Riemannian metrics on the moduli space of GHMC anti-de Sitter structures
aa r X i v : . [ m a t h . DG ] A p r RIEMANNIAN METRICS ON THE MODULI SPACE OF GHMCANTI-DE SITTER STRUCTURES
ANDREA TAMBURELLI
Abstract.
We first extend the construction of the pressure metric to the de-formation space of globally hyperbolic maximal Cauchy-compact anti-de Sitterstructures. We show that, in contrast with the case of the Hitchin components,the pressure metric is degenerate and we characterize its degenerate locus. Wethen introduce a nowhere degenerate Riemannian metric adapting the work ofQiongling Li on the
SL(3 , R ) -Hitchin component to this moduli space. We provethat the Fuchsian locus is a totally geodesic copy of Teichmüller space endowedwith a multiple of the Weil-Petersson metric. Contents
Introduction 11. Pressure metric on GH ( S )
22. A non-degenerate Riemannian metric on GH ( S ) Introduction
Let S be a closed, connected, oriented surface of negative Euler characteristic.The aim of this short note is to introduce two Riemannian metrics on the deforma-tion space GH ( S ) of convex co-compact anti-de Sitter structures on S × R . Theseare the geometric structures relevant for the study of pairs of conjugacy classes ofrepresentations ρ L,R : π ( S ) → P SL(2 , R ) that are faithful and discrete.In recent years, much work has been done in order to understand the geometry of GH ( S ) ([Tam19b], [Tam20a], [Tam19a], [Tam20b], [Tam19c], [Ouy19]). It turns outthat many of the phenomena described in the aformentioned papers have analogouscounterparts in the theory of Hitchin representations in SL(3 , R ) ([LZ18], [Lof19],[Lof07], [Lof04], [DW15], [TW20], [OT19]). Pushing this correspondence even fur-ther, we explain in this paper how to construct two Riemannian metrics in GH ( S ) following analogous constructions known for the SL(3 , R ) -Hitchin component. Date : April 16, 2020. GH ( S ) The first Riemannian metric we define is the pressure metric introduced by Brigde-man, Canary, Labourie and Sambarino ([BCLS15], [BCS18]) for the Hitchin compo-nents, and inspired by previous work of Bridgeman ([Bri10]) on quasi-Fuchsian rep-resentations and McMullen’s thermodynamic interpretation ([McM08]) of the Weil-Petersson metric on Teichmüller space. Although the construction of the pressuremetric in GH ( S ) can be carried out analogously, we show that, unlike in the Hitchincomponents, the pressure metric is degenerate and we characterize its degeneratelocus: Theorem A.
The pressure metric on GH ( S ) is degenerate only at the Fuchsianlocus along pure bending directions. Here, the Fuchsian locus in GH ( S ) consists of pairs of discrete and faithful rep-resentations of π ( S ) that coincide up to conjugation, and pure bending directionscorrespond to deformations of representations away from the Fuchsian locus thatare analogs of bending deformations for quasi-Fuchsian representations in P SL(2 , C ) ([Tam19b]).The second Riemannian metric we define follows instead the contruction of Li onthe SL(3 , R ) -Hitchin component ([Li16]) and it is based on the introduction of apreferred ρ -equivariant scalar product in R for a given ρ ∈ GH ( S ) . The main resultis the following: Theorem B.
This Riemannian metric is nowhere degenerate in GH ( S ) and restrictsto a multiple of the Weil-Petersson metric on the Fuchsian locus, which, moreover,is totally geodesic. Pressure metric on GH ( S ) In this section we adapt the construction of the pressure metric on the Hitchincomponent ([BCLS15], [BCS18]) to the deformation space of globally hyperbolicmaximal Cauchy-compact anti-de Sitter manifolds. We will show that the pressuremetric is degenerate at the Fuchsian locus along “pure bending” directions.1.1.
Background on anti-de Sitter geometry.
We briefly recall some notions ofanti-de Sitter geometry that will be used in the sequel.The -dimensional anti-de Sitter space AdS is the local model of Lorentzian man-ifolds of constant sectional curvature − and can be defined as the set of projectiveclasses of time-like vectors of R endowed with a bilinear form of signature (2 , .We are interested in a special class of spacetimes locally modelled on AdS , in-troduced by Mess ([Mes07]), called Globally Hyperbolic Maximal Cauchy-compact(GHMC). This terminology comes from physics and indicates that these spacetimescontain an embedded space-like surface that interesects any inextensible causal curvein exactly one point. From a modern mathematical point of view ([DGK18]), wecan describe these manifolds as being convex co-compact anti-de Sitter manifoldsdiffeomorphic to S × R , where S is a closed surface of genus at least . This IEMANNIAN METRICS ON GH ( S ) means that, identifying the fundamental group of S with a discrete subgroup Γ of Isom (AdS ) ∼ = P SL(2 , R ) × P SL(2 , R ) via the holonomy representation hol : π ( S ) → P SL(2 , R ) × P SL(2 , R ) , the group Γ acts properly disontinuously and co-compactlyon a convex domain in AdS .We denote by GH ( S ) the deformation space of globally hyperbolic maximal Cauchy-compact anti-de Sitter structures on S × R . It turns out that the holonomy of aGHMC anti-de Sitter manifold into P SL(2 , R ) × P SL(2 , R ) is faithful and discrete ineach factor. Moreover, we have a homeomorphism between GH ( S ) and the product T ( S ) × T ( S ) of two copies of the Teichmüller space of S ([Mes07]). In particular, eachsimple closed curve γ ∈ π ( S ) is sent by the holonomy representation ρ = ( ρ L , ρ R ) to a pair of hyperbolic isometries of H , which preserves a space-like geodesic in theconvex domain of discontinuity of ρ in AdS , on which ρ ( γ ) acts by translation by ℓ ρ ( γ ) = 12 ( ℓ ρ L ( γ ) + ℓ ρ R ( γ )) . We will refer to ℓ ρ ( γ ) as the translation length of the isometry ρ ( γ ) (see [Tam19b]).We will say that the holonomy ρ : π ( S ) → P SL(2 , R ) × P SL(2 , R ) of a GHMCanti-de Sitter structure is Fuchsian if, up to conjugation, its left and right projectionscoincide.1.2. Background on thermodynamical formalism.
