitchin map. We prove Theorem 2.2.1.

3 Cross ratios and the boundary at infinity

3.1 The boundary at infinity

Let ∂_{∞}π_{1}(S) be the boundary at infinity of π_{1}(S). We recall that ∂_{∞}π_{1}(S) is a circle with a Hölder structure and is equipped with an action of π_{1}(S) by Hölder homeomorphisms. Up to equivariant Hölder homeomorphisms, this action can be characterised by the following two properties

• every orbit is dense,

• every nontrivial element of π_{1}(S) has exactly two fixed points: one attractive and one repulsive.

If one fixes an uniformisation of the universal cover of the surface equipped with a complex structure, then ∂_{∞}π_{1}(S) can be identified with the real projective line ℝℙ^{1} considered as the boundary of the Poincaré disk model.

3.2 Cross ratios

Let X be a metric space equipped with an action of a group Γ by Hölder homeomorphisms. Let

X^{4∗} = {(x, y, z, t) ∈ X^{4} | x ≠ t and y ≠ z}.

We equip X^{4∗} with the diagonal action of Γ. In the sequel, our main examples are X = ∂_{∞}π_{1}(S) equipped with the natural action of π_{1}(S) by Hölder homeomorphisms, or variants of that.

Definition 3.2.1 [Cross ratio] A cross ratio on X is a real valued Γ-invariant Hölder function 𝔹 on X^{4∗} which satisfies the following rules

𝔹(x, y, z, t) = 𝔹(z, t, x, y),  (1)
𝔹(x, y, z, t) = 0 ⇔ x = y or z = t,  (2)
𝔹(x, y, z, t) = 1 ⇔ x = z or y = t,  (3)
𝔹(x, y, z, t) = 𝔹(x, y, z, w)𝔹(x, w, z, t),  (4)
𝔹(x, y, z, t) = 𝔹(x, y, w, t)𝔹(w, y, z, t).  (5)

The classical cross ratio b on ℝℙ^{1}, defined in an affine chart by

b(x, y, z, t) = \frac{(x − y)(z − t)}{(x − t)(y − z)},

is an example of a cross ratio with respect to the action of PSL(2, ℝ).

Definition 3.2.2 [Period] Let 𝔹 be a cross ratio on ∂_{∞}π_{1}(S) and γ be a nontrivial element in π_{1}(S). The period ℓ_{𝔹}(γ) is

ℓ_{𝔹}(γ) = log |𝔹(γ^{−}, γy, γ^{+}, y)|,  (6)

where γ^{+} and γ^{−} are respectively the attracting and repelling fixed points of γ on ∂_{∞}π_{1}(S) and y is any element of ∂_{∞}π_{1}(S) different from γ^{+} and γ^{−}.

Relation (4) and the invariance under the action of γ imply that ℓ_{𝔹}(γ) does not depend on y. Moreover, by Equation (1), ℓ_{𝔹}(γ) = ℓ_{𝔹}(γ^{−1}).

For more information and examples on a related notion see the work of Otal and Ledrappier in [37, 33]. For other applications to representations of surface groups, see [29, 32].

3.3 Periods and lengths

The next proposition compares periods with length of geodesics.

Proposition 3.3.1 We fix a hyperbolic metric on S. For every nontrivial γ in π_{1}(S), let λ(γ) be the length of the closed geodesic associated to γ for this hyperbolic metric. Let 𝔹 be a cross ratio. Then there exists a positive constant A, depending only on the cross ratio and the choice of the hyperbolic metric, such that or all nontrivial element γ in π_{1}(S)

\frac{1}{A}λ(γ) ⩽ ℓ_{𝔹}(γ) ⩽ Aλ(γ).

The idea of the proof is to define compatible flows on the space of oriented triples of pairwise distinct points of ∂_{∞}π_{1}(S) and study their periodic orbits.

3.3.1 Compatible flows on the space of oriented triples

Recall first that the orientation on S induces an orientation on ∂_{∞}π_{1}(S).

Definition 3.3.2 [Oriented triples] We denote by ∂_{∞}π_{1}(S)^{3+} the space of oriented triples of pairwise distinct points of ∂_{∞}π_{1}(S).

The quotient ∂_{∞}π_{1}(S)^{3+}/π_{1}(S) is compact and homeomorphic to the unitary tangent bundle of the surface S for any auxiliary metric.

Definition 3.3.3 [Compatible flow] A continuous flow {ϕ_{t}}_{t∈ℝ} on ∂_{∞}π_{1}(S)^{3+} is compatible if

• every homeomorphism ϕ_{t} is π_{1}(S)-equivariant,

• every homeomorphism ϕ_{t} acts without fixed points,

• for every real number t, for every triple (x, z, y) in ∂_{∞}π_{1}(S)^{3+}, there exists u in ∂_{∞}π_{1}(S) such that ϕ_{t}(x, z, y) = (x, u, y).

If we identify ∂_{∞}π_{1}(S)^{3+} with the unit tangent bundle of S for some hyperbolic metric, then a compatible flow is nothing but a reparametrisation of the geodesic flow.

Definition 3.3.4 [Period for a flow] For notational convenience, given a compatible flow {ϕ_{t}}_{t∈ℝ}, we define for every real number t the map

ϕ̂_{t} : ∂_{∞}π_{1}(S)^{3+} → ∂_{∞}π_{1}(S),

by the condition that (x, ϕ̂_{t}(x, z, y), y) = ϕ_{t}(x, z, y).

If γ is a nontrivial element of π_{1}(S), we then define the period of γ with respect to {ϕ_{t}}_{t∈ℝ} to be the least positive number ℓ_{ϕ}(γ) such that

ϕ̂_{ℓ_{ϕ}(γ)}(γ^{−}, y, γ^{+}) = γ(y).

We first prove

Proposition 3.3.5 Let {ϕ_{t}}_{t∈ℝ} and {ψ_{t}}_{t∈ℝ} be two compatible flows. Then there exists a constant K such that for all nontrivial element γ of π_{1}(S)

ℓ_{ψ}(γ) ⩽ Kℓ_{ϕ}(γ).

Proof: There exists a positive continuous π_{1}(S)-invariant positive function Θ defined on ∂_{∞}π_{1}(S)^{3+} such that

ψ̂_{1}(x, y, z) = ϕ̂_{Θ(x,y,z)}(x, y, z).

Since ∂_{∞}π_{1}(S)^{3+}/π_{1}(S) is compact, it follows that there exists a constant A such that for all oriented triple (x, y, z)

0 < Θ(x, y, z) ⩽ A.

Then we have

ℓ_{ψ}(γ) ⩽ Aℓ_{ϕ}(γ) + A.

By compactness of ∂_{∞}π_{1}(S)^{3+}/π_{1}(S), there exists a constant l such that for all nontrivial element γ of π_{1}(S), we have

ℓ_{ϕ}(γ) ⩾ l > 0.

Therefore it follows that

ℓ_{ψ}(γ) ⩽ (A + A/l)ℓ_{ϕ}(γ).

Q.e.d.

3.3.2 Proof of Proposition 3.3.1

We show that every cross ratio is associated to a compatible flow and thereby conclude the proof of Proposition 3.3.1.

Proposition 3.3.6 Let 𝔹 be a cross ratio. Let x and y be two distinct elements of ∂_{∞}π_{1}(S). Let I be one of the connected components of ∂_{∞}π_{1}(S) \ {x, y}. Let z be an element of I. Then the map from I to ℝ given by

φ : t → log(𝔹(x, t, y, z)),

is a homeomorphism.

Proof: By the definition of cross ratio, if (x, s, t, y) is cyclically oriented, then 𝔹(x, s, y, t) is greater than 1. In particular 𝔹(x, t, y, z) is positive if z and t belong to I and thus φ is well defined.

