Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

In this note we want to relate the Weil-Petersson metric on Teichm¨uller space to the boundary correspondence between the actions on the boundary of Fuchsian groups. Consider the space of Riemann metrics g on a compact surface V with negative Euler characteristic. This can be endowed with a number of natural Riemannian metrics. Of particular interest is the Weil-Petersson metric, whose definition was proposed by Weil in 1958 based on earlier work of Petersson, cf. [18]. There is a particularly intuitive equivalent definition of the Weil-Petersson metric using the second derivative of lengths of typical (closed) geodesics due to Thurston and Wolpert [19].


Introduction
In this note we want to relate the Weil-Petersson metric on Teichmüller space to the boundary correspondence between the actions on the boundary of Fuchsian groups. Consider the space of Riemann metrics g on a compact surface V with negative Euler characteristic. This can be endowed with a number of natural Riemannian metrics. Of particular interest is the Weil-Petersson metric, whose definition was proposed by Weil in 1958 based on earlier work of Petersson, cf. [18]. There is a particularly intuitive equivalent definition of the Weil-Petersson metric using the second derivative of lengths of typical (closed) geodesics due to Thurston and Wolpert [19]. Definition 1.1 (Thurston-Wolpert [19]) Let g λ , − < λ < , be a (non-constant) smooth family of Riemann metrics. Then Weil-Petersson metric is given by where the summations are over closed geodesics γ whose lengths l g 0 (γ ) with respect to the metric g 0 are less than or equal to T .
We can associate to the metrics g λ a family of Fuchsian groups λ acting on the unit disk D such that (V, g λ ) is isometric to D/ λ . Moreover, there exists a family of conjugating homeomorphisms π λ : ∂D → ∂D on the boundary satisfying γ λ • π λ = π λ • γ 0 for corresponding elements γ 0 ∈ 0 and γ λ ∈ λ . In [10], McMullen gave an interesting interpretation of · WP in terms of the Hausdorff dimension dim H (π λ (l)) of the image π λ (l) of Lebesgue measure l on ∂D.

be a (non-constant) smooth family of Riemann metrics. Then Weil-Petersson metric is given by
where g ≥ 2 is the genus of M.
We will complement this result by giving an alternative definition which depends on the dimension of sets, rather than the dimension of measures. We first recall the well known "rigidity" result that if the boundary map π λ is not linear fractional, then it must be singular [17] (i.e. there exists a set E ⊂ ∂D such that l(E) = 0 and l(∂D\π λ (E)) = 0). In fact, there is even the following stronger result of Bishop and Steger (cf. [2], Theorem 3, and [3]).
Clearly, the boundary map π λ : ∂D → ∂D is differentiable almost everywhere, with zero derivative, by monotonicity of the map. In the remaining set of zero measure, we denote by N λ the set of points of non-differentiability, then these have dimension satisfying 0 Our main result is the following.

be a (non-constant) smooth family of Riemann metrics. Then Weil-Petersson metric is given by
where g ≥ 2 is the genus of M.
Our approach also makes a connection with the Manhattan curve of Burger [6] and the work of Schwartz and Sharp [14] on comparing lengths of closed geodesics with respect to different metrics.

The Manhattan curve and the correlation number
Let g λ , − < λ < , be a (non-constant) smooth family of Riemann metrics on the surface V . Given λ > 0, say, we recall the definition of the Manhattan curve of Burger for the two metrics g 0 and g λ . Given a free homotopy class γ we let l 0 (γ ) and l λ (γ ) denote the lengths of the associated closed geodesics for each of these metrics.

Definition 2.1 We can consider the convex set
The boundary is called the Manhattan curve for the metrics g 0 and g λ .
We can parameterize the Manhatten curve by a curve χ λ : R → R, i.e., Fig. 1 The Manhattan curve restricted to 0 ≤ t ≤ 1. It can be used to define D λ = χ λ (t λ ) + t λ where t = t λ is where the perpendicular to the curve χ λ has slope 1 Remark 2. 2 We could equally well define the Manhattan curve in terms of two variable Poincaré series or zeta function associated to the pair of metrics.
Some simple and well known properties of the curve are summarised in the next lemma (cf. [6], [16]). (The final statement follows from the fact that, for any Riemann metric g, lim T →+∞ T −1 log Card{γ : l g (γ ) ≤ T } = 1.)
If g 0 = g λ then we see by Lemma 2.3 that we can choose a unique value 0 < t λ < 1 such that ∂χ λ ∂t t=t λ = −1 (Fig. 1). The connection with the quantity in Theorem 1.4 comes from the following identification.
Recall that D λ denotes the Hausdorff dimension of the set where the boundary map π λ : ∂D → ∂D which conjugates the actions of 0 and λ is not differentiable.