Let X be a Riemannianmanifold. A smooth flow φ = ( φ t ) t ∈ R is Anosov if there is a flow-invariant splitting T X = E s ⊕ E ⊕ E u , where E is the bundle parallel to the flow and, for t ≥ , thedifferential dφ t exponentially contracts E s and exponentially expands E u . We saythat φ is topologically transitive if it has a dense orbit.Given a periodic orbit a for the flow φ , we denote by ℓ ( a ) its period. Let f : X → R be a positive Hölder function. It is possible ([BCLS15]) to reparametrize the flow φ and obtain a new flow φ f with the property that each closed orbit a has period ℓ f ( a ) := Z ℓ ( a )0 f ( φ s ( x )) ds x ∈ a . We define • the topological entropy ([Bow72]) of f as h ( f ) = lim sup T → + ∞ log( | R T ( f ) | ) T where R T = { a closed orbit of φ | ℓ f ( a ) ≤ T } ; • the topological pressure ([BR75]) of a Hölder function g (not necessarilypositive) as P ( g ) = lim sup T → + ∞ T log X a ∈ R T e ℓ g ( a ) . IEMANNIAN METRICS ON GH ( S ) These two notions are related by the following result:
Lemma 1.1 ([Sam14]) . Let φ be a topologically transitive Anosov flow on X and let f : X → R be a positive Hölder function. Then P ( − hf ) = 0 if and only if h = h ( f ) . Consider then the space P ( X ) = { f : X → (0 , + ∞ ) | f Hölder , P ( f ) = 0 } and its quotient H ( X ) by the equivalence relation that identifies positive Hölderfunctions with the same periods. The analitic regularity of the pressure ([PP90],[Rue78]) allows to define the pressure metric on T f P ( X ) as k g k P = − d dt P ( f + tg ) | t =0 ddt P ( f + tg ) | t =0 . Theorem 1.2 ([PP90], [Rue78]) . Let X be a Riemannian manifold endowed with atopologically transitive Anosov flow. Then the pressure metric on H ( X ) is positivedefinite. In particular, given a one parameter family of positive Hölder functions f t : X → R ,the functions Φ( t ) = − h ( f t ) f t describe a path in P ( X ) by Lemma 1.1 and k ˙Φ k P = 0 if and only if ˙Φ has vanishing periods, hence if and only if ddt | t =0 h ( f t ) ℓ f t ( a ) = 0 forevery closed orbit a for φ .1.3. Pressure metric on GH ( S ) . We apply the above theory to the unit tangentbundle X = T S of a hyperbolic surface ( S, ρ ) endowed with its geodesic flow φ = φ ρ . Here, ρ : π ( S ) → P SL(2 , R ) is a fixed Fuchsian representation that definesa marked hyperbolic metric on S . We also fix an identification of the universal cover ˜ S with H , and, consequently, of the Gromov boundary ∂ ∞ π ( S ) of the fundamentalgroup with S . The following facts are well-known from hyperbolic geometry: Proposition 1.3 ([BCS18]) . If ρ, η : π ( S ) → P SL(2 , R ) are two Fuchsian repre-sentations, then there is a unique ( ρ, η ) -equivariant Hölder homeomorphism ξ ρ,η : ∂ ∞ H → ∂ ∞ H that varies analytically in η . Proposition 1.4 ([BCS18]) . For every Fuchsian representation η , there is a positiveHölder function f η : X → R with period ℓ f η ( γ ) coinciding with the hyperbolic length ℓ η ( γ ) of the closed geodesic γ for the hyperbolic metric induced by η . Moreover, f η varies analytically in η . Corollary 1.5.
For every ρ = ( η L , η R ) ∈ GH ( S ) , there exists a positive Hölderfunction f ρ : X → R such that ℓ f ρ ( γ ) = ℓ ρ ( γ ) for every simple closed curve γ ∈ π ( S ) .Proof. Recall that ℓ ρ ( γ ) = ( ℓ η L ( γ ) + ℓ η R ( γ )) , thus it is sufficient to choose f ρ = ( f η L + f η R ) . Moreover, f ρ varies analytically in ρ by Proposition 1.4. (cid:3) IEMANNIAN METRICS ON GH ( S ) We can then introduce the termodynamic mapping:
Φ : GH ( S ) −→ P ( X ) ρ
7→ − h ( f ρ ) f ρ . By pulling-back the pressure metric via Φ , we obtain a semi-definite metric on GH ( S ) ,which we still call pressure metric. Proposition 1.6.