We now prove that φ is injective. Suppose that φ(s) = φ(t). This implies that

𝔹(x, t, y, s) = \frac{𝔹(x, t, y, z)}{𝔹(x, s, y, z)} = e^{φ(t)−φ(s)} = 1.

Hence s = t by the definition of a cross ratio. The same proof shows that φ is increasing. It follows that φ(I) is an interval ]α, β[.

We now prove that β = +∞. Assume on the contrary that β is finite. By definition

lim_{t→y} log(𝔹(x, t, y, z)) = β.

Choose an auxiliary compatible flow {ψ_{t}}_{t∈ℝ} on ∂_{∞}π_{1}(S)^{3+}. Since (x, t, ψ̂_{1}(x, t, y), y) is cyclically oriented, we have

lim_{t→y} ψ̂_{1}(x, t, y) = y.

Hence,

lim_{t→y} 𝔹(x, t, y, ψ̂_{1}(x, t, y)) = lim_{t→y} (\frac{𝔹(x, t, y, z)}{𝔹(x, ψ̂_{1}(x, t, y), y, z)}) = 1.  (7)

Now choose a sequence {t_{n}}_{n∈ℕ} converging to y. Since the quotient ∂_{∞}π_{1}(S)^{3+}/π_{1}(S) is compact, there exists a sequence {γ_{n}}_{n∈ℕ} of elements in π_{1}(S) and an oriented triple (X, Y, T) of pairwise distinct elements in ∂_{∞}π_{1}(S) such that

lim_{n→∞} (γ_{n}(x), γ_{n}(y), γ_{n}(t_{n})) = (X, Y, T) ∈ ∂_{∞}π_{1}(S)^{3+} .

If follows from Assertion (7) that

𝔹(X, T, Y, ψ̂_{1}(X, T, Y)) = lim_{n→∞} 𝔹(γ_{n}(x), γ_{n}(t_{n}), γ_{n}(y), ψ̂_{1}(γ_{n}(x), γ_{n}(t_{n}), γ_{n}(y))
= lim_{n→∞} 𝔹(γ_{n}(x), γ_{n}(t_{n}), γ_{n}(y), γ_{n}(ψ̂_{1}(x, t_{n}, y))
= lim_{n→∞} 𝔹(x, t_{n}, y, ψ̂_{1}(x, t_{n}, y))
= 1.

This yields the conclusion that T = ψ̂_{1}(X, T, Y), which contradicts the fact that ψ_{1} has no fixed points. A similar argument yields α = −∞. Therefore φ is a homeomorphism. Q.e.d.

Proposition 3.3.7 There exists a compatible flow {ϕ_{t}}_{t∈ℝ} on ∂_{∞}π_{1}(S)^{3+} such that log(𝔹(x, z, y, ϕ̂_{t}(x, z, y))) = t.

Proof: By Proposition 3.3.6, the family of homeomorphisms {ϕ_{t}}_{t∈ℝ} is well defined. The multiplicative cocycle relation

𝔹(x, y, z, t).𝔹(x, t, z, u) = 𝔹(x, y, z, u),

implies that {ϕ_{t}}_{t∈ℝ} is a one-parameter group. Q.e.d.

Proposition 3.3.1 is now a consequence of Propositions 3.3.7 and 3.3.5.

4 Representations and cross ratios

We shall in the sequel distinguish between homomorphism from a group to an other and representation which we consider as a class of homomorphism up to conjugation. Whenever a property is defined for a homomorphism, this definition will be extended to representation whenever the property is invariant by conjugation.

We explain that our two favourite classes of representations from π_{1}(S) are associated to cross ratios whose periods can be computed from holonomy. We also recall that these representations are reductive.

4.1 Hitchin representations

Definition 4.1.1 [Deformation] Let ρ_{0} and ρ_{1} be representations of π_{1}(S) with values in a topological group G. A deformation of ρ_{0} into ρ_{1} is a family of representations {ρ_{t}}_{t∈[0,1]} such that for every element γ of π_{1}(S), the map t → ρ_{t}(γ) is continuous.

Definition 4.1.2 [Hitchin homomorphisms] Following [30], a Fuchsian homomorphism from π_{1}(S) to PSL(n, ℝ) is a homomorphism ρ which factors as ρ = ι∘ρ_{0}, where ρ_{0} is a convex-cocompact injective homomorphism with values in PSL(2, ℝ) and ι is an irreducible homomorphism from PSL(2, ℝ) to PSL(n, ℝ).

A Hitchin homomorphism is a homomorphism that may be deformed into a Fuchsian homomorphism.

Definition 4.1.3 [Reductive homomorphism] A homomorphism is reductive if the Zariski closure of its image is a reductive group.

Later we will show that every Hitchin homomorphism is reductive. In [26], Hitchin studies the moduli space of reductive Hitchin representations.

In [30], we showed the following result.

Theorem 4.1.4 Let ρ be a reductive Hitchin representation. Let γ be a nontrivial element of π_{1}(S). Then ρ(γ) is ℝ-split.

4.1.1 Reductivity

Recall that a subgroup of (or a homomorphism with values in) PSL(n, ℝ) is irreducible if it does not preserve any proper subspace of ℝ^{n}. We show the following result.

Proposition 4.1.5 Every Hitchin representation is irreducible.

Proof: In [30], Lemma 10.1, using elementary observations on Higgs bundle, we show that every reductive Hitchin representation is irreducible. We now explain that every Hitchin representation is reductive, hence irreducible.

Obviously the set of irreducible homomorphisms is open since its complement is closed. Hence, to conclude the proof, it suffices to show that the set of irreducible Hitchin homomorphisms is closed.

Let ρ be a limit of Hitchin homomorphisms. Let G be the Zariski closure of ρ(π_{1}(S)). Let N be the nilradical of G. Let R = G/N be the reductive part of G – i.e the Levi component. We identify R with a subgroup of G so that G = N ⋊ R. Let π be the projection from G to R.

We first prove that π ∘ρ is also a limit of reductive homomorphisms. Indeed, there exists an element h in the centraliser of R such that for all u in N,

lim_{n→∞} h^{−n}uh^{n} = 1.

It follows that

π ∘ ρ = lim_{n→∞} h^{−n}ρh^{n}.

In particular, π ∘ ρ is also a limit of Hitchin homomorphisms and hence a Hitchin homomorphism itself. By construction, π ∘ ρ is reductive and hence irreducible.

We now prove by contradiction that N is trivial. Assume the contrary. Then, since N is unipotent, the set of vectors fixed by N is a proper subspace of ℝ^{n}. This set is fixed by R and hence by π ∘ ρ, from which it follows that π ∘ ρ is not irreducible. Hence we obtain a contradiction.

We have just shown that N is trivial. By definition ρ is reductive, and hence irreducible. Q.e.d.

I owe this argument to F. Paulin.

4.1.2 Cross ratios

In [29], we showed how to associate a cross ratio to every Hitchin representation. More precisely, we showed the following:

Theorem 4.1.6 Let ρ be a Hitchin representation. Then there exists a cross ratio 𝔹 on ∂_{∞}π_{1}(S)^{4∗} such that for every nontrivial element γ of π_{1}(S), the period of γ is given by

ℓ_{𝔹}(γ) = log (|\frac{λ_{max}(ρ(γ))}{λ_{min}(ρ(γ))}|) .  (8)

Here λ_{max}(ρ(γ)) and λ_{min}(ρ(γ)) are respectively the eigenvalues of of ρ(γ) with the maximum and minimum modulus.

Using O. Guichard’s work [23], we also proved a converse of this statement (see [29]).

4.2 Symplectic Anosov structures

In [7], together with Burger, Iozzi and Weinhard, we studied maximal representations from surface groups to PSp(2n, ℝ). We first recall some definitions and notation from [8].

4.2.1 The symplectic structure

In this section, we normalise the symplectic form on the associated symmetric space M and construct the Toledo invariant.

