Second variation of Selberg zeta functions and curvature asymptotics

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichm\"uller space. We then use this formula to determine the asymptotic behavior as $s \to \infty$ of the second variation. As a consequence, we determine the signature of the Hessian of $\log Z(s)$ for sufficiently large $s$. As a further consequence, the asymptotic behavior of the second variation of $\log Z(s)$ shows that the Ricci curvature of the Hodge bundle $H^0(\mathcal K^m_t)\mapsto t$ over Teichm\"uller space agrees with the Quillen curvature up to a term of exponential decay, $O(s^2 e^{-l_0 s}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.


Introduction
Selberg was one of many mathematicians for whom investigating the Riemann hypothesis would lead to deep results of broad interest, not only in analytic number theory but also in many other neighboring fields. To wit, Selberg's trace formula was one of the main inspirations of the Langlands program. Although the Selberg trace formula has been applied in a multitude of settings, the Selberg zeta function is nonetheless shrouded with a certain mystique, because it is defined in terms of quantities which are in general incomputable, namely the set of lengths of closed geodesics on a Riemannian manifold, 1 − e −l(γ)·(s+k) .
Here, the geometric setting is a compact Riemann surface, X, of genus g ≥ 2, equipped with the hyperbolic metric of curvature −1. Let Γ be the fundamental group of X. We may then fix X as the quotient of the upper half plane, H = {z = x + iy, y > 0}, by Γ, so that X = Γ\H. Then Prim(Γ) in (1.1) is the set of conjugacy classes of primitive hyperbolic elements γ of Γ, which is in canonical bijection with the set of primitive closed geodesics, and ℓ(γ) is the geodesic length of the associated conjugacy class of γ. It is clear from (1.1) that the Selberg zeta function is intimately linked to the Riemannian geometry of X. What is perhaps not so obvious is that it is also closely connected to the complex structure of X. To describe this, we fix S, the so-called model surface of genus g ≥ 2. The Teichmüller space T = T g of surfaces of genus g is the set of equivalence classes [(Σ, ϕ)], where Σ is a Riemann surface, and ϕ : S → Σ is a diffeomorphism, known as a marking. On each such surface, Σ, there is a unique Riemannian metric which has constant curvature −1, however in this notation the Riemannian metric is suppressed. The equivalence relation identifies if there is an isometry I : Σ 1 → Σ 2 such that I and ϕ 2 • ϕ −1 1 are isotopic. Hence, from the Riemannian geometric perspective, these two surfaces are identical, in that they are topologically the same, and they are equipped with the same Riemannian metric.
One may also consider T from a complex analytic perspective, in the spirit of Ahlfors and Bers [2]. For this purpose, we recall that a Beltrami differential, µ, is a Γ invariant ∂ z ⊗ dz tensor on H, thus we write µ = µ(z)∂ z ⊗ dz. It is harmonic if µ(z) = φ(z)y 2 , and φ = φ(z)dz 2 is a Γ invariant holomorphic quadratic differential. A deep result due to Ahlfors and Bers [2] states that tangent vectors in T t for a point t = [(X, ϕ)] ∈ T are represented by harmonic Beltrami differentials. To see this, for a harmonic Beltrami differential µ, let f µ be the solution of the Beltrami equation where L is the lower half plane in C. For a fixed Beltrami differential µ of (supremum) norm 1, let ǫ be a small complex number. Consider the Beltrami equation for εµ with solution f εµ . Then, for sufficiently small ε, f εµ defines a Fuchsian group Γ ε = f εµ Γ(f εµ ) −1 . The Riemann surfaces X ε = Γ ε \H define a curve in T with X 0 = X. Hence, we identify unit tangent vectors in T t with harmonic Beltrami differentials of unit norm. Each of these in turn defines a local one parameter family of Riemann surfaces, X ε . In this way we compute the variation of quantities defined on the surface X corresponding to the point t ∈ T in the directions corresponding to these harmonic Beltrami differentials of unit norm. The stage is now set to present our main results.
1.1. Main results. Our first main result generalizes [14,Theorem 1.1.2] in which Gon computes a formula for the first variation of the log of the Selberg zeta function; this may be compared with our Proposition 4. The variation is computed, in both our setting and that of Gon, by viewing the Selberg zeta function as a function on Teichmüller space, T , defined for t ∈ T as the Selberg zeta function on the corresponding Riemann surface equipped with the hyperbolic Riemannian metric of constant curvature −1. Above z γ (s) andz γ (s) are as in (3.15) and (3.17), respectively.