The restriction of the pressure metric to the Fuchsian locus in GH ( S ) is a constant multiple of the Weil-Petersson metric.Proof. Let ρ t = ( η t , η t ) be a path on the Fuchsian locus. Then Φ( ρ t ) = − h ( f ρ t ) f ρ t = − h ( f ρ t ) f η t = − f η t , where in the last step we used the fact that the entropy of a Fuchsian representationis ([Tam20a], [GM16]). Therefore, d Φ( ˙ ρ ) = − ˙ f η and the result follows from([McM08]). (cid:3) Definition 1.7.
Let ρ = ( η, η ) be a Fuchsian representation. We say that a tangentvector w ∈ T ρ GH ( S ) is a pure bending direction if w = ( v, − v ) for some v ∈ T η T ( S ) . Lemma 1.8.
The pressure metric on GH ( S ) is degenerate on the Fuchsian locusalong pure bending directions.Proof. Let ρ t = ( η L,t , η
R,t ) be a path in GH ( S ) such that ρ is Fuchsian and ˙ ρ = ddt | t =0 ρ t = ( v, − v ) for some v ∈ T η T ( S ) . By definition of the pressure metric andTheorem 1.2, we have k d Φ( ˙ ρ ) k = 0 if and only if ddt | t =0 h ( ρ t ) ℓ ρ t ( γ ) = 0 for everyclosed geodesic γ on S . By the product rule and the fact that the entropy is maximaland equal to at the Fuchsian locus ([Tam20a], [CTT19]), we get ddt | t =0 h ( ρ t ) ℓ ρ t ( γ ) = ddt | t =0 ℓ ρ t ( γ )= 12 (cid:18) ddt | t =0 ℓ η L,t ( γ ) + ddt | t =0 ℓ η R,t ( γ ) (cid:19) = 12 ( dℓ η ( v ) + dℓ η ( − v )) = 0 . (cid:3) Remark . As remarked in [Bri10], we note that, along a general path ρ t ∈ GH ( S ) ,the condition ddt | t =0 h ( ρ t ) ℓ ρ t ( γ ) = 0 for every closed geodesic γ is equivalent to theexistence of a constant k ∈ R such that ddt | t =0 ℓ ρ t ( γ ) = kℓ ρ ( γ ) . In fact, k = − h ( ρ ) ddt | t =0 h ( ρ t ) . IEMANNIAN METRICS ON GH ( S ) Lemma 1.10.
Let v ∈ T ρ GH ( S ) be a non-zero vector. If there exists k ∈ R suchthat (1.1) ddt | t =0 ℓ ρ t ( γ ) = kℓ ρ ( γ ) for every closed geodesic γ , then k = 0 or ρ is Fuchsian.Proof. We show that if ρ is not Fuchsian, then k is necessarily . The proof followsthe line of [Bri10, Lemma 7.4]. Let v = ( v , v ) and ρ t = ( ρ ,t , ρ ,t ) . Choose simpleclosed curves α and β in S . Up to conjugation we can assume that A i ( t ) = ρ i,t ( α ) = (cid:18) λ i ( t ) 00 λ i ( t ) − (cid:19) , where we denoted by λ i ( t ) the largest eigenvalue of the hyperbolic isometry ρ i,t ( α ) .Let B i ( t ) = ρ i,t ( β ) = (cid:18) a i ( t ) b i ( t ) c i ( t ) d i ( t ) (cid:19) such that det( B i ( t )) = 1 and tr( B i ( t )) > . Notice that b i ( t ) c i ( t ) = 0 because B i ( t ) is hyperbolic and A i ( t ) and B i ( t ) have different axis. For every n ≥ , we considerthe matrices C i,n ( t ) = A ni ( t ) B i ( t ) = ρ i,t ( γ n ) = (cid:18) λ i ( t ) n a i ( t ) λ i ( t ) n b i ( t ) λ i ( t ) − n c i ( t ) λ i ( t ) − n d i ( t ) (cid:19) associated to some closed curves γ n on S . The eigenvalues µ i,n of C i,n ( t ) satisfy log( µ i,n ( t )) = n log( λ i ( t )) + log( a i ( t )) + λ i ( t ) − n (cid:18) a i ( t ) d i ( t ) − a i ( t ) (cid:19) + O ( λ i ( t ) − n ) as n → + ∞ . Applying Equation (1.1) to the curves γ n , we obtain ddt | t =0 ℓ ρ t ( γ n ) − kℓ ρ ( γ n )= n log( λ λ ) ′ − kn log( λ λ )+ log( a a ) ′ − k log( a a ) − n (cid:20) λ − n − λ ′ (cid:18) a d − a (cid:19) + λ − n − λ ′ (cid:18) a d − a (cid:19)(cid:21) + λ − n (cid:20)(cid:18) a d − a (cid:19) ′ − k (cid:18) a d − a (cid:19)(cid:21) + λ − n (cid:20)(cid:18) a d − a (cid:19) ′ − k (cid:18) a d − a (cid:19)(cid:21) + o ( λ − ni ) where all derivatives and all functions are intended to be taken and evaluated at t = 0 . The term n log( λ λ ) ′ − kn log( λ λ ) vanishes by assumption because ℓ ρ t ( α ) = 12 ( ℓ ρ ,t ( α ) + ℓ ρ ,t ( α )) = log( λ ( t ) λ ( t )) IEMANNIAN METRICS ON GH ( S ) and Equation (1.1) holds for the curves α . Taking the limit of the above expressionas n → + ∞ , we deduce that log( a a ) ′ − k log( a a ) = 0 . Because ρ is not Fuchsian,we can assume to have chosen α and β so that λ (0) > λ (0) . Then if we multiplythe equation above by λ n n and take the limit as n → + ∞ , we deduce that λ ′ = 0 .Similarly, multiplying by λ n n , we find that λ ′ = 0 . Therefore, ddt | t =0 ℓ ρ t ( α ) = ddt | t =0 log( λ ( t ) λ ( t )) = 0 , hence k = 0 . (cid:3) Theorem 1.11.