We denote by 𝔤 the Lie algebra of G = PSp(2n, ℝ) and we identify 𝔤 with the Lie algebra of Killing vector fields on M. For every m in M, let 𝔨_{m} be the Lie algebra of the stabiliser K_{m} of m in G. We finally identify T_{m}M with the orthogonal of 𝔨_{m} in 𝔤 with respect to the Killing form.

We choose a continuous map ∂_{θ} from M to 𝔤 such that, for every m in M, ∂_{θ}(m) is a generator of the centre of 𝔨_{m} verifying

exp(s∂_{θ}(m)) = 1 ⇔ s ∈ 2πℤ.

Observe that ∂_{θ} is well defined up to sign. The complex structure on T M is then given by the following map from T M to itself

J : A ↦ [∂_{θ}, A].

We normalise the Killing form ⟨ , ⟩ so that ∥∂_{θ}∥ = 1.

Definition 4.2.1 [Canonical complex structure] The canonical symplectic structure ω of M is given, for all tangent vectors X and Y, by

ω(X, Y) = ⟨[X, Y], ∂_{θ}⟩.

4.2.2 The Toledo invariant

The homomorphism

det : K_{m} → 𝕋,

is a degree n map when restricted to the centre of K_{m}. It follows that n.ω is the curvature of a line bundle L.

The following inequality is fundamental.

|τ(ρ)| ⩽ n|χ(S)|.

For n = 1, this is the Milnor-Wood Inequality [36]. This specific inequality for the symplectic group is due to V. Turaev [43]. It has been extended to other Hermitian symmetric spaces (see in particular [14, 41, 42, 9, 8]). We shall restrict ourselves to PSp(2n, ℝ) although the discussion extends to the general case as well.

Definition 4.2.2 [Maximal representation] A maximal representation is a representation, say ρ, for which

|τ(ρ)| = n|χ(S)|.

4.2.3 Reductivity

Using bounded cohomology techniques, Burger, Iozzi and Wienhard prove in [8]:

Theorem 4.2.3 Every maximal representation is reductive.

4.2.4 Cross ratios and maximal representations: the main result

For every A in PSp(2n, ℝ), let {λ_{i}}_{1⩽i⩽2n} be the eigenvalues (with multiplicities) of A ordered so that

|λ_{1}| ⩽ |λ_{2}| … ⩽ |λ_{2n}|.

We define

c(A) = ∏_{i=n+1}^{i=2n} |λ_{i}|.

The main result of this paragraph is the following.

Theorem 4.2.4 Let ρ be a maximal symplectic representation. Then there exists a cross ratio 𝔹 on ∂_{∞}π_{1}(S) such that

ℓ_{𝔹}(ρ(γ)) = 2 log c(ρ(γ)).

We prove this theorem in Paragraph 4.2.8.

4.2.5 Positivity

Let L(E) be the Grassmannian of Lagrangian spaces in a vector space E equipped with a symplectic form ω.

Definition 4.2.5 A triple of Lagrangian spaces (F, G, L) is positive if • F ⊕ L = E and

• ω(u_{f}, u_{l}) is positive for every pair of vectors (u_{f}, u_{l}) in F × L such that u_{f} + u_{l} belongs to G,

Definition 4.2.6 [Positive curve] An oriented curve ξ from the circle 𝕋 to L(E) is positive if (ξ(x), ξ(y), ξ(z)) is positive for every oriented triple of pairwise distinct points (x, y, z) in 𝕋.

With Burger, Iozzi and Wienhard, we proved the following result (see [7])

Theorem 4.2.7 Let E be a symplectic vector space. Let ρ be a maximal symplectic homomorphism from π_{1}(S) to P Sp(E). Then there exists a positive Hölder ρ-equivariant curve ξ from ∂_{∞}π_{1}(S) to L(E).

Furthermore, ξ(γ^{+}) (respectively ξ(γ^{−})) is generated by the eigenvectors of ρ(γ) corresponding to the eigenvalues of absolute value greater than 1 (resp. smaller than 1).

4.2.6 The cross ratio of four Lagrangian spaces

Let (L_{1}, L_{2}, L_{3}, L_{4}) be a quadruple of Lagrangian spaces in a symplectic space of dimension 2n. We suppose that L_{4} is transverse to L_{1} and that L_{2} is transverse to L_{3}. Let l^{1}, l^{2}, l^{3} and l^{4} be bases of L_{1}, L_{2}, L_{3} and L_{4} respectively. For every pair (a, b) ∈ {1, 2, 3, 4}^{2}, we consider the n × n matrix

A_{l^{a},l^{b}} = (ω(l_{i}^{a}, l_{j}^{b})).

We observe that for every endomorphism g of L^{a} whose matrix in the basis l^{a} is G then

A_{g(l^{a}),l^{b}} = G.A_{l^{a},l^{b}}.  (9)

Similarly

A_{l^{a},l^{b}} = −A_{l^{b},l^{a}}^{t}.

We now define

B(ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}) = \frac{det(A_{l^{1},l^{2}}). det(A_{l^{3},l^{4}})}{det(A_{l^{1},l^{4}}). det(A_{l^{3},l^{2}})}.

By Assertion (9), B(ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}) depends only on (L_{1}, L_{2}, L_{3}, L_{4}). Hence we can define

B(L_{1}, L_{2}, L_{3}, L_{4}) = B(ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}).

The following proposition follows easily from the definition

Proposition 4.2.8 We have

B(L_{1}, L_{2}, L_{3}, L_{4})B(L_{1}, L_{4}, L_{3}, L_{5}) = B(L_{1}, L_{2}, L_{3}, L_{5}),  (10)
B(L_{1}, L_{2}, L_{3}, L_{4}) = B(L_{2}, L_{1}, L_{4}, L_{3}).  (11)

Finally, if (L, U, V) are generic,

B(L, U, L, V) = 1,  (12)
B(L, L, U, V) = 0.  (13)

4.2.7 Cross ratios and positivity

Proposition 4.2.9 If the triples of Lagrangian spaces (E, F_{1}, G) and (E, F_{2}, G) are positive, then

B(E, F_{1}, G, F_{2}) > 0.

If moreover the triple (F_{1}, F_{2}, G) is positive, then

B(E, F_{1}, G, F_{2}) > 1.

Proof: We assume that (E, F_{1}, G) and (E, F_{2}, G) are positive. Let p be the projection onto G along E. Let q_{i} be the quadratic form on F_{i} defined by

q_{i}(u) = ω(p(u), u).

Since (E, F_{i}, G) is positive, it follows that q_{i} is positive definite. By simultaneous orthogonalisation, we can choose an basis f^{i} of F_{i}, which is orthogonal for q_{i}, such that p(f^{1}) = p(f^{2}) = g. Let

e^{i} = (1 − p)(f^{i}) = f^{i} − g

be the corresponding bases of E. We then have

A_{e^{i},f^{j}} = A_{e^{i},g} = −A_{g,f^{i}},

and also

ω(e_{i}^{j}, g_{k}) = q(f_{i}^{j}, f_{k}^{j}) = ω(g_{i}, f_{k}^{j}).

Furthermore let

λ_{i} = \frac{q(f_{i}^{2}, f_{i}^{2})}{q(f_{i}^{1}, f_{i}^{1})},

then

e_{i}^{2} = λ_{i}e_{i}^{1}.

It follows that

B(E, F_{1}, G, F_{2}) = \frac{det(A_{e^{1},f^{1}}). det(A_{g,f^{2}})}{det(A_{e^{1},f^{2}}). det(A_{g,f^{1}})}
= \frac{det(A_{e^{1},g}). det(−A_{e^{2},g})}{det(A_{e^{1},g}). det(−A_{e^{1},g})}
= ∏_{i} λ_{i} > 0.

Finally, assume that (F_{1}, F_{2}, G) is positive. Let π be the projection onto G along F_{1}. Recall that

f_{i}^{2} = e_{i}^{2} + g_{i} = λ_{i}e_{i}^{1} + g_{i} = λ_{i}f_{i}^{1} + (1 − λ_{i})g_{i}.