Proposition 2.4 We have that D
Proof This follows from a result of Jordan et al. [8]. The actions of 0 and λ on ∂D may be modelled by expanding Markov maps T 0 : ∂D → ∂D and T λ : ∂D → ∂D, respectively [1], [15]. These maps are conjugated by π λ : ∂D → ∂D, which is Hölder continuous. Apart from a finite number of exceptions, there is a natural one-to-one correspondence between the periodic orbits of T 0 and closed geodesics on (V, g 0 ) (and hence the free homotopy classes on V ). Furthermore, if we define Hölder continuous functions f 0 : ∂D → R and f λ : ∂D → R by .) It now follows from standard thermodynamic formalism that χ λ is defined implicitly by where P is the pressure function associated to the transformation T 0 . Noting that no nontrivial linear combination a f 0 + b f λ has the form u • T 0 − u for u : ∂D → R continuous (which follows from the Independence Lemma in [14], for example), we can apply Theorem 1.1 of [8] to assert that D λ = χ λ (t λ ) + t λ , as required. (Actually, to carry out the analysis in [8], it is convenient to introduce and work with a subshift of finite type which models T 0 . We discuss related symbolic dynamics further in the next section.) There is an interesting connection between the value D λ and the relative lengths of corresponding closed geodesics on V which gives another interpretation to D λ . Let γ denote a free homotopy class on V and denote by l 0 (γ ) and l λ (γ ) the lengths of the corresponding geodesics with respect to the metrics g 0 and g λ . Given δ > 0 and T > 0 we let Schwartz and Sharp [14] showed that λ (T, δ) has a well-defined exponential growth rate independent of the choice of δ. Subsequently, Sharp [16] showed that this growth rate, called the correlation number, is equal to χ λ (t λ ) + t λ and hence it is equal to the dimension D λ Combing these observations, the main result of Schwartz and Sharp [14] takes the following form.

Proposition 2.5
For any δ > 0, we have that Moreover, there exists a constant C = C(δ) > 0 such that

Thermodynamic formalism and geodesic flows
In this section we shall introduce the technical machinery which will be used in the sequel to prove Theorem 1.4. Given a Riemann metric · g on V , we can model the geodesic flow φ with respect to the norm · g , by a suspension flow over a subshift of finite type. Furthermore, for a family of metrics g λ , − < λ < , by structural stability we can use the same subshift for each flow, only varying the suspension function.
We begin by defining subshifts of finite type and suspension flows. Let A be a k × k matrix with entries 0 and 1. We shall always assume that A is aperiodic (i.e. that A n has positive entries for some n ≥ 0). Let where x = (x n ) n∈Z and y = (y n ) n∈Z . The subshift of finite type σ : X A → X A is defined by (σ x) n = x n+1 , for n ∈ Z, and is a homeomorphism with respect to the given metric. Given a strictly positive Hölder continuous function r : (σ x, 0). We define the suspension flow σ r t : X r A → X r A by σ r t (x, u) = (x, u + t), subject to the identification. Associated to the shift σ and the flow σ r t is a body of definitions and results known as thermodynamic formalism. We now introduce what we will need in the sequel. If f : X A → R is a Hölder continuous function then we can define the pressure P( f ) by where h(μ) is the entropy of σ with respect to the measure μ. There is a unique σ -invariant probability measure realizing the above supremum, denoted by μ f and called the Gibbs measure for f . Fixing the Hölder exponent, we recall some results on the pressure functional P : C α (X A , R) → R on the Banach space of α-Hölder continuous functions.
where D g denotes the directional derivative We now consider the corresponding statements for flows. For a Hölder continuous function F : X r A → R, we define the pressure P(F) by P(F) = sup h(m) + F dm : m is a σ r -invariant probability measure .
In particular, there is a unique σ r -probability measure realizing the above supremum, denoted by m F and called the Gibbs measure for F. The connection between the pressure and Gibbs measure for F and f is given by the following. [5], see also [11], Proposition 6.1) Given a Hölder continuous function F : X r A → R we can associate a Hölder continuous function f :

Lemma 3.2 (Bowen-Ruelle
We may also define pressure and Gibbs states with respect to a geodesic flow φ (g) t : T 1 V → T 1 V for Hölder continuous functions F : T 1 V → R in an analogous way.
The connection between the geodesic flow and the suspension flow is given by the following.