Let v = ( v L , v R ) ∈ T ρ GH ( S ) be a non-zero tangent vector such that k d Φ( v ) k = 0 . Then ρ is Fuchsian and v is a pure bending direction.Proof. Let ρ t be a path in GH ( S ) such that ρ = ρ and ρ t is tangent to v . If ρ = ( η, η ) is Fuchsian, then, combining Remark 1.9 with the fact that the entropy is maximaland equal to at the Fuchsian locus, we get ddt | t =0 h ( ρ t ) ℓ ρ t ( γ ) = ddt | t =0 ℓ ρ t ( γ ) for every simple closed geodesic γ in S . Therefore, ddt | t =0 ℓ ρ t ( γ ) = 12 ( dℓ η ( γ )( v L ) + dℓ η ( γ )( v R )) from which we deduce that v R = − v L , because { dℓ η ( γ ) } γ generates T ∗ η T ( S ) . Hence, v is a pure-bending direction.We are thus left to show that ρ is necessarily Fuchsian. Suppose it is not and denotewith ρ L = ρ R the projections of ρ . By the previous lemma ℓ ′ ρ ( γ ) = ddt | t =0 ℓ ρ t ( γ ) = 0 for every simple closed geodesic γ in S . Moreover, we have shown in the proofof Lemma 1.10 that if ℓ ρ L ( γ ) = ℓ ρ R ( γ ) then ℓ ′ ρ L ( γ ) = ℓ ′ ρ R ( γ ) = 0 . Otherwise, ℓ ′ ρ L ( γ ) = − ℓ ′ ρ R ( γ ) . Exploiting the isomorphism SL(2 , R ) × SL(2 , R ) ∼ = SO (2 , , wefind that the matrix ρ t ( γ ) is conjugated to exp (cid:18)
12 diag( ℓ ρ L,t + ℓ ρ R,t , ℓ ρ L,t − ℓ ρ R,t , ℓ ρ R,t − ℓ ρ L,t , − ℓ ρ L,t − ℓ ρ R,t ) (cid:19) , thus d tr( ρ ( γ ))( v ) = ddt | t =0 tr( ρ t ( γ )) = 0 for every simple closed geodesic γ . Because ρ is generic in the sense of [BCLS15,Proposition 10.3], the differentials of traces { d tr( ρ ( γ )) } γ generate T ∗ ρ GH ( S ) and wemust have v = 0 . (cid:3) IEMANNIAN METRICS ON GH ( S ) A non-degenerate Riemannian metric on GH ( S ) In this section we define a non-degenerate Riemannian metric on GH ( S ) followingLi’s construction ([Li16]) for the SL(3 . R ) -Hitchin component.2.1. Preliminaries.
In this section we identify GH ( S ) with a connected componentof the space of representations Hom( π ( S ) , SO (2 , / SO (2 , via the holonomymap. Recall that by Mess’ parametrization ([Mes07]), this component is smooth anddiffeomorphic to T ( S ) × T ( S ) . This allows to identify the tangent space T ρ GH ( S ) at ρ ∈ GH ( S ) with the cohomology group H ( S, so (2 , Ad ρ ) , where so (2 , Ad ρ denotes the flat so (2 , bundle over S with holonomy Ad ρ . Explicitly, so (2 , Ad ρ = ( ˜ S × so (2 , / ∼ where (˜ x, v ) ∼ ( γ ˜ x, Ad ρ ( γ )( v )) for any γ ∈ π ( S ) , x ∈ ˜ S and v ∈ so (2 , .In order to define a Riemannian metric on GH ( S ) is thus sufficient to introduce anon-degenerate scalar product on H ( S, so (2 , Ad ρ ) . Let us assume for the momentthat we have chosen an inner product ι on the bundle so (2 , Ad ρ and a Riemannianmetric h on S . A Riemannian metric in cohomology follows then by standard Hodgetheory that we recall briefly here. The Riemannian metric h and the orientation on S induce a scalar product h· , ·i on the space A p ( S ) of p -forms on S , which allows todefine a Hodge star operator ∗ : A p ( S ) → A − p ( S ) by setting α ∧ ( ∗ β ) = h α, β i h dA h . This data gives a bi-linear pairing ˜ g in the space of so (2 , Ad ρ -valued -forms asfollows: ˜ g ( σ ⊗ φ, σ ′ ⊗ φ ′ ) = Z S ι ( φ, φ ′ ) σ ∧ ( ∗ σ ′ ) , where σ, σ ′ ∈ A ( S ) and φ, φ ′ are sections of so (2 , Ad ρ .Given ρ ∈ GH ( S ) , we denote by ρ ∗ the contragradient representation (still into SO (2 , ) defined by ( ρ ∗ ( γ ) L )( v ) = L ( ρ − ( γ ) v ) for every v ∈ R and L ∈ R ∗ =Hom( R , R ) . The flat bundle so (2 , Ad ρ ∗ is dual to so (2 , Ad ρ and the innerproduct ι induces an isomorphism ([Rag72]) so (2 , Ad ρ → so (2 , Ad ρ ∗ defined by setting ( A )( B ) = ι ( A, B ) for A, B ∈ so (2 , . This extends naturally to an isomorphism A p ( S, so (2 , Ad ρ ) → A p ( S, so (2 , Ad ρ ∗ ) . Consequently, we can introduce a coboundary map δ : A p ( S, so (2 , Ad ρ ) → A p − ( S, so (2 , Ad ρ ) IEMANNIAN METRICS ON GH ( S ) by setting δ = − ( − ∗ − d ∗ , and then a Laplacian operator ∆ : A p ( S, so (2 , Ad ρ ) → A p ( S, so (2 , Ad ρ ) given by ∆ = dδ + δd . A -form ξ is said to be harmonic if ∆ ξ = 0 , or, equivalently,if dξ = δξ = 0 . We have an orthogonal decomposition A ( S, so (2 , Ad ρ ) = Ker(∆) ⊕ Im( d ) ⊕ Im( δ ) and by the non-abelian Hodge theory ([Rag72]) every cohomology class contains aunique harmonic representative. Therefore, the bi-linear pairing ˜ g induces a scalarproduct in cohomology by setting g : H ( S, so (2 , Ad ρ ) × H ( S, so (2 , Ad ρ ) → R ([ α ] , [ β ]) ˜ g ( α harm , β harm ) , where α harm and β harm are the harmonic representatives of α and β .2.2. Definition of the metric.