It follows that

ω(π(f_{i}^{2}), (1 − π)(f_{i}^{2})) = ω((1 − λ_{i})g_{i}, λ_{i}f_{i}^{1})
= (1 − λ_{i})λ_{i}ω(g_{i}, f_{i}^{1})
= λ_{i}(λ_{i} − 1)q_{1}(f_{i}^{1}, f_{i}^{1}).

Hence the positivity of (F_{1}, F_{2}, G) implies that λ_{i} > 1 and therefore

B(E, F_{1}, G, F_{2}) = ∏_{i} λ_{i} > 1.

Q.e.d.

Finally, we have the following:

Proposition 4.2.10 Let S be a symplectic automorphism preserving two transverse Lagrangian spaces E and F. Then, for every Lagrangian space G, we have

B(E, G, F, S(G)) = \frac{det(S|_{E})}{det(S|_{F})} = det(S|_{E})^{2}.

Proof: Let S be a symplectic transformation. Let e be a basis of a space K invariant by S. We have

det(A_{e,S(l)}) = det(A_{S^{−1}e,l}) = \frac{det(A_{e,l})}{det(S|_{K})} ,

from which the formula follows. Q.e.d.

4.2.8 Cross ratios and maximal representations

We now prove Theorem 4.2.4. Let ρ be a maximal symplectic homomorphism. Let ξ be the positive map from ∂_{∞}π_{1}(S) to L(ℝ^{2n}) associated to ρ by Theorem 4.2.7. By Proposition 4.2.9 the following formula (using the notation of Paragraph 4.2.6) defines a cross ratio on ∂_{∞}π_{1}(S)

𝔹(x, y, z, t) = B(ξ(x), ξ(y), ξ(z), ξ(t)).

Furthermore, by Proposition 4.2.10, we have

ℓ_{𝔹}(γ) = 2 log det (ρ(γ)|_{ξ(γ^{+})}).

The Theorem follows from this.

5 The energy functional and the minimal area

Let M be a Hadamard manifold– i.e complete, nonpositively curved and simply connected. Let

ρ : π_{1}(S) ⟶ Iso(M)

be a representation from π_{1}(S) to the group of isometries of M and let M_{ρ} be the associated M bundle over S. We denote by F_{ρ} the space of ρ-equivariant smooth mappings from the universal cover S̃ of S to M:

F_{ρ} = {f : S̃ → M | f ∘ γ = ρ(γ) ∘ f}.

The space F_{ρ} is canonically identified with the space Γ(S, M_{ρ}) of smooth sections of M_{ρ}.

5.1 The energy of a map and the energy functional

Let ⟨, ⟩ be the metric on M. Let f be an element of F_{ρ}. Let J be a complex structure on S lifted to S̃. Let u and v be tangent vectors in S̃, the following expression defines an exterior differential 2-form on S̃

⟨df ∧ df ∘J⟩(u, v) := \frac{1}{2} (⟨Tf(v), Tf(Ju)⟩ − ⟨Tf(u), Tf(Jv)⟩) .

Notice that ⟨df ∧ df ∘J⟩ is π_{1}(S)-invariant, hence defines an exterior differential 2-form on S also denoted by ⟨df ∧ df ∘J⟩.

Definition 5.1.1 [Energy of a map] The energy of f with respect to J is the following real number

Energy(J, f) = ∫_{S}⟨df ∧ df ∘J⟩.

The definition above is slightly nonstandard and restricted to dimension 2. We use it in order to emphasise the conformal invariance of the energy. Observe that for any diffeomorphism ϕ of S isotopic to the identity and lifted to a diffeomorphism Φ of S̃ we have

Energy(J, f) = Energy(Φ^{∗}J, f ∘ Φ).  (14)

Definition 5.1.2 [Energy functional on Teichmüller space] Let ρ be a representation of π_{1}(S) in Iso(M). The energy functional associated to ρ is the real valued function on the Teichmüller space T (S) of S defined by

e_{ρ} : J → e_{ρ}(J) := inf{Energy(J, f) | f ∈ F_{ρ}}.

In the definition above, we have used that e_{ρ}(J) only depends on the isotopy class of J which is a consequence of Equation (14).

5.2 Harmonic mappings

By definition, a harmonic mapping is a critical point of the energy. Whenever ρ is reductive, the existence of a ρ-equivariant harmonic mapping is guaranteed by Corlette’s Theorem in the context of symmetric spaces [11]. Note that [27] gives an alternative simpler proof which works in the general context of Hadamard manifolds:

Theorem 5.2.1 [Corlette]. Let ρ be a reductive representation from π_{1}(S) into a connected semi-simple real Lie group G without compact factor and with trivial centre. Let M be the associated symmetric space. Then there exists a ρ-equivariant harmonic mapping f from S to M. Furthermore this mapping is unique up to an isometry of M and minimises the energy.

The definition of the energy extends to maps from higher dimensional manifolds – although in that case it is not anymore a conformal invariant – and the above statement holds in this general context.

As a byproduct of the proof of this theorem, together with a simple application of the implicit function Theorem, the energy functional is a smooth function on Teichmüller space. Combining Corlette’s Theorem with Propositions 4.2.3, 4.1.5, we deduce the following result.

Proposition 5.2.2 Let ρ be a Hitchin or maximal symplectic representation. Let J be a complex structure on S. Then there exists a unique (up to isometries) ρ-equivariant harmonic mapping f_{ρ,J}. Moreover

Energy(J, f_{ρ,J}) = e_{ρ}(J).

5.3 Minimal area

Let f be an element of F_{ρ} and let R(f) be the open set of points x in S for which T_{x}f is injective. We observe that the induced bilinear form f^{∗}(g_{M}) is invariant under π_{1}(S) and defines a metric on R(f). We define the area of f to be the area of R(f) with respect to this metric, i.e.

Area(f) = area_{f^{∗}(g_{M})}(R(f)).

We recall that

Area(f) ⩽ Energy(J, f),  (15)

with equality if and only if f is conformal with respect to J. Finally one can find a sequence of complex structure {J_{n}}_{n∈ℕ} on S so that

Area(f) = lim_{n→∞} Energy(J_{n}, f),  (16)

Definition 5.3.1 [Minimal area] The minimal area of ρ is

MinArea(ρ) = inf{e_{ρ}(J) | J ∈ T (S)}.

It follows from Assertions (15) and (16) that

MinArea(ρ) = inf{Area(f) | f ∈ F_{ρ}}.

We also recall the classical results of Sacks–Uhlenbeck [39][38] and Schoen–Yau [40]:

Theorem 5.3.2 [Sacks–Uhlenbeck], [Schoen–Yau] Let J be a point in Teichmüller space which is a critical point of the energy. Let f_{J} be a mapping such that

Energy(J, f) = MinArea(ρ).

Then f is harmonic and conformal.

6 Displacing representations and the energy functional

In Definition 6.1.2, we introduce the notion of displacing homomorphism. In Theorem 6.1.3, we prove that Hitchin and maximal symplectic representations are displacing and finally show in Theorem 6.2.1 that the energy functional associated to a displacing representation is a proper map.

6.1 Displacing representations

Definition 6.1.1 [Displacement function] Let γ be an isometry of a metric space M. The displacement of γ is

d(γ) = inf_{x∈M} d(x, γ(x)).

If M is the Cayley graph of a group Γ with set of generators G and word length ∥ ∥_{G}, then

d(γ) = inf_{η∈Γ} ∥ηγη^{−1}∥_{G}.