To avoid cumbersome notation, we have not included the explicit formulas for the first and second variations of the lengths of closed geodesics in the second variational formula above. These are contained in §3.1, Propositions 2 and 3, respectively.
Using a result of Wolpert, [35], we are able to show that the kernel of the linear map has (complex) dimension precisely equal to 3g − 4, for each γ. For each t ∈ T corresponding to the Riemann surface X = X t , we define the subspace Thus, we have the orthogonal decomposition with respect to the Weil-Petersson metric, We have then Corollary 1. The Hessian∂∂ log Z(s) is asymptotically diagonalized under the decomposition More precisely, for The remainder, In particular∂ µ ∂ µ log Z(s) is non-degenerate and of signature As a second consequence, we prove that the Ricci curvature Ric (m) (µ, µ) of the Hodge bundle H 0 (K m t ) → t ∈ T over the Teichmüller space, T , agrees with the Quillen curvature up to a term of exponential decay. In particular we obtain the full expansion of the Ricci curvature in m.
Corollary 2. The Ricci curvature, Ric (m) (µ, µ) of the Hodge bundle H 0 (K m t ) → t ∈ T over the Teichmüller space, T , has the following expansion, Here, µ 2 W P is the square of the Weil-Petersson norm of µ. The remainder The corollary shows that the Ricci and Quillen curvatures agree up to an exponentially small remainder term. This improves, in the case of Riemann surfaces, the result of Ma-Zhang [21] where the first two terms are found. In this sense, our result can be seen as a variational version of the result of Bismut-Vasserot [9] on the asymptotics of analytic torsion. It is also closely related to the curvature of the Quillen metric on Teichmüller space which has been studied in the general context of holomorphic families of Kähler manifolds [8].

Related works.
To the best of our knowledge, the first result on plurisubharmonicity of naturally defined functions on Teichmüller space appeared in [37]. There, Wolpert showed that the logarithm of a certain finite sum of geodesic lengths is plurisubharmonic as a function on Teichmüller space. He has built upon and generalized those results in [38,32]. More recently, Axelsson & Schumacher [4], using Kähler geometric methods, established the plurisubharmonicity of each geodesic length function. They obtained this result as a corollary to formulas they demonstrated for the first and the second variations of the geodesic length as a function on Teichmüller space.
Closely related to our work is that of Gon [14]. That context is more general because the underlying model surface in the definition of Teichmüller space is of type (g, n), that is genus g and punctured at n points. Our work is only for (g, 0), so that our surfaces are not punctured. The main result of [14] expresses the variation ∂ µ log Z Γ (s) as a sum over conjugacy classes of primitive hyperbolic elements of certain quantities depending on local higher zeta functions and periods of the automorphic forms over the closed geodesic. The formula was obtained with the help of Takhtajan and Zograf's integral expression for the first variation of the Selberg zeta formula, Above, F s is a certain Poincaré series constructed from a second derivative of the resolvent kernel Q s (z, z ′ ) of the Laplacian on the surface. More precisely, where e denotes the identity element. Using the explicit formulas for Q s (z, γz ′ ), Gon was able to express ∂ µ log Z Γ (s) via the sum of local Selberg zeta functions [14, Theorem 1.
Our method is different. Instead of exploiting the integral formula of Zograf and Takhtajan, we show that it is possible to differentiate the definition of the Selberg zeta function, directly, as long as it converges. This serves our purposes well, because we are interested in arguments for which the product (1.1) converges. Next, we use the result of Axelsson and Schumacher [4] for the variation of the length of a geodesic. Interestingly, although our method is different, we arrive at the same formula as Gon. Moreover, our approach gives a geometric interpretation for the automorphic forms appearing in Gon's formula: the automorphic forms correspond to the variations of the lengths of the geodesics. Furthermore, with our method we are able to calculate the second variation of the Selberg zeta function as a sum over primitive closed geodesics of the surface.