As explained before, in order to define a Riemann-ian metric on GH ( S ) it is sufficient to define a Riemannian metric h on S and ascalar product ι on so (2 , Ad ρ .Let us begin with the metric h on S . Given ρ ∈ GH ( S ) , we denote by M ρ theunique GHMC anti-de Sitter manifold with holonomy ρ , up to isotopy. It is well-known that M ρ contains a unique embedded maximal (i.e. with vanishing meancurvature) surface Σ ρ ([BBZ07]). A natural choice for h is thus the induced metricon Σ ρ .As for the scalar product ι , we first introduce a scalar product in R that is closelyrelated to the maximal surface and its induced metric. Lifting the surface Σ ρ to theuniversal cover, we can find a ρ -equivariant maximal embedding ˜ σ : ˜ S → d AdS ⊂ R ,where [ AdS denotes the double cover of AdS consisting of unit time-like vectors in R endowed with a bi-linear form of signature (2 , . For any ˜ x ∈ ˜ S , we thus havea frame of R formed by the unit tangent vectors u (˜ x ) and u (˜ x ) to the surfaceat ˜ σ (˜ x ) , the time-like unit normal vector N (˜ x ) at ˜ σ (˜ x ) and the position vector ˜ σ ( x ) . We can define a scalar product ι ˜ x on R depending on the point ˜ x ∈ ˜ S bydeclaring the frame { u (˜ x ) , u (˜ x ) , ˜ σ ( x ) , N (˜ x ) } to be orthonormal for ι ˜ x . Because so (2 , ⊂ gl (4 , R ) ∼ = R × R ∗ , the inner product ι ˜ x induces an inner product on so (2 , and, consequently, on the trivial bundle ˜ S × so (2 , over ˜ S . This descendsto a metric ι on so (2 , Ad ρ by setting ι p ( φ, φ ′ ) := ι ˜ x ( ˜ φ ˜ x , ˜ φ ′ ˜ x ) for some ˜ x ∈ π − ( p ) where p ∈ S , π : ˜ S → S is the natural projection and ˜ φ ˜ x , ˜ φ ′ ˜ x are lifts of φ, φ ′ to thetrivial bundle ˜ S × so (2 , evaluated at ˜ x . Because ι ˜ x is ρ -equivariant, it is easy tocheck (see [Li16]) that ι p does not depend on the choice of ˜ x ∈ π − ( p ) and thus ι isa well-defined metric on the flat bundle so (2 , Ad ρ . IEMANNIAN METRICS ON GH ( S ) The following lemma is useful for computations with this metric:
Lemma 2.1 ([Li16]) . Assume that we have a matrix representation H of the innerproduct ι ˜ x at a point ˜ x ∈ π − ( p ) with respect to the canonical basis of R . Then ι p ( A, B ) = tr( A t H − BH ) for A, B ∈ so (2 , . Restriction to the Fuchsian locus.