The displacement function is explicit in the case of M = PSL(n, ℝ)/SO(n, ℝ). Let A be an element in PSL(n, ℝ). Let {λ_{i}}_{1⩽i⩽n} be the eigenvalues of A, then up to a multiplicative constant that depends on the normalisation of the metric on M

d(A) = \sqrt{∑_{i=0}^{n} ( log |λ_{i}|)^{2}}.  (17)

Definition 6.1.2 [Displacing homomorphism]. Let M be a metric space. A homomorphism ρ from a finitely generated group Γ to Iso(M) is displacing if for every finite generating set G of Γ there exist positive constants A and B such that such that for every γ in Γ

d(ρ(γ)) ⩾ A. inf_{η∈Γ} ∥ηγη^{−1}∥_{G} − B.

where ∥γ∥_{G} is the word length of the element γ of Γ.

In other words, the displacement of a displacing representation is roughly controlled by the displacement in the Cayley graph. Alternatively, in the case Γ = π_{1}(S), the representation ρ is displacing if for every hyperbolic metric g on S, there exist positive constants A and B such that for every nontrivial element γ in π_{1}(S)

d(ρ(γ)) ⩾ A.λ_{g}(γ) − B.

Here λ_{g}(γ) is the length of the closed geodesic, with respect to g, representing the nontrivial element γ.

6.1.1 Examples of displacing representations

Representations in Iso(M) are displacing for cocompact groups. The same holds for convex-cocompact groups whenever M is Hadamard. The use of cross ratios allow us to identify other examples of displacing representations.

Theorem 6.1.3 Every Hitchin representation is displacing. Every maximal symplectic representation is displacing.

Proof: Let ρ be a Hitchin representation. Let 𝔹 be the cross ratio associated to ρ by Theorem 4.1.6. Let {λ_{i}}_{1⩽i⩽n} be the eigenvalues of the element ρ(γ). We order the eigenvalues so that |λ_{j}| ⩾ |λ_{i}| if j > i. Combining Equations (8) and (17), we obtain

d(ρ(γ)) = \sqrt{∑_{i=1}^{i=n} (log |λ_{i}|)^{2}}

⩾ \frac{1}{\sqrt{n}} ∑_{i=1}^{i=n} |log |λ_{i}||

⩾ \frac{1}{\sqrt{n}} (|log |λ_{n}|| + |log |λ_{1}||) .

Since |λ_{n}| ⩾ 1 ⩾ |λ_{1}|, we get

d(ρ(γ)) ⩾ \frac{1}{\sqrt{n}} log \frac{|λ_{n}|}{|λ_{1}|} = \frac{1}{\sqrt{n}}ℓ_{𝔹}(γ).

By Theorem 3.3.1, there exist positive constants A and B such that

ℓ_{𝔹}(γ) ⩾ A.λ(γ) − B,

where λ(γ) is the length of the geodesic associated to γ in some auxiliary hyperbolic metric. It follows that every Hitchin representation is displacing.

For maximal symplectic representations, we have a very similar argument. Let ρ be such a representation and let 𝔹 be the cross ratio associated to it by Proposition 4.2.4. Since the injection i from PSp(2n, ℝ) into PSL(2n, ℝ) gives rise to a totally geodesic embedding of the corresponding symmetric spaces, it suffices to show that i ∘ ρ is displacing. Let {λ_{i}}_{1⩽i⩽2n} be the eigenvalues, with multiplicities, of the element ρ(γ). We order them so that

|λ_{1}| ⩽ |λ_{2}| … ⩽ |λ_{2n}|.

We observe that λ_{i}λ_{2n+1−i} = 1 from which it follows that for i less than n, |λ_{i}| ⩽ 1 ⩽ |λ_{2n+1−i}|. Then, we have

d(ρ(γ)) = \sqrt{∑_{i=1}^{i=2n} (log |λ_{i}|)^{2}}

⩾ \frac{1}{\sqrt{n}} ∑_{i=1}^{i=2n} |log |λ_{i}||

⩾ \frac{1}{n} log |\frac{∏_{i=n+1}^{i=2n} λ_{i}}{∏_{i=1}^{i=n} λ_{i}} |

⩾ \frac{1}{\sqrt{n}} log ( ∏_{i=n+1}^{i=2n} |λ_{i}|^{2}) = \frac{1}{\sqrt{n}}ℓ_{𝔹}(γ).

The result follows from this last inequality. Q.e.d.

6.2 Energy of displacing representations

The main result of this section, proved in Paragraph 6.2.2, is the following.

Theorem 6.2.1 Let ρ be a displacing representation from π_{1}(S) to the isometry group of a Hadamard manifold. Then the energy functional e_{ρ}, defined from T (S) to ℝ, is proper.

In particular, we recover as a corollary the result of W. Goldman and R. Wentworth [21] that the energy functional is proper for convex-cocompact representations.

Corollary 6.2.2 Let ρ be a Hitchin or maximal representation. Then there exists a complex structure J_{0} on S and a J_{0}-conformal harmonic ρ-equivariant mapping f defined from the universal cover of S to the corresponding symmetric space such that

Area(f) = Energy(J_{0}, f) = MinArea(ρ).

Proof: By Proposition 6.1.3 and the previous theorem, there exists a complex structure J_{0} on S which achieves the minimum of the energy functional. By Proposition 5.2.2, there exists a ρ-harmonic mapping f, such that

Energy(J_{0}, f) = e_{ρ}(J_{0}) = MinArea(ρ).

We conclude the proof by applying Theorem 5.3.2 of Sachs–Uhlenbeck and Schoen–Yau. Q.e.d.

6.2.1 The intersection is proper

Let g and g_{0} be two hyperbolic metrics on S. Let US and U_{0}S be the associated unit tangent bundles, with geodesic flows {ϕ_{t}}_{t∈ℝ} and {ϕ_{t}^{0}}_{t∈ℝ} generated by X and X_{0} respectively. Let μ and μ_{0} be the corresponding Liouville measures normalised to be probability measures. We know that these two geodesic flows are orbit conjugate. In other words, there exist a homeomorphism F from US to US_{0} and a positive continuous function ψ_{(g,g_{0})} on US such that F is differentiable along X and DF (ψ_{(g,g_{0})}X) = X_{0}.

Definition 6.2.3 [intersection] The intersection of g and g_{0} is

inter(g, g_{0}) = ∫_{US} ψ_{(g,g_{0})}dμ.

The following proposition is a classical result. For the sake of completeness since we could not find a good reference for it, we include a sketchy proof. For a less down to earth point of view and for extra information on intersections, we refer to Francis Bonahon’s original article ([1]) or to Curt McMullen’s notes ([35]).

Proposition 6.2.4 Fixing g_{0}, the function g ↦ inter(g, g_{0}) is a proper map from T (S) to ℝ.

Proof: We denote by [γ] the free homotopy class of a closed curve γ. By definition, the intersection of two homotopy classes of closed curves c_{1} and c_{2} in the compact surface S is

inter(c_{1}, c_{2}) = inf{♯(γ_{1} ∩ γ_{2}) | [γ_{i}] = c_{i}}.

If γ_{1} and γ_{2} are geodesics with distinct support for a negatively curved metric g, then

inter([γ_{1}], [γ_{2}]) = ♯(γ_{1} ∩ γ_{2}).

We denote by G(S) the set of closed geodesics in S. Let η be a simple closed curve. Using a tubular neighbourhood of η, we see that there exists a constant C(η, g), depending only on the isotopy class of η and the metric g, such that for every closed geodesic γ we have

inter([η], [γ]) ⩽ C(η, g)λ_{g}([γ]).  (18)

Now let g be a hyperbolic metric and let

G_{L} = {γ ∈ G(S) | λ_{g}(γ) ⩽ L}.

According to the equirepartition of closed geodesics due to R. Bowen in [2], for every continuous function f on US the following formula relates the integral of f with respect to the Liouville measure for g with its average along closed geodesics

∫_{US} fdμ_{g} = lim_{L→∞} (\frac{1}{♯(G_{L})} ∑_{γ∈G_{L}} \frac{∫_{γ} fdt}{λ_{g}(γ)}).  (19)

Hence, if g_{0} is another hyperbolic metric

inter(g, g_{0}) = lim_{L→∞} (\frac{1}{♯(G_{L})} ∑_{γ∈G_{L}} \frac{λ_{g_{0}}(γ)}{λ_{g}(γ)}).