Consequently, this method is applicable not only for studying the Selberg zeta function, but also to study all other functions that are defined in an analogous way. In particular, our techniques apply equally well to functions which are defined as a sum or product, over primitive closed geodesics, of quantities depending on the lengths of closed geodesics. To illustrate the utility of our method, we compute in §4.2 variational results for: the trace of the squared resolvent, the Ruelle zeta function, the zeta-regularized determinant of the Laplacian, and the hierarchy of higher Selberg zeta functions. For the definition of these zeta functions, see (2.14). These form a real-parameter family Z(s, t) of zeta functions which generalize the notion of the Selberg zeta function. Interestingly, the local version of these higher Selberg zeta functions also appear in Gon's formula. Here, we prove directly using their definitions a variational formula for the whole hierarchy which relates the variation of Z(s, t) to that of Z(s, t ′ ) for t ′ < t.
We would also like to mention related results obtained by Fay [12], who considered Selberg zeta functions twisted by a representation of Γ. It may be possible to generalize our results to that setting as well, if the key elements in the proofs are amenable to suitable adaptations.
1.3. Key elements in the proofs. Initially, we prove the first and second variational formulas, Proposition 4 and Theorem 1, respectively, by differentiating the definition of the logarithm of the Selberg zeta function and verifying convergence. As we proceed directly using the sum over closed geodesics, we must compute the first and second variation of the length of each closed geodesic. To compute the first variation, although this would follow from [4], we compute in a more classical and direct way using Gardiner's formula, obtaining an equivalent but superficially different formula. However, unlike the first variational formula of [4], one can read-off the terms in Gon's formula directly from our Proposition 2.
The asymptotics of the second variation of log Z(s) are obtained by locating the dominant term in our formula as s → ∞. When s = m ≥ 0 is an integer, the variation is the difference of the Ricci-curvature of the Hodge bundle H 0 (K m ) over the Teichmüller space and the Quillen curvature. The Quillen curvature is wellknown and is given by a second degree polynomial in m. We therefore obtain the full expansion of the Ricci-curvature. Apart from the case of abelian varieties this seems the first case where a full expansion of the Ricci-curvature has been obtained.
1.4. Further developments. In a subsequent paper [13] we shall demonstrate an integral formula for the second variation which holds for ℜ(s) > 1, in the spirit of the integral formula of Takhtajan & Zograf, [40, Teorema 2] for s = m ∈ N. We shall use this formula to define the Ricci curvature, Ric (m) (µ, µ), for non-integer m. The Teichmüller space and the Hodge bundle H 0 (K m ) can be formulated in the general setup of relative ample line bundles for fibrations of Kähler manifolds [5]. In a recent preprint [30] we have been able to prove a generalization of (3) in this general setup.
1.5. Organization. In the next section we gather definitions and notations and demonstrate the requisite preliminaries. In §3 we prove the first variational formula as well as estimates and a convergence result which will be used to justify convergence of the second variational formulas demonstrated in §4. The asymptotics of the second variation for large s are computed in §5, which are then used to compute the asymptotics of the Ricci curvature. We conclude with an investigation of the special cases m = 1, 2. Finally, in the appendix we provide a calculation of the Hilbert Schmidt norm of the squared resolvent. Although the formula is known, our particular method of calculation is not contained in the literature to the best of our knowledge and therefore may be interesting or useful. This calculation is used to compute the variation of the Hilbert-Schmidt norm in §4.
1.6. Acknowledgements. We are very grateful to Bo Berndtsson for several inspiring discussions and to Steve Zelditch for clarifying some concepts in Teichmüller theory. We thank also Werner Müller for drawing our attention to the reference [12]. We appreciate stimulating discussions with Dennis Eriksson and a careful reading of the paper by Magnus Goffeng and Xueyan Wan.

Preliminaries
We take this opportunity to fix notations and prepare the technical tools required for our proofs.
2.1. Hodge and∂ Laplacians on L 2 m,l (X). We briefly recall a few known results for certain Laplace operators on Riemann surfaces which shall be important ingredients in the proofs of our results. Each point in Teichmüller space T = T g is canonically identified with a compact Riemann surface, X of genus g ≥ 2, which admits a unique Riemannian metric of constant curvature −1. Then, X is identified with the quotient Γ\H, where Γ is the fundamental group of X, and H is the upper half plane in C. The hyperbolic metric in Euclidean coordinates on the upper half plane is given by Let ∆ 0 = −y 2 (∂ 2 x + ∂ 2 y ) be the Laplace operator on scalar functions. We note that For this reason, there are different normalizations of the Laplacian by different authors. Particularly relevant to our work is the definition in [28], who define the Laplacian as Let K be the holomorphic cotangent bundle andK the anti-holomorphic cotangent bundle over X. We also use the standard notation that K −k = (K k ) * is the dual bundle of K k . The scalar product on sections of K k ⊗K l is given by The integration above is with respect to the Euclidean measure, dA = dxdy, and is taken over a fundamental domain of X. In the case of functions, we note that k = l = 0, and we may simply write the integral of a function ϕ on X as X ϕ, suppressing the ρdxdy. Let L 2 k,l (X) be the corresponding Hilbert space. We write L 2 and ′′ =∂ * ∂ be the corresponding Laplace operators. The Chern connection ∇ = ∇ ′ +∂ can be extended to sections of K m ⊗K as (0, 1)-forms with values in K m . Note that ′′ =∂ * ∂ +∂∂ * =∂∂ * on (0, 1)-forms.