In order to compute the restriction ofthe metric g to the Fuchsian locus, we need to understand the induced metric on theequivariant maximal surface and find a matrix representation of the inner product ι .If ρ ∈ GH ( S ) is Fuchsian, the representation preserves a totally geodesic space-likeplane in AdS . Realizing explicitly (the double cover of) anti-de Sitter space as d AdS = { x ∈ R | x + x − x − x = − } , we can assume, up to post-composition by an isometry, that ρ preserves the hyper-boloid H = { x ∈ R | x + x − x = − x = 0 } , which is isometric to the hyperbolic plane H = { z ∈ C | Im( z ) > } : an explicitisometry ([Li16]) being f : H → H ⊂ R (2.1) ( x, y ) (cid:18) xy , x + y − y , x + y + 12 y , (cid:19) . (2.2)The respresentation ρ : π ( S ) → SO (2 , factors then through the standard copyof SO (2 , inside SO (2 , , which is isomorphic to P SL(2 , R ) via the map ([KZ17]) Φ : P SL(2 , R ) → SO (2 , < SO (2 , (2.3) (cid:18) a bc d (cid:19) ad + bc ac − bd ac + bd ad − cd a − b − c + d a + b − c − d ab + cd a − b + c − d a + b + c + d
00 0 0 1 . (2.4)The map Φ induces a Lie algebra homomorphism, still denoted by Φ , given by Φ : sl (2 , R ) → so (2 , (2.5) (cid:18) a bc − a (cid:19) c − b c + b b − c a b + c a (2.6)It follows that if ρ ( π ( S )) = Γ < SO (2 , , then the maximal surface Σ ρ is realizedby H / Γ and is isometric to the hyperbolic surface H / Φ − (Γ) .Let us now turn our attention to the scalar product ι on so (2 , Ad ρ . Recallthat ι is determined by a family of inner products ι ˜ x on R depending on ˜ x ∈ ˜ S , IEMANNIAN METRICS ON GH ( S ) which is obtained by declaring the frame { u (˜ x ) , u (˜ x ) , ˜ σ (˜ x ) , N (˜ x ) } orthonormal.If we identify the universal cover of ˜ S with H , the map f gives an explicit ρ -equivariant maximal embedding of ˜ S into d AdS . Therefore, the coordinates of thevectors tangent and normal to the embedding with respect to the canonical basis of R can be explicitly computed and the following matrix representation H of ι z canbe obtained for any z ∈ H ([Li16, Corollary 6.5]): H = x y + 1 x ( x + y − y − x ( x + y +1) y x ( x + y − y ( x + y − y + 1 − ( x + y − x + y +1)2 y − x ( x + y +1) y − ( x + y − x + y +1)2 y ( x + y +1) y − with H − = x y + 1 x ( x + y − y x ( x + y +1) y x ( x + y − y ( x + y − y + 1 ( x + y − x + y +1)2 y x ( x + y +1) y ( x + y − x + y +1)2 y ( x + y +1) y − . Together with Lemma 2.1 we obtain the following:
Corollary 2.2 ([Li16]) . For any z ∈ H , after extending the definition of Lemma2.1 to A, B ∈ so (4 , C ) by ι z ( A, B ) = tr( A t H − BH ) , we have ι z (cid:16) Φ (cid:16) − z z − z (cid:17) , Φ (cid:16) − z z − z (cid:17)(cid:17) = 16 y . The last ingredient we need in order to describe the restriction of g to the Fuchsianlocus is an explicit realization of the tangent space to the Fuchsian locus inside T GH ( S ) . Lemma 2.3.
Let ρ ∈ GH ( S ) be a Fuchsian representation.(i) The tangent space at ρ to the Fuchsian locus is spanned by the cohomology classof φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17) , where φ ( z ) dz is a holomorphic quadratic differentialon Σ ρ .(ii) the so (2 , Ad ρ -valued -forms φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17) are harmonic representa-tives in their own cohomology class.Proof. (i) Let ρ ′ = Φ − ( ρ ) be the corresponding Fuchsian representation in P SL(2 , R ) .The claim follows from the fact ([Gol84]) that the tangent space to Teichmüller spaceis generated by the sl (2 , R ) Ad ρ ′ -valued -forms φ ( z ) dz ⊗ (cid:16) − z z − z (cid:17) and thus the tan-gent space to the Fuchsian locus is generated by the inclusion of H ( S, sl (2 , R ) Ad ρ ′ ) inside H ( S, so (2 , Ad ρ ) induced by the map Φ .(ii) We need to show that φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17) is d -closed and δ -closed. The firstfact has been proved in [Li16, Lemma 6.6]. As for δ -closedness, we will follow the IEMANNIAN METRICS ON GH ( S ) lines of the aformentioned lemma. From the definition of δ , it is enough to show that d ∗ ( (cid:16) φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17)(cid:17) = 0 . By linearity (cid:16) φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17)(cid:17) = z φ ( z ) dz ⊗ )) − φ ( z ) dz ⊗ )) − zφ ( z ) dz ⊗ (cid:16) Φ (cid:16) / − / (cid:17)(cid:17) . We then want to calculate )) , )) and (cid:16) Φ (cid:16) / − / (cid:17)(cid:17) . We choosea basis for so (2 , given by E = Φ( ) = (cid:18) − (cid:19) E = Φ (cid:16) / − / (cid:17) = (cid:18) (cid:19) E = Φ( ) = (cid:18) − (cid:19) E = (cid:18) (cid:19) E = (cid:18) (cid:19) E = (cid:18) − (cid:19) The map so (2 , Ad ρ → so (2 , Ad ρ ∗ is defined by setting ( A )( B ) = ι ( A, B ) thus A = X i =1 ι ( A, E i ) E ∗ i , where E ∗ i satisfies E ∗ i ( E j ) = ( if i = j otherwise . Applying Lemma 2.1 to compute ι ( E i , E j ) , we obtain the following E = 4 y ( E ∗ − x E ∗ + xE ∗ ) E = 4 y ( xE ∗ − x ( x + y ) E ∗ + ( x + y ) E ∗ ) E = 4 y ( − x E ∗ + ( x + y ) E ∗ − x ( x + y ) E ∗ ) . Putting everything together, we get d ∗ ( (cid:16) φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17)(cid:17) = d ∗ [ z φ ( z ) dz ⊗ E − φ ( z ) dz ⊗ E − zφ ( z ) dz ⊗ E ]= d ∗ [ − φ ( z ) dz ⊗ E ∗ + 4 z φ ( z ) dz ⊗ E ∗ − zφ ( z ) dz ⊗ E ∗ ] . Because z is a conformal coordinate for the induced metric on the maximal surface,from the definition of the Hodge star operator we see that ∗ dx = dy and ∗ dy = − dx , IEMANNIAN METRICS ON GH ( S ) hence, extending the operator to complex -forms by complex anti-linearity (i.e. ∗ ( iα ) = − i ∗ ¯ α ) we see that ∗ φ ( z ) dz = iφ ( z ) d ¯ z . Therefore, d ∗ [ − φ ( z ) dz ⊗ E ∗ + 4 z φ ( z ) dz ⊗ E ∗ − zφ ( z ) dz ⊗ E ∗ ]= d [ − iφ ( z ) d ¯ z ⊗ E ∗ + 4 iφ ( z )¯ z d ¯ z ⊗ E ∗ − iφ ( z )¯ zd ¯ z ⊗ E ∗ ] = 0 because φ ( z ) is holomorphic. As a consequence φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17) is d -closed and δ -closed, hence it is harmonic. (cid:3) We can finally prove one of the main results of the section:
Theorem 2.4.