Furthermore, if η is a simple closed geodesic, then we also have

λ_{g}(η) = lim_{L→∞} (\frac{1}{♯(G_{L})} ∑_{γ∈G_{L}} \frac{inter(η, γ)}{λ_{g}(γ)}).  (20)

Combining Equation (20) and Inequality (18), we obtain that for every simple closed geodesic η,

λ_{g}(η) ⩽ C(η, g_{0})inter(g, g_{0}).

But we can find a finite set of simple closed curves A such that the function

λ_{A} : g → ∑_{η∈A} inf_{c∈[η]} λ_{g}(c)

is proper on Teichmüller space. The statement thus follows from the following inequality

λ_{J}(g) ⩽ inter(g, g_{0}) ∑_{η∈A} C(η, g_{0}).

Q.e.d.

6.2.2 The energy functional is proper

We now prove Theorem 6.2.1 by adapting a beautiful argument of C. Croke and A. Fathi ([12]). We use the notation of the previous paragraph. Let ρ be a displacing representation. Let J be a complex structure on S and let g be the associated hyperbolic metric whose area form is dσ and Liouville measure – normalised to be a probability measure – is dμ. Let g_{0} be a fixed hyperbolic metric. We now prove that there exists a constant K dependent on g_{0} but independent of J such that

e_{ρ}(J) ⩾ K(inter(g, g_{0}))^{2}.

Let f an element of F_{ρ}. We consider the function from US to ℝ

h : u → ∥Tf(u)∥.

From the definition of the energy, we have

Energy(J, f) = \frac{1}{2} ∫_{S} trace(Tf^{∗}Tf)dσ = 2π|χ(S)| ∫_{US} h^{2}dμ.

By the Cauchy-Schwarz inequality

Energy(J, f) ⩾ 2π|χ(S)| (∫_{US} h dμ)^{2} .

Let γ be a closed orbit of the geodesic flow of g and let λ_{g}(γ) be the length of γ with respect to g. We denote also by γ the corresponding conjugacy class in the fundamental group. Let c be the curve from the interval [0, λ_{g}(γ)] to M defined by

c : t → f(γ̃(t)),

where γ̃ is a lift of γ in S̃. Then

∫_{γ} h dt = length(c)
⩾ d(c(0), ρ(γ)(c(0))
⩾ Aλ_{g_{0}}(γ) − B = A ∫_{γ} ψ_{(g,g_{0})}dt − B.

From Equation (19), it follows that for every f

\sqrt{\frac{Energy(J, f)}{2π|χ(S)|}} ⩾ ∫_{US} h dμ = lim_{L→∞} (\frac{1}{♯(G_{L})} ∑_{γ∈G_{L}} \frac{∫_{γ} hdt}{λ_{g}(γ)})

⩾ A lim_{L→∞} (\frac{1}{♯(G_{L})} ∑_{γ∈G_{L}} \frac{∫_{γ} ψ_{(g,g_{0})}dt}{λ_{g}(γ)})

−B lim_{n→∞} (\frac{1}{♯(G_{L})} ∑_{γ∈G_{L}} \frac{1}{λ_{g}(γ)})

⩾ A ∫_{US} ψ_{(g,g_{0})}dμ = A inter(g, g_{0}).

Hence

e_{ρ}(J) ⩾ 2π|χ(S)|A^{2}(inter(g, g_{0}))^{2}.

Finally, by Proposition 6.2.4, the function g ↦ inter(g, g_{0}) is proper. Hence, the energy functional e_{ρ} is proper.

6.3 Mapping class group and displacing representations

6.3.1 Point set topology

We recall some elementary point set topology. Let X be a topological space. We define an equivalence relation on X as follows: we say that x ∼ y, if for any continuous function f from X to a Hausdorff topological space, we have f(x) = f(y). We denote by X^{♯} the quotient X/ ∼. Observe that

Proposition 6.3.1 The space X^{♯} is Hausdorff. Moreover, we have a morphism from the group of homeomorphisms of X to the group of homeomorphisms of X^{♯}. Finally, every continuous map from X to a Hausdorff space factors through X^{♯}.

Proof: Let us denote by [x] the equivalence class of a point x in X. For any continuous function f from X to a Hausdorff space, the preimage of any point is a union of equivalence classes. This implies the last statement of the assertion. Moreover, by definition, if [x] ≠ [y], there exists a continuous function f with values in a Hausdorff space so that f(x) ≠ f(y). Then the preimage of disjoint neighbourhoods of f(x) and f(y) are disjoint neighbourhoods of [x] and [y]. Q.e.d.

Let G be a topological group acting continuously on a topological space M.

Definition 6.3.2 The action of G on M is proper if the map from G × M to M × M defined by (g, m) → (gm, m) is proper, or equivalently if, for any compact K in M, the set

G_{K} = {g ∈ G | gK ∩ K ≠ ∅}

is a compact of G.

This definition immediately implies the following result.

Proposition 6.3.3 Assume that the topological group G acts continuously on the topological spaces M and N. Assume that the action on N is proper. Assume that there exists a continuous G-equivariant map from M to N. Then the action of G on M is proper.

This notion of a proper action is mainly interesting under hypotheses on M whenever one is interested in the quotient space: for instance the quotient of a locally compact Hausdorff space by a proper action is Hausdorff. Moreover in the sequel, we shall only consider the case of continuous actions on Hausdorff spaces.

6.3.2 Proper actions of the mapping class group

Let Hom_{disp}(π_{1}(S), Iso(M)) be the space of displacing homomorphisms from π_{1}(S) to Iso(M). Let us define

Rep_{disp}^{♯}(π_{1}(S), Iso(M)) = [Hom_{disp}(π_{1}(S), Iso(M))/ Iso(M)]^{♯}

Observe that M(S) acts continuously on Rep^{♯}(π_{1}(S), Iso(M)).

Proposition 6.3.4 The mapping class group M(S) acts properly on the space Rep_{disp}^{♯}(π_{1}(S), Iso(M)).

Proof: Let ℝ^{π_{1}(S)} be the space of maps from π_{1}(S) to ℝ. We equip ℝ^{π_{1}(S)} with the product topology. We fix a hyperbolic metric on S and for every γ in π_{1}(S) we denote by λ(γ) the length of the closed geodesic associated to γ. Let

ℝ_{hyp}^{π_{1}(S)} = {ℓ ∈ ℝ^{π_{1}(S)} | ∃A, B > 0, ∀γ ∈ π_{1}(S), ℓ(γ) ⩾ Aλ(γ) − B.}

For every divergent sequence {g_{n}}_{n∈ℕ} of elements of M(S), there exists an element γ in π_{1}(S) such that after extracting a subsequence

lim_{n→∞} λ(g_{n}(γ)) = ∞.

It follows that M(S) acts properly on ℝ_{hyp}^{π_{1}(S)}. Let

𝔖 : \{ \begin{matrix}Hom_{disp}(π_{1}(S), Iso(M)) & → & ℝ^{π_{1}(S)}, \\ ρ & ↦ & \{d(ρ(γ))\}_{γ∈π_{1}(S)}. \end{matrix}

Observe that 𝔖 is a continuous map, invariant under conjugation and equivariant with respect to the action of the mapping class group. Therefore 𝔖 yields the existence of a continuous M(S)-equivariant map from Rep_{disp}^{♯}(π_{1}(S), Iso(M))

to ℝ_{hyp}^{π_{1}(S)}. Thus, by Proposition 6.3.3, the properness of the action of M(S) on

ℝ_{hyp}^{π_{1}(S)} implies the properness of the action of M(S) on Rep_{disp}^{♯}(π_{1}(S), Iso(M)). Q.e.d.