Proof. The Chern connection ∇ ′ (also called Maass operator) is given by .
. This completes the proof. 2.2. Holomorphic analytic torsion. Holomorphic analytic torsion, or∂-torsion, as Ray & Singer originally introduced in their pioneering work [22], is a complex analogue of analytic torsion. To define it, let D p,q be the set of C ∞ complex (p, q)forms on X. Then the exterior differential d splits We note that defined in this way, as in [22], the eigenvalues of this operator are non-positive. Thus, they defined the associated spectral zeta function, This zeta function may also be expressed in terms of the Mellin transform of the heat trace. In this way, using the short time asymptotic expansion of the heat trace, one can prove that ζ Dp,q extends to a meromorphic function in C which is regular at s = 0. One may therefore make the following Definition 1. For each p = 0, . . . , N , where N is the complex dimension of X, the holomorphic analytic torsion, T p (X), is defined by In our case, N = 1, and so there are two holomorphic analytic torsions, It is well known that the non-zero eigenvalues of D 0,1 coincide with those of D 0,0 = ∆ 0 as do those of D 1,1 [7]. We therefore have 2.3. The curvature of vector bundles on Teichmüller space. For each point t ∈ T = T g , the Teichmüller space of marked surfaces of genus g, we denote the corresponding Riemann surface as X t . When it is clear from context, we may simply write X. The holomorphic tangent vectors at each t ∈ T , µ ∈ T In this way, we have an anti-complex linear identification between T (1,0) t (T ) with the holomorphic quadratic differential H 0 (X t , K 2 ), namely the dual of T Let m ≥ 1. The map H 0 (X t , K m ) → t ∈ T can be used to define a holomorphic Hermitian vector bundle over the Teichmüller space T . More precisely, p : X → T is a smooth proper holomorphic fibration of complex manifolds of (complex) dimensions 3g − 2 and 3g − 3, respectively, such that for each point t ∈ T , the fiber over t is the surface, X t . In the language of [6], Y = T , and the relative (complex) dimension, n = 1. We then consider the holomorphic line bundle L → X such that Xt . It is well known that L → X is equipped with a smooth metric of positive curvature; the positivity follows from [36,Lemma 5.8]. Moreover, Wolpert also showed in that work that X is equipped with a Kähler metric.
The direct image sheaf of the relative canonical bundle twisted with L, is then associated to the vector bundle, E, over T , with fibers The Kodaira-Spencer map at a point t ∈ T is a map from the holomorphic tangent space to the first Dolbeault cohomology group, H 0,1 (X t , T 1,0 (X t )) of X t with values in the holomorphic tangent space of X t . The image of this map is known as the Kodaira-Spencer class, K t . This class has a natural action on u ∈ E t , which is denoted by K t · u. If k t is a vector-valued (0, 1) form in K t , then In this setting, we note that the harmonic Beltrami differential, µ, is in the Kodaira-Spencer class, K t , so we may take k t = µ above. Then, the action µ·u is well-defined defined via k t · u.
We recall the curvature formula of Berndtsson [5,6] for general relative direct image bundle specified to our case above. Interestingly, it is precisely the curvature of this bundle which shall appear in our second variational formula for the logarithm of the Selberg zeta function at integer points. Fix X = X t and denote Similarly for an element u ∈ E t , |u| 2 is also a well-defined pointwise function.
Letting {u j } dm j=1 be an orthonormal basis of H 0 (K m ), the Ricci curvature, is given by where the norm is the same as in (2.8). For m ≥ 2, the Ricci curvature Proof. The results are proved in [5,6] for general setup and we specify them to our case, and we also show how the result in [27] for our case is a consequence of the general results. The fibration is now the Teichmüller curve, denoted X above; X is the natural fiber space over Teichmüller space, T = T g . The fiber, X t , for t ∈ T is the Riemann surface X = X t . The line bundle L = K m−1 whose restriction on each surface X is K m−1 X . The metric, indicated by e −φ in [6] on K is in our case ρ −1 = y 2 on each fiber X. More precisely, on each fiber, X the metric is ρ(z)|dz| 2 , with ρ(z) = y −2 , defined via the identification of X = Γ\H. The harmonic Beltrami differential µ here is a representative of the element k t in the Kodaira-Spencer class as described above. The first formula (2.8) for m = 1 is now an immediate consequence of [6, Theorem 1.1].
To state the curvature formula in [6, Theorem 1.2] for m ≥ 2 we recall that the metric on K m−1 is e −ψ = y 2m−2 , with ψ = (m − 1)φ. The complex gradient V ψ , of ψ with respect to the fixed potential φ (see [6, p. 1213]) can be chosen as V ψ = V φ . The corresponding Kodaira-Spencer class k (m−1)φ t is represented by∂V φ restricted to the fiber space and is thus also k t , so that Here, ξ is the L 2 -minimal solution of the∂ equation, The equation for ξ is now solved by and finally using again Lemma 1. This completes the proof.
Remark 1. The sum j |u| 2 j in I (m) is the Bergman kernel, its expansion [20] for large m can be used to compute the expansion of Ric (m) (µ, µ). However we shall estimate the 2nd variation of the Selberg zeta function and find the expansion of Ric (m) (µ, µ) as a corollary. Indeed the explicit formula for Z(s) contains much detailed geometric data. We note also that in [27] the second term The discrepancy with our formula is due to our definition on the norms on the L 2 -spaces of form (m, 1) and thus our formula for ∇ ′ * ; indeed conceptually it might be better to the an abstract uniform definition for all the forms starting with a fixed Chern form √ −1∂∂φ.
2.4. The Weil-Petersson Metric on Teichmüller space. We introduce the Weil-Petersson metric following [29], and we take the opportunity to recall as so nicely done there the origins of this metric. Petersson introduced an inner product on the spaces of modular forms of arbitrary weight in the context of number theory.
Observing that modular forms of weight two are precisely holomorphic quadratic differentials, André Weil remarked in a letter to Lars Ahlfors that Petersson's inner product should give rise to a Riemannian metric on Teichmüller space. This is indeed the case, and Ahlfors went on to prove [1] that the holomorphic sectional curvature and the Ricci curvature of T with respect to this metric, known as the Weil-Petersson metric, are both negative. However, this metric is not complete, which was demonstrated by Wolpert [33].
At each point t ∈ T , we may identify the holomorphic tangent vectors at t with harmonic Beltrami differentials. The Weil-Petterson metric is defined by This defines a Hermitian metric on T , which taking the real part defines a Riemannian metric on T , but we shall only be interested in the Hermitian Weil-Petersson metric on T .
2.5. Zeta functions and zeta-regularized determinant. Let det(∆ 0 +s(s−1)) be the zeta-regularized determinant of the Laplacian on scalars. This is defined through the spectral zeta function. For ℜ(s) > 1 and ℜ(z) > 1, this spectral zeta function is defined by where {λ k } k∈N are the eigenvalues of ∆ 0 . In the special case s = 1, the sum above is only taken over the non-zero eigenvalues of ∆ 0 . It is well known that the spectral zeta function admits a meromorphic extension to z ∈ C which is regular at z = 0. The determinant is then defined to be exp(−ζ ′ (0)). This determinant is closely related to the Selberg zeta function (1.1). By [24, Theorem 1] the determinant, det(∆ 0 + s(s − 1)), and the Selberg zeta function, Z(s), are related by Above, g is the genus, Γ 2 (s) is the Barnes double gamma function, defined by the canonical product Above, A is the Glaisher-Kinkelin constant. 1 In the special case s = 1, we have We shall also consider the Ruelle zeta function. For ℜ(s) > 1, the Ruelle zeta function is Above, Prim(Γ) denotes the set of conjugacy classes of primitive hyperbolic elements in Γ. The last type of zeta functions which are relevant to our present work are the higher Selberg zeta functions, the first of which was introduced and studied by Kurokawa and Wakayama [19]: More generally, this notion of higher Selberg zeta function was generalized in [