The metric g on GH ( S ) restricts on the Fuchsian locus to a constantmultiple of the Weil-Petersson metric on Teichmüller space.Proof. By Lemma 2.3, it is sufficient to show that ˜ g (cid:16) φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17) , ψ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17)(cid:17) = h φ, ψ i W P , where here we are extending ˜ g to an hermitian metric on the space of so (4 , C ) Ad ρ -valued -forms. From the definition of ˜ g and Corollary 2.2 we have ˜ g (cid:16) φ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17) , ψ ( z ) dz ⊗ Φ (cid:16) − z z − z (cid:17)(cid:17) = R e Z S ι z (cid:16) Φ (cid:16) − z z − z (cid:17) , Φ (cid:16) − z z − z (cid:17)(cid:17) φ ( z ) dz ∧ ∗ ( ψ ( z ) dz )= R e Z S ι z (cid:16) Φ (cid:16) − z z − z (cid:17) , Φ (cid:16) − z z − z (cid:17)(cid:17) φ ( z ) dz ∧ ( iψ ( z ) d ¯ z )= R e Z S iφ ( z ) ψ ( z ) y dz ∧ d ¯ z = 32 h φ, ψ i W P . (cid:3) Our next goal is to show that the Fuchsian locus is totally geodesic. To thisaim it is sufficient to find an isometry of ( GH ( S ) , g ) that fixes the Fuchsian locus.In Mess’ parametrization of GH ( S ) as T ( S ) × T ( S ) , there is a natural involutionthat swaps left and right representations, thus fixing pointwise the Fuchsian locus.Identifying SL(2 , R ) × SL(2 , R ) with SO (2 , , this corresponds to conjugation by Q = diag( − , − , , − ∈ O (2 , . Therefore, we introduce the map q : GH ( S ) → GH ( S ) ρ QρQ − and show that this is an isometry for the metric g .We first need to compute the induced map in cohomology q ∗ : H ( S, so (2 , Ad ρ ) → H ( S, so (2 , Ad q ( ρ ) ) . IEMANNIAN METRICS ON GH ( S ) It is well-known (see e.g. [Gol84]) that a tangent vector to a path of representations ρ t is a -cocycle, that is a map u : π ( S ) → so (2 , satisfying u ( γγ ′ ) − u ( γ ′ ) = Ad( ρ ( γ )) u ( γ ′ ) . It is then clear that, if u is a cocycle tangent to ρ , then QuQ − is a -cocycletangent to q ( ρ ) . A -cocycle represents a cohomology class in H ( π ( S ) , so (2 , which is isomorphic to H ( S, so (2 , Ad ρ ) via H ( S, so (2 , Ad ρ ) → H ( π ( S ) , so (2 , σ ⊗ φ ] u σ ⊗ φ : γ Z γ σ ⊗ φ . Lemma 2.5.
For any σ ∈ A ( S ) and for any section φ of so (2 , Ad ρ , we have q ∗ [ σ ⊗ φ ] = [ σ ⊗ QφQ − ] Proof.
It is sufficient to show that u σ ⊗ QφQ − = Qu σ ⊗ φ Q − . This follows because,for any γ ∈ π ( S ) Z γ σ ⊗ QφQ − = Q (cid:18)Z γ σ ⊗ φ (cid:19) Q − . (cid:3) By an abuse of notation, we will still denote by q ∗ the map induced by q at the levelof so (2 , Ad ρ -valued -forms. Our next step is to show that q ∗ preserves the metric ˜ g . Lemma 2.6.
For any σ, σ ′ ∈ A ( S ) and for any sections φ and φ ′ of so (2 , Ad ρ ,we have ˜ g ( q ∗ ( σ ⊗ φ ) , q ∗ ( σ ′ ⊗ φ ′ )) = ˜ g ( σ ⊗ φ, σ ⊗ φ ) . Proof.
Given ρ ∈ GH ( S ) , we denote by M ρ the GHMC anti-de Sitter manifold withholonomy ρ . Because M ρ and M QρQ − are isometric via the map induced in thequotients by Q : d AdS → d AdS , the minimal surfaces Σ ρ and Σ QρQ − are isometricas well. In particular, their induced metric h and h q coincide on every ˜ x ∈ ˜ S .Moreover, if ˜ σ : ˜ S → d AdS is the ρ -equivariant maximal embedding, then Q ˜ σ is QρQ − -equivariant and still maximal. We deduce that if H is a matrix representationof the ρ -equivariant inner product ι ˜ x on ˜ S × so (2 , , then H q = Q t HQ is the matrixrepresentation of the QρQ − -equivariant inner product ι q ˜ x . Therefore, noting that IEMANNIAN METRICS ON GH ( S ) Q = Q t = Q − , for any φ and φ ′ sections of so (2 , Ad ρ and for any p ∈ S , we have ι p ( φ, φ ′ ) = ι ˜ x ( A, B ) taking ˜ φ ˜ x = A , ˜ φ ′ ˜ x = B and ˜ x ∈ π − ( p )= tr( A t H − BH ) by Lemma 2.1 = tr( Q t ( Q t ) − A t Q t ( Q t ) − QQ − H − ( Q t ) − Q t Q − QBQ − Q ( Q t ) − Q t HQQ − )= tr( Q t ( q ∗ ( A )) t ( H q ) − q ∗ ( B ) H q Q − )= tr(( q ∗ ( A )) t ( H q ) − q ∗ ( B ) H q Q − Q t )= tr(( q ∗ ( A )) t ( H q ) − q ∗ ( B ) H q )= ι q ˜ x ( q ∗ ( A ) , q ∗ ( B )) = ι qp ( q ∗ ( φ ) , q ∗ ( φ ′ )) by Lemma 2.5.We can now compute ˜ g ( σ ⊗ φ, σ ′ ⊗ φ ′ ) = Z S ι ( φ, φ ′ ) σ ∧ ( ∗ σ ′ )= Z S ι ( φ, φ ′ ) h σ, σ ′ i h dA h = Z S ι q ( q ∗ ( φ ) , q ∗ ( φ ′ )) h σ, σ ′ i h q dA h q = ˜ g ( q ∗ ( σ ⊗ φ, σ ′ ⊗ φ ′ ) which shows that q ∗ is an isometry for the Riemannian metrics on the bundles A ( S, so (2 , Ad ρ ) and A ( S, so (2 , Ad q ( ρ ) ) (cid:3) In order to conclude that q : GH ( S ) → GH ( S ) is an isometry for g , it is sufficientnow to show that the map q ∗ preserves harmonicity of forms. Lemma 2.7.
The map q ∗ : H ( S, so (2 , Ad ρ ) → H ( S, so (2 , Ad q ( ρ ) ) sends har-monic forms to harmonic forms.Proof. Let P i σ i ⊗ φ i be the harmonic representative in its cohomology class. Thisis equivalent to saying that d ( P i σ i ⊗ φ i ) = 0 and δ ( P i σ i ⊗ φ i ) = 0 . We need toshow that these imply d ( P i σ i ⊗ Qφ i Q − ) = 0 and δ ( P i σ i ⊗ Qφ i Q − ) = 0 , as well.The condition d ( P i σ i ⊗ Qφ i Q − ) = 0 easily follows by linearity of d .As for δ -closedness, by definition of δ , we have δ ( P i σ i ⊗ φ i ) = 0 if and only if d ∗ P i σ i ⊗ φ i ) = d ∗ ( P i σ i ⊗ φ i ) = 0 . Let us denote by q the analogous oper-ator defined on so (2 , Ad q ( ρ ) -valued -forms. Let { E j } j =1 be the basis of so (2 , introduced in the proof of Lemma 2.3 and denote by { E ∗ j } j =1 its dual. By definitionof and q we have A = X j =1 ι ( A, E j ) E ∗ j and q A = X j =1 ι q ( A, E j ) E ∗ j , IEMANNIAN METRICS ON GH ( S ) where, as in Lemma 2.6, we denoted by ι q the inner product on so (2 , Ad q ( ρ ) . Hence, d ∗ ( P i σ i ⊗ φ i ) = 0 if and only if d ∗ X i σ i ⊗ X j =1 ι ( φ i , E j ) E ∗ j = 0 , which implies that(2.7) d ∗ X i σ i ι ( φ i , E j ) ! = 0 for every j = 1 , . . . , .Therefore, using that ι q ( QAQ − , QBQ − ) = ι ( A, B ) for every A, B ∈ so (2 , , wehave d ∗ X i σ i ⊗ q Qφ i Q − ! = d ∗ X i σ i ⊗ X j =1 ι q ( Qφ i Q − , E j ) E ∗ j = d ∗ X i σ i ⊗ X j =1 ι q ( Qφ i Q − , QQ − E j QQ − ) E ∗ j = d ∗ X i σ i ⊗ X j =1 ι ( φ i , Q − E j Q ) E ∗ j . A straightforward computation shows that Q − E Q = − E Q − E Q = − E Q − E Q = − E Q − E Q = E Q − E Q = E Q − E Q = − E thus d ∗ ( P i σ i ⊗ P j =1 ι ( φ i , Q − E j Q ) E ∗ j ) = 0 , because, up to a sign, the coefficientsof E ∗ j coincide with those in Equation 2.7 for j = 1 , and the coefficient of E ∗ isswapped with that of E ∗ in Equation 2.7. Hence, d ∗ P i σ i ⊗ Qφ i Q − ) = 0 , andthen δ ( P i σ i ⊗ Qφ i Q − ) = 0 , as required. (cid:3) Combining the above result with Lemma 2.6, by definition of the metric g on GH ( S ) we obtain the following: Theorem 2.8.
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