Remark: The Hausdorff space Rep_{disp}^{♯}(π_{1}(S), Iso(M)) is locally compact when M is a symmetric space: indeed Hom(π_{1}(S), Iso(M)) is locally compact. We do not know however under which conditions on M such a result holds.

Our main goal is the following result.

Corollary 6.3.5 The mapping class group M(S) acts properly on the spaces Rep_{H}(π_{1}(S), PSL(n, ℝ)) and Rep_{T} (π_{1}(S), PSp(2n, ℝ)). Moreover the quotient spaces are Hausdorff.

Proof: We write the proof only in the case of PSL(n, ℝ), the other case being similar. Let M be the symmetric space of PSL(n, ℝ) and observe that PSL(n, ℝ) is a subgroup of Iso(M). Let

Rep_{H}^{♯}(π_{1}(S), PSL(n, ℝ)) = [Hom_{H}((π_{1}(S), PSL(n, ℝ))/PSL(n, ℝ)]^{♯}.

From Theorem 6.1.3, it follows that Hom_{H}(π_{1}(S), PSL(n, ℝ)) consists only of displacing homomorphisms. Since the induced map from Rep_{H}^{♯}(π_{1}(S), PSL(n, ℝ)) to Rep_{disp}^{♯}(π_{1}(S), Iso(M)) is an M(S)-equivariant continuous map, it follows from the previous proposition that the action of M(S) on Rep_{H}^{♯}(π_{1}(S), PSL(n, ℝ)) is proper.

We already know that Rep_{H}(π_{1}(S), PSL(n, ℝ)) is Hausdorff, this implies by the construction before Proposition 6.3.1 that

Rep_{H}(π_{1}(S), PSL(n, ℝ)) = Rep_{H}^{♯}(π_{1}(S), PSL(n, ℝ)).

Hence the first part of the statement follows. The last part follows from the local compactness of the spaces under consideration. Q.e.d.

7 The Toledo invariant and the minimal area

We now restrict ourselves to the case of maximal symplectic representations.

Theorem 7.0.6 Let ρ be a representation of π_{1}(S) in PSp(2n, ℝ). Then

\frac{n}{2π} MinArea(ρ) ⩾ |τ(ρ)|.

Moreover, if ρ is maximal and the inequality above is an equality, then ρ is diagonal. In this situation, there exists a unique minimal equivariant immersion, realised by a totally geodesic embedding, of S̃ in the symmetric space of PSp(2n, ℝ).

We begin with a lemma which uses the identification of tangent vectors to elements of the Lie algebra of Killing vector fields as in the beginning of Paragraph 4.2.1.

Lemma 7.0.7 Let M be an irreducible Hermitian symmetric space of noncompact type. Let ω be its symplectic form. Let (u, v) be an orthonormal pair of tangent vectors at a point m of M. Let κ be the sectional curvature of the plane generated by (u, v), then

|ω(u, v)| ⩽ \sqrt{−κ}.  (21)

Moreover the equality occurs exactly whenever the Lie bracket [u, v] generates the centre of the Lie algebra of the stabiliser of m in the isometry group of M.

Proof: In the notation of Paragraph 4.2.1, the symplectic structure is given by

ω(u, v) = ⟨[u, v], ∂_{θ}⟩ ⩽ \sqrt{−⟨[u, v], [u, v]⟩}.

The curvature tensor satisfies R(u, v)w = [[u, v], w]. The result then follows by

\sqrt{−⟨[u, v], [u, v]⟩} ⩽ \sqrt{−κ}.

Q.e.d.

We can now prove Theorem 7.0.6.

Proof: The first point is immediate. Let ω be the Kähler form on the associated symmetric space X. Let (u, v) be an orthonormal system in T_{x}X. Then, ω(u, v) ⩽ 1 with equality if and only if the plane generated by (u, v) is complex. It follows that for every ρ-equivariant mapping f, we have

\frac{n}{2π} Area(f) ⩾ \frac{n}{2π} ∫_{S} f^{∗}(ω) = τ(ρ).  (22)

Moreover, if the equality in (22) holds for an immersion f, then there exists an invariant complex structure on S̃ for which f is a holomorphic map.

Assume now that ρ is maximal. According to Theorem 6.2.2, there exist a complex structure J on S and a J-conformal harmonic mapping f such that

Area(f) = MinArea(ρ).

Let f be such a conformal harmonic mapping. We know by Proposition 2.4 and Example (3) of the article by Gulliver, Osserman and Royden [24] that f is a branched minimal immersion. Let x_{1}, …, x_{n} be the branch points of order k_{1}, …, k_{n} respectively. Let Ŝ = S \ {x_{1}, …, x_{n}}. Denote by κ the curvature of the metric f^{∗}g_{X} on Ŝ. Note that

\frac{1}{2π} ∫Ŝ κdμ − ∑_{i}(k_{i} − 1) = χ(S).

Let κ_{f} be the sectional curvature of the 2-plane Tf(T Ŝ) and let B be the second fundamental form of f. By the Gauss equation

κ = κ_{f} − ∥B∥ ⩽ κ_{f}.

Finally, assume that τ(ρ) = \frac{n}{2π} MinArea(ρ). Let μ be the measure of area of the metric f^{∗}g_{X}. We have by Inequality (21)

2π|τ(ρ)| = n |∫Ŝ f^{∗}ω|
⩽ n ∫Ŝ \sqrt{−κ_{f}}dμ

⩽ n\sqrt{Area(f) ∫Ŝ −κ_{f}dμ}

⩽ \sqrt{2nπτ(ρ)}\sqrt{∫Ŝ −κ_{f}dμ}.

It follows that

\frac{1}{n}|τ(ρ)| ⩽ −\frac{1}{2π} ∫Ŝ κ_{f}dμ
⩽ −\frac{1}{2π} ∫Ŝ κdμ − \frac{1}{2π} ∫Ŝ ∥B∥dμ
⩽ −χ(S) − ∑_{i}(k_{i} − 1) − \frac{1}{2π} ∫Ŝ ∥B∥dμ

⩽ \frac{1}{n}|τ(ρ)| − ∑_{i}(k_{i} − 1) − \frac{1}{2π} ∫Ŝ ∥B∥dμ.

As a first consequence, we see that k_{i} = 1 for all i. In other words, f is an immersion. Moreover B vanishes everywhere. This means that f is totally geodesic. It follows from the equality case in Lemma 7.0.7 that f is associated to an embedding of PSL(2, ℝ) in PSp(2n) whose Lie algebra contains the centre of the Lie algebra of the maximal compact subgroup. Hence ρ is diagonal. Q.e.d.

Corollary 7.0.8 Let ρ be a maximal representation. Assume that there exists a holomorphic equivariant map f from S to the associated symmetric space of PSp(2n, ℝ). Then ρ is diagonal and f is totally geodesic.

Proof: If f is holomorphic then f is minimal and Area(f) = ∫_{S} f^{∗}ω. By Inequality (22), it follows that MinArea(ρ) = τ(ρ). Hence by the previous theorem, ρ is diagonal and f is totally geodesic. Q.e.d.

8 The Hitchin map

8.1 Representations and holomorphic differentials

We first recall that representations from π_{1}(S) to a semi-simple connected real Lie group G, without compact factor and with trivial centre, give rise to holomorphic differentials by a construction quite similar to the basic construction in Chern-Weil theory. The construction that we now describe associates to every reductive representation from π_{1}(S) to G, to every complex structure J on S and to every Ad(G)-invariant polynomial q of degree n on the Lie algebra of G, a holomorphic n-ic differential on (S, J).