Lemma 3.3 We can associate to a smoothly varying family of Riemann metrics g λ , − < λ < , a subshift of finite type σ : X A → X A and smoothly varying families of roof functions r λ and a family of Hölder continuous surjections p
• p λ , ∀t ∈ R; 2. p λ is bounded-to-one; and 3. p λ is a measure-theoretic isomorphism with respect to the Gibbs measures for F • p λ and F, for any Hölder continuous function F : Proof For each fixed value of λ, this is a classical result due to Ratner [12] and Bowen [4]. The behaviour as λ varies follows from the structural stability of the geodesic flow [7].

Example 3.4
If we let f = −r 0 then dm 0 = dμ −r 0 × dt/ r 0 dμ −r 0 is the measure of maximal entropy for the suspension flow. In particular, p −1 0 m 0 is the measure of maximal entropy for φ Moreover, since the metric g 0 has constant curvature this is the same as the normalized Liouville measure (the product of the g 0 -area on V and Lebesgue measure on each fibre).
Given a smooth family g λ , − < λ < , of Riemann metrics on V , we can write (1) lies in the tangent space for Teichmüller space) and r λ = r 0 + λr (1) The following gives a simple, slightly technical, but useful, symbolic characterization of the tangent space to the Teichmüller space of V and of the Weil-Petersson metric itself. = σ 2 (r (1) ) where g ≥ 2 is the genus of the surface.

Differentiability of the dimension function D λ
In this section we prove Theorem 1.4 (restated below as Theorem 4.3). We begin by characterising the Manhattan curve in terms of the pressure function defined in the preceding section. Moreover, χ λ has an analytic dependence on λ ∈ (− , ).
Proof The first part is due Lalley [9]. The second part follows easily by the Implicit Function Theorem.

Lemma 4.2 We have that
for all 0 ≤ t ≤ 1.
Proof Using the analytic dependence of r λ and χ λ (t) on λ ∈ (− , ) we can expand and for each 0 < t < 1. Substituting these expansions into P(−tr 0 − χ λ (t)r λ ) = 0 and using the second order expansion of the pressure function gives r 0 + χ 0 (t)r (1) − χ 0 (t) r (2) for all 0 ≤ t ≤ 1. From the λ term we see that, since r 0 dμ −r 0 > 0, In particular, differentiating (4.1) with respect to t now gives We can next consider the λ 2 term and use (4.1) to get which can be rearranged to give for every 0 ≤ t ≤ 1, using Proposition 3.5.
Finally we can prove our main in result, Theorem 1.4, which we now restate.

Theorem 4.3 We have that
.
Also, using (4.3), we have that so that ∂t λ /∂λ remains bounded as λ → 0. Combining these observations, we conclude from (4.5) that To complete the proof of Theorem 1.4 we need to show that lim λ→0 t λ exists and is equal to 1/2. Since ∂t λ /∂λ is bounded as λ → 0, we have for some M ≥ 0. Hence, t λ is a uniformly continuous function of λ, so that t 0 := lim λ→0 t λ exists, and, furthermore, t λ = t 0 + O(λ). For λ sufficiently close to 0 we can approximate χ λ (t) by using (5.2). Differentiating and evaluating at t = t λ gives the equation Cancelling the constant term and dividing by λ 2 , we see that t 0 = 1/2, as required. The result then follows by Proposition 2.4. In particular, when F = 0 we see that ζ g,0 (s) = Z g (s + 1)/Z g (s) where is the Selberg zeta function which converges for Re(s) > 1 and has a simple zero at s = 1.
Let us now consider the choice F : T 1 V → R given by F(v) = λg (1) (v, v) then we have a characterization of the Weil-Petersson metric in terms of the second derivative of the residue of Z g λ (s): (s − 1) ∂ 2 log ζ g 0 ,λg (1) The formulation of the · 2 WP in terms of the Selberg zeta function, combined with Ruelle's Grothendieck determinant approach to Z g (s) [13] leads, in principle, to a very efficient method for numerically computing the Weil-Petersson metric.
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