Geodesic Coordinates for the Pressure Metric at the Fuchsian Locus
aa r X i v : . [ m a t h . DG ] O c t Geodesic Coordinates for the Pressure Metric at the FuchsianLocus
Xian Dai
Abstract
We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsianlocus for the pressure metric in the Hitchin component H ( S ) of surface group representa-tions into P SL (3 , R ).The proof consists of the following elements: we compute first derivatives of the pressuremetric using the thermodynamic formalism. We invoke a gauge-theoretic formula to com-pute first and second variations of reparametrization functions by studying flat connectionsfrom Hitchin’s equations and their parallel transports. We then extend these expressionsof integrals over closed geodesics to integrals over the two-dimensional surface. Symmetriesof the Liouville measure then provide cancellations, which show that the first derivatives ofthe pressure metric tensors vanish at the Fuchsian locus. The Weil-Petersson metric on Teichm¨uller space is a central object in classical Teichm¨ullertheory. Quite a bit is known about it: It is a negatively curved real analytic K¨ahler metricwith isometry group induced from the extended mapping class group (Ahlfors [1], Tromba [36],Masur-Wolf [24]). Although it is not complete (Wolpert [37], Chu [10]), it resembles a completenegative curved metric and shares many similar nice properties (Wolpert [37],[38]).In recent years, considerable attention has focused on higher rank Teichm¨uller spaces ([13],[15], [19]). It is natural to seek metric structures on these spaces with the hope that structurewill reflect important properties of the spaces. To that end, Bridgeman, Canary, Labourie andSambarino in [8] have extended the Weil-Petersson metric from Teichm¨uller space to an analyticRiemannian metric by techniques from thermodynamic formalism, called the pressure metricon Hitchin components. The Hitchin component H n ( S ), defined by Hitchin in [15] is a specialcomponent of the representation space of the fundamental group of a closed surface S of genus g ≥ P SL ( n, R ). In particular, the Teichm¨uller space T ( S ), identified as representationsinto P SL (2 , R ), embeds in this component and is called the Fuchsian locus. To define thepressure metric, we associate a geodesic flow to each Hitchin representation and describe thesereparametrized geodesic flows by some H¨older functions called reparametrization functions. Ourpressure metric is defined on the tangent space of a Hithchin component by taking the varianceof the first variation of reparametrization functions that record the infinitesimal change of therepresentations.Bridgeman, Canary, Labourie and Sambarino have proved that the pressure metric in factrestricts to a multiple of the Weil-Petersson metric on the Fuchsian locus and is invariant under1he action of the mapping class group. Despite this nice coincidence, very little is presentlyknown about the pressure metric. Our goal in this paper is to investigate some variational C properties of the pressure metric using the tools from thermodynamic formalism. Some C properties of the pressure metric have recently been identifed by Labourie and Wentworth in[20]. In particular, they show, when restricted to the Fuchsian locus, the pressure metric isproportional to a Petersson-type pairing for variation given by holomorphic differentials.One may be curious to what extent that the pressure metric in Hitchin components resemblesWeil-Petersson geometry. Inspired by Ahlfors’ work in [1] that the Bers coordinates are geodesicfor Weil-Petersson metric, we will show that for one particular case of Hitchin component H ( S ),similar coordinates are geodesic for the pressure metric near the Fuchsian locus. We expectsimilar results will hold for the general case of Hitchin components H n ( S ).We will find and evaluate expressions for the derivatives of the pressure metric at theFuchsian locus for the case of P SL (3 , R ) and its Hitchin component H ( S ).The coordinates we choose are very natural in the setting of Hitchin components from aHiggs bundle perspective. Picking ( q , · · · , q g − ) to be a basis for H ( X, K ) over R and( q g − , · · · , q g − ) to be a basis for H ( X, K ) over R , every element of H ( S ) corresponds tosome m ( ξ ) = ξ q + · · · ξ l q l with ξ = ( ξ , · · · , ξ l ) ∈ R l and l = 16 g − ξ i are coordinate functions and the coordinate system is realized by the Hitchin parametriza-tion H ( S ) ∼ = H ( X, K ) L H ( X, K ). The Hitchin parametrization is given by the Hitchinfibration and the Hitchin section which are defined by Hitchin in [15] and will be explained inthe next section.We will show Theorem 1.1.
Let S be a closed oriented surface with genus g ≥ . For any point σ ∈ T ( S ) ⊂H ( S ) , let X be the Riemann surface corresponding to σ . Then the Hitchin parametrization H ( X, K ) L H ( X, K ) provides geodesic coordinates for the pressure metric at σ . More explicitly, if we denote components of the pressure metric at σ as g ij ( σ ) with respectto the coordinates given by Hitchin parametrization, then ∂ k g ij ( σ ) = 0 for all possible i, j, k ranging from 1 to 16 g − C properties. On the other hand, reparametriza-tion functions and their variations need to be understood via their Higgs bundle invariants. Wenow outline some important ingredients of our computations and proofs.Since there are two types of tangential directions in H ( S ), directions given by quadraticdifferentials and directions given by cubic differentials (corresponding roughly to directionsalong the Fuchsian locus and transverse to it respectively), the derivatives of the metric tensorwill be divided into different cases according to this distinction. • The vanishing of a few types of first derivatives of the metric tensor follows easily from the2eometric facts that the Fuchsian locus is a totally geodesic embedding into the Hitchincomponent and that the Bers coordinates on Teichm¨uller space are geodesic. • On the other hand, to compute the bulk of the components, we need to invoke ther-modynamic formalism to obtain an explicit formula for first derivatives of the pressuremetric. We find the formula of the first variation of the pressure metric by computing thirdderivatives of pressure functions using the theory of the Ruelle operator. This expressioninvolves first and second variations of the reparametrization functions. • We start from studying first and second variations of reparametrization functions on closedgeodesics. Because vectors tangent to periodic geodesics are dense in tangent bundles ofhyperbolic surfaces, the computation of first and second variations of the reparametriza-tion functions on closed geodesics can be extended to the unit tangent bundle after anargument that the natural extensions are H¨older functions. • To study the first variation of reparametrization functions on closed geodesics, we recalla gauge theoretic formula from [20]. We then interpret the resulting formula as defininga system of homogeneous ordinary differential equations which we proceed to solve. • Finding the second variation of the reparametrization functions is equivalent to under-standing the first variation of our gauge theoretic formula from the previous paragraph.The difficulty here is describing how projections onto the eigenvectors for the holonomymap vary when we have a family of representations in the Hitchin component. Indeed, itturns out that we need to understand the variation of all of the eigenvectors of our holon-omy map. We interpret this problem in terms of solving a system of non-homogeneousordinary differential equations with suitable boundary conditions which we then proceedto solve. • For some types of metric tensors that involve both the tangential directions and transversedirections to the Fuchsian locus, analyzing flat connections associated to these directionsrequire understanding the corresponding harmonic metrics that are solutions of Hitchin’sequations. The harmonic metrics become no longer diagonalizible when leaving the Fuch-sian locus along these mixed directions. We break up the infinitesimal version of Hitchin’sequation system and obtain nine scalar equations. We analyze them by maximum princi-ples and Bochner techniques to compute second variations of reparametrization functions. • The evaluation of first derivatives of the pressure metric can be lifted to the Poincar´e diskfollowing an idea from [20]. Here is where it becomes important that we are taking firstderivatives of the pressure metric rather than zero derivatives of the pressure metric. Inparticular, we find formulas involving iterated integrals of these holomorphic differentials.Specifying a point on the unit tangent bundle, we can identify the Poincar´e disk as ourcoordinate chart and write down the analytic expansions of our holomorphic differentialson this chart. Using geodesic flow invariance and rotational invariance of the Liouvillemeasure, we find that no nonzero coefficients of our analytic expansions remain afterintegration.We conjecture this theorem is true for H n ( S ) for n ≥ Conjecture 1.1.
Let S be a closed oriented surface with genus g ≥ and n ≥ . For anypoint σ ∈ T ( S ) ⊂ H n ( S ) , let X be the Riemann surface corresponding to σ , the Hitchinparametrization i = n L i =2 H ( X, K i ) provides geodesic coordinates for the pressure metric at σ . tructure of the article: The article is organized as follow. In section 2, we recall somefundamental results from the theory of thermodynamic formalism and reparametrizations ofgeodesic flows. We define the pressure metric. We also introduce Higgs bundles and Hitchindeformation for defining our coordinates in Hitchin components. Section 3 is devoted to pre-liminary proofs. We compute the formula for third derivatives of the pressure function usingthermodynamic formalism machinery. We also include the gauge-theoretic formula given byLabourie and Wentworth in [20]. Both are significant for the proof of the main theorem. Insection 4, we start the proof of the main theorem. We divide the components of first derivativesof metric tensor into several types and show they are zero following the steps explained above.
Acknowledgement
The author would like to thank her advisor, Michael Wolf, for his helpand kind support. The weekly meetings were an important source of encouragement and guid-ance. The author also wants to thank Martin Bridgeman for his warm introduction to thepressure metric. The author also wants to express her appreciation to Siqi He, Qiongling Li,Andrea Tamburelli and Siran Li for the helpful conversation with them. Finally the authoracknowledges support from U.S. National Science Foundation grants and Geometric structuresand Representation Varieties (the GEAR network). The paper would not have been possiblewithout this support.
In this section, we develop the notation and background material we will need. We begin insection 2.1 with a discussion of reparametrization of geodesic flows. Then in section 2.2 we recallthe elements of thermodynamic formalism we will need and finally in section 2.3, we concludewith some notation from the theory of Higgs bundles which arises in our arguments.Let S be a closed oriented surface with genus g ≥
2. We will define all the concepts forintroducing the pressure metric in the context of Hitchin components H n ( S ). The reader canfind more general version in [8]. The Hitchin components H n ( S ) will be briefly introduced insection 2.3.Equip S with a complex structure J so that X = ( S, J ) is a Remain surface and thus a pointin Teichm¨uller Space. Let σ be the hyperbolic metric in the conformal class of X . We denotethe unit tangent bundle of X with respect to σ by U X and Φ the geodesic flow on (
X, σ ). In this subsection, we introduce how we reparametrize the geodesic flow Φ by reparametriza-tion functions. In particular, we introduce Livˇsic’s theorem and geodesic flows for Hitchinrepresentations.Suppose f : U X → R is a positive H¨older function and a a closed orbit. We will reparametrizethe flow Φ by the function f so that for the new flow Φ f , the flow’s direction remains the sameeverywhere but the speed of the flow changes. In particular, for a Φ-periodic orbit a , denotingits period with respect to Φ by l ( a ), we want the period of a for the new flow Φ f to be thefollowing: 4 f ( a ) = Z l ( a )0 f (Φ s ( x ))d s, where x is any point on a .This leads to the following definition of reparametrization. Definition 2.1.
Let f : U X → R be a positive H¨older continuous function. We define thereparametrization of Φ by f to be the flow Φ f on U X such that for any ( x, t ) ∈ U X × R , Φ ft ( x ) = Φ αf ( x,t ) ( x ) , where κ f ( x, t ) = R t f (Φ s ( x )))d s and α f : U X × R → R satisfies α f ( x, κ f ( x, t )) = t. Remark 2.1.
Suppose O is the set of periodic orbits of Φ . If a ∈ O , then its period as a Φ ft periodic orbit is l f ( a ) because Φ fl f ( a ) ( x ) = Φ α f ( x,l f ( a )) ( x ) = Φ l ( a ) ( x ) = x. We introduce Livˇsic cohomology classes, originally established by Livˇsic [22]. Livˇsic coho-mologous H¨older functions turn out to reparametrize a flow in “equivalent” ways.Let C h ( U X ) denote the set of real-valued H¨older functions on
U X . Definition 2.2.
For f, g ∈ C h ( U X ) , we say they are Livˇsic cohomologous if there exists aH¨older continuous function V : U X → R that is differentiable in the flow’s direction such that f ( x ) − g ( x ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 V (Φ t ( x )) . If f is Livˇsic cohomologous to g , then we will denote it as f ∼ g . We have the following important properties of Livˇsic cohomologous functions:1. If f and g are Livˇsic cohomologous then they have the same integral over any Φ-invariantmeasure.This is because R UX V (Φ t ( x ))d m = Const for any Φ-invariant measure m and any t ∈ R .2. If f and g are positive and Livˇsic cohomologous, then the reparametrized flows Φ f andΦ g are H¨older conjugate, i.e. there exists a H¨older homeomorphism h : U X → U X suchthat, for all x ∈ U X and t ∈ R , h (Φ ft ( x )) = Φ gt ( h ( x )) . See ([17] Prop.19.2.8). 5he procedure of reparametrizing geodesic flows can be applied to Hitchin components H n ( S ) and provides reparametrization functions as codings for for representations (see [8], [9]).We can associate to each (conjugacy class of) Hitchin representation ρ ∈ H n ( S ) a geodesic flowΦ ρ . This flow relates H n ( S ) to thermodynamic formalism. In particular, it has the followingproperties: • Φ ρ is an Anosov flow. • There exists a H¨older function f ρ : U X −→ R + , called the reparametrization function of ρ , such that the reparametrized flow Φ f ρ of Φ is H¨older conjugate to Φ ρ . • The period of the orbit associated to [ γ ] ∈ π ( S ) is log Λ γ ( ρ ), where Λ γ ( ρ ) is the spectralradius of ρ ( γ ), i,e, the largest modulus of the eigenvalues of ρ ( γ ). Next we will introduce some concepts arising from the thermodynamic formalism needed forour proofs. The introduction of most of the material here can also be found in [8]. After theintroduction, we will define the pressure metric on Hitchin components.As usual, we let Φ denote the geodesic flow on a hyperbolic surface (
X, σ ). We denote by M Φ the set of Φ-invariant probability measures on U X . Recall l ( a ) denotes the period of theperiodic point a with respect to Φ. Let R T = { a closed orbit of Φ | l ( a ) ≤ T } . Definition 2.3.
The topological entropy of Φ is defined as: h (Φ) = lim sup T −→∞ log R T T .
Recall for a H¨older function f : U X → R , we denote l f ( a ) = Z l ( a )0 f (Φ s ( x ))d s. Definition 2.4.
The topological pressure (or simply pressure) of a continuous function f : U X → R with respect to Φ is defined by P (Φ , f ) = lim sup T −→∞ T log X a ∈ R T e l f ( a ) . Remark 2.2.
From this definition, we see the pressure of a function f only depends on theperiods of f , i.e. the collection of numbers { l f ( a ) } for any a ∈ O . From Livˇsic’s Theorem, weconclude the pressure of a function only depends on Livˇsic cohomologous class. In statistical mechanics, suppose we are given a physical system with different possible states i = 1 , · · · , n and the energies of these states are E , E · · · , E n with probability p j that state j occurs. When energy is fixed, the principle “nature maximizes entropy h ” says the entropy6 ( p , · · · , p n ) = P ni =1 − p i log p i of the distribution will be maximized with right choices of p i .However when the physical system is put in contact with a much larger “heat source” whichis at a fixed temperature T and energy is allowed to pass between the original system andthe heat source, “nature minimizes the free energy” will instead apply by reaching the “Gibbsdistribution”. The free energy is E − kT h , where k is a physical constant, h is the entropy and E = P ni =1 p i E i is the average of energy. In the thermodynamic formalism, energy potentials E i of different states are encoded by continuous functions and “Gibbs distributions” for discreteprobability spaces are generalized to equilibrium states. The principle “nature minimizes freeenergy” motivates the following. Proposition 2.1. (Variational principle.)Denoting the measure-theoretic entropy of Φ with respect to a measure m ∈ M Φ as h (Φ , m ) ,the (topological) pressure of a continuous function f : U X → R satisfies P (Φ , f ) = sup m ∈M Φ ( h (Φ , m ) + Z UX f d m ) . In particular, the topological entropy is the supremum of all measure-theoretic entropy, P (Φ ,
0) = sup m ∈M Φ ( h (Φ , m )) = h (Φ) . Remark 2.3.
One can also take Proposition 2.1 as definitions of pressure and topologicalentropies.
We shall omit the background geodesic flow Φ in the notation of pressure and simply write P ( · ) = P (Φ , · ) Definition 2.5.
A measure m ∈ M Φ on U X such that P ( f ) = h (Φ , m ) + Z UX f d m is called an equilibrium state of f . Proposition 2.2. (Bowen-Ruelle [6]) For any H¨older function f : U X → R , with respect tothe geodesic flow Φ , there exists a unique equilibrium state for f , denoted as m f . Moreover, m f is ergodic. Remark 2.4.
We observe from the definition of equilibrium states that if f − g is Livˇsic coho-mologous to a constant, then f and g have the same equilibrium states. Definition 2.6.
The equilibrium state m for f = 0 is called a probability measure of maximalentropy. It is also called the Bowen-Margulis measure of Φ . We also denote it as m Φ . Itsatisfies P (0) = P (Φ ,
0) = h (Φ , m Φ ) = h (Φ) . Remark 2.5.
For H n , the Liouville measure is a probability measure of maximal entropy forgeodesic flows (see [16] section 2). Thus in our case of hyperbolic surface ( X, σ ) , denoting theLiouville measure as m L ∈ M Φ , we have m L = m Φ . Given f a positive H¨older continuous function on U X , denoting h ( f ) = h (Φ f ) to be thetopological entropy of the reparametrized flow Φ f , we have the following lemma that allows usto “normalize” a H¨older function to have pressure zero.7 emma 2.3. (Sambarino [33], Bowen-Ruelle [6]) P ( − hf ) = 0 if and only if h = h ( f ) = h (Φ f ) . Potrie and Sambarino show, in the Hitchin component H n ( S ), the topological entropy ismaximized only along the Fuchsian locus. In particular, it is a constant on the Fuchsian locus. Theorem 2.4. (Potrie-Sambarino [30]) If ρ ∈ H n ( S ) , then h ( ρ ) ≤ n − . Moreover, if h ( ρ ) = n − , then ρ lies in the Fuchsian locus. We start to define variance and covariance which will be important. The convergence ofthem for mean zero functions is classical.
Definition 2.7.
For g a H¨older continuous function on U X with mean zero with respect to m f (i.e. R UX g d m f = 0 ), the variance of g with respect to f is defined as: Var( g, m f ) = lim T →∞ T Z UX (cid:18)Z T g (Φ s ( x ))d s (cid:19) d m f ( x ) . (2.1) Definition 2.8.
For g , g H¨older continuous functions on
U X with mean zero with respect to m f (i.e. R UX g d m f = R UX g d m f = 0 ), the covariance of g , g with respect to f is defined as: Cov ( g , g , m f ) = lim T →∞ T Z UX (cid:18)Z T g (Φ s ( x ))d s (cid:19) (cid:18)Z T g (Φ s ( x ))d s (cid:19) d m f ( x ) . (2.2)Note these expressions are finite: Proposition 2.5.
For g , g H¨older continuous function on
U X with mean zero with respectto m f , the covariance of g and g is finite: Cov ( g , g , m f ) < ∞ . The convergence is guaranteed by decay of correlations (see [25]).
Definition 2.9.
We define an operator P m : C h ( U X ) → C h ( U X ) associated to a probabilitymeasure m on U X to be: P m ( g )( x ) = g ( x ) − Z UX g d m. We will use the notation m ( g ) = R UX g d m for a probability measure m .The following corollary will be useful: Corollary 2.1.
It suffices to have m f ( g ) = 0 and m f ( g ) < ∞ to guarantee the convergenceof covariance and Cov ( g , g , m f ) = Cov ( g , P m f ( g ) , m f ) < ∞ . (2.3) Proof of Corollary 2.1. T Z UX (cid:18)Z T g (Φ s ( x ))d s (cid:19) (cid:18)Z T g (Φ s ( x )) − P m f ( g (Φ s ( x )))d s (cid:19) d m f ( x )= 1 T Z UX (cid:18)Z T g (Φ s ( x ))d s (cid:19) (cid:18)Z T m f ( g )d s (cid:19) d m f ( x )= m f ( g ) Z UX Z T g (Φ s ( x ))d s d m f ( x ) As m f ( g ) is a constant= m f ( g ) Z T Z UX g (Φ s ( x ))d m f ( x ) ds By Fubini theorem= m f ( g ) Z T Z UX g ( x )d m f ( x ) ds m f is Φ-invariant=0Let T → ∞ , we obtain the desired result. The same applies to the case m f ( h ) = 0 , m f ( h ) < ∞ . We will also need the following characterization of covariance for later use. Definition 2.10. (Pollicott [29]) For g , g H¨older continuous function with mean zero withrespect to m f (i.e. R UX g d m f = R UX g d m f = 0 ), the covariance of g , g may also be writtenas: Cov ( g , g , m f ) = lim T →∞ Z UX g ( x ) Z T − T g (Φ s ( x ))d s ! d m f ( x ) . Proof.
This proof is from [29].
Cov ( g , g , m f ) = lim T →∞ T Z UX (cid:18)Z T g (Φ s ( x ))d s (cid:19) (cid:18)Z T g (Φ s ( x ))d s (cid:19) d m f ( x )= lim T →∞ T Z UX Z T − T g (Φ s ( x ))d s ! Z T − T g (Φ s ( x ))d s ! d m f ( x ) m f is Φ-invariant= lim T →∞ Z T − T Z UX g (Φ t ( x )) 1 T Z T − T g (Φ s ( x ))d s ! d m f ( x )d t Because m ∈ M Φ , the following does not vary with s . Const. = lim T →∞ Z T − T Z UX g (Φ t ( x )) g (Φ s ( x ))d m f ( x )d t ∀ s ∈ R = lim T →∞ Z T − T Z UX g (Φ t ( x )) 1 S Z S − S g (Φ s ( x ))d s ! d m f ( x )d t Average over s ∈ (cid:20) − S , S (cid:21) = lim S →∞ lim T →∞ Z T − T Z UX g (Φ t ( x )) 1 S Z S − S g (Φ s ( x ))d s ! d m f ( x )d t = lim T →∞ Z T − T Z UX g (Φ t ( x )) 1 T Z T − T g (Φ s ( x ))d s ! d m f ( x )d t = Cov ( g , g , m f ) 9n particular, setting s = 0 gives Cov ( g , g , m f ) = lim T →∞ Z T − T Z UX g (Φ t ( x )) g (( x ))d m f ( x )d t Rearranging the integrals gives the desired result.Higher correlation and higher covariance are introduced for Anosov diffemorphism in [18].For geodesic flows, we define
Definition 2.11.
For g , g , g H¨older continuous functions with mean zero with respect to m f ,we define the higher covariance as follows, Cov ( g , g , g , m f ) = lim T →∞ T Z UX Z T g (Φ t ( x ))d t Z T g (Φ t ( x ))d t Z T g (Φ t ( x ))d t d m f ( x ) . equivalently, Cov ( g , g , g , m f ) = lim T →∞ Z UX g ( x ) Z T − T g (Φ s ( x ))d s ! Z T − T g (Φ s ( x ))d s ! d m f ( x ) . This equivalence is clear from the proof of equivalent Definition 2.10. The convergence of
Cov ( h , h , h , m ) is guaranteed by “exponential multiple mixing” for geodesic flow on nega-tively curved compact surfaces (see Pollicott’s note [27]). These definitions will be used laterwhen we introduce first derivatives of the pressure metric.We use the general notation in the sequel: ∂ s f (0) = dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 f ( s ) , ∂ ss f (0) = d ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 f ( s ) (2.4)If there are more than one parameter, e.g. f ( s , s , · · · , s k ) and k ≥
2, then we specify theindexes that we are taking derivatives of: ∂ s i ··· ,s ij f (0) = ∂ j f ( s , s , · · · , s k ) ∂s i · · · ∂s i j (cid:12)(cid:12)(cid:12)(cid:12) s = s = ··· =0 (2.5) Theorem. (Parry-Pollicott [26], McMullen [25]) Let f s be a smooth family of functions in C h ( U X ) , then we have1. The first derivative of P ( f s ) at s = 0 is given by d P ( f s ) ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Z UX ∂ s f d m f , (2.6)
2. If the first derivative is zero, then d P ( f s ) ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 = V ar ( ∂ s f , m f ) + Z UX ∂ ss f d m f , (2.7)
3. If the first derivative is zero, then
V ar ( ∂ s f , m f ) = 0 if and only if ∂ s f is Livˇsic cohomologous to zero. emark 2.6. If f ( s, t ) is a smooth two parameter family in C h ( U X ) , then ∂ P ( f ( s, t )) ∂t∂s (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = Cov ( P mf (0) ( ∂ s f (0)) , P mf (0) ( ∂ t f (0)) , m f (0) ) + Z UX ∂ st f (0)d m f (0) . (2.8)Define P ( U X ) to be the set of pressure zero H¨older functions on
U X , i.e. P ( U X ) = { f ∈ C h ( U X ) : P ( f ) = 0 } . The tangent space of P ( U X ) at f is the set T f P ( U X ) = ker d f P = { h ∈ C h ( U X ) | Z UX h d m f = 0 } We define a pressure semi-norm on the tangent space of P ( U X ) at f , by letting Definition 2.12.
The pressure semi-norm of g ∈ T f P ( U X ) is defined as: h g, g i P = − V ar ( g, m f ) R UX f dm f . One notices for g ∈ T f P ( U X ), the variance
V ar ( g, m f ) = 0 if and only if g is Livˇsiccohomologous to 0, i.e. g ∼ In this subsection we introduce all the notation from the theory of Higgs bundles that will arisein our arguments. We also introduce a coordinate system on the Hitchin component at the endof the section.Recall S is a closed oriented surface with genus g ≥ X = ( S, J ) is a Riemann surface.
Definition 2.13.
A rank n Higgs bundle over X is a pair ( E, Φ) where E is a holomorphicvector bundle of rank n and Φ ∈ H ( X, End ( E ) ⊗ K ) is called a Higgs field. A SL ( n, C ) -Higgsbundle is a Higgs bundle ( E, Φ) satisfying det E = O and Tr Φ = 0 . Definition 2.14.
1. A Higgs bundle ( E, Φ) is (semi)stable if every proper Φ -invariant holomorphic subbundle F of E satisfies deg( F )rank( F ) ( ≤ ) < deg( E )rank( E ) .
2. A semi-stable Higgs bundle ( E, Φ) is polystable if it decomposes as a direct sum of stableHiggs bundles. Theorem 2.6.
It is classical that for a holomorphic vectot bundle E with holomorphic struc-ture ¯ ∂ E and a Hermitian metric H , there exists a unique connection ∇ ¯ ∂ E ,H , called the Chernconnection, such that . ∇ , ∂ E ,H = ¯ ∂ E .2. ∇ ¯ ∂ E ,H is unitary. We will from now on restrict our interest to degree zero Higgs bundles.
Theorem 2.7. (Hitchin[14], Simpson[34]) Let ( E, Φ) be a rank n , degree zero Higgs bundle on X . Then E admits a Hermitian metric H satisfying Hitchin’s equation if and only if ( E, Φ) ispolystable. Hitchin’s equation is F ¯ ∂,H + [Φ , Φ ∗ H ] = 0 . (2.9) where F ¯ ∂,H is the curvature of the Chern connection ∇ ¯ ∂ E ,H , and Φ ∗ H is the Hermitian adjointof Φ . Remark 2.7.
Define a connection D H on ( E, Φ , H ) as D H = ∇ , ∂ E ,H + Φ + Φ ∗ H . (2.10) D H is flat if and only if the Hitchin’s equation is satisfied. Definition 2.15. • The space of gauge equivalence classes of polystable SL ( n, C ) Higgs bundles is called themoduli space of SL ( n, C ) -Higgs bundles and is denoted by M Higgs ( SL ( n, C )) . • The space of gauge equivalence classes of reductive flat SL ( n, C ) conections is called thede Rham moduli space and is denoted by M deRham ( SL ( n, C )) . Remark 2.8.
The Hitchin-Simpson Theorem gives a 1-1 correspondence between M Higgs ( SL ( n, C )) and M deRham ( SL ( n, C )) from the above remark. It is also called Hitchin-Kobayashi correspon-dence. Definition 2.16.
Given a basis of SL ( n, C ) -invariant homogeneous polynomials p i of degree i on sl ( n, C ) for ≤ i ≤ n , the Hitchin fibration is a map from the moduli space of SL ( n, C ) -Higgsbundles over X to the direct sum of holomorphic differentials p : M Higgs ( SL ( n, C )) −→ i = n M i =2 H ( X, K i ) , ( E, Φ) ( p (Φ) , · · · , p n (Φ)) . Definition 2.17.
A Hitchin section s of the Hitchin fibration is a map from i = n L i =2 H ( X, K i ) backto M Higgs ( SL ( n, C )) . For q = ( q , q , · · · , q n ) ∈ i = n L i =2 H ( X, K i ) , we define s ( q ) to be a Higgsbundle E = K n − L K n − · · · L K − n with its Higgs field Φ( q ) = r q r r q r r r q · · · n − Q i =1 r i q n − n − Q i =1 r i q n r q r r q · · · · · · n − Q i =2 r i q n − r q r r q · · · .... . . . . . . . . . . . ... r n − q r n − r n − q r n − q : E → E ⊗ K here r i = i ( n − i )2 .Such a Higgs field Φ( q ) is denoted as ˜ e + q e + q e + · · · q n e n − and K is a holomorphicline bundle with its square to be the canonical line bundle K . For the construction of Hitchinsection and related Lie theory, see [15]. The notation for e i we use here can be found in [21]. Remark 2.9.
Hitchin shows the Higgs bundles in the image of the Hitchin section have holon-omy in SL ( n, R ) . The representation space of these Higgs bundles form a connected componentof the SL ( n, R ) -representation variety, called the Hitchin component H n ( S ) . The isomorphismbetween H n ( S ) and i = n L i =2 H ( X, K i ) yields a parametrization of Hitchin component H n ( S ) . Wecall i = n L i =2 H ( X, K i ) the Hitchin base. In particular, the tangent space at Fuchsian point X isidentified with the Hitchin base. Fixing E = K n − L K n − · · · L K − n , we consider the following map as an infinitesimalchange of a family of Higgs fields Φ ǫ associated to q . φ : i = n M i =2 H ( X, K i ) → Ω , ( X, sl ( n , R )) φ ( q ) = n X i =2 q i ⊗ e i − . In particular, the infinitesimal change of a family of flat connections (2 .
10) in M deRham ( SL ( n, C ))associated to q defines an isomorphism of i = n L i =2 H ( X, K i ) with the tangent space of the Hitchincomponent T X H n ( S ) that coincides with the infinitesimal version of Hitchin parametrization.We define the deformation of flat connections as follows, Definition 2.18.
At the Fuchsian point X , we define our Hitchin deformation associated to q to be ψ ( q ) := φ ( q ) + λ ( φ ( q )) , where λ is an anti-involution for the split real form of sl ( n , C ) . It is explained in [2], [15]. This type of deformation will be the tangential objects we consider for the pressure metric.
Remark 2.10.
The Hitchin parametrization in Remark 2.9 gives a coordinate system for H n ( S ) based at X . More explicitly, given a basis { q i } i = li =1 of i = n L i =2 H ( X, K i ) with l = 2( n − g − ,the coordinate system is given by m ( ξ ) = ξ q + · · · ξ l q l , where ξ = ( ξ , · · · , ξ l ) ∈ R l . Because of the isomorphism between H n ( S ) and i = n L i =2 H ( X, K i ) ,the vector ξ = ( ξ , · · · , ξ l ) provides local parameters on H n ( S ) and the function ξ i : H n ( S ) → R is a coordinate function for ≤ i ≤ l . .4 The pressure metric on Hitchin components We define the pressure metric for Hitchin components H n ( S ) in this subsection and state someknown results about it.Recall H ( U X ) is the space of pressure zero H¨older functions modulo Livˇsic coboundaries.We relates H ( U X ) to the Hitchin component H n ( S ) by the following thermodynamic mapping. Definition 2.19.
The thermodynamic mapping
Ψ : H n ( S ) −→ H ( U X ) from a Hitchin compo-nent H n ( S ) to the space H ( U X ) of Livˇsic cohomology classes of pressure zero H¨older functionson U X is defined as Ψ( ρ ) = [ − h ( ρ ) f ρ ] , where h ( ρ ) = h ( f ρ ) = h (Φ f ρ ) is the topological entropy of the reparametrized flow Φ f ρ . The mapping Ψ admits local analytic lifts to the space P ( U X ) of pressure zero H¨olderfunctions. In particular, the map ˜Ψ : H n ( S ) −→ P ( U X ) given by ˜Ψ( ρ ) = − h ( ρ ) f ρ is ananalytic local lift of Ψ. This enables us to pull back the pressure form on P ( U X ) to obtain apressure form on H n ( S ).We will from now on denote f Nρ = − h ( ρ ) f ρ to be the normalized reparametrization function.Given an analytic family { ρ s } s ∈ ( − , of (conjugatcy class of) representations in Hitchin com-ponent H n ( S ), we denote ˙ ρ = ∂ s ρ = ∂ s ρ s | s =0 . Let { f ρ s } s ∈ ( − , be associated reparametriza-tion functions, we pull back the pressure form on P ( U X ) as follow: h ˙ ρ , ˙ ρ i P = h d ˜Ψ( ˙ ρ ) , d ˜Ψ( ˙ ρ ) i P = h ∂ ( − h ( ρ s ) f ρ s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 , ∂ ( − h ( ρ s ) f ρ s ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 i P = h ∂ s ( f Nρ s ) | s =0 , ∂ s ( f Nρ s ) | s =0 i P = − V ar ( ∂ s ( f Nρ s ) | s =0 , m f Nρ ) R UX f Nρ dm f Nρ It is proved in [8] the pull back pressure form is nondegenrate and thus defines a Riemanninanmetric on H n ( S ): Definition 2.20.
Suppose { ρ s } s ∈ ( − , and { η t } t ∈ ( − , are two analytic families of (conjugacyclasses of ) representations in Hitchin component H n ( S ) such that ρ = η , the pressure metricfor ˙ ρ , ˙ η ∈ T ρ H n ( S ) is defined as: h ˙ ρ , ˙ η i P = − Cov ( ∂ s ( f Nρ s ) | s =0 , ∂ s ( f Nη s ) | s =0 , m f Nρ ) R UX f Nρ dm f Nρ For similicity, later we will also denote ∂ s ( f Nρ s ) | s =0 = ∂ s f Nρ and ∂ s ( f Nη s ) | s =0 = ∂ s f Nη . Theprinciple is, we always first normalize a family of reparametrization functions to be pressurezero and then take derivatives.Because of the identification of i = n L i =2 H ( X, K i ) with the tangent space of the Hitchin com-ponent T X H n ( S ), our Hitchin deformation ψ ( q ) introduced in Definition 2.18 can be thought14s tangent vectors in T X H n ( S ). With this understood, we introduce the following importantresults from [20] by Labourie and Wentworth:Let q i be a holomorphic differential of degree k on X and let ψ ( q i ) be the associated Hitchindeformation. Labourie and Wentworth in [20] show the pressure metric satisfies: h ψ ( q i ) , ψ ( q i ) i P = C ( n, k ) h q i , q i i X , where C ( n, k ) > σ and h q i , q i i X is the Petersson pairing: h q i , q i i X = Z X q i ¯ q i σ − k ( z ) dσ with d σ = σ ( z )d x ∧ d y denoting the area form for the hyperbolic metric σ .If q i , q j are holomorphic differentials of the same degree, then h ψ ( q i ) , ψ ( q j ) i P = 14 [ h ψ ( q i + q j ) , ψ ( q i + q j ) i P − h ψ ( q i − q j ) , ψ ( q i − q j ) i P ]= C ( n, k )4 h q i + q j , q i + q j i X − C ( n, k )4 h q i − q j , q i − q j i X = C ( n, k ) h q i , q j i X If q i , q j are holomorphic differentials of different degrees on X , Labourie and Wentworth showin [20] that h ψ ( q i ) , ψ ( q j ) i P = 0 (2.11)We denote the pressure metric components with respect to the coordinates introduced inRemark 2.10 as g ij . Equivalently, the metric tensor g ij ( ξ ) means that the pressure metric h· , ·i P is evaluated at ξ with tangential vectors parallel to q i -axis and q j -axis. In particular,at the point X , we have g ij (0) = g ij ( σ ) = h ψ ( q i ) , ψ ( q j )) i P . It is always possible to choose anorthonormal basis { q i } with respect to our pressure metric from the vector space i = n L i =2 H ( X, K i )so that g ij ( δ ) = δ ij . We will include proofs of some propositions here that will be important for the proof of themain theorem in the next section. In subsection 3.1, we will introduce Ruelle operator andRuelle-Perron-Frobenius Theorem. These are important tools to compute third derivatives ofthe pressure functions given in Proposition 3.4. In subsection 3.2, we include the gauge-theoreticproof of a formula from [20] that will be crucial in computing variations of reparametrizationfunctions in the next section.
Bowen and Ruelle’s work ([3], [4], [6]) guarantee that many of the results in the thermody-namic formalism proved for subshifts of finite type by Ruelle operator still hold for Axiom A15iffeomorphisms and Axiom A flows. We adopt this idea of simplifying the rather complicatedobject ”flow” by discretizing it and studying a relative simple object ”shift” given by symboliccoding. We compute in this subsection the formula of third derivatives of pressure functions inProposition 3.4 for subshifts of finite types by the method of the Ruelle operator and generalizeit to our setting of geodesic flows. The reader can find an introduction for modelling hyperbolicdiffeomorphisms by subshifts of finite type and modelling hyperbolic flows by suspension flowsthrough Markov partition and symbolic dynamics in ([5], section 3, 4) and ([26], Appendix III).We start with a cursory introduction to the elements of thermodynamic formalism for sub-shifts of finite types. A complete description is in [25] and [26].
Definition 3.1.
Let A be a k × k matrix of zeros and ones, we define the associated two-sidedshift of finite type (Σ , σ A ) where Σ is the set of sequences Σ = { x = ( x n ) ∞ n = −∞ : x n ∈ { , · · · , k } , n ∈ Z , A ( x n , x n +1 ) = 1 } and σ A : Σ → Σ is defined by σ A ( x ) = y , where y n = x n +1 . If instead, we consider x = ( x n ) ∞ n =0 with the same restriction given by matrix A and σ ( x ) = y , i.e. y n = x n +1 for n ≥
0, then we obtain a one-sided shift of finite type.The set { , · · · , k } is equipped with the discrete topology and the two-sided (one-sided)shift space Σ A is equipped with the associated product topology.Given α ∈ (0 , d α ( x, y ) = α N where N is the largest non-negative integer such that x i = y i for | i | < N .Similarly, we have a metric d α defined for one-sided shift space.We let C (Σ) be the space of real-valued continuous functions on Σ and C α (Σ) be the spaceof real-valued H¨older functions on Σ with H¨older exponent α with respect to d α .The two-sided (one-sided) shift of finite type (Σ , σ A ) is called a subshift of finite type if σ A is topological transitive.We define the pullback operator on C α (Σ) by ( σ ∗ A f )( y ) = f ( σ A ( y )). Similarly to Definition2.2, we define Definition 3.2. f and f in C α (Σ) are (Livˇsic) cohomologous if f − f = f − σ ∗ A f . for some f ∈ C α (Σ) . From now on, we assume our subshift of finite type (Σ , σ A ) to be one-sided unless otherwisespecified. Definition 3.3.
Given w ∈ C α (Σ) , the Ruelle operator (or transfer operator) on f ∈ C α (Σ) isdefined by L w ( f )( x ) = X σ A ( y )= x e w ( y ) f ( y ) . We have that the following holds for Ruelle operator L w .16 heorem 3.1. (Ruelle-Perron-Frobenius)Suppose (Σ , σ A ) is topological mixing (i.e. A Mi,j > ∀ i, j , for some M > , also calledirreducible and aperiodic) and w ∈ C α (Σ) , then1. There is a simple maximal positive eigenvalue ρ ( L w ) of L w : C α (Σ) → C α (Σ) with acorresponding strictly positive eigenfunction e ψ : L w ( e ψ ) = ρ ( L w ) e ψ .
2. The remainder of the spectrum of L w (excluding ρ ( L w ) ) is contained in a disc of radiusstrictly smaller than ρ ( w ) .3. There is a unique probability measure m w on Σ so that L w ∗ µ w = e ψ µ w . The pressure P ( w ) of w , which can be defined in an analogous way as the pressure offunctions on U X by variational principle 2.1, turns out to be related to the spectral radius ofthe Ruelle operator: P ( w ) = log ρ ( L w ) (see ([5] Thm.1.22).Associated to µ w is another measure m w = e ψ µ w . It is called the equilibrium measure of w . It is an σ A -invariant and ergodic probability measure and satisfied L w ∗ m w = m w .We will from now on assume P ( w ) = 0. As pressure functions and equilibrium measuresdepend only on cohomology class, we can modify w by a coboundary so that L w (1) = 1 and µ w = m w . One notices this implies L w ( σ ∗ A f ) = f .Fixing m w , we denote an inner product < f , f > := R Σ f f d m w on the Banach space C α (Σ).For the convenience of notation, we also denote S n ( f, x ) = n − P i =0 f ( σ iA x ).The following lemmas will be useful. Lemma 3.2. (McMullen, [25] Thm.3.2, Thm.3.3)For any g ∈ C (Σ) and f ∈ C α (Σ) with R Σ f d m w = 0 , we have lim n →∞ h g, S n ( f ) /n i = V ar ( f, m w ) Z Σ g d m w = 0 where V ar ( f, m w ) = lim n →∞ n h S n ( f ) , S n ( f ) i . Lemma 3.3.
For any f ∈ C α (Σ) with R Σ f d m w = 0 , lim n →∞ n Z Σ ( S n ( f )) d m w < ∞ . Proof.
This proof is similar to Thm.3.3 of [25].1 n Z Σ ( S n ( f )) d m = 1 n n − X i =0 n − X j =0 n − X k =0 h f ◦ σ iA · f ◦ σ jA , f ◦ σ kA i k > j > i , h f ◦ σ iA · f ◦ σ jA , f ◦ σ kA i = h σ ∗ iA ( f · f ◦ σ j − iA ) , σ ∗ iA ( f ◦ σ k − iA ) i = h f · f ◦ σ j − iA , f ◦ σ k − iA i σ A -invariance of m w = h f, f ◦ σ j − iA · f ◦ σ k − iA i = h f, σ ∗ ( j − i ) A ( f · f ◦ σ k − jA ) i = hL j − iw ( f ) , f · f ◦ σ Ak − j i L w ( σ ∗ A f ) = f and L w ∗ m w = m w = h f · L j − iw ( f ) , f ◦ σ Ak − j i We define a projection operator on C α (Σ) by P m w ( h )( x ) = h ( x ) − R Σ h d m w . Because P m w ( h )has mean zero with respect to m w . The spectrum of the operator T w = L w ◦ P m w lies in a diskof radius r < h h , h ◦ σ i = h T w ( h ) , h i (3.1)whenever h or h has mean zero.Because f is mean zero with respect to m w . T w ( f ) = L w ( f ). Moreover, we have h f · L j − iw ( f ) , f ◦ σ Ak − j i = h f · T j − iw ( f ) , f ◦ σ Ak − j i = h T k − jw ( f · T j − iw ( f )) , f i by equation (3.1) ≤ (cid:13)(cid:13)(cid:13) T k − j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T j − i (cid:13)(cid:13) k f k ≤ Cr k − i for some C > T is the operator norm.Thus 1 n X ≤ i Suppose (Σ , σ A ) is a two-sided shift of finite type. Given a roof function r : Σ → R + , the suspension flow of (Σ , σ A ) under r is the quotient space Σ r = { ( x, t ) ∈ Σ × R : 0 ≤ t ≤ r ( x ) , x ∈ Σ } / ( x, r ( x )) ∼ ( σ A ( x ) , equipped with the natural flow σ rA,s ( x, t ) = ( x, t + s )Any σ A -invariant probability measure m on Σ induces a natural σ rA,s -invariant probabilitymeasure on Σ r d m r = d m d t R Σ r d m . (3.2)This correspondence gives a bijection between σ A -invariant probability measures and σ rA,s -invariant probability measures.Bowen shows in [3] the construction of Markov partitions for Axiom A diffeomorphism. Hethen shows how to model Axiom A flows via the Markov partition and symbolic dynamics in[4]. We illustrate the version of this celebrated result in our context (see also [31]): the geodesicflow Φ admits a Markov coding (Σ A , π, r ) where (Σ A , σ A ) is a topological mixing two-sided shiftof finite type, the roof function r : Σ A → R + is a H¨older continuous and the map π : Σ A → U X is also H¨older continuous. The suspension flow σ rA,t models Φ t effectively in the following sense: • π is surjective; • π is one-one on a set of full measure (for any ergodic measure of full support) and on aresidual set; • π is bounded-one; • πσ rA,t = Φ t π for all t ∈ R .Now we are able to state and prove the major proposition in this subsection. Proposition 3.4. Let F s be a smooth family in C h ( U X ) such that P ( F ) = 0 and ∂ s P ( F s ) | s =0 .Then d P ( F s )d s (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Z UX ∂ s F ( x )d m F ( x )+ lim r →∞ r Z UX Z r ∂ s F (Φ t ( x ))d t Z r ∂ ss F (Φ t ( x ))d t d m F ( x )+ Z UX (cid:18)Z r ∂ s F (Φ t ( x ))d t (cid:19) d m F ( x ) ! . (3.3) In particular, if F ( u, v, w ) is a smooth three-parameter family of H¨older functions on U X such that P ( F (0 , , and all of the first variations of P ( F ( u, v, w )) are zero, then P ( F ( u, v, w )) ∂u∂v∂w (cid:12)(cid:12)(cid:12)(cid:12) u = v = w =0 = Z UX ∂ u ∂ v ∂ w F (0)( x )d m F (0) ( x )+ lim r →∞ r Z UX (cid:18)Z r ∂ u F (0)(Φ t ( x ))d t (cid:19) (cid:18)Z r ∂ v F (0)(Φ t ( x ))d t (cid:19) (cid:18)Z r ∂ w F (0)(Φ t ( x ))d t (cid:19) d m F (0) ( x )+ Z UX (cid:18)Z r ∂ u F (0)(Φ t ( x ))d t (cid:19) (cid:18)Z r ∂ vw F (0)(Φ t ( x ))d t (cid:19) d m F (0) ( x )+ Z UX (cid:18)Z r ∂ v F (0)(Φ t ( x ))d t (cid:19) (cid:18)Z r ∂ uw F (0)(Φ t ( x ))d t (cid:19) d m F (0) ( x )+ Z UX (cid:18)Z r ∂ w F (0)(Φ t ( x ))d t (cid:19) (cid:18)Z r ∂ uv F (0)(Φ t ( x ))d t (cid:19) d m F (0) ( x ) ! . (3.4) Proof. The proof proceeds in two steps. In the first step, we find a formula for third derivativesof pressure functions for topological mixing shifts of finite type. In the second step, we showhow the computation can be carried to geodesic flows through symbolic coding and suspensionflows. • Step 1.The computation of first and second derivatives of pressure functions for aperiodic shiftsof finite type are shown in Parry and Pollicott’s book [26] by Ruelle operator. We willgive a computation of the third derivative by the same method and then generalize it toour flow case.Let (Σ A , σ A ) be a (either one-sided or two-sided) shift of finite type that is topologicalmixing. We assume f s is a smooth family of functions on C α (Σ A ) such that P ( f ) = 0and ∂ s P ( f s ) | s =0 . We will prove ∂ s P ( f s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = lim n →∞ n Z X ( S n ( ∂ s f )) d m f + lim n →∞ n Z X S n ( ∂ s f ) S n ( ∂ ss f )d m f + Z X ∂ s f d m f . (3.5)Any H¨older function on a two-sided shift space is colomologous to a H¨older functiondepending only on the corresponding one-sided shift space (see [26], Proposition 1.2). Itsuffices to prove equation (3.5) for one-sided shifts of finite type. We assume (Σ A , σ A ) isone-sided and f s is a smooth family of H¨older function (with possibly a different H¨olderexponent from α ) on Σ A .We change f in its cohomology class so that L f (1) = 1.Following the method in [26], let Q ( s ) be a projection-valued function Q ( s ) which isanalytic in s and satisfies L f s Q ( s ) = Q ( s ) L f s . Let w ( s ) : Σ A → R be w ( s )( x ) := Q ( s ) · 1. So L f s w ( s ) = e P ( f s ) w ( s ) (3.6)and w (0)( x ) = Q (0) · s = 0. ∂ s ( X σ A y = x e S n ( f s )( y ) w ( s )( y )) | s =0 = ∂ s ( e n P ( f s ) w ( s )) | s =0 . (3.7)Notice P ( f ) = 0, ∂ s P ( f s ) | s =0 = 0 and R UX ∂ s f d m f = 0. Integrating both sides ofequation (3.7) with respect to m f yields,3 n∂ ss P ( f s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 Z X ∂ s w (0)d m f + n∂ s P ( f s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Z X S n ( ∂ s f )d m f + 3 Z X ( S n ( ∂ s f ) + S n ( ∂ s f )) ∂ s w (0)d m f +3 Z X S n ( ∂ s f ) w ss (0)d m f + 3 Z X S n ( ∂ s f ) S n ( ∂ ss f )d m f + Z X S n ( ∂ s f ) d m f . Divide by n and take n → ∞ . From ergodicity of m f , we may evaluate two of theresulting terms:lim n →∞ n Z X S n ( ∂ s f ) ∂ ss w (0)d m f = Z X ∂ s f d m f Z X ∂ ss w (0)d m f = 0 . lim n →∞ n Z X S n ( ∂ s f ) ∂ s w (0)d m f = Z X ∂ s f d m f Z X ∂ s w (0)d m f . We also notice that by applying Lemma 3.2 and the formula for second derivatives ofpressure functions, we have the following equality ∂ ss P ( f s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 Z X ∂ s w (0)d m f = lim n →∞ n Z X S n ( ∂ s f ) ∂ s w (0)d m f + lim n →∞ n Z X S n ( ∂ s f ) ∂ s w (0)d m f Therefore we obtain a formal expression ∂ s P ( f s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = lim n →∞ n Z X ( S n ( ∂ s f )) d m f + lim n →∞ n Z X S n ( ∂ s f ) S n ( ∂ ss f )d m f + Z X ∂ s f d m f We observe each term of the right-hand side converges. lim n →∞ n R X S n ( ∂ s f ) S n ( ∂ ss f )d m f < ∞ is guaranteed by Corollary 2.1 and lim n →∞ n R X ( S n ( ∂ s f )) d m f < ∞ has been shown inLemma 3.3. • Step 2.We now explain how we obtain the flow version of the above formula.Suppose F s is a smooth family of functions in C h ( U X ) such that P ( F s ) = 0. Wehave a topological mixing Markov coding (Σ A , π, r ) for U X . Because of the conjugacy πσ rA,t = Φ t π between geodesic flow and the suspension flow of (Σ A , π, r ), it suffices toprove equation (3.3) for F s ◦ π : Σ A,r → R on suspension space with pull back measure π ∗ m F . For simlicity, we still denote F s ◦ π as F s and π ∗ m F as m F .We then want to reduce the problem of proving equation (3.3) for suspension flows toproving it for subshifts of finite type. We construct a function ˆ F s : Σ A → R from thefunction F s on the suspension space as:ˆ F s ( x ) = Z r ( x )0 F s ( x, t )d t. (3.8)21s F s and r are H¨older on Σ A,r and Σ A respectively, the function ˆ F s is clearly H¨older.Denoting the set of σ rA -invariant probability measures as M σ rA and the set of σ A -invariantprobability measures as M σ A , we have: P ( σ rA,t , F s ) = sup m r ∈M σrA ( h ( σ rA, , m r ) + Z Σ A,r F s d m r )= sup m ∈M σA h ( σ A , m ) + R Σ A F s d m R Σ A r d m . Let c s = P ( σ rA,t , F s ), we have the following relation between the pressure function of F s and the pressure function of ˆ F s (also see [6]) P ( σ A , ˆ F s − c s r ) = 0 (3.9)Denote ∂ s c = ∂ s ( c s ) | s =0 and ∂ ss c = ∂ s ( c s ) | s =0 .We have the assumption ∂ s c = 0. Without loss of generality, we can also assume ∂ ss c = 0. Otherwise we consider the family of functions ˜ F s := F s − s ∂ ss c . Clearly ∂ s P ( ˜ F s ) | s =0 = ∂ s P ( ˜ F s ) | s =0 = 0 and ∂ s P ( ˜ F s ) | s =0 = ∂ s P ( F s ) | s =0 .Now let’s take the third s -derivative of equation (3.9) with the assumptions ∂ s c = ∂ ss c =0. By equation (3.5),0 = ∂ s P ( ˆ F s − c s r ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = lim n →∞ n Z Σ A ( S n ( ∂ s ˆ F )) d m ˆ F + lim n →∞ n Z Σ A S n ( ∂ s ˆ F ) S n ( ∂ ss ˆ F )d m ˆ F + Z Σ A ( ∂ s ˆ F − ∂ s c r )d m ˆ F . This yields ∂ s c = ∂ s P ( σ rA,t , F s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ( Z Σ A r d m ˆ F ) − ∂ s P ( σ A , ˆ F s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 . Therefore proving equation (3.3) for F s is equivalent to proving the followinglim r →∞ r Z Σ A,r ( Z r ∂ s F d t ) d m F + lim r →∞ r Z Σ A,r Z r ∂ s F d t Z r ∂ ss F d t d m F + Z Σ A,r ∂ s F d m F =( Z Σ A r d m ˆ F ) − (cid:18) lim n →∞ n Z Σ A ( S n ( ∂ s ˆ F )) d m ˆ F + lim n →∞ n Z Σ A S n ( ∂ s ˆ F ) S n ( ∂ ss ˆ F )d m ˆ F + Z Σ A ∂ s ˆ F d m ˆ F (cid:19) . Each term on the left is actually equal to the corresponding term on the right. We showhere how to obtainlim r →∞ r Z Σ A,r ( Z r ∂ s F ( σ rt ( y ))d t ) d m F ( y ) = ( Z Σ A r d m ˆ F ) − lim n →∞ n Z Σ A ( S n ( ∂ s ˆ F )( x )) d m ˆ F ( x )(3.10)The other two terms follow a similar analysis.22o see equation (3.10),we begin by noting the following identity ([28]) where we denote y = ( x, u ), ∂ s F ( σ rA,t ( x, u )) = X n ∈ Z Z r ( σ nA x )0 ∂ s F ( σ nA x, v ) δ ( u + t − v − r n ( x ))d v, where r n ( x ) = r ( x ) + r ( σ A x ) + · · · + r ( σ n − A x ) for n > r ( x ) = 0 and r − n ( x ) = − ( r ( σ A − x ) + · · · + r ( σ − nA x )) for n ≥ r →∞ r Z Σ A,r ( Z r ∂ s F ( σ rA,t ( y ))d t ) d m F ( y )= Z ∞−∞ Z ∞−∞ Z Σ A,r ∂ s F ( y ) ∂ s F ( σ rA,t ( y )) ∂ s F ( σ rA,v ( y ))d m F ( y )d t d v =( Z Σ A r d m ˆ F ) − Z ∞−∞ Z ∞−∞ Z Σ A Z r ( x )0 ∂ s F ( x, u ) ∂ s F ( σ rA,t ( x, u )) ∂ s F ( σ rA,v ( x, u ))d u d m ˆ F ( x )d t d v =( Z Σ A r d m ˆ F ) − X m,n ∈ Z Z Σ A d m ˆ F ( x ) Z r ( x )0 ∂ s F ( x, u )d u Z r ( σ nA x )0 ∂ s F ( σ nA x, v )d v Z r ( σ mA x )0 ∂ s F ( σ mA x, v )d v =( Z Σ A r d m ˆ F ) − X m,n ∈ Z Z Σ A ∂ s ˆ F ( x ) ∂ s ˆ F ( σ nA x ) ∂ s ˆ F ( σ mA x )d m ˆ F ( x )=( Z Σ A r d m ˆ F ) − lim n →∞ n Z Σ A ( S n ( ∂ s ˆ F )( x )) d m ˆ F ( x ) . We therefore obtain a suspension flow version of equation (3.5) for F s .The arguments for three-parameters families are analogous.Next we introduce a formula for taking derivatives of integrals over varying measures bytools of thermodynamic formalism. This formula will be very useful in later proofs. Lemma 3.5. Suppose { f s } s ∈ ( − , is a smooth family of pressure zero H¨older functions over U X and suppose { m fs } s ∈ ( − , is the associated family of equilibrium states. Suppose furthermorethat { w s } s ∈ ( − , is another smooth family of H¨older functions over U X . Then ∂ s (cid:18)Z UX w s d m fs (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Cov ( w , ∂ s f , m f ) + Z UX ∂ s w d m f (3.11)23 roof. ∂ s (cid:18)Z UX w s d m fs (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ∂ s (cid:18) ∂ P ( f s + tw s ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 by formula (2.6)= ∂ P ( f s + tw s ) ∂s∂t (cid:12)(cid:12)(cid:12)(cid:12) s = t =0 = Cov ( P m f ( w ) , P m f ( ∂ s f ) , m f ) + Z UX ∂ s w d m f by equation (2.8)= Cov ( P m f ( w ) , ∂ s f , m f ) + Z UX ∂ s w d m f = Cov ( w , ∂ s f , m f ) + Z UX ∂ s w d m f by Corollary 2.1 In [20], Labourie and Wentworth show the variation of reparametrization functions can beexpressed by a gauge-theoretical formula. We include the formula and its proof here for com-pleteness. We add some assumptions which are natural for our case of Hitchin components H n ( S ).We consider ( E, H ) a rank n Hermitian bundle over the surface S equipped with a Rieman-nian metric g . We let γ be a closed curve on S with arc length parametrization γ ( t ). Suppose D A is a flat connection on E so that the holonomy of it has distinct eigenvalues along γ .Suppose λ γ is one eigenvalue with a corresponding eigenline L γ and H γ is the complementaryhyperplane stablized by the holonomy. We denote by L γ ( t ) the line generated by the paral-lel transports of L γ along γ at time t, by H γ ( t ) the hyperplane generated by supplementaryeigenvectors and by π ( t ) the projection on L γ ( t ) along H γ ( t ). Then we have Proposition 3.6. (Labourie-Wentworth, [20] )For D As a smooth one parameter family of flat connections, we have a unique smooth func-tion λ γ ( s ) so that for s small enough, λ γ ( s ) is the eigenvalue of the holonomy of D A s with λ γ (0) = λ γ . Moreover, d log λ γ ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − Z l γ T r ( ∂ s D A ( t ) · π ( t )) dt, (3.12) where ∂ s D A ( t ) := ∂D As ( t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 .Proof. We prove equation 3.12 here.Let { g s } be a family of gauge transformation acting on { D A s } with g = id . Denote thenew connection one-forms ˜ A s := g ∗ s A s . We first prove: Z l γ T r ( ∂ s D A ( t ) · π ( t )) dt = Z l γ T r ( ∂ s D ˜ A ( t ) · π ( t )) dt, ∂ s D ˜ A ( t ) := ∂D ˜ As ( t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 .Taking a derivative of ˜ A s := g ∗ s A s at s = 0 yields ∂ s D ˜ A = ∂ s D A + D A ˙ g where ˙ g denoting ∂g s ∂ s (cid:12)(cid:12)(cid:12)(cid:12) s =0 is a section of End ( E ) and the connection D A acts on ˙ g as D A ˙ g = d ˙ g + [ A , ˙ g ].We want to show Z l γ T r (( D A ˙ g ) π ) dt = 0We start from proving that π is a D A -parallel section in End ( E ). Given any section v ∈ Γ( E ), we can write it as a linear combination of eigenvectors of holonomy. Set v ( t ) = P ni =1 a i ( t ) e i ( t ) where e i ( t ) satisfies parallel transport equation D A e i = 0 with boundary con-ditions e i ( l γ ) = λ iγ e i (0) , || e i (0) || = 1. In particular, we assume λ γ = λ γ and L γ ( t ) is generatedby e ( t ). Then( D A π )( v ) = [ D A , π ] v = D A ( πv ) − π ( D A v )= D A ( a ( t ) e ( t )) − π ( n X i =1 ( da i ( t ) e i ( t ) + a i ( t ) D A e i ( t )))= da ( t ) e ( t ) − da ( t ) e ( t )= 0Thus Z l γ ( T r ( ∂∂t ˙ g · π )) dt = Z l γ T r ( ∂∂t ( ˙ gπ )) dt = Z l γ T r ( D A ( ˙ gπ )) dt since T r ([ A , ˙ gπ ]) = 0= Z l γ T r ([ D A , ˙ gπ ]) dt Notice ˙ gπ ∈ Γ( End ( E ))= Z l γ T r ([ D A , ˙ g ] π + ˙ g [ D A , π ]) dt = Z l γ T r (( D A ˙ g ) π + ˙ g ( D A π )) dt action of a connection on Γ( End ( E ))= Z l γ T r (( D A ˙ g ) π ) dt Since D A π = 0So Z l γ T r (( D A ˙ g ) π ) dt = Z l γ ddt ( T r ( ˙ g · π )) dt = T r ( ˙ g ( l γ ) π ( l γ )) − T r ( ˙ g (0) π (0)) = 025s s varies, the eigenline L sγ ( t ) corresponding to λ γ ( s ) varies according to s and so isthe supplementary hyperplane H sγ ( t ) . By picking suitable gauges { g s } , we can assume, for˜ A s := g ∗ s A s , the eigenlines ˜ L sγ ( t ) and supplementary hyperplanes ˜ H sγ ( t ) satisfy ˜ L sγ ( t ) = L γ ( t )and ˜ H sγ ( t ) = H γ ( t ).Without lose of generality, we assume D A s is itself the connection after suitable gauge andthe set { e si } are eigenvectors for A s with e s corresponding to L sγ . Thus we have the followingequations. ( D A s e si ( t ) = 0 e si ( l γ ) = λ iγ ( s ) e si (0)In particular, we can assume ( D A s e s ( t ) = 0 , e s ( l γ ) = λ γ ( s ) e s (0) e s ( t ) = c s ( t ) e ( t ) , e s (0) = e (0)So e s ( l γ ) = c s ( l γ ) e ( l γ ) = c s ( l γ ) λ γ (0) e (0) = λ γ ( s ) e s (0) = λ γ ( s ) e (0)and thus c s ( l γ ) = λ γ ( s ) λ γ (0) and c ( l γ ) = 1. Notice H ( e ( t ) , D A s e ( t )) H ( e ( t ) , e ( t )) = H ( e ( t ) , D A s ( e s ( t ) c s ( t ) )) H ( e ( t ) , e s ( t ) c s ( t ) ) = ∂ t ( c s ( t ) ) c s ( t ) = − ∂ (log c s ( t )) ∂t So Z l γ T r ( ∂ s D A π ) dt = Z l γ H ( e ( t ) , ∂ s D A e ( t )) H ( e ( t ) , e ( t )) dt = − Z l γ ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:18) ∂ (log c s ( t )) ∂t (cid:19) dt = − d log λ γ ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) s =0 We first restate our main theorem. Recall theorem 1.1 Theorem 1.1. Let S be a closed oriented surface with genus g ≥ . For any point σ ∈ T ( S ) ⊂H ( S ) , let X be the Riemann surface corresponding to σ . Then the Hitchin parametrization H ( X, K ) L H ( X, K ) provides geodesic coordinates for the pressure metric at σ . We want to show ∂ k g ij ( σ ) = 0 for the pressure metric components g ij with respect to thecoordinates introduced before in Remark 2.10 for all possible i, j, k .26 .1 Some geometrical observation In this subsection, we conclude some derivatives of metric tensors vanish by some geometricobservation. Starting from the next section, we will develop a general method to compute firstderivatives of the pressure metric by the thermodynamic formalism.From now on, we restrict ourselves to the Hitchin component H ( S ). Suppose { q i } isa basis of holomorphic differentials in H ( X, K ) L H ( X, K ) and suppose { ψ ( q i ) } is theassociated Hitchin deformation given in Definition 2.18. Recall we use the notation g ij ( σ ) = h ψ ( q i ) , ψ ( q j )) i P to emphasize the metric tensor is evaluated at σ ∈ T ( S ). We also assume g ij ( δ ) = δ ij .Futhermore, instead of using only the English letters i, j, k to denote arbitrary holomor-phic differentials of degree 2 and 3, we let the English letters i, j, k to only refer to quadraticdifferentials q i , q j , q k ∈ H ( X, K ) from now on. Therefore the corresponding Hitchin deforma-tions ψ ( q i ) , ψ ( q j ) , ψ ( q k ) are tangential directions to Fuchsian locus in T X H ( S ). We also usethe Greek letters α, β, γ to refer to cubic differentials q α , q β , q γ ∈ H ( X, K ). Then the corre-sponding Hitchin deformation ψ ( q α ) , ψ ( q β ) , ψ ( q γ ) are normal directions to the Fuchsian locusin T X H ( S ) with respect to the pressure metric.With above notation understood, we have in total six types of first derivatives of metrictensors that need to be considered: ∂ k g ij , ∂ j g iα , ∂ α g ij , ∂ i g αβ , ∂ β g iα , ∂ γ g αβ . Our goal is to provethey all vanish.We first notice the following facts.1. ∂ k g ij ( σ ) = 0.To see this, note that the pressure metric is a constant multiple of the Weil-Peterssonmetric on Teichm¨uller space T ( S ). The Bers coordinates are geodesic ([1]) for the Weil-Petersson metric imply for the pressure metric: ∂ k g ij ( σ ) = 0.2. ∂ j g αi ( σ ) = 0 = ⇒ ∂ α g ij ( σ ) = 0.The contragredient involution κ : P SL (3 , R ) → P SL (3 , R ) given by κ ( g ) = ( g − ) t inducesan involution ˆ κ on H ( S ) by ˆ κ ( ρ )( γ ) = κ ( ρ ( γ )). Because ˆ κ is an isometry of H ( S ) withrespect to the pressure metric and the fixed points set of ˆ κ is T ( S ), the Fuchsian locusis in fact totally geodesic in H ( S )( see [7]). So for ˜ ∇ the Levi-Civita connection of thepressure metric and any X, Y ∈ T σ T ( S ), we haveΠ( X, Y ) = ( ˜ ∇ X Y ) ⊥ = 0 . (4.1)Thus the Christoffel symbols for connection ˜ ∇ satisfy: Γ αij ( σ ) = 0 and becauseΓ αij = 12 g βα ( ∂ j g iβ + ∂ i g jβ − ∂ β g ji ) since g kα ( σ ) = 0 , g kα ( σ ) = 0= 12 g αα ( ∂ j g iα + ∂ i g jα − ∂ α g ji ) since g αβ = σ αβ It suffices to know ∂ j g iα ( σ ) = 0 and ∂ i g jα ( σ ) = 0 to conclude ∂ α g ij ( σ ) = 0.3. ∂ β g αα ( σ ) = 0 = ⇒ ∂ γ g αβ ( σ ) = 0. 27 i g αα ( σ ) = 0 = ⇒ ∂ i g αβ ( σ ) = 0. This is because ∂ γ g αβ = 12 ( ∂ γ g α + β,α + β − ∂ γ g αα − ∂ γ g ββ ) ∂ i g αβ = 12 ( ∂ i g α + β,α + β − ∂ i g αα − ∂ i g ββ )The remaining four cases left to prove are as follows.(i) ∂ β g αα ( σ ) = 0 ;(ii) ∂ i g αα ( σ ) = 0;(iii) ∂ j g αi ( σ ) = 0;(iv) ∂ β g αi ( σ ) = 0.We will have a general method to prove them. We first give a general formula for first derivativesof the pressure metric in next subsection. The computation for the model case ∂ β g αα ( σ ) willbe shown in Section 5 and Section 6. THe other three cases will be in Section 7. This subsection is devoted to a formula of first derivatives of pressure metric. We also prove wehave some freedom to choose representatives for variations of reparametrization functions fromthe Livˇsic cohomologous class.Suppose { ρ ( u, v, w ) } ( u,v,w ) ∈{ ( − , } is an analytic three-parameter family of representationsin the Hitchin component H n ( S ) with base point ρ (0 , , ∈ T ( S ) corresponding to X andsuppose { f ρ ( u,v,w ) } ( u,v,w ) ∈{ ( − , } are associated reparametrization functions. For simplicity ofnotation, we denote the renormalized reparametrization functions as F ( u, v, w ) = f Nρ ( u,v,w ) = − h ( ρ ( u, v, w )) f ρ ( u,v,w ) We also denote F (0) = F (0 , , 0) and ρ (0) = ρ (0 , , h ( ρ (0)) = 1 (See Theorem 2.4). Since Φ ρ (0) = Φ, thereparametrization function f ρ (0) can be chosen to be 1 in the Livˇsic cohomologous class. There-fore one can choose F (0) = − F (0) is important. Lemma 4.1. The equilibrium state m F (0) for F (0) is the Liouville measure m L .Proof. Recall Remark 2.4 and Remark 2.5, the Liouville measure m L is the Bowen-Margulis28easure m Φ for Φ which is the unique equilibrium state m for f = 0. Because F (0) = − P (Φ , F (0)) = sup m ∈M Φ ( h (Φ , m ) + Z UX F (0)d m )= sup m ∈M Φ ( h (Φ , m ) − m ( U X ))= sup m ∈M Φ ( h (Φ , m )) − h (Φ , m Φ ) − . This shows m F (0) = m = m L .We list some porperties of the Liouville measure here which will be repeatedly used for ourproofs later. • The measure m L is preserved by the geodesic flow Φ, i.e. m L ∈ M Φ (see [35]) • The measure m L is rotationally invariant on U X , i.e. ( e iθ ) ∗ m L = m L where e iθ acts on U X by usual multiplication.The Liouville measure m L is the normalized Riemannian measure on U X . The volumeelement (Riemannian volume form) is generated by the canonical metric (Sasaki metric)on U X . Proposition 4.2. The first derivatives of the pressure metric at ρ (0) satisfy ∂ w h ∂ u ρ (0 , , w ) , ∂ v ρ (0 , , w ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w =0 = lim r →∞ r Z UX Z r ∂ u F (0)d t Z r ∂ v F (0)d t Z r ∂ w F (0)d t d m + Z UX Z r ∂ u F (0)d t Z r ∂ wv F (0)d t d m + Z UX Z r ∂ v F (0)d t Z r ∂ wu F (0)d t d m ! , where the flow Φ t ( x ) is omitted for simplicity.Proof. Starting from the Fuchsian point ρ (0), along the ray parameterized by { (0 , , w ) } w ∈ ( − , ,the pressure metric h· , ·i P : T (0 , ,w ) H n ( S ) × T (0 , ,w ) H n ( S ) −→ R satisfies: h ∂ u ρ (0 , , w ) , ∂ v ρ (0 , , w ) i P = − Cov( ∂ u F (0 , , w ) , ∂ v F (0 , , w ) , m F (0 , ,w ) ) R UX F (0 , , w )d m F (0 , ,w ) = − ∂ v ∂ u P ( F (0 , , w )) − R UX ∂ uv F (0 , , w )d m F (0 , ,w ) R UX F (0 , , w )d m F (0 , ,w ) equation (2.8)We first notice R UX F (0)d m = − ∂ w (cid:12)(cid:12)(cid:12)(cid:12) w =0 ( Z UX F (0 , , w )d m F (0 , ,w ) ) = Cov ( F (0) , ∂ w F (0) , m ) + Z UX ∂ w F (0)d m = 029herefore, ∂ w h ∂ u ρ (0 , , w ) , ∂ v ρ (0 , , w ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w =0 = ∂ w ∂ v ∂ u P ( F (0)) − ∂ w Z UX ∂ uv F (0)d m F (0 , ,w ) (cid:12)(cid:12)(cid:12)(cid:12) w =0 = ∂ w ∂ v ∂ u P ( F (0)) − Cov ( ∂ uv F (0) , ∂ w F (0) , m ) − Z UX ∂uvwF (0)d m by equation (3.11)= lim r →∞ r Z UX Z r ∂ u F (0)d t Z r ∂ v F (0)d t Z r ∂ w F (0)d t d m + Z UX Z r ∂ u F (0)d t Z r ∂ vw F (0)d t d m + Z UX Z r ∂ v F (0)d t Z r ∂ wu F (0)d t d m ! by equation (3.4) Proposition 4.3. The formula for the first derivatives of the pressure metric in Proposition4.2 only depends on the Livˇsic class of each component function: ∂ u F (0) , ∂ v F (0) , ∂ w F (0) , ∂ wv F (0) , and ∂ wu F (0) .Proof. Suppose h , h and h are mean-zero H¨older functions with respect to the Liouvillemeasure m and suppose h is Livˇsic cohomologous to ˜ h . We want to show Cov ( h , h , m ) = Cov (˜ h , h , m )and Cov ( h , h , h , m ) = Cov (˜ h , h , h , m )As h is Livˇsic cohomologous to ˜ h , there exists a H¨older function V : U X → R of class C along the flow direction such that h ( x ) − ˜ h ( x ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 V (Φ t ( x )).This is, Cov ( h − ˜ h , h , m )= lim T →∞ Z UX ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! d m ( x )Note because m ∈ M Φ ,lim T →∞ Z UX V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! d m ( x ) = Const. ∀ t ∈ R (4.2)Let t be small and fix T >> t , again since m is Φ-invariant, ∂∂t Z UX V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! d m ( x )= ∂∂t Z UX V ( x ) Z T − t − T − t h (Φ s ( x ))d s ! d m ( x )= Z UX V ( x ) (cid:16) h (Φ − T − t ( x )) − h (Φ T − t ( x )) (cid:17) d m ( x )30ow, geodesic flows on hyperbolic surfaces exhibit exponential decay of correlations with respectto the Liouville measure m L = m (see [11], [32]). Suppose V, h , h , h have some H¨olderexponent 0 < γ ≤ 1. Then Z UX V ( x ) (cid:16) h (Φ − T − t ( x )) (cid:17) d m ( x ) ≤ Ce − α ( T + t ) k V k γ k h k γ Z UX V ( x ) (cid:16) h (Φ T − t ( x )) (cid:17) d m ( x ) ≤ Ce − α ( T − t ) k V k γ k h k γ with some constants C, α > γ .Thus we see when T → ∞ , the partial t -derivative ∂ t R UX V (Φ t ( x )) (cid:18)R T − T h (Φ s ( x ))d s (cid:19) d m ( x )uniformly converges to 0 for t close to 0. This guarantee we can exchange the limit and thederivative in the following, Cov ( h − ˜ h , h , m )= lim T →∞ Z UX ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! d m ( x )= ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 lim T →∞ Z UX V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! d m ( x ) ! = 0 . We turn to Cov ( h − ˜ h , h , h , m )= lim T →∞ Z UX ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! Z T − T h (Φ s ( x ))d s ! d m ( x ) . Again since m is Φ-invariant, ∂∂t Z UX V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! Z T − T h (Φ s ( x ))d s ! d m ( x ) (4.3)= ∂∂t Z UX V ( x ) Z T − t − T − t h (Φ s ( x ))d s ! Z T − t − T − t h (Φ s ( x ))d s ! d m ( x )= Z UX V ( x ) (cid:16) h (Φ − T − t ( x )) − h (Φ T − t ( x )) (cid:17) Z T − t − T − t h (Φ s ( x ))d s ! d m ( x )+ Z UX V ( x ) (cid:16) h (Φ − T − t ( x )) − h (Φ T − t ( x )) (cid:17) Z T − t − T − t h (Φ s ( x ))d s ! d m ( x ) . Write Z UX V ( x ) h (Φ − T − t ( x )) Z T − t − T − t h (Φ s ( x ))d s ! d m ( x )= Z T − t − T − t Z UX V ( x ) h (Φ s ( x )) h (Φ − T − t ( x ))d m ( x )d s (4.4)31otice V ( x ) h (Φ s ( x )) is also a H¨older function with the exponent γ . Thus, for the insideintegral in (4.4), we have by the decay of correlation ([11]) Z UX V ( x ) h (Φ s ( x )) h (Φ − T − t ( x ))d m ( x ) ≤ Ce − α ( T + t ) k V · h ◦ Φ s k γ k h k γ There exsits C > k V · ( h ◦ Φ s ) k ≤ C for all s ∈ R . Therefore Z T − t − T − t Z UX V ( x ) h (Φ s ( x )) h (Φ − T − t ( x ))d m ( x )d s ≤ CC T e − α ( T + t ) k h k γ Other terms are similar.One therefore conclude when T → ∞ , the partial t -derivative in (4.3) uniformly convergesto 0 as t tends to 0. So Cov ( h − ˜ h , h , h , m )= ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 lim T →∞ Z UX V (Φ t ( x )) Z T − T h (Φ s ( x ))d s ! Z T − T h (Φ s ( x ))d s ! d m ( x ) ! = 0 . This finishes the proof. In this and the next sections, we consider the model case ∂ β g αα ( σ ). Note the treatment of thiscase will involve all the steps needed for the other cases. This justifies the expositional decisionthat we consider it here first and in isolation.In this case, we are given parameters ( u, v ) ∈ { ( − , } with (conjugacies classes of) repre-sentations { ρ ( u, v ) } in H ( S ) corresponding to { (0 , uq α + vq β ) } ⊂ H ( X, K ) L H ( X, K ) byHitchin parametrization (see Remark 2.9). In particular, at the Fuchsian point ρ (0) = X , weidentify ∂ u ρ (0 , 0) with ψ ( q α ) and ∂ v ρ (0 , 0) with ψ ( q β ), where ψ ( q α ) and ψ ( q β ) are the Hitchindeformation given in Definition 2.18. We suppose { f ρ ( u,v ) } is an associated two-parameter familyof reparametrization functions. By Proposition 4.2, the formula for ∂ β g αα ( σ ) is ∂ β g αα ( σ ) = ∂ v h ∂ u ρ (0 , v ) , ∂ u ρ (0 , v ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v =0 = lim r →∞ r "Z UX (cid:18)Z r ∂ u f Nρ (0) d t (cid:19) Z r ∂ v f Nρ (0) d t d m + 2 Z UX Z r ∂ u f Nρ (0) d t Z r ∂ uv f Nρ (0) d t d m . Because ∂ u h ( ρ ( u, ∂ v h ( ρ (0 , v )) = 0 on Fuchsian locus T ( S ) by Theorem 2.4, thevariations of reparametrization functions that need to be computed are the following:(i) ∂ u f Nρ (0) = − ∂ u f ρ (0) ; 32ii) ∂ v f Nρ (0) = − ∂ v f ρ (0) ;(iii) ∂ uv f Nρ (0) = − ∂ uv h ( ρ (0)) − ∂ uv f ρ (0) .Before proceeding to compute (i), (ii) and (iii), we explain our general strategies to computevariations of reparametrization functions. Definition 5.1. A compact Riemannian manifold M is said to have the DPG property (densityof periodic geodesics) if the vectors tangent to periodic geodesics in M are dense in T M , thetangent bundle of M . Example. If the geodesic flow of a compact manifold M is Anosov, then M has the DPGproperty. In our case, X has the DPG property. Our computation will be based on Proposition 3.6 and tools from Higgs bundles theory. Letus first set up our Higgs bundles.In the component H ( S ) we are considering, the rank-3 holomorphic vector bundle is fixedas E = K L O L K − . Associated to a representation ρ in H ( S ) is a Hermitian metric H on E that solves Hitchin’s equation (2.9) and a flat connection D H = ∇ ¯ ∂ E ,H + Φ + Φ ∗ H where ∇ ¯ ∂ E ,H is the chern connection (see Theorem 2.6).Given a parameter s ∈ ( − , { ρ s } in H ( S ). On the one hand, there is a family of flat connections { D H ( s ) } given by equation (2.10) assiociated to { ρ s } . On the other hand, there is a family ofreparametrization functions { f ρ s } s ∈ ( − , assiociated to { ρ s } from the thermodynamical pointof view. Recall our notation (2.4), (2.5). For a family of flat connection { D H ( s ) } , we denote ∂ s D H (0) = ∂∂ s (cid:12)(cid:12)(cid:12)(cid:12) s =0 D H ( s ) . and for a family of reparametrization functions { f ρ s } , we denote ∂ s f ρ = ∂∂ s (cid:12)(cid:12)(cid:12)(cid:12) s =0 f ρ s . By Proposition 3.6 and Livˇsic’s Theorem, we know the H¨older function − Tr (cid:0) ∂ s D H (0) π (cid:1) ( x )and ∂ s f ρ ( x ) are in the same Livˇsic cohomology class. Recall our notation in Definition 2.2, ∂ s f ρ ( x ) ∼ − Tr (cid:0) ∂ s D H (0) π (cid:1) ( x ) . (5.1)Proposition 4.3 allows us to consider first and second variations of reparametrization functionsin terms of Livˇsic cohomology class instead of individual functions. From now on, for first andsecond variations of reparametrization functions, we will no longer distinguish cohomologouselements.Because X has the DPG property, to recover the information of ∂ s f ρ , it suffices to com-pute Tr (cid:0) ∂ s D H (0) π (cid:1) on each closed geodesic. Similarly, to compute the second variations ofreparametrization functions, it suffices to compute them on each closed geodesic.Now we start to give a complete computation of the first and second variations of reparametriza-tion functions for the case ∂ β g αα ( σ ). The steps of our argument are divided into differentsubsections: 33. We set up coordinates adapted to the closed geodesics we model and conclude propertiesof affine metrics on these geodesics.2. We construct a homogeneous ODE arising from the parallel transport equation for thebase flat connection at ρ (0) = σ ∈ T ( S ) and obtain the first variation of reparametrizationfunctions.3. We consider a family of parallel transport equations associated to a family of flat con-nections by solving Hitchin’s equations based at ρ (0) = σ ∈ T ( S ). The variation of thisfamily of parallel transport equations at σ gives rise to some nonhomogeneous ODEs andyields solutions to second variations of reparametrization functions on the closed geodesicswe consider.4. We extend our computation from the closed geodesics to the surface. In this subsection, we set up coordinates adapted to the closed geodesics we model and concludesome properties of affine metrics on these geodesics. They will be used in computation of firstand second variations of reparametrization functions in the following sections.The convention we use for a Hermitian metric H on E is: it is C -linear in the secondvariable and conjugate-linear in the first variable. Suppose on a coordinate chart ( U, z ), thebundle E = K L O L K − is trivialized as E | U ∼ = U × C . Locally we have a holomorphicframe ( s , s , s ) on U . With respect to the local holomorphic frame and our convention of theHermitian metric, the (1 , ∇ ¯ ∂ E ,H is H − ∂H . The Hermitianconjugate is Φ ∗ H = H − ¯Φ t H . The connection one form A of the flat connection D H is thus A = H − ∂H + Φ + Φ ∗ H . Associated to representations { ρ ( u, v ) } are a two-parameter family of flat connections { D H ( u,v ) } .We will study their connection one-forms in holomorphic frames with respect to some carefullychosen coordinates on the surface X .When the Higgs field is Φ( u, v ) = uq α + vq β , Baraglia proves the Hermitian metric H ( u, v ) that solves Hitchin’s equation (2.9) is diagonal(see[2]). Following Baraglia’s notation in [2], we denote the Hermitian metric as H ( u, v ) = e u,v ) .We have H ( u, v ) = h ( u, v ) − h ( u, v ) , where h = h ( u, v ) is a section of ¯ K ⊗ K andΩ( u, v ) = − ω ( u, v ) 0 00 0 00 0 ω ( u, v ) ω ( u, v ) = log h ( u, v ).We denote the corresponding flat connection by D H ( u,v ) = ∇ ¯ ∂ E ,H ( u,v ) +Φ( u, v )+Φ( u, v ) ∗ H ( u,v ) .The connection one-form A ( u, v ) ∈ Γ( T ∗ X ⊗ EndE ) is thus A ( u, v ) = − ∂ω ( u, v ) h ( u, v ) uq α + vq β h ( u, v ) h − ( u ¯ q α + v ¯ q β ) 1 2 ∂ω ( u, v ) (5.2)In fact 2 h ( u, v ) is an affine metric for some hyperbolic affine sphere in the conformal classof σ (see [2]).We write down the equation that h ( u, v ) satisfies. Suppose we are considering z -coordinateand suppose the affine metric in this coordinate is e ψ ( u,v,z ) | dz | . Suppose futhermore the hy-perbolic metric in this coordinate is σ = e δ ( z ) | dz | . Then h = h ( u, v, z ) d ¯ z ⊗ dz = 12 e ψ ( u,v,z ) d ¯ z ⊗ dz = 12 e φ ( u,v,z )+ δ ( z ) d ¯ z ⊗ dz (5.3)We notice here φ is actually a globally well-defined function that does not depend on coordinate z . For simplicity of notation, we omit u, v variables and write h ( z ) = h ( u, v, z ) and ψ ( z ) = ψ ( u, v, z ).The Hitchin’s equation (2.9), also the integrability condition for affine sphere (see [23]) that theaffine metric satisfies, is2 ∂ ¯ ∂ω + ( uq α + vq β )( u ¯ q α + v ¯ q β ) e − ω − e ω = 0 . (5.4)Equivalently, in the z -coordinate, this is ψ z ¯ z + 4 | uq α + vq β | e − ψ − e ψ = 0 . (5.5)We have the following observation from equation (5.5): • When ( u, v ) = (0 , ψ z ¯ z − e ψ = 0.This implies φ = φ (0 , 0) = 0. The affine metric e ψ ( z ) | dz | = e δ ( z ) | dz | = σ is indeed thehyperbolic metric of constant curvature − • Taking u -derivative or v -derivative of equation (5.4) at ( u, v ) = (0 , 0) yields∆ σ φ u − e φ φ u = 0 , ∆ σ φ v − e φ φ v = 0 . The notation we adopt for Laplacian is ∆ σ = ∂ ¯ z ∂ z σ .Therefore, at ( u, v ) = (0 , φ = φ (0 , 0) = 0 implies φ u = φ u (0 , 0) = 0 and φ v = φ v (0 , 0) = 0. 35e now choose a special coordinate system that facilitates the study of holonomy problemson a closed geodesic. Suppose γ ( t ) is any closed geodesic on the Riemann surface X . Writtenin the z -coordinate, it is γ ( t ) = z ( t ) = Re γ ( t ) + i Im γ ( t )and ˙ γ ( t ) ∂∂t = (Re ˙ γ ( t ) + i Im ˙ γ ( t )) ∂∂z + (Re ˙ γ ( t ) − i Im ˙ γ ( t )) ∂∂ ¯ z . In particular, we can model γ ( t ) on a strip S = { x + iy | | y | < π } with the hyperbolic metric ds = | dz | cos y and γ ( t ) = ( t, γ is called a Fermi coordinate and satisfiesRe ˙ γ ( t ) = 1, Im ˙ γ ( t ) = 0. Thus it’s easy to check on γ one has γ ∗ ds = | dz | and δ ( z ) = 0.The variable t is then the arc length parameter for our choice of coordinates. Thereforeif one denotes ˙ γ (0) = x ∈ U X , then ˙ γ ( t ) = Φ t ( x ). We will always assume ˙ γ (0) = x in ourdiscussion.In the end of this subsection, we conclude some important properties we have for the affinemetric after setting up the coordinates. At ( u, v ) = (0 , γ satisfies ψ ( z ) = φ ( z ) + σ ( z ) = σ ( z ) = 0 ,ψ z ( z ) = σ z ( z ) = 0 ,ψ u ( z ) = φ u ( z ) = 0 ,ψ v ( z ) = φ v ( z ) = 0 . In this subsection, we construct a homogeneous ODE arising from the parallel transport equa-tion for the base flat connection at σ ∈ T ( S ) and obtain first variations of reparametrizationfunctions.We first explain our notation. For q i = q i ( z ) dz any quadratic differential and q α = q α ( z ) dz any cubic differential, we also use q i and q α to denote H¨older functions on unit tangent bundle U X as follows. We let q i : U X → C and q α : U X → C be q i ( x ) := q i ( z ) dz ( x, x ) = q i ( z )( dz ( x )) (5.6) q α ( x ) := q α ( z ) dz ( x, x, x ) = q α ( z )( dz ( x )) (5.7)First variations of reparametrization functions are described in the following proposition. Proposition 5.1. The first variations of reparametrization functions ∂ u f ρ (0) : U X → R and ∂ v f ρ (0) : U X → R for our model case ∂ β g αα ( σ ) satisfy − ∂ u f ρ (0) ( x ) ∼ Re q α ( x ) , − ∂ v f ρ (0) ( x ) ∼ Re q β ( x ) . where the notation ∼ has been introduced in Definition 2.2 π and the first derivative of flat connections. To computethem, we consider the parallel transport equation for D H (0) and the holonomy problem for thebase point ρ (0) ∈ T ( S ) . We start to construct the parallel transport equations and explainhow to solve them with the coordinates introduced in last section. After these discussions, wewill be able to compute first variation and prove Proposition 5.1 at the end of this section.The parallel transport equation for connection D H (0) on the closed geodesic γ is as follows, D H (0) , ˙ γ V = 0 , (5.8)where V ∈ Γ( E ) is a parallel section with boundary conditions: V ( l γ ) = λ i ( γ, ρ (0)) V (0) . (5.9)Here λ i ( γ, ρ (0)) is one of the eigenvalues for holonomy of D H (0) on γ for i = 1 , , 3. We want towrite the equation (5.8) on a specific holomorphic frame which can be constructed as follows.We cover γ by m charts { ( U i , z i ) } mi =1 so that z i : U i → z i ( U i ) ⊂ C is a diffeomorphism for1 ≤ i ≤ m . We assume our holomorphic bundle E is trivialized on each U i . Furthermore weassume the transition map on every overlap is either the identity or a hyperbolic translationviewed on the universal cover D . Since dz i is a local holomorphic section of K on U i and ∂∂z i is a local holomorphic section of K − on U i , we can define a local holomorphic frame s i = ( s i , s i , s i ) for E = K L O L K − on U i , where s i = dz i and s i = 1 and s i = ∂∂z i . Setting( U m +1 , z m +1 ) = ( U , z ) and s m +1 j = s j , this yields a well-defined holomorphic frame for γ because on each overlap and for j = 1 , , 3, we have s ij = s i +1 j on γ | U i ∩ γ | U i +1 with 1 ≤ i ≤ m .We will simply write the holomorphic frame on γ as s j for j = 1 , , 3. With respect to thisframe given by ( s , s , s ), the parallel transport equation for V ( t ) = P i =1 V i ( t ) s i ( t ) becomes ∂ t V ( t ) V ( t ) V ( t ) + 01 0 V ( t ) V ( t ) V ( t ) = 0 . There are three eigenvalues for this ordinary differential equation system: λ ( γ, ρ (0)) = e l γ , λ ( γ, ρ (0)) = 1 and λ ( γ, ρ (0)) = e − l γ . The solutions for V (assuming norm 1 at the startingpoint with respect to the Hermitian metric H (0)), denoted as e i corresponding to λ i ( γ ) for i = 1 , , 3, are e = √ e t − , e = 12 − , e = √ e − t . We note at the Fuchsian point ρ (0) ∈ T ( S ), the eigenvectors e , e and e are orthogonal.In our holomorphic frame, the projection π (0) = π ( ρ (0)) can be computed as π (0) = 12 − 12 14 − − − . 37e write q α = q α ( z ) dz and q β = q β ( z ) dz with respect to the Fermi coordinate z and recall x = ˙ γ (0). Then in our holomorphic frame ( s , s , s ), ∂ u D H (0) ( x ) = q α ( z )0 0 04¯ q α ( z ) 0 0 , ∂ v D H (0) ( x ) = q β ( z )0 0 04¯ q β ( z ) 0 0 . With the above information, we are able to prove Proposition 5.1. Proof of Proposition 5.1. For x = ˙ γ (0), we know q α ( x ) = q α ( z )( dz ( x )) = q α ( z ) with respect to the Fermi coordinate z . Similarly, we have q β ( x ) = q β ( z ).By formula (5.1), we obtain ∂ u f ρ (0) ( x ) ∼ − Tr (cid:0) ∂ u D H (0) π (0) (cid:1) ( x ) = − Tr (cid:0) ∂ u D H (0) ( x ) π (0) (cid:1) = − Re q α ( x ) ,∂ v f ρ (0) ( x ) ∼ − Tr (cid:0) ∂ v D H (0) π (0) (cid:1) ( x ) = − Tr (cid:0) ∂ v D H (0) ( x ) π (0) (cid:1) = − Re q β ( x ) . The above formulas are valid on a dense subset of U X as the closed geodesic γ is arbi-trarily chosen and so is the starting point x tangential to γ . Because − Tr (cid:0) ∂ u D H (0) π (0) (cid:1) ( x )and − Tr (cid:0) ∂ v D H (0) π (0) (cid:1) ( x ) are H¨older functions and because of the DPG property, we obtainProposition 5.1 for ∂ β g αα ( σ ). After computing the first variations of reparametrization functions, we continue computing thesecond variation of the reparametrization functions ∂ uv f ρ (0) in this and next subsection. Withour formula (5.1), we have ∂ uv f ρ (0) ∼ − ∂ v (Tr (cid:0) ∂ u D H (0) π (0) (cid:1) )= − Tr ∂ D H (0) ∂u∂v π (0) ! − Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) . In this subsection, we compute ∂ uv f ρ (0) along a closed geodesic. We study variation of holomonyproblems along a closed geodesic and construct associated inhomogeneous ODEs. In the nextsubsection, we extend the computation of ∂ uv f ρ (0) to the whole surface.The computation of ∂ v (Tr (cid:0) ∂ u D H (0) π (0) (cid:1) ) in this subsection along a closed geodesic is di-vided into computations of Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) and of Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) : • Compute Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) . 38ith the holomorphic frames and Fermi coordinates set up as before, one obtains on γ∂ uv D H (0) ( x ) = − ( ψ z ) uv ( z ) ψ uv ( z ) 00 0 ψ uv ( z )0 0 ( ψ z ) uv ( z ) . Thus Tr ∂ D H (0) ∂u∂v ( x ) π (0) ! = − ψ uv ( z ) . More explicitly, we have Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) : U X → R satisfiesTr ∂ D H (0) ∂u∂v π (0) ! ( x ) = − ψ uv ( z ( p ( x ))) = − φ uv ( p ( x )) , where p : U X → X is the projection from the unit tangent bundle to our surface and z is the Fermi coordinate we choose evaluating at the point p ( x ) ∈ X . We remark here theaffine metric ψ is always real and φ = ψ − σ does not depend on coordinates we choose. • Compute Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) .To study ∂ v π (0) takes some effort. We set u = 0 and take a family of flat connections { D H ( v ) } with connection one forms A (0 , v ) (recall equation (5.2)). Associated to each ofthem is a parallel transport equation along the closed geodesic γ on ( S, σ ): D H ( v ) , ˙ γ V ( v, t ) = 0 (5.10)with the assumption k V ( v, k H (0) = 1.In [19], Labourie proves images of every Hitchin representation are purely loxodromic.For ρ (0 , v ) in H ( S ), we know ρ (0 , v )( γ ) has distinct eigenvalues: λ ( γ, v ) > λ ( γ, v ) >λ ( γ, v ). The holonomy problem for ρ (0 , v ) has three distinct eigenvectors which areparallel sections { e i ( v, t ) } i =1 along γ ( t ). Each section V ( v, t ) = e i ( v, t ) satisfies equation(5.10). In addition to the norm 1 condition at starting point : k V ( v, k H (0) = 1, wealso impose another boundary condition in order to guarantee these are eigenvectors. Theboundary conditions are for i = 1 , , k e i ( v, k H (0) = 1;(ii) e i ( v, l γ ) = λ i ( γ, v ) e i ( v, E along γ mentioned, theholomorphic frame ( s , s , s ) and the frame spanned by eigenvectors ( e , e , e ). One theone hand, we can write our holomorphic frames as linear combinations of eigenvectors: s i ( t ) = P j =1 a ij ( v, t ) e j ( v, t ) for i=1,2,3. On the other hand, we can write the eigenvectors aslinear combinations of our holomorphic frames: e j ( v, t ) = P k =1 e jk ( v, t ) s k ( t ) for j = 1 , , s , s , s ), the projection onto e along the hy-perplane spanned by ( e , e ) in matrix form is, π ( v, t ) = (cid:2) π ( v, t ) s ( t ) , π ( v, t ) s ( t ) , π ( v, t ) s ( t ) (cid:3) = (cid:2) a ( v, t ) e ( v, t ) , a ( v, t ) e ( v, t ) , a ( v, t ) e ( v, t ) (cid:3) = a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) a ( v, t ) e ( v, t ) . To understand ∂ v π (0), we need to know ∂ v e (0) and ∂ v α i (0) for i = 1 , , 3. One cancheck in the holomorphic frame,Tr (cid:0) ∂ u D A (0) ∂ v π (0) (cid:1) = q α ( ∂ v a (0) e (0)+ a (0) ∂ v e (0))+4¯ q α ( ∂ v a (0) e (0)+ a (0) ∂ v e (0)) , (5.11)where e (0) and e (0) are known. Thus we need to compute ∂ v e (0)) and ∂ v a (0) and ∂ v a (0).We first show how to obtain ∂ v e (0)). ∂ v e (0 , t ) is the solution of a inhomogeneous ODEsystem arising from taking v -derivative for a family of parallel transport equations (5 . v = 0, ∂ t ∂ v e (0 , t ) ∂ v e (0 , t ) ∂ v e (0 , t ) + 01 0 ∂ v e (0 , t ) ∂ v e (0 , t ) ∂ v e (0 , t ) = − √ e t q β (Φ t ( x ))02 q β (Φ t ( x )) . with boundary conditions H ( ∂ v e (0 , , e (0 , ∂ v e (0 , l γ ) = − e l γ (cid:18)Z l γ Re q β (Φ s ( x ))d s (cid:19) e (0 , 0) + e l γ ∂ v e (0 , . The boundary conditions arise from taking v -derivative for boundary conditions ( i ) and( ii ) of the parallel transport equation (5.10) that the maximum eigenvector e satisfies.With these boundary conditions, we solve ∂ v e ( t ) ∂ v e ( t ) ∂ v e ( t ) = − √ R t e s (cosh( t − s ) Re q β + i Im q β )d s √ R t e s (sinh( t − s ) Re q β d s −√ R t e s (cosh( t − s ) Re q β − i Im q β )d s + − √ ( e l γ − − R l γ e s − t Re q β d s − √ i ( e l γ − − R l γ e s Im q β d s − √ ( e l γ − − R l γ e s − t Re q β d s − √ ( e l γ − − R l γ e s − t Re q β d s + √ i ( e l γ − − R l γ e s Im q β d s . Here q β refers to q β (Φ s ( x )) defined in notation (5.7).40e continue to compute ∂ v α (0) and ∂ v α (0). Combining e j ( v, t ) = P k =1 e jk ( v, t ) s k ( t )and s i ( t ) = P j =1 a ij ( v, t ) e j ( v, t ) gives j =3 X j =1 a ij ( v, t ) e jk ( v, t ) = σ ik . (5.12)Recall e jk (0 , t ) are known: e (0 , t ) = e (0 , t ) e (0 , t ) e (0 , t ) = √ e t − ,e (0 , t ) = e (0 , t ) e (0 , t ) e (0 , t ) = 12 − ,e (0 , t ) = e (0 , t ) e (0 , t ) e (0 , t ) = √ e − t . Then one obtains a (0 , t ) = a a a a a a a a a = √ e − t − √ e t − √ e − t √ e t √ e − t √ e t . Taking the v -derivative of equation (5.12) at v = 0, j =3 X j =1 ∂ v a ij (0 , t ) e jk (0 , t ) + j =3 X j =1 a ij (0 , t ) ∂ v e jk (0 , t ) = 0 . Solutions of ∂ v a ij (0 , t ) can be expressed in terms of ∂ v e (0 , t ), ∂ v e (0 , t ) and ∂ v e (0 , t ). Wehave just solved ∂ v e . Similarly, ∂ v e (0) and ∂ v e (0) are solutions of another two systems ofnonhomogeneous ODEs deduced from equation (5.10). We now proceed to solve ∂ v e (0 , t )and ∂ v e (0 , t ).1. For ∂ v e (0 , t ), we have ∂ t ∂ v e (0 , t ) ∂ v e (0 , t ) ∂ v e (0 , t ) + 01 0 ∂ v e (0 , t ) ∂ v e (0 , t ) ∂ v e (0 , t ) = − q β (Φ t ( x ))02 q β (Φ t ( x )) with boundary conditions H ( ∂ v e (0 , , e (0 , ∂ v e (0 , l γ ) = 2 (cid:18)Z l γ Re q β (Φ s ( x ))d s (cid:19) e (0 , 0) + ∂ v e (0 , . 41. For ∂ v e (0 , t ), we get ∂ t ∂ v e (0 , t ) ∂ v e (0 , t ) ∂ v e (0 , t ) + 01 0 ∂ v e (0 , t ) ∂ v e (0 , t ) ∂ v e (0 , t ) = − √ e − t q β (Φ t ( x ))02 q β (Φ t ( x )) with boundary conditions H ( ∂ v e (0 , , e (0 , ∂ v e (0 , l γ ) = − e − l γ (cid:18)Z l γ Re q β (Φ s ( x ))d s (cid:19) e (0 , 0) + e − l γ ∂ v e (0 , . We obtain solutions respectively as follows ∂ v e ( t ) ∂ v e ( t ) ∂ v e ( t ) = − R t Re q β + i cosh( t − s ) Im q β d s R t i sinh( t − s ) Im q β d s R t Re q β − i cosh( t − s ) Im q β d s + − i R l γ Im q β ((1 − e l γ ) − e l γ + t − s ) + (1 − e − l γ ) − e − l γ − t + s ))d si R l γ Im q β ((1 − e l γ ) − e l γ + t − s ) − (1 − e − l γ ) − e − l γ − t + s ))d s − i R l γ Im q β ((1 − e l γ ) − e l γ + t − s ) + (1 − e − l γ ) − e − l γ − t + s ))d s and ∂ v e ( t ) ∂ v e ( t ) ∂ v e ( t ) = − √ R t e − s (cosh( t − s ) Re q β + i Im q β )d s √ R t e − s sinh( t − s ) Re q β d s −√ R t e − s (cosh( t − s ) Re q β − i Im q β )d s + − √ ( e − l γ − − R l γ e t − s Re q β d s − √ i ( e − l γ − − R l γ e − s Im q β d s √ ( e − l γ − − R l γ e t − s Re q β d s − √ ( e − l γ − − R l γ e t − s Re q β d s + √ i ( e − l γ − − R l γ e − s Im q β d s , where again q β in the solutions again refers to q β (Φ s ( x )) defined in notation (5.7).We are therefore able to solve ∂ v a ij (0 , t ) from ∂ v e (0 , t ), ∂ v e (0 , t ) and ∂ v e (0 , t ). For aclosed geodesic γ of length l γ starting from ˙ γ (0) = x , we compute from equation (5.11),Tr (cid:0) ∂ u D A (0) ∂ v π (0) (cid:1) (Φ t ( x ))= Re q α (Φ t ( x )) Z t ( e t − s ) − e s − t ) ) Re q β (Φ s ( x ))d s +2 Im q α (Φ t ( x )) Z t ( e t − s − e s − t ) Im q β (Φ s ( x ))d s + Re q α (Φ t ( x )) Z l γ ( e t − s ) e − l γ − − e s − t ) e l γ − q β (Φ s ( x ))d s +2 Im q α (Φ t ( x )) Z l γ ( e t − s e − l γ − − e s − t e l γ − q β (Φ s ( x ))d s. (5.13)42n particular, at t = 0,Tr (cid:0) ∂ u D A (0) ∂ v π (0) (cid:1) ( x )= Re q α ( x ) Z l γ ( e − s e − l γ − − e s e l γ − q β (Φ s ( x ))d s +2 Im q α ( x ) Z l γ ( e − s e − l γ − − e s e l γ − q β (Φ s ( x ))d s. (5.14) Remark 5.1. One may notice every point on the closed geodesic γ plays equivalent roles.We can always let y = Φ t ( x ) be the initial point of our γ and set up boundary conditionsfor our ODEs based at y instead of x . The solution of this new ODE system is equation (5.14) treating y = Φ t ( x ) as the initial point. It is in fact the same as starting from x andobtain Tr (cid:0) ∂ u D A (0) ∂ v π (0) (cid:1) (Φ t ( x )) from equation (5.13) . We will extend our result for second variations of reparametrization functions on closedgeodesics to the Riemann surface X in this subsection.Denoting the subset of U X that consists of all unit tangent vectors to closed geodesics as W , we define a function ψ : W × R + −→ R by ψ ( x, r ) = Re q α ( x ) Z r ( e − s e − r − − e s e r − q β (Φ s ( x ))d s +2 Im q α ( x ) Z r ( e − s e − r − − e s e r − q β (Φ s ( x ))d s. Given x ∈ W , we denote the closed geodesic that x is tangential to as γ x with length l γ x .Fixing x , we want to show ψ ( x, r ) attains the same value when r is any positive integermultiple of l γ x , Lemma 5.2. ψ ( x, kl γ x ) = ψ ( x, l γ x ) ∀ x ∈ W, ∀ k ∈ Z + .Proof. For any k ∈ Z + , we have Z kl γx ( e − s e − kl γx − − e s e kl γx − q β (Φ s ( x ))d s = k X i =1 Z il γx ( i − l γx ( e − s e − kl γx − − e s e kl γx − q β (Φ s ( x ))d s = 1 e − kl γx − k X i =1 Z il γx ( i − l γx e − s Re q β (Φ s ( x ))d s − e kl γx − k X i =1 Z il γx ( i − l γx e s Re q β (Φ s ( x ))d s = Z l γx ( e − s e − l γx − − e s e l γx − q β (Φ s ( x ))d s. Similar arguments hold for R l γx ( e − s e − lγx − − e s e lγx − ) Im q β (Φ s ( x ))d s .Thus we obtain ψ ( x, kl γ x ) = ψ ( x, l γ x ). Remark 5.2. This equality is clear if one understands that ψ ( x, kl γ ) is the solution of theholonomy problem that goes around our closed geodesic γ k -times with the same boundaryconditions. x to Φ l γx ( x ), we view x as our midpoint and consider our flowfrom Φ − lγx ( x ) to x and then from x to Φ lγx ( x ). From this point of view, we can write ψ ( x, l γ x ) as ψ ( x, l γ x ) = Re q α ( x ) Z lγx ( e − s e − l γx − − e s e l γx − q β (Φ s ( x ))d s + Re q α ( x ) Z − lγx ( e s e − l γx − − e − s e l γx − q β (Φ s ( x ))d s +2 Im q α ( x ) Z lγx ( e − s e − l γx − − e s e l γx − q β (Φ s ( x ))d s +2 Im q α ( x ) Z − lγx ( e s e − l γx − − e − s e l γx − q β (Φ s ( x ))d s. The above also holds if we replace l γ x by kl γ x . We consider defining a function η : W → R from ψ ( x, kl γ x ) by taking k → ∞ , η ( x ) = − Re q α ( x ) Z ∞ e − s Re q β (Φ s ( x ))d s − Re q α ( x ) Z −∞ e s Re q β (Φ s ( x ))d s − q α ( x ) Z ∞ e − s Im q β (Φ s ( x ))d s − q α ( x ) Z −∞ e s Im q β (Φ s ( x ))d s. (5.15)It turns out we have the following interesting result. The two variable function ψ ( x, l γ x )can be expressed independent of l γ x as, Proposition 5.3. For any x ∈ W , ψ ( x, l γ x ) = η ( x ) .Proof. Suppose max x ∈ UX {| Re q α ( x ) | , | Im q α ( x ) | , | Re q β ( x ) | , | Im q β ( x ) |} = M . Then notice | ψ ( x, kl γ x ) − η ( x ) |≤ M Z ∞ klγx e − s d s + 4 M Z ∞ klγx e − s d s +2 M Z klγx (cid:12)(cid:12)(cid:12)(cid:12) e − s e − kl γx − − e s e kl γx − e − s (cid:12)(cid:12)(cid:12)(cid:12) d s +4 M Z klγx (cid:12)(cid:12)(cid:12)(cid:12) e − s e − kl γx − − e s e kl γx − e − s (cid:12)(cid:12)(cid:12)(cid:12) d s −→ k → ∞ Thus we obtain for any x ∈ W , ψ ( x, l γ x ) = lim k →∞ ψ ( x, kl γ x ) = η ( x ) . Proposition 5.4. We extend η ( x ) to be a function defined on U X by equation (5.15) .Then η ( x ) : U X → R is a H¨older function.Proof. We consider the canonical (Sasaki) metric h· , ·i on U X . This Riemannian metricinduces a distance function d on U X . We want to show there exists some constants0 < r < C > | η ( x ) − η ( y ) | ≤ Cd ( x, y ) r for any given x, y ∈ U X .44e first want to show for the hyperbolic surface ( S, σ ), we have some control on thegeodesic flow as follows, d (Φ s ( x ) , Φ s ( y )) ≤ N e s d ( x, y ) , for some constant N > x, y ∈ U X , we will set l = d ( x, y ) and let t ∈ [0 , l ] be the arc-length parameter.We want to construct Γ( s, t ) = γ t ( s ) to be a smooth variation through geodesics, i.e. forfixed t , each γ t ( s ) is a unit speed geodesic. We let x = ∂ s γ (0), y = ∂ s γ (0) and consider G ( t ) as a unit speed geodesic on U X connecting G (0) = x to G ( l ) = y . For each fixed t , we define ∂ s γ t (0) := ∂ s Γ(0 , t ) = G ( t ) and then obtain a unit speed geodesic γ t ( s ) withthe initial vector ∂ s γ t (0) = G ( t ).We have the Jacobi equation for J ( s, t ) = ∂ t Γ( s, t ): D s J ( s, t ) − J ( s, t ) = 0 . We note d ( x, y )= Z l k ∂ t G ( t ) k d t = Z l k ∂ t Γ(0 , t ) k d t + Z l k D t ∂ s Γ(0 , t ) k d t definition of Sasaki metric= Z l k ∂ t Γ(0 , t ) k d t + Z l k D s ∂ t Γ(0 , t ) k d t symmetric lemma: D s ∂ t Γ = D t ∂ s Γ= Z l k J (0 , t ) k d t + Z l k D s J (0 , t ) k d t. Similarly d (Φ s ( x ) , Φ s ( y )) = Z l k J ( s, t ) k d t + Z l k D s J ( s, t ) k d t. For each γ t , we have two orthonormal parallel sections ˙ γ t ( s ) and E t ( s ) where E t ( s ) isparallel normal vector field along γ . We can write J ( s, t ) as a linear combination of ˙ γ t ( s )and E t ( s ): J ( s, t ) = u t ( s ) ˙ γ t ( s ) + v t ( s ) E t ( s ). The Jacobi equation yields u t ( s ) = a ( t ) cosh( s ) + b ( t ) sinh( s ) ,v t ( s ) = a ( t ) cosh( s ) + b ( t ) sinh( s ) . Therefore h J (0 , t ) , J (0 , t ) i = a ( t ) + a ( t ) . h J ( s, t ) , J ( s, t ) i = | a ( t ) cosh( s ) + b ( t ) sinh( s ) | + | a ( t ) cosh( s ) + b ( t ) sinh( s ) | . h D s J (0 , t ) , D s J (0 , t ) i = b ( t ) + b ( t ) . h D s J ( s, t ) , D s J ( s, t ) i = | b ( t ) cosh( s ) + a ( t ) sinh( s ) | + | b ( t ) cosh( s ) + a ( t ) sinh( s ) | . For s ≥ 0, we conclude k J ( s, t ) k ≤ e s ( k J (0 , t ) k + k D s J (0 , t ) k ) ≤ e s ( k J (0 , t ) k + k D s J (0 , t ) k ) k D s J ( s, t ) k ≤ e s ( k J (0 , t ) k + k D s J (0 , t ) k ) . Thus d (Φ s ( x ) , Φ s ( y )) = Z l k J ( s, t ) k d t + Z l k D s J ( s, t ) k d t ≤ Z l √ e s ( k J (0 , t ) k + k D s J (0 , t ) k )d t =2 √ e s d ( x, y ) . Recall our notation in (5.7). Because Re q α ( x ), Re q β ( x ) and Im q α ( x ), Im q β ( x ) are H¨olderfunctions, we can assume they all have H¨older exponents r and there’s a constant C > | Re q α ( x ) − Re q α ( y ) | ≤ Cd ( x, y ) r , | Re q β ( x ) − Re q β ( y ) | ≤ Cd ( x, y ) r , | Im q α ( x ) − Im q α ( y ) | ≤ Cd ( x, y ) r , | Im q β ( x ) − Im q β ( y ) | ≤ Cd ( x, y ) r . Also recall max x ∈ UX {| Re q α ( x ) | , | Im q α ( x ) | , | Re q β ( x ) | , | Im q β ( x ) |} = M .Thus − Re q α ( x ) Z ∞ e − s Re q β (Φ s ( x ))d s + Re q α ( y ) Z ∞ e − s Re q β (Φ s ( y ))d s = − (Re q α ( x ) − Re q α ( y )) Z ∞ e − s Re q β (Φ s ( x ))d s + Re q α ( y ) Z ∞ e − s (Re q β (Φ s ( y )) − Re q β (Φ s ( x )))d s ≤ CM d ( x, y ) r Z ∞ e − s d s + M Z ∞ e − s Cd (Φ s ( x ) , Φ s ( y )) r d s ≤ CM d ( x, y ) r + M C (2 √ r − r d ( x, y ) r . Similarly, − q α ( x ) Z ∞ e − s Im q β (Φ s ( x ))d s + 2 Im q α ( y ) Z ∞ e − s Im q β (Φ s ( y ))d s = − q α ( x ) − Im q α ( y )) Z ∞ e − s Im q β (Φ s ( x ))d s +2 Im q α ( y ) Z ∞ e − s (Im q β (Φ s ( y )) − Im q β (Φ s ( x )))d s ≤ CM d ( x, y ) r + 2 M Z ∞ e − s Cd (Φ s ( s ) , Φ s ( y )) r d s ≤ CM d ( x, y ) r + 2 M Z ∞ e − s C (2 √ r e rs d ( x, y ) r d s =2 CM d ( x, y ) r + 2 M C (2 √ r − r d ( x, y ) r . q α ( x ) Z −∞ e s Re q β (Φ s ( x ))d s = Re q α ( − x ) Z ∞ e − s Re q β (Φ s ( − x ))d s Im q α ( x ) Z −∞ e s Im q β (Φ s ( x ))d s = Im q α ( − x ) Z ∞ e − s Im q β (Φ s ( − x ))d s. Thus | η ( x ) − η ( y ) | ≤ C d ( x, y ) r holds for 0 < r < C .We finally can state our main proposition in these two subsection about second variationsof reparametrization functions on U X . Proposition 5.5. The second variation of reparametrization functions ∂ uv f ρ (0) : U X → R for our model case ∂βg αα ( σ ) satisfies ∂ uv f ρ (0) ( x ) ∼ φ uv ( p ( x ))+ Re q α ( x ) Z ∞ e − s Re q β (Φ s ( x ))d s + Re q α ( x ) Z −∞ e s Re q β (Φ s ( x ))d s +2 Im q α ( x ) Z ∞ e − s Im q β (Φ s ( x ))d s + 2 Im q α ( x ) Z −∞ e s Im q β (Φ s ( x ))d s, where we recall φ is defined in equation (5.3) and p : U X → X is the projection from theunit tangent bundle U X to our Riemann surface X .Proof. We have done most of necessary elements for this proof in previous estimates.We assemble everything together here. Because Tr (cid:0) ∂ u D A (0) ∂ v π (0) (cid:1) (Φ t ( x )) is a H¨olderfunction, and because it equals to the H¨older function η ( x ) (5.15) on a dense subset of U X . We conclude it coincides everywhere with η ( x ) on U X . We obtain ∂ uv f ρ (0) ∼ − ∂ u (Tr (cid:0) ∂ v D H (0) π (0) (cid:1) )= − Tr ∂ D H (0) ∂u∂v π (0) ! − Tr (cid:0) ∂ v D H (0) ∂ u π (0) (cid:1) = 12 φ uv ( p ( x ))+ Re q α ( x ) Z ∞ e − s Re q β (Φ s ( x ))d s + Re q α ( x ) Z −∞ e s Re q β (Φ s ( x ))d s + 2 Im q α ( x ) Z ∞ e − s Im q β (Φ s ( x ))d s + 2 Im q α ( x ) Z −∞ e s Im q β (Φ s ( x ))d s. where we recall here φ = ψ − σ is a globally well-defined function defined in (5.3) evaluatingat the point p ( x ) ∈ X and p : U X → X is the projection from the unit tangent bundle toour surface. After the computation of first and second variations of reparametrization functions on U X inthe last two section, we are able to evaluate ∂ β g αα ( σ ). Our goal in this section is to show thefollowing, 47 roposition 6.1. For σ ∈ T ( S ) , ∂ β g αα ( σ ) = 0 . Let’s first write down the expression for ∂ β g αα ( σ ), ∂ β g αα ( σ ) = ∂ v h ∂ u ρ (0 , v ) , ∂ u ρ (0 , v ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v =0 = lim r →∞ r "Z UX (cid:18)Z r ∂ u f Nρ (0) d t (cid:19) Z r ∂ v f Nρ (0) d t d m + 2 Z UX Z r ∂ u f Nρ (0) d t Z r ∂ uv f Nρ (0) d t d m = lim r →∞ r Z UX (cid:18)Z r Re q α (Φ t ( x ))d t (cid:19) Z r Re q β (Φ t ( x ))d t d m + lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r − ∂ uv h ( ρ (0)) − ∂ uv f ρ (0) (Φ t ( x ))d t d m = I + IIHere the first term is denoted as I and the second term is denoted as II. The formula for ∂ uv f ρ (0) is given in Proposition (5.5).We aim to prove both I and II are zero for Proposition 6.1. The following lemma will becrucial. Lemma 6.2. For any t, s ∈ R , we have Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x ) = 0 . (6.1) Z UX Re q α ( x ) Im q α (Φ t ( x )) Im q β (Φ s ( x ))d m ( x ) = 0 . (6.2)We transfer the problem of evaluating the integrals in equation (6.1) and equation (6.2) toanalyzing the Fourier coefficients of holomorphic differentials. The key to show the integrals arezero is to use the symmetries properties of the Liouville measure m = m L and homogeneity ofholomorphic differentials viewed as functions on U X .Before we start our proof, we first explain the coordinates we will use to do the computation.Following an idea from [20], we take Poincar´e disk as our charts. Pick a point x ∈ U X . Weidentify the universal cover of ( X, σ ) with D by the unique isometry that takes π ( x ) ∈ X to0 ∈ D and identify the vector x ∈ U X with vector (1 , ∈ T D .We express our holomorphic differentials in these coordinates. For the holomorphic cubicdifferential q α , it has the following analytic expansion in the coordinate based on x , q α,x ( z ) = dz ∞ X n =1 a n ( x ) z n . Recall the hyperbolic distance d H in the Poincar´e disk model satisfies, d H (0 , Re iθ ) = r ( R ) = 12 log (cid:18) R − R (cid:19) . Thus ∂∂ r = (1 − R ) ∂∂R and dz ( ∂∂r ) (cid:12)(cid:12)(cid:12)(cid:12) Re iθ = (1 − R ) e iθ . q α,x ( z ) := Re (cid:0) q α,x ( z )( ∂∂r , ∂∂r , ∂∂r ) (cid:1) , one hasRe q α (Φ r ( e iθ x )) = ˜ q α,x ( Re iθ ) = Re ∞ X n =0 a n ( x ) R n (1 − R ) e i ( n +3) θ ! . (6.3)In particular, when r = 0, lim R → dz ( ∂∂r ) (cid:12)(cid:12)(cid:12)(cid:12) Re iθ = e iθ . Therefore,Re q α ( e iθ x ) = ˜ q α,x (0 · e iθ ) = lim R → Re (cid:18) q α,x ( Re iθ )( ∂∂r , ∂∂r , ∂∂r ) (cid:19) = Re( a ( x ) e i θ ) . (6.4)Suppose the coefficients of the analytic expansion for q β are b n , thenRe q β (Φ r ( e iθ x )) = ˜ q β,x ( Re iθ ) = Re ∞ X n =0 b n ( x ) R n (1 − R ) e i ( n +3) θ ! . (6.5)For the convenience of computation later for other cases, we also write down here two analyticexpansions for holomorphic quadratic differentials q i , q j with coefficients c n and d n respectively.Re q i (Φ r ( e iθ x )) = ˜ q i,x ( Re iθ ) = Re ∞ X n =0 c n ( x ) R n (1 − R ) e i ( n +2) θ ! . (6.6)Re q j (Φ r ( e iθ x )) = ˜ q j,x ( Re iθ ) = Re ∞ X n =0 d n ( x ) R n (1 − R ) e i ( n +2) θ ! . (6.7) Proof of Lemma 6.2. We begin with showing equation (6.1).The proof of it will be divided into two cases: • t ≥ s ≥ • t < s < s = t and s = t . We observe some symmetries in these two situations andargue from these symmetries that equation (6.1) holds for the first case. We then apply theresults for the first case to the second case by flow invariant properties of m L . Equation (6.2)then follows easily from equation (6.1) once we find the relation between them.Since m = m L is rotationally invariant, i.e. ( e iθ ) ∗ m L = m L , we have Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x )= 12 π Z π Z UX Re q α ( e iθ x ) Re q α (Φ t ( e iθ x )) Re q β (Φ s ( e iθ x ))d m ( x )d θ. 49. We restrict ourselves to the case t, s ≥ t ( T ) = log (cid:16) T − T (cid:17) and s ( S ) = log (cid:16) S − S (cid:17) . We first consider t > s > θ -variable, in terms of the analytic expansion, we get Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x )= 14 ∞ X n =0 (cid:18) Z UX Re( a a n ¯ b n +3 )d m T n (1 − T ) S n +3 (1 − S ) + Z UX Re( a ¯ a n +3 b n )d m T n +3 (1 − T ) S n (1 − S ) (cid:19) . (6.8)We denote A n = R UX Re( a a n ¯ b n +3 )d m and B n = R UX Re( a ¯ a n +3 b n )d m . To show equa-tion (6.1) holds for t, s ≥ 0, it suffices to prove for n ≥ A n = B n = 0 . (6.9)If t = 0 or s = 0, equation (6.1) is equivalent to the follows which are included in equation(6.9): A = B = 0 . To prove equation (6.9), we consider two special cases of equation (6.1): flow time s = t and s = t .By the Φ t -invariance of m , flow time s = t satisfies Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ t ( x ))d m ( x )= Z UX Re q α (Φ − t ( x )) Re q α ( x ) Re q β ( x )d m ( x ) . A convenient observation is flowing from x backwards for time t is the opposite of flowingforwards for time t from − x , i.e. Φ − t ( x ) = − Φ t ( − x ). Let y = − x and notice ( e iπ ) ∗ m = m , we have Z UX Re q α (Φ − t ( x )) Re q α ( x ) Re q β ( x )d m ( x )= − Z UX Re q α (Φ t ( y )) Re q α ( y ) Re q β ( y )d m ( y ) . Therefore Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ t ( x ))d m ( x )= − Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β ( x )d m ( x ) . This implies ∞ X n =0 (cid:18) A n + B n (cid:19) T n +3 (1 − T ) = − B T (1 − T ) . (6.10)50he coefficient of T yields A + 2 B = 0 (6.11)Similarly when flow time s = t . We let y = − x and again use the fact ( e iπ ) ∗ m = m . Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ t ( x ))d m ( x )= Z UX Re q α (Φ − t ( x )) Re q α ( x ) Re q β (Φ − t ( x ))d m ( x )= − Z UX Re q α (Φ t ( − x )) Re q α ( − x ) Re q β (Φ t ( − x ))d m ( x )= − Z UX Re q α (Φ t ( y )) Re q α ( y ) Re q β (Φ t ( y ))d m ( y ) . Thus R UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ t ( x ))d m ( x ) = 0.Recall t ( T ) = log (cid:16) T − T (cid:17) and s = log (cid:16) S − S (cid:17) . In the case s = t , we have T = SS +1 . Theanalytic expansion for R UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ t ( x ))d m ( x ) = 0 with condition T = SS +1 simplifies to ∞ X n =0 ( A n ( S + 1) + 8 B n )( 2 S S + 1 ) n = 0 . Denote W = S S +1 where 0 < W < . Then the above is equivalent to ∞ X n =0 ( A n ∞ X k =0 12 ( k + 1)( k + 2) W k + 8 B n )2 n W n = 0 . This give relations2 n +3 B n + n X k =0 ( n − k + 1)( n − k + 2)2 k − A k = 0 , n ≥ n = 0, combining with equation (6.10), we obtain A = B = 0. Then equation(6.10) with right hand side zero yields A n + B n = 0 for all n ∈ N . This fact combiningwith the above formula gives A n = B n = 0 and equation (6.1) holds for t, s ≥ t < s < 0, there are three cases we need to discuss. • If t ≤ s and t < 0, then as m is Φ t -invariant, Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x )= Z UX Re q α (Φ − t ( x )) Re q α ( x ) Re q β (Φ s − t ( x ))d m ( x ) . This is the same as s, t ≥ • If s < t ≤ 0, then 51 UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x )= Z UX Re q α (Φ − t ( x )) Re q α ( x ) Re q β (Φ s − t ( x ))d m ( x )= − Z UX Re q α (Φ − t ( x )) Re q α ( x ) Re q β (Φ t − s ( − x ))d m ( x ) = 0 . This is from the observation that the analytic expansion of Re q β (Φ r ( − e iθ x )) basedat x for r > q β (Φ r ( − e iθ x )) = Re q β (Φ r ( e i ( θ + π ) x )) = ˜ q β,x ( Re i ( θ + π ) ) = Re ∞ X n =0 b n ( x ) R n (1 − R ) e i ( n +3)( θ + π ) ! . and that for n ≥ e − i ( n +6) π Z UX Re( a a n ¯ b n +3 )d m = 0 ,e i ( n +3) π Z UX Re( a ¯ a n +3 b n )d m = 0 . • If s < ≤ t , then we consider Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x )= Z UX Re q α (Φ t ( − x )) Re q α ( x ) Re q β (Φ t − s ( − x ))d m ( x ) = 0 . The argument is essentially the same as other cases. This finishes the proof ofequation (6.1).Equation (6.2) follows easily from equation (6.1) because of the facts that for all t, s ∈ R ,Re (cid:18) Z UX Re q α ( x ) q α (Φ t ( x )) q β (Φ s ( x ))d m ( x ) (cid:19) = Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q β (Φ s ( x ))d m ( x ) − Z UX Re q α ( x ) Im q α (Φ t ( x )) Im q β (Φ s ( x ))d m ( x ) . and Z UX Re q α ( x ) q α (Φ t ( x )) q β (Φ s ( x ))d m ( x ) = 0 . (6.12)This is easy to see from the fact that R π Re( a ( x ) e i θ ) e i ( n +3) θ e i ( m +3) θ d θ = 0 for all n, m ≥ t, s > Z UX Re q α ( x ) q α (Φ t ( x )) q β (Φ s ( x ))d m ( x )= 12 π Z π Z UX Re q α ( e iθ x ) q α (Φ t ( e iθ x )) q β (Φ s ( e iθ x ))d m ( x )d θ = 12 π X m,n ≥ Z UX Z π Re( a ( x ) e i θ ) a n ( x ) T n (1 − T ) e i ( n +3) θ ! b m ( x ) S m +3 (1 − S ) e i ( m +3) θ ! d θ d m ( x )=0 52he argument for t ≤ s ≤ t > s > − Φ − t ( − x ) = Φ t ( x ) and − e iθ x = e i ( θ + π ) x . We conclude equation (6.12)holds for all t, s ∈ R and thus equation (6.2) holds.With these preliminaries accomplished, we can now prove Proposition 6.1. Proof of Proposition 6.1. We start to show I = II = 0.I = 0 reduces to equation (6.1) of Lemma (6 . 2) if we take r → ∞ for the following1 r Z UX (cid:18)Z r Re q α (Φ t ( x ))d t (cid:19) Z r Re q β (Φ t ( x ))d t d m = 1 r Z r Z r Z r Z UX Re q α (Φ t ( x )) Re q α (Φ s ( x )) Re q β (Φ µ ( x ))d m d µ d t d s Fubini’s theorem= 1 r Z r Z r Z r Z UX Re q α (Φ t − s ( x )) Re q α ( x ) Re q β (Φ µ − s ( x ))d m d µ d t d s since m is Φ t -invariant=0 . We next look into II.II = lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r − ∂ uv h ( ρ (0)) − ∂ uv f ρ (0) (Φ t ( x ))d t d m = − lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r ∂ uv h ( ρ (0))d t d m − lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r T r ( ∂ D A (0) ∂u∂v π (0))(Φ t ( x ))d t d m + lim r →∞ r Z UX Z r Re q α (Φ t ( x )) T r ( ∂ v D A (0) ∂ u π (0))(Φ t ( x ))d t d m . There are three terms here. Since ∂ uv h ( ρ (0)) is a constant, the first term islim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r ∂ uv h ( ρ (0))d t d m = lim r →∞ ∂ uv h ( ρ (0)) Z UX Z r Re q α (Φ t ( x ))d t d m . Z UX Z r Re q α (Φ t ( x ))d t d m = Z r Z UX Re q α (Φ t ( x ))d m d t = Z r Z UX Re q α ( x )d m d t since m is Φ t -invariant= 12 π Z r Z UX Z π Re q α ( e iθ x )d θ d m d t since m is rotationally invariant= r π Z UX Z π Re( a ( x ) e i θ )d θ d m =0 . The second term in II is − lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r Tr ∂ D A (0) ∂u∂v π (0) ! (Φ t ( x ))d t d m = lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r φ uv (Φ t ( x ))d t d m , recalling that φ is a globally well-defined function on X (see formula (5.3)).12 φ uv ( p (Φ t ( x ))) = 12 φ uv ( p (Φ t ( e iθ x ))) . So 1 r Z UX Z r Re q α (Φ t ( x ))d t Z r φ uv (Φ t ( x ))d t d m = 1 r Z UX Z π Z r Re q α (Φ t ( e iθ x ))d t Z r φ uv ( p (Φ t ( e iθ x )))d t d θ d m = 1 r Z UX Z π Z r Re q α (Φ t ( e iθ x ))d t Z r φ uv ( p (Φ t ( x )))d t d θ d m = 1 r Z r Z r Z UX φ uv ( p (Φ t − s ( x ))) Z π Re q α ( e iθ x )d θ d m d s d t. Again by the fact R π Re q α ( e iθ x )d θ = R π Re( a ( x ) e i θ )d θ = 0, we concludelim r →∞ r Z UX Z r Re q α d t Z r Tr ∂ D A (0) ∂u∂v π (0) ! d t d m = 0 . It remains to show lim r →∞ r Z UX Z r Re q α d t Z r Tr (cid:0) ∂ v D A (0) ∂ u π (0) (cid:1) d t d m = 054his islim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r η (Φ t ( x ))d t d m = − lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r Re q α (Φ µ ( x )) Z ∞ e − s Re q β (Φ µ + s ( x ))d s d µ d m + Z UX Z r Re q α (Φ t ( x ))d t Z r Re q α (Φ µ ( x )) Z −∞ e s Re q β (Φ µ + s ( x ))d s d µ d m + Z UX Z r Re q α (Φ t ( x ))d t Z r q α (Φ µ ( x )) Z ∞ e − s Im q β (Φ µ + s ( x ))d s d µ d m + Z UX Z r Re q α (Φ t ( x ))d t Z r q α (Φ µ ( x )) Z −∞ e s Im q β (Φ µ + s ( x ))d s d µ d m (cid:19) . We have estimates for the following tail terms,1 r Z UX Z r Re q α (Φ t ( x ))d t Z r Re q α (Φ µ ( x )) Z ∞ r e − s Re q β (Φ µ + s ( x ))d s d µ d m + 1 r Z UX Z r Re q α (Φ t ( x ))d t Z r Re q α (Φ µ ( x )) Z − r −∞ e s Re q β (Φ µ + s ( x ))d s d µ d m ≤ M r r Z ∞ r e − s d s = 2 M re − r r →∞ −−−→ . The other two tail terms with integrals involving Im q α and Im q β also go to zeros for the samereason. So in factlim r →∞ r Z UX Z r Re q α d t Z r Tr (cid:0) ∂ v D A (0) ∂ u π (0) (cid:1) d t d m = − lim r →∞ r Z UX Z r Re q α (Φ t ( x ))d t Z r Re q α (Φ µ ( x )) Z r e − s Re q β (Φ µ + s ( x ))d s d µ d m + Z UX Z r Re q α (Φ t ( x ))d t Z r Re q α (Φ µ ( x )) Z − r e s Re q β (Φ µ + s ( x ))d s d µ d m + Z UX Z r Re q α (Φ t ( x ))d t Z r q α (Φ µ ( x )) Z r e − s Im q β (Φ µ + s ( x ))d s d µ d m + Z UX Z r Re q α (Φ t ( x ))d t Z r q α (Φ µ ( x )) Z − r e s Im q β (Φ µ + s ( x ))d s d µ d m (cid:19) . Similar to I, the above equals to 0 reduces to equation (6.2). This finishes our proof of Propo-sition (6.1) and so concludes the discussion of the model case ∂ β g αα ( σ ). We will show in this section the proofs of the remaining three cases, i,e. ∂ i g αα ( σ ) = 0, ∂ j g αi ( σ ) =0 and ∂ β g αi ( σ ) = 0. They provide a complete proof of Theorem 1.1. ∂ i g αα ( σ ) In this case, given parameters ( u, v ) ∈ { ( − , } , we obtain a family of (conjugacy classes of)representations { ρ ( u, v ) } in H ( S ) corresponding to { ( vq i , uq α ) } ⊂ H ( X, K ) L H ( X, K )55y the Hitchin parametrization. In particular, we have ∂ u ρ (0 , 0) is identifed with ψ ( q α ) and ∂ v ρ (0 , 0) is identified with ψ ( q i ). The formula for ∂ i g αα ( σ ) is ∂ i g αα ( σ ) = ∂ v h ∂ u ρ (0 , v ) , ∂ u ρ (0 , v ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v =0 = lim r →∞ r (cid:20)Z UX ( Z r ∂ u f Nρ (0) d t ) Z r ∂ v f Nρ (0) d t d m + 2 Z UX Z r ∂ u f Nρ (0) d t Z r ∂ uv f Nρ (0) d t d m (cid:21) . where the first and second variations are(i) ∂ u f Nρ (0) = − ∂ u f ρ (0) ;(ii) ∂ v f Nρ (0) = − ∂ v f ρ (0) ;(iii) ∂ uv f Nρ (0) = − ∂ uv h ( ρ (0)) − ∂ vu f ρ (0) ; We compute first and second variations for the case of ∂ i g αα ( σ ) in this subsection.We have Higgs fields Φ( u, v ) = vq i uq α vq i . Following the steps and methods for our model case ∂ β g αα ( σ ) in section 5, we prove in thissubsection Proposition 7.1. The first variation of reparametrization functions ∂ u f ρ (0) : U X → R and ∂ v f ρ (0) : U X → R for the case ∂ i g αα ( σ ) satisfy ∂ u f ρ (0) ( x ) ∼ − Req α ( x ) ,∂ v f ρ (0) ( x ) ∼ Req i ( x ) . and the second variation of reparametrization function ∂ vu f ρ (0) : U X → R for the case ∂ i g αα ( σ ) satisfies ∂ uv f ρ (0) ( x ) ∼ 12 Re y ( x ) − q α ( x ) (cid:18)Z ∞ Im q i (Φ s ( x )) e − s d s + Z −∞ Im q i (Φ s ( x )) e s d s (cid:19) . where p : U X → X is the projection from the unit tangent bundle U X to our Riemann surface X . Understanding a section of End ( E ) as a linear map on each fiber of E = K L O L K − over a point of X , the element y is the component of the section Y = H − ∂ uv H that takes K to O . As a function on U X , y transforms as y ( e iθ x ) = e − iθ y ( x ) .Proof. The computation of first variations are the same as the model case which is omit-ted here. The computation of the second variation of reparametrization functions ∂ uv f ρ (0) ∼− Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) − Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) is again divided into computation of Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) and computation of Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) . 56 Compute Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) . The major difference between the case ∂ i g αα ( σ ) and ∂ β g αα ( σ ) is the computation of thisterm. As before, our flat connection is D H ( u,v ) = ∇ ¯ ∂ E ,H ( u,v ) + Φ( u, v ) + Φ( u, v ) ∗ H ( u,v ) .For the computation of ∂ u f ρ (0) and ∂ v f ρ (0) , when u = 0 or v = 0, the harmonic metric H ( u, v ) is diagonal and one obtains ∂ u D H (0) = q α q α ∂ v D H (0) = q i q i q i q i However ,when u = 0 , v = 0 both hold, the harmonic metric H ( u, v ) corresponding to ourHiggs field Φ( u, v ) is not diagonal. The computation of ∂ D H (0) ∂u∂v requires an analysis of theHitchin’s equations.We start from a family of Hitchin’s equations F D H ( u,v ) + [Φ( u, v ) , Φ( u, v ) ∗ H ( u,v ) ] = 0 (7.1)We take u, v -derivatives of Hitchin’s equations (7.1) at u, v = 0, ∂ u ∂ v | u,v =0 ( F D H ( u,v ) + [Φ( u, v ) , Φ( u, v ) ∗ H ( u,v ) ]) = 0 . (7.2)We consider taking H − ∂ vu H as a variable. We define Y = H − ∂ vu H = y y y y y y y y y .Y = H − ∂ uv H is a section of End ( E ).We now work with local coordinates and local trivialization. When varing the u, v realparameters, the holomorphic structure of our bundle E does not change. Thus fixinga local holomorphic frame for all { ( u, v ) } , the Chern connection one-form under thisframe compatible with the Hermitian metric H ( u, v ) is A ( u, v ) = H ( u, v ) − ∂H ( u, v ). Thecurvature term in our holomorphic frame is F D H ( u,v ) = dA ( u, v ) + A ( u, v ) ∧ A ( u, v ) = ¯ ∂ ( H ( u, v ) − ∂H ( u, v )) . The section Y ∈ Γ( End ( E )) in a local holomorphic frame has the following properties:(i) Tr( Y ) = 0.(ii) H ( u, v ) ∗ = H ( u, v ). Also because u, v are real parameters, we have ∂ uv H = ∂ uv ( H ∗ ) =( ∂ uv H ) ∗ and Y ∗ = HY H − .we can express ∂ D H (0) ∂u∂v in terms of Y on γ . With respect to the local holomorphic frameintroduced in the model case adapted to Fermi coordinate, we have ∂H = 0 on γ . So ∂ uv ( D H (0) ) | γ = ∂ uv ( H ( u, v ) − ∂H ( u, v ) + Φ( u, v ) + Φ( u, v ) ∗ H ( u,v ) ) | u,v =0 = − Y H − ∂H + H − ∂HY + ∂Y + Φ ∗ H Y − Y Φ ∗ H = ∂Y + Φ ∗ H Y − Y Φ ∗ H . (7.3)57e want to simplify equation (7.2) as a equation about Y and then solve Y from equation(7.2).Before we continue, we first fix some notation. We will denote H = H (0 , , Φ = Φ(0 , ,∂ u H = ∂H ( u, v ) ∂u (cid:12)(cid:12)(cid:12)(cid:12) u,v =0 ,∂ v H = ∂H ( u, v ) ∂v (cid:12)(cid:12)(cid:12)(cid:12) u,v =0 ,∂ uv H = ∂H ( u, v ) ∂u∂v (cid:12)(cid:12)(cid:12)(cid:12) u,v =0 . As a generalization of the classic result of Ahlfors, the first variation of the harmonicmetric vanishes at the Fuchsian point (see [20], Thm.3.5.1). In particular, ∂ u H = ∂ v H = 0 . Taking H − ∂ uv H as a variable, one can verify from equation (7.2) that0 = ¯ ∂∂ ( H − ∂ uv H ) − H − ∂H ∧ ¯ ∂ ( H − ∂ uv H ) − ¯ ∂ ( H − ∂ uv H ) ∧ H − ∂H + ¯ ∂ ( H − ∂H ) H − ∂ uv H − H − ∂ uv H ¯ ∂ ( H − ∂H )+[ ∂ u Φ , ( ∂ v Φ) ∗ H ] + [ ∂ v Φ , ( ∂ u Φ) ∗ H ] + [Φ , [ − H − ∂ uv H, Φ ∗ H ]] (7.4)Equation (7.4) can be simplified by the following observation.¯ ∂ ( H − ∂H ) H − ∂ uv H − H − ∂ uv H ¯ ∂ ( H − ∂H ) + [Φ , [ − H − ∂ uv H, Φ ∗ H ]]=[ H − ∂ uv H, [Φ , Φ ∗ H ]] − [Φ , [ H − ∂ uv H, Φ ∗ H ]] Hitchin equation=[[ H − ∂ uv H, Φ] , Φ ∗ H ] Jacobi IdentityAs Y = H − ∂ uv H , this yields¯ ∂∂Y + [Φ ∗ H , [ Y, Φ]] − H − ∂H ∧ ¯ ∂Y − ¯ ∂Y ∧ H − ∂H = − [ ∂ u Φ , ( ∂ v Φ) ∗ H ] − [ ∂ v Φ , ( ∂ u Φ) ∗ H ] . (7.5)The PDE system (7.5) in local holomorphic frames is equivalent to the following ninescalar equations about y ij .1. ¯ ∂∂y + h ( y − y ) = 0;2. ¯ ∂∂y + h ( y − y + y ) = 0;3. ¯ ∂∂y + h ( y − y ) = 0;4. ¯ ∂∂y + h ( y − y ) + h − ∂h ¯ ∂y = h − q i q α ;5. ¯ ∂∂y + h ( y − y ) + h − ∂h ¯ ∂y = − h − q i q α ;6. ¯ ∂∂y + h ( y − y ) − h − ∂h ¯ ∂y = h − q α q i ;7. ¯ ∂∂y + h ( y − y ) − h − ∂h ¯ ∂y = − h − q α q i ;58. ¯ ∂∂y + 2 h − ∂h ¯ ∂y = 0;9. ¯ ∂∂y + 2 hy − h − ∂h ¯ ∂y = 0 . From porperty (ii) of Y , one can thus verify (4) is equivalent to (6). (5) is equivalent to(7). (8) is equivalent to (9). Thus it suffices to consider the following six equations. – ¯ ∂∂y + h ( y − y ) = 0; – ¯ ∂∂y + h ( y − y + y ) = 0; – ¯ ∂∂y + h ( y − y ) = 0; – ¯ ∂∂y + h ( y − y ) + h − ∂h ¯ ∂y = h − q i q α ; – ¯ ∂∂y + h ( y − y ) + h − ∂h ¯ ∂y = − h − q i q α ; – ¯ ∂∂y + 2 h − ∂h ¯ ∂y = 0 . We first take a look at the first three equations. We deduce from them¯ ∂∂ ( y + y + y ) = 0;¯ ∂∂ ( y − y ) − h ( y − y ) = 0;¯ ∂∂ ( y + y ) + h (2 y − ( y + y )) = 0 . As Y = H − ∂ uv H is a section of End ( E ), the components y ii ∈ Γ( O ) are acturally justfunctions on the surface X for i = 1 , , 3. Recall our notation ∆ σ = ∂ z ∂ ¯ z σ and the fact h = h (0 , 0) = σ , the above equations can be written independent of coordinate chartson our surface as follows, ∆ σ ( y + y + y ) = 0;∆ σ ( y − y ) − y − y ) = 0;∆ σ ( y + y ) + 2(2 y − ( y + y )) = 0 . We have the following observations, – From the first equation, we obtain y + y + y = C where C is a constant. – Since all eigenvalues of ∆ σ should be non-positive, the second equation can hold onlywhen y − y = 0. – The third equation is ∆ σ ( y + y ) − y + y ) = − C . By a maximum principleargument, one gets y + y = C .Thus property (i) of Y gives y = y = y = 0 . We then continue on the other three equations. From them, we deduce¯ ∂∂ ( y + y ) + h − ∂h ¯ ∂ ( y + y ) = 0;¯ ∂∂ ( y − y ) − h ( y − y ) + h − ∂h ¯ ∂ ( y − y ) = 2 h − q i q α ;¯ ∂∂y + 2 h − ∂h ¯ ∂y = 0 . Let w = y + y . We want to compute ∆ h k w k h where the h -norm k . k h is defined as: k s k h = h − i s ¯ s for a section s ∈ Γ( K i ) and i ∈ Z . 59ecause h = h (0 , 0) = σ and σ = e δ ( z ) | dz | is a hyperbolic metric with curvature K ( σ ) = − ∆ σ (log σ ) = − 1. we have h satisfies¯ ∂∂h = ∂h ¯ ∂hh + 12 h . (7.6)Note w ∈ Γ( K − ). The metric h induces a Chern connection ∇ h on K − and in our localholomorphic frames, one has formula: ∇ h, (1 , w = ∂w + h − ∂hw. One recognizes ∇ h, (1 , w is a section of Ω (1 , ( K − ) = Γ( O ). Therefore, (cid:13)(cid:13)(cid:13) ∇ h, (1 , w (cid:13)(cid:13)(cid:13) h = ( ∂w + h − ∂hw )( ∂w + h − ∂hw ) (7.7)Combining equation (7.6) and equation (7.7) gives∆ h k w k h = 4 ¯ ∂∂ ( hw ¯ w ) h = 2 k w k h + 4 k ∂ ¯ w k h + 4 (cid:13)(cid:13)(cid:13) ∇ h, (1 , w (cid:13)(cid:13)(cid:13) h ≥ . This is an inequality independent of coordinates valid on the Riemann surface. By amaximum principle argument, k w k h must be a constant M . If M = 0, then 0 = ∆ h ( M ) ≥ M > M = 0 and y + y = 0.We have similar arguments for ¯ ∂∂y + 2 h − ∂h ¯ ∂y = 0. We begin with computing∆ h k y k h .Since y is a section of Γ( K − ), in lcoal holomorphic frames, the Chern connection ∇ h, induced from h in this case acts as ∇ h, (1 , y = ∂y + h − ∂ ( h ) y .We obtain ∆ h k y k h = 4 ¯ ∂∂ ( h y y ) h = k y k h + 4 (cid:13)(cid:13) ¯ ∂y (cid:13)(cid:13) h + 4 (cid:13)(cid:13)(cid:13) ∇ h, (1 , y (cid:13)(cid:13)(cid:13) h ≥ . Similar to the argument for w , this leads to y = 0.We conclude up to this point that Y = H − ∂ vu H ∈ Γ( End ( E )) in our local frame is ofthe form Y = H − ∂ uv H = hy y − hy − y with ¯ ∂∂y − hy + h − ∂h ¯ ∂y = h − q i q α . With respect to the Fermi coordinate, we have h ( z ) = and ∂ z h = 0 on γ . Also, we know Y ∗ = HY H − , we finally obtain on γ from equation (7.3) ,Tr ∂ D H (0) ∂u∂v π (0) ! ( x ) = Tr ∂ D H (0) ∂u∂v ( x ) π (0) ! = − 12 Re y ( x ) . emark 7.1. We remark here y ( x ) = y ( z ) where x = ˙ γ (0) is the starting point of γ .Recall y is the component of Y ∈ Γ( End ( E )) taking K to O and y ( z ) is y evaluatingat p ( x ) in the trivialization given by the holomorphic frame adapted to the Fermi coordinate z for γ .In particular, if we consider another closed geodesic γ starting from γ ′ (0) = e iθ x withits Fermi coordinate around γ to be w , then y ( e iθ x ) = y ( w ) . We have y ( w ) = y ( z ) dwdz = y ( z ) e iθ .Because of the DPG property of our geodesic flow, we can extend y to be everywheredefined on U X . We conclude as a function on U X , y transfers in the following way: y ( e iθ x ) = e − iθ y ( x ) . This finishes the computation of Tr (cid:16) ∂ D H (0) ∂u∂v π (0) (cid:17) on U X . We now move to Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) which together provides an expression for second variation of reparametrization functions. • Compute Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) . We haveTr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) ;= q α ( ∂ v a (0) e (0) + a (0) ∂ v e (0)) + 4¯ q α ( ∂ v a (0) e (0) + a (0) ∂ v e (0)) . Similar to the model case ∂ β g αα , here ∂ v e (0) = y is the solution of a nonhomogeneousODE system which arises from taking a v -derivative on the system of parallel transportequation (5.10) at v = 0: ∂ t y ( t ) y ( t ) y ( t ) + 01 0 y ( t ) y ( t ) y ( t ) = √ e t q i (Φ t ( x )) − q i (Φ t ( x ))2 q i (Φ t ( x )) . with boundary conditions H ( y (0) , e (0 , y ( l γ ) = e l γ (cid:18)Z l γ q i (Φ s ( x ))d s (cid:19) e (0 , 0) + e l γ y (0) . The boundary conditions are set up based on the same consideration as the case of ∂ β g αα ( σ ). The solution is ∂ v e ( t ) ∂ v e ( t ) ∂ v e ( t ) = √ R t ( e t Re q i + ie s Im q i )d s −√ R t e t Re q i d s √ R t ( e t Re q i − ie s Im q i )d s + √ ( e l γ − − R l γ ie s Im q i d s −√ e l γ − − R l γ ie s Im q i d s . ∂ v e (0) and ∂ v e (0) by this method. It turns out thatTr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) (Φ t ( x ))=2 Im q α (Φ t ( x )) Z t ( e s − t − e t − s ) Im q i (Φ s ( x ))d s +2 Im q α (Φ t ( x )) Z l γ ( e s − t e l γ − − e t − s e − l γ − q i (Φ s ( x ))d s. We therefore obtain, for a closed geodesic γ of length l γ starting from ˙ γ (0) = x ,Tr( ∂ u D H (0) ∂ v π (0))( x )=2 Im q α ( x ) Z l γ ( e s e l γ − − e − s e − l γ − q i (Φ s ( x ))d s. Similar to our model case of g αα,β ( σ ), one can define a function η : W → R , η ( x ) = 2 Im q α ( x ) (cid:18)Z ∞ e − s Im q i (Φ s ( x ))d s + Z −∞ e s Im q i (Φ s ( x ))d s (cid:19) . and we verify that η ( x ) is H¨older such that Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) ( x ) ≡ η ( x ) on U X .We conclude ∂ uv f ρ (0) ( x ) ∼ − ∂ v (Tr (cid:0) ∂ u D H (0) π (0) (cid:1) )( x )= 12 Re y ( x ) − q α ( x ) (cid:18)Z ∞ e − s Im q i (Φ s ( x ))d s + Z −∞ e s Im q i (Φ s ( x ))d s (cid:19) . This finishes the proof of Proposition 7.1. Remark 7.2. Instead of starting from the first variation of reparametrization functions ∂ u f ρ (0) ( x ) ∼− Tr (cid:0) ∂ u D H (0) π (0) (cid:1) )( x ) , we can take the first variation of reparametrization functions to be ∂ v f ρ (0) ( x ) ∼ − Tr (cid:0) ∂ v D H (0) π (0) (cid:1) )( x ) by formula (5.1) and consider: ∂ vu f ρ (0) ( x ) ∼ − ∂ u (Tr (cid:0) ∂ v D H (0) π (0) (cid:1) )( x )= − Tr ∂ D H (0) ∂v∂u π (0) ! ( x ) − Tr (cid:0) ∂ v D H (0) ∂ u π (0) (cid:1) ( x ) . By the same method, we get Tr (cid:0) ∂ v D H (0) ∂ u π (0) (cid:1) (Φ t ( x ))=2 Im q i (Φ t ( x )) Z t ( e s − t − e t − s ) Im q α (Φ s ( x ))d s +2 Im q i (Φ t ( x )) Z l γ ( e s − t e l γ − − e t − s e − l γ − q α (Φ s ( x ))d s. One can verify by Fubini’s theorem, Z l γ Tr (cid:0) ∂ u D H (0) ∂ v π (0) (cid:1) (Φ t ( x ))d t = Z l γ Tr (cid:0) ∂ v D H (0) ∂ u π (0) (cid:1) (Φ t ( x ))d t. This coincides with the fact that ∂ v (Tr (cid:0) ∂ u D H (0) π (0) (cid:1) )( x ) and ∂ u (Tr (cid:0) ∂ v D H (0) π (0) (cid:1) )( x ) shouldbe in the same Livˇsic class by Livˇsic’s Theorem. .1.2 Evaluation on Poincar´e disk With the computation in last section, we have ∂ i g αα ( σ ) = ∂ v h ∂ u ρ (0 , v ) , ∂ u ρ (0 , v ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v =0 = lim r →∞ r (cid:20)Z UX ( Z r Req α d t ) Z r − Req i d t d m − Z UX Z r Req α d t Z r ∂ uv f Nρ (0) d t d m (cid:21) , where ∂ uv f Nρ (0) = − ∂ uv h ( ρ (0)) − ∂ vu f ρ (0) and ∂ uv f ρ (0) ( x ) ∼ 12 Re y ( x ) − q α ( x ) (cid:18)Z ∞ Im q i (Φ s ( x )) e − s d s + Z −∞ Im q i (Φ s ( x )) e s d s (cid:19) . We show in this subsection: Proposition 7.2. For σ ∈ T ( S ) , ∂ i g αα ( σ ) = 0 . The argument for this proposition boils down to the following lemma. Lemma 7.3. We have the following holds for any t, s ∈ R , Z UX Re q i ( x ) Re q α (Φ t ( x )) Re q α (Φ s ( x ))d m ( x ) = 0 . (7.8) Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q i (Φ s ( x ))d m ( x ) = 0 . (7.9) Z UX Re q α ( x ) Im q α (Φ t ( x )) Im q i (Φ s ( x ))d m ( x ) = 0 . (7.10) Proof. The proof of this lemma is basically the same as the proof of Lemma 6.2 except thatflow time s = t tells us nothing in this case. We instead choose flow time to be the followingthree special cases: s = t , s = 2 t and s = 3 t . We recall our analytic expansions for q i and q α are: q i (Φ r ( e iθ x )) = ∞ X n =0 c n ( x ) R n (1 − R ) e i ( n +2) θ ! ,q α (Φ r ( e iθ x )) = ∞ X n =0 a n ( x ) R n (1 − R ) e i ( n +3) θ ! . We have when t, s > Z UX Re q i ( x ) Re q α (Φ t ( x )) Re q α (Φ s ( x ))d m ( x )= 12 π Z π Z UX Re q i ( e iθ x ) Re q α (Φ t ( e iθ x )) Re q α (Φ s ( e iθ x ))d m ( x )d θ = 14 ∞ X n =0 Z UX Re( c a n ¯ a n +2 )d m T n S n (1 − T ) (1 − S ) ( S + T ) . t = s > 0. Then Z UX Re q i ( x ) Re q α (Φ t ( x )) Re q α (Φ t ( x ))d m ( x )= Z UX Re q i (Φ − t ( x )) Re q α ( x ) Re q α ( x )d m ( x )= Z UX Re q i ( − Φ t ( − x )) Re q α ( − x ) Re q α ( − x )d m ( x )= Z UX Re q i (Φ t ( y )) Re q α ( y ) Re q α ( y )d m ( y ) with y = − x The analytic expansions of left and right hand sides of the above equation gives12 ∞ X n =0 Z UX Re( c a n ¯ a n +2 )d m T n (1 − T ) T = 14 Z UX Re( a a ¯ c )d m (1 − T ) T (7.11)We denote C n = R UX Re( c a n ¯ a n +2 )d m and D n = R UX Re( a a n ¯ c n +4 )d m for n ≥ 0. Weproceed to prove C n = 0 for n ≥ T and T yield the following equations C = 0 , C − C = D . On the other hand, if we consider s = 2 t and s = 3 t . They lead to the following twoequations Z UX Re q i ( x ) Re q α (Φ t ( x )) Re q α (Φ t ( x ))d m ( x )= − Z UX Re q i (Φ t ( x )) Re q α ( x ) Re q α (Φ t ( − x ))d m ( y )and Z UX Re q i ( x ) Re q α (Φ t ( x )) Re q α (Φ t ( x ))d m ( x )= − Z UX Re q i (Φ t ( x )) Re q α ( x ) Re q α (Φ t ( − x ))d m ( y ) . When s = 2 t , we have S = TT +1 and S = 2 T + O ( T ). When s = 3 t , we have S = T + T T +1 and S = 3 T + O ( T ).Compare coefficients of T of the analytic expansions of above two equations and use therelations S = 2 T + O ( T ) and S = 3 T + O ( T ) to obtain D = 0. Therefore from equation647.11) we conclude C n = 0 for n ≥ t, s > 0. For s ≤ t ≤ ∂ β g αα ( σ ) case. We omit it here.Equation (7.9) then follows from equation (7.8) by a Φ t -invariance argument of m . Toprove equation (7.10), we just need the following Z UX Re q α ( x ) q i (Φ t ( x )) q i (Φ s ( x ))d m ( x ) = 0 ∀ t, s ∈ R The argument is the same as the argument for Lemma 6.2. This finishes the proof of Lemma7.3. Proof of Proposition 7.2. We begin by showing the following is zero by evaluation the integral on Poincar´e disk.lim r →∞ r Z UX Z r ∂ u f Nρ (0) d t Z r Tr ∂ D H (0) ∂u∂v π (0) ! d t d m = 0 . Recall from the last subsection that y is the solution of ¯ ∂∂y − hy + h − ∂h ¯ ∂y = h − q i q α .Because q i and q α are real analytic and because h = h (0 , 0) = σ is also real analytic, we know y is real analytic by analytic elliptic regularity theory ([12]).As discussed before, the function y on U X transfers as y ( e iθ x ) = e − iθ y ( x ). Similarlyto the model case of g αα,β , we write the real analytic expansion for y in the coordinates givenby Poincar´e disck model based on x , y ,x ( z ) = ∂∂z X n,m ≥ b n,m ( x ) z n ¯ z m . Denote ˜ y ,x ( z ) := Re ( y ,x ( z )( dr )). Recall r ( R ) = log (cid:16) − R R (cid:17) . one has y (Φ r ( e iθ x )) = ˜ y ,x ( Re iθ ) = Re X n,m ≥ b n,m ( x ) R n + m (1 − R ) − e i ( n − m − θ ! . Thus lim r →∞ r Z UX Z r ∂ u f Nρ (0) d t Z r T r ( ∂ D H (0) ∂u∂v π (0))d t d m = lim r →∞ r Z UX Z r Req α (Φ t ( x ))d t Z r Re y (Φ t ( x ))d t d m = lim r →∞ r Z r Z r Z UX Re q α (Φ t ( x )) Re y (Φ s ( x ))d m d t d s = lim r →∞ r Z r Z r Z UX Re q α (Φ t − s ( x )) Re y ( x )d m d t d s. When µ = t − s ≥ Z UX Re q α (Φ µ ( x )) Re y ( x )d m = 12 π Z UX Z π Re q α (Φ µ ( e iθ x )) Re y ( e iθ x )d θ d m . R π Re( e − iθ b , ) Re( a n e i ( n +3) θ )d θ = 0 for ∀ n ≥ µ ≤ q α (Φ − µ ( − x )) = − Re q α (Φ µ ( x )).Therefore we concludelim r →∞ r Z UX Z r ∂ u f Nρ (0) d t Z r Tr ∂ D H (0) ∂u∂v π (0) ! d t d m = 0 . Arguments for other terms in ∂ i g αα ( σ ) to be equal to zero are analogous to the model case of ∂ β g αα ( σ ). They all reduce to Lemma 7.3. We therefore finish the proof of Proposition 7.2. ∂ j g αi ( σ ) The proofs for the case of ∂ j g αi ( σ ) in this subsection and the case of ∂ β g αi ( σ ) in the nextsubsection are basically the same as the cases for ∂ β g αα ( σ ) and ∂ i g αα ( σ ). Although there’s nomore new ingredients in the proofs, we include them here for completeness.For ∂ j g αi ( σ ), we have three parameters { ( u, v, w ) } ∈ { ( − , } . The representations { ρ ( u, v, w ) } in H ( S ) corresponds to { ( vq i + wq j , uq α ) } ⊂ H ( X, K ) L H ( X, K ) by Hitchinparametrization. In particular, we have ∂ u ρ (0 , , 0) is identified with ψ ( q α ) and ∂ v ρ (0 , , 0) isidentified with ψ ( q i ). Also ∂ w ρ (0 , , 0) is identified with ψ ( q j ). The formula for ∂ j g αi ( σ ) is ∂ j g αi ( σ ) = ∂ w h ∂ u ρ (0 , , w ) , ∂ v ρ (0 , , w ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w =0 = lim r →∞ r Z UX Z r ∂ u f Nρ (0) d t Z r ∂ v f Nρ (0) d t Z r ∂ w f Nρ (0) d t d m + Z UX Z r ∂ u f Nρ (0) d t Z r ∂ vw f Nρ (0) d t d m + Z UX Z r ∂ v f Nρ (0) d t Z r ∂ uw f Nρ (0) d t d m ! where the first and second variations are(i) ∂ u f Nρ (0) = − ∂ u f ρ (0) .(ii) ∂ v f Nρ (0) = − ∂ v f ρ (0) .(iii) ∂ uw f Nρ (0) = − ∂ uw h ( ρ (0)) − ∂ uw f ρ (0) .(iv) ∂ vw f Nρ (0) = − ∂ vw h ( ρ (0)) − ∂ vw f ρ (0) . Our Higgs fields in this case areΦ( u, v, w ) = vq i + wq j uq α vq i + wq j ∂ β g αα ( σ ) and ∂ β g αα ( σ ), we have Proposition 7.4. The first variations of reparametrization functions ∂ u f ρ (0) : U X → R , ∂ v f ρ (0) : U X → R and ∂ w f ρ (0) : U X → R for the case ∂ j g αi ( σ ) satisfy ∂ u f ρ (0) ( x ) ∼ − Req α ( x ) ,∂ v f ρ (0) ( x ) ∼ Req i ( x ) ,∂ w f ρ (0) ( x ) ∼ Req j ( x ) . and the second variations of reparametrization functions ∂ uw f ρ (0) : U X → R and ∂ vw f ρ (0) : U X → R satisfy ∂ uw f ρ (0) ∼ 12 Re y ( x ) − q α ( x ) (cid:18)Z ∞ Im q i (Φ s ( x )) e − s d s + Z −∞ Im q i (Φ s ( x )) e s d s (cid:19) ∂ vw f ρ (0) ( x ) ∼ φ vw ( p ( x )) − q i ( x ) (cid:18)Z ∞ Im q j (Φ s ( x )) e − s d s + Z −∞ Im q j (Φ s ( x )) e s d s (cid:19) where p : U X → X and y are defined as before.Proof. The computation of the first variations are the same as the model case. For the secondvariations of reparametrization functions, we have computed ∂ uw f ρ (0) in the ∂ i g αα ( σ ) case: ∂ uw f ρ (0) ∼ 12 Re y ( x ) − q α ( x ) (cid:18)Z ∞ Im q i (Φ s ( x )) e − s d s + Z −∞ Im q i (Φ s ( x )) e s d s (cid:19) The computation of ∂ vw f ρ (0) ∼ Tr (cid:16) ∂ D A (0) ∂w∂v π (0) (cid:17) + Tr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) is divided into compu-tation of Tr (cid:16) ∂ D A (0) ∂w∂v π (0) (cid:17) and computation of Tr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) . • Compute Tr (cid:16) ∂ D H (0) ∂v∂w π (0) (cid:17) we set u = 0, the Higgs field isΦ( v, w ) = vq i + wq j 01 0 vq i + wq j The harmonic metric H ( v, w ) is diagonalizable and the computation of ∂ vw D H (0) is thesame as the model case of ∂ β g αα ( σ ).With respect to the notation defined in the model case of ∂ β g αα ( σ ), one obtains:Tr ∂ D H (0) ∂v∂w π (0) ! ( x ) = − ψ vw ( z ( p ( x ))) = − φ vw ( p ( x ))Where p : U X → X is the projection from the unit tangent bundle to our surface and z is the Fermi coordinate we choose evaluating at the point p ( x ) ∈ X . • Compute Tr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) ∂ v D H (0) and ∂ w π (0) have been computed in ∂ i g αα ( σ ) case. One can checkTr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) (Φ t ( x ))= q i ( ∂ w a (0) e (0) + a (0) ∂ w e (0) + ∂ w a (0) e (0) + a (0) ∂ w e (0))+2¯ q i ( ∂ w a (0) e (0) + a (0) ∂ w e (0) + ∂ w a (0) e (0) + a (0) ∂ w e (0))=2 Im q i (Φ t ( x )) Z t Im q j (Φ s ( x ))( e t − s − e s − t )d s +2 Im q i (Φ t ( x )) Z l γ Im q j (Φ s ( x ))( e t − s e − l γ − − e s − t e l γ − s In particular, Tr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) ( x )=2 Im q i ( x ) Z l γ Im q j (Φ s ( x ))( e − s e − l γ − − e s e l γ − s Similar to the cases of ∂ β g αα ( σ ) and ∂ i g αα ( σ ), one can then define a function η : U X → R η ( x ) = 2 Im q i ( x ) (cid:18)Z ∞ Im q j (Φ s ( x )) e − s d s + Z −∞ Im q j (Φ s ( x )) e s d s (cid:19) and verify that η ( x ) is H¨older such that Tr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) ( x ) ≡ η ( x ) on U X .We finally obtain ∂ vw f ρ (0) ( x ) ∼ − Tr ∂ D H (0) ∂v∂w π (0) ! ( x ) − Tr (cid:0) ∂ v D H (0) ∂ w π (0) (cid:1) ( x )= 12 φ vw ( p ( x )) − q i ( x ) (cid:18)Z ∞ Im q j (Φ s ( x )) e − s d s + Z −∞ Im q j (Φ s ( x )) e s d s (cid:19) We show in this subsection Proposition 7.5. For σ ∈ T ( S ) , ∂ j g αi ( σ ) = 0 . For the same reasoning as before, the proof of the above proposition reduces to the followinglemma. Lemma 7.6. We have the following holds for any t, s ∈ R , Z UX Re q i ( x ) Re q j (Φ t ( x )) Re q α (Φ s ( x ))d m ( x ) = 0 (7.12) Z UX Re q i ( x ) Im q j (Φ t ( x )) Im q α (Φ s ( x ))d m ( x ) = 0 (7.13) Z UX Re q α ( x ) Re q i (Φ t ( x )) Re q j (Φ s ( x ))d m ( x ) = 0 (7.14) Z UX Re q α ( x ) Im q i (Φ t ( x )) Im q j (Φ s ( x ))d m ( x ) = 0 (7.15)68 roof. We just need to show equation (7.12). Equations (7.13), (7.14) and (7.15) follow easilyusing methods we developed in the former cases.We start from a special case of equation (7.12) that q i = q j Z UX Re q i ( x ) Re q i (Φ t ( x )) Re q α (Φ s ( x ))d m ( x ) = 0 ∀ t, s ∈ R (7.16)The proof of this case is an analogy of the case ∂ β g αα ( σ ) because of the following observationsfor flow time s = t and s = t . Z UX Re q i ( x ) Re q i (Φ t ( x )) Re q α (Φ t ( x ))d m ( x )= − Z UX Re q i ( x ) Re q i (Φ t ( x )) Re q α ( x )d m ( x )and Z UX Re q i ( x ) Re q i (Φ t ( x )) Re q α (Φ t ( x ))d m ( x ) = 0For t, s > 0, recall our analytic expansions given in (6.3) and (6.6) lead to Z UX Re q i ( x ) Re q i (Φ t ( x )) Re q α (Φ s ( x ))d m ( x )= 12 π Z π Z UX Re q i ( e iθ x ) Re q i (Φ t ( e iθ x )) Re q α (Φ s ( e iθ x ))d m ( x )d θ = 14 ∞ X n =0 (cid:18) Z UX Re( c c n ¯ a n +1 )d m T n (1 − T ) S n +1 (1 − S ) + Z UX Re( c ¯ c n +3 a n )d m T n +3 (1 − T ) S n (1 − S ) (cid:19) Denoting E n = R UX Re( c c n ¯ a n +1 )d m and F n = R UX Re( c ¯ c n +3 a n )d m . We argue for n ≥ E n = F n = 0 (7.17)The case t = 0 or s = 0 of equation (7.16) is included in the n = 0 case of equation (7.17).When flow time s = t , we have ∞ X n =0 (cid:18) E n T n +1 (1 − T ) + F n T n +3 (1 − T ) (cid:19) = − F T (1 − T ) (7.18)69his implies E = 0 E = − F When flow time s = t , we obtain ∞ X n =0 (cid:18) E n T n (1 − T ) S n +1 (1 − S ) + F n T n +3 (1 − T ) S n (1 − S ) (cid:19) = 0where T = S S .It simplifies to ∞ X n =0 ( E n + 8 F n S ( S + 1) )( 2 S S + 1 ) n = 0Let W = S S , we have ∞ X n =0 ( E n ∞ X k =0 ( k + 1) W k + 8 F n W )(2 W ) n = 0This gives relations E =0 n X k =0 k ( n − k + 1) E k + 2 n +2 F n − =0 n ≥ E = F = 0. Therefore the right hand side of equa-tion(7.18) is zero and we obtain from it E n +1 + F n = 0 for n ≥ 0. Combining it with (7.19)and by an induction argument, one concludes E n = F n = 0. This proves equation (7.17) for s, t ≥ 0. The case s, t < q i = q j , Z UX Re q i ( x ) Re q i (Φ t ( x )) Re q α (Φ s ( x ))d m = 0 Z UX Re q j ( x ) Re q j (Φ t ( x )) Re q α (Φ s ( x ))d m = 0 Z UX Re( q i + q j )( x ) Re( q i + q j )(Φ t ( x )) Re q α (Φ s ( x ))d m = 0Therefore for all t, s ∈ R , Z UX Re q i ( x ) Re q j (Φ t ( x )) Re q α (Φ s ( x ))d m + Z UX Re q j ( x ) Re q i (Φ t ( x )) Re q α (Φ s ( x ))d m = 0(7.20)70ecall the analytic expansion for q j is given in (6.7). Consider t, s > Z UX Re q i ( x ) Re q j (Φ t ( x )) Re q α (Φ s ( x ))d m ( x )= 12 π Z π Z UX Re q i ( e iθ x ) Re q j (Φ t ( e iθ x )) Re q α (Φ s ( e iθ x ))d m ( x )d θ = 14 ∞ X n =0 (cid:18) Z UX Re( c d n ¯ a n +1 )d m T n (1 − T ) S n +1 (1 − S ) + Z UX Re( c ¯ d n +3 a n )d m T n +3 (1 − T ) S n (1 − S ) (cid:19) Denoting G n = R UX Re( c d n ¯ a n +1 )d m and H n = R UX Re( c ¯ d n +3 a n )d m . We want to show G n = H n = 0 for n ≥ m be an integer and m ≥ 2. Let flow time s = mt . Observe we have Z UX Re q i ( x ) Re q j (Φ t ( x )) Re q α (Φ mt ( x ))d m ( x )= Z UX Re q i (Φ − t ( x )) Re q j ( x ) Re q α (Φ ( m − t ( x ))d m ( x )= − Z UX Re q j ( y ) Re q i (Φ t ( y )) Re q α (Φ − ( m − t ( y ))d m ( y ) y = − x and Φ t ( − y ) = − Φ − t ( y )= Z UX Re q i ( y ) Re q j (Φ t ( y )) Re q α (Φ − ( m − t ( y ))d m ( y ) by equation (7.20)= − Z UX Re q j ( x ) Re q i (Φ t ( x )) Re q α (Φ mt ( x ))d m ( x ) exchange the roles of q i and q j = − Z UX Re q i ( x ) Re q j (Φ t ( x )) Re q α (Φ ( m − t ( − x ))d m ( x )When s = mt , we have S = S ( m ) = (1+ T ) m − (1 − T ) m (1+ T ) m +(1 − T ) m = mT + O ( T ). From the analyticexpansion ∞ X n =0 (cid:18) G n T n S ( m ) n +1 (1 − S ( m ) ) + G n e inπ T n S ( m − n +1 (1 − S ( m − ) (cid:19) = − ∞ X n =0 (cid:18) − H n e inπ T n +3 S ( m − n (1 − S ( m − ) + H n T n +3 S ( m ) n (1 − S ( m ) ) (cid:19) The coefficients of T and T and T yield the following respectively G = 0( m − ( m − ) G = 0( m + ( m − ) G = − (2 m − H + (6 m − H The cases m = 2, m = 3 and m = 4 together give H = H = G = 0. By induction, assuming G k = H k − = 0 for 1 ≤ k < n , the coefficient of T n +1 gives( m n +1 + e inπ ( m − n +1 ) G n = ( e i ( n − π ( m − n − − m n − ) H n − 71e conclude G n = H n = 0 for n ≥ m . This finishes the proof ofequation (7.12) for t, s > 0. Equation (7.12) for t ≤ s ≤ ∂ β g αi ( σ ) This is the last case. In this case, the representations { ρ ( u, v, w ) } in H ( S ) corresponds to { ( vq i , uq α + wq β ) } ⊂ H ( X, K ) L H ( X, K ) by Hitchin parametrization. Our metric tensoris ∂ β g αi ( σ ) = ∂ w h ∂ u ρ (0 , , w ) , ∂ v ρ (0 , , w ) i P !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w =0 = lim r →∞ r Z UX Z r ∂ u f Nρ (0) d t Z r ∂ v f Nρ (0) d t Z r ∂ w f Nρ (0) d t d m + Z UX Z r ∂ u f Nρ (0) d t Z r ∂ vw f Nρ (0) d t d m + Z UX Z r ∂ v f Nρ (0) d t Z r ∂ uw f Nρ (0) d t d m ! where the first and second variations are(i) ∂ u f Nρ (0) = − ∂ u f ρ (0) ;(ii) ∂ v f Nρ (0) = − ∂ v f ρ (0) ;(iii) ∂ uw f Nρ (0) = − ∂ uw h ( ρ (0)) − ∂ uw f ρ (0) ;(iv) ∂ vw f Nρ (0) = − ∂ vw h ( ρ (0)) − ∂ vw f ρ (0) . Our Higgs fields in this case areΦ( u, v, w ) = vq i uq α + wq β vq i Proposition 7.7. The first variations of reparametrization functions ∂ u f ρ (0) : U X → R and ∂ v f ρ (0) : U X → R for the case ∂ β g αi ( σ ) satisfy ∂ u f ρ (0) ( x ) ∼ − Req α ( x ) ,∂ v f ρ (0) ( x ) ∼ Req i ( x ) ,∂ w f ρ (0) ( x ) ∼ − Req β ( x ) . nd the second variations of reparametrization functions ∂ uw f ρ (0) : U X → R and ∂ vw f ρ (0) : U X → R satisfy ∂ uw f ρ (0) ( x ) ∼ φ uw ( p ( x ))+ Re q α ( x ) Z ∞ e − s Re q β (Φ s ( x ))d s + Re q α ( x ) Z −∞ e s Re q β (Φ s ( x ))d s +2 Im q α ( x ) Z ∞ e − s Im q β (Φ s ( x ))d s + 2 Im q α ( x ) Z −∞ e s Im q β (Φ s ( x ))d s,∂ vw f ρ (0) ( x ) = ∂ wv f ρ (0) ( x ) ∼ 12 Re y ( x ) − q β ( x ) (cid:18)Z ∞ Im q i (Φ s ( x )) e − s d s + Z −∞ Im q i (Φ s ( x )) e s d s (cid:19) . where p : U X → X and y are defined as before.Proof. All of the computations have been done in the former cases. We show in this subsection Proposition 7.8. For σ ∈ T ( S ) , ∂ β g αi ( σ ) = 0 . For the same reasoning as before, the proof of the above proposition reduces to the followinglemma. Lemma 7.9. We have the following holds for any t, s ∈ R , Z UX Re q α ( x ) Re q β (Φ t ( x )) Re q i (Φ s ( x ))d m ( x ) = 0 (7.21) Z UX Im q α ( x ) Im q β (Φ t ( x )) Re q i (Φ s ( x ))d m ( x ) = 0 (7.22) Z UX Re q α ( x ) Im q β (Φ t ( x )) Im q i (Φ s ( x ))d m ( x ) = 0 (7.23) Proof. We just need to show equation (7.21). Equations (7.22) and (7.23) follow easily similarto formal cases.From the computation of ∂ i g αα ( σ ), we know Z UX Re q α ( x ) Re q α (Φ t ( x )) Re q i (Φ s ( x ))d m = 0 Z UX Re q β ( x ) Re q β (Φ t ( x )) Re q i (Φ s ( x ))d m = 0 Z UX Re( q α + q β )( x ) Re( q α + q β )(Φ t ( x )) Re q i (Φ s ( x ))d m = 0We deduce Z UX Re q α ( x ) Re q β (Φ t ( x )) Re q i (Φ s ( x ))d m + Z UX Re q β ( x ) Re q α (Φ t ( x )) Re q i (Φ s ( x ))d m = 073imilar to ∂ j g αi ( σ ), we consider s = mt for m ∈ N and m ≥ 2. We observe Z UX Re q α ( x ) Re q β (Φ t ( x )) Re q i (Φ mt ( x ))d m = Z UX Re q α ( x ) Re q β (Φ t ( x )) Re q i (Φ ( m − t ( − x ))d m We recall the Poinc´are disk model and our analytic expansion for q α , q β , q i in (6.3), (6.5) and(6.7). For t, s ≥ 0, the analytic expansio Z UX Re q α ( x ) Re q β (Φ t ( x )) Re q i (Φ s ( x ))d m ( x )= 12 π Z π Z UX Re q α ( e iθ x ) Re q β (Φ t ( e iθ x )) Re q i (Φ s ( e iθ x ))d m ( x )d θ = 14 ∞ X n =0 (cid:18) Z UX Re( a b n ¯ c n +4 )d m T n (1 − T ) S n +4 (1 − S ) + Z UX Re( a ¯ b n +2 c n )d m S n (1 − S ) T n +2 (1 − T ) (cid:19) Denoting I n = R UX Re( a b n ¯ c n +4 )d m and J n = R UX Re( a ¯ b n +2 c n )d m for n ≥ I n = J n = 0 . When s = mt , we have S = S ( m ) = (1+ T ) m − (1 − T ) m (1+ T ) m +(1 − T ) m = mT + O ( T ). The analytic expansionsgive ∞ X n =0 (cid:18) I n T n S ( m ) n +4 (1 − S ( m ) ) − I n e inπ T n S ( m − n +4 (1 − S ( m − ) (cid:19) = ∞ X n =0 (cid:18) − J n T n +2 S ( m ) n (1 − S ( m ) ) + J n e inπ T n +2 S ( m − n (1 − S ( m − ) (cid:19) The coefficients of T yield the following respectively( m − ( m − ) I = − (2 m − J + (4 m − J The cases m = 2 and m = 3 and m = 4 gives I = J = J = 0. By induction, assuming I k = J k +1 = 0 for 1 ≤ k < n , the coefficient of T n +4 gives( m n +4 − e inπ ( m − n +4 ) I n = ( e i ( n +1) π ( m − n +1 − m n +1 ) J n +1 We conclude I n = J n = 0 for n ≥ m . This finishes the proof ofequation (7.21) for t, s > 0. Equation (7.21) for t ≤ s ≤ ∂ β g αα ( σ ) = 0, (ii) ∂ i g αα ( σ ) = 0, (iii) ∂ j g αi ( σ ) = 0 and (iv) ∂ β g αi ( σ ) = 0in consecutive sections. This finishes the proof of our Theorem 1.1.74 eferences [1] L. V. Ahlfors. Some remarks on Teichm¨uller’s space of Riemann surfaces. Ann. of Math.(2) , 74:171–191, 1961.[2] D. Baraglia. 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