By Corlette’s Theorem ([11] and [27] for an alternative simpler proof), there exists a ρ-equivariant harmonic mapping f from S to the symmetric space M associated to G. Moreover this mapping is unique up to an isometry of M. We define

Hom_{red}(π_{1}(S), G)

to be the space of reductive homomorphisms from π_{1}(S) to G and

Rep_{red}(π_{1}(S), G) = Hom_{red}(π_{1}(S), G)/G.

8.1.1 Harmonic maps on surfaces

We begin with a standard observation on harmonic maps on surfaces. Let S be a Riemann surface whose complex structure is denoted by J. Let f be a smooth map from S to a smooth manifold M. Let

Ω^{1}(S, f^{∗}T M)

be the space of one-forms on S with values in the pull back vector bundle f^{∗}T M and let

Ω_{ℂ}^{1}(S, f^{∗}T M ⊗_{ℝ} ℂ)

be the space of complex linear one-forms on S with values in the complexified vector bundle f^{∗}T M_{ℂ} = f^{∗}T M ⊗_{ℝ} ℂ. For every one-form ω on S with values in f^{∗}T M, we denote by ω_{ℂ} its complexification. This complexification is defined for every tangent vector u in S by

ω_{ℂ}(u) = ω(u) − iω(Ju).

The map ω → ω_{ℂ} is a linear map from Ω^{1}(S, f^{∗}T M) to Ω_{ℂ}^{1}(S, f^{∗}T M_{ℂ}).

Definition 8.1.1 [holomorphic one-form] An element β of Ω_{ℂ}^{1}(S, f^{∗}T M_{ℂ}) is holomorphic if

∇_{Ju}β = i∇_{u}β.

We consider the tangent map Tf of f as a one-form on S with values in the pullback bundle by f of T M. Then, we have the following classical observation.

Proposition 8.1.2 The map f is harmonic if and only if Tf_{ℂ} is holomorphic.

Proof: Indeed, f is harmonic if and only if for every X

∇_{X}Tf(X) + ∇_{JX}Tf(JX) = 0.

Since ∇_{X}Tf(Y) is symmetric in X and Y, the condition above is equivalent to

∇_{X}Tf(Y) + ∇_{JX}Tf(JY) = 0,

for all X and Y in T S. This turns out to be equivalent to

∇_{JX}Tf(Y) − ∇_{X}Tf(JY) = 0,

for all X and Y in T S. On the other hand, by definition

(∇_{JX}Tf_{ℂ} − i∇_{X}Tf_{ℂ})(Y) = (∇_{JX}Tf(Y) − ∇_{X}Tf(JY))
− i(∇_{X}Tf(Y) + ∇_{JX}Tf(JY)).

The statement follows from these remarks. Q.e.d.

8.1.2 Holomorphic differentials

Let M be a Riemannian manifold. Let p be a parallel section of (T^{∗}M)^{⊗k}. We denote by p_{ℂ} the parallel section of (T^{∗}M_{ℂ})^{⊗k} characterised by

p_{ℂ}|_{(T M)^{⊗k}} = p.

Let f be a map from a Riemann surface S to M. Then we have the following easy observation.

Proposition 8.1.3 Let β be an element in Ω_{ℂ}^{1}(S, f^{∗}T M_{ℂ}). Suppose that β is holomorphic. Then p_{ℂ}(β, β, …, β) is a holomorphic differential of degree k.

As a specific example of this construction, we can take p = g, the Riemannian metric on M. If f is harmonic by Propositions 8.1.1 and 8.1.3

H(f) := g_{ℂ}(T f_{ℂ}, T f_{ℂ})

is a quadratic differential, which is called the Hopf differential of f. Observe that we have the following result.

Proposition 8.1.4 Let f be a harmonic map from a surface S to M. Then the Hopf differential of f vanishes if and only if f is minimal.

Proof: Indeed, the quadratic differential g_{ℂ}(T f_{ℂ}, T f_{ℂ}) vanishes if and only if f is conformal and hence minimal. Q.e.d.

8.1.3 Commutative Chern-Weil Theory

When M is the symmetric space associated to G = Iso(M), then every Ad(G)- invariant symmetric k-multilinear form P on the Lie algebra of G gives rise naturally to a parallel k-tensor field function P̌ on M. Indeed such a P gives naturally rise to a G-invariant tensor field P̌ on M. But on a symmetric space, any tensor field invariant under isometries is parallel.

Let J be a complex structure on S and P be a symmetric Ad(G)-invariant multilinear form P of degree k on G. Let Q(k, J) be the space of holomorphic k-differentials on S equipped with the complex structure J. Combining the above constructions, we define a map F_{P,J} from Rep_{red}(π_{1}(S), G) to Q(k, J) by

ρ ↦ F_{P,J}(ρ) := P̌_{ℂ}(Tf_{ℂ}, …, Tf_{ℂ}),

where f is a ρ-equivariant harmonic mapping from S to M given by Corlette’s Theorem. The uniqueness part of Corlette’s result shows that F_{P,J} is well defined.

When G = PSL(n, ℝ), let σ_{k} be the symmetric polynomial of degree k seen as a homogeneous function of degree k on the Lie algebra 𝔤 of G. There exists a unique k-multilinear symmetric Ad(G)-invariant form p_{k} on 𝔤 so that

p_{k}(A, …, A) = σ_{k}(A).

Notice that up to a multiplicative constant F_{p_{2}} is the metric on M. We define the map

ξ_{J} = ⨁_{k=2}^{k} F_{p_{k},J}.

We can now state Hitchin’s Theorem [26].

Theorem 8.1.5 [Hitchin] The map ξ_{J} is a homeomorphism from the space of Hitchin representations Rep_{H}(π_{1}(S), PSL(n, ℝ)) to Q(2, J) ⊕ … ⊕ Q(n, J).

As in the introduction, we define the Hitchin map

H \{ \begin{matrix}E^{(n)} & → & Rep_{H}(π_{1}(S), PSL(n, ℝ)), \\ (J, ω) & ↦ & ξ_{J}^{−1}(ω). \end{matrix}

This map is equivariant with respect to the mapping class group action. The following result is now immediate.

Theorem 8.1.6 The Hitchin map is surjective.

Proof: Let ρ be a Hitchin representation. By Corollary 6.2.2, there exists a complex structure J on S and a ρ equivariant conformal harmonic mapping f with respect to J. It follows by Proposition 8.1.4 that the quadratic differential F_{p_{2},J} vanishes. This shows that the Hitchin map is surjective. Q.e.d.

8.1.4 The normal bundle to the space of Fuchsian representations

We conclude with a partial result. The energy functional associated to a faithful cocompact representation ρ with values in PSL(2, ℝ) is the same – up to a multiplicative constant only depending on n – as the energy functional eρ̌ associated to the Fuchsian representation ρ̌ = ι ∘ ρ with values in PSL(n, ℝ). Hence the energy functional eρ̌ has a unique strict minimum. Therefore the same holds for representations which are closed to being Fuchsian. It follows that the Hitchin map is a diffeomorphism from a small neighbourhood of the zero section onto its image. This implies that the normal bundle of the space of Fuchsian representations in the Hitchin component can be identified – equivariantly with respect to the action of the mapping class group – with E^{(n)}.

9 Comments and extensions

We conclude this article with two comments

1. The theory of Hitchin representations extends to all real split groups. Similarly, the theory of maximal representations extends to all isometry groups of Hermitian symmetric spaces. It is quite natural to conjecture that the constructions of this article extend to these more general cases. The fact that these representations are (at least conjecturally) Anosov representations is certainly meaningful from this point of view. However, one cannot expect all the results here to extend to all Anosov representations since one can construct Anosov representations which are not reductive. It also remains a puzzle to understand the algebraic conditions under which Anosov representations are associated to cross ratios; this is a crucial argument in our paper.

2. For maximal symplectic representations in other components than Hitchin’s, the map from the space of equivariant minimal surfaces to the space of representation is surjective for the same reason that apply to the Hitchin component. However, in that case, the structure of a generic fibre is mysterious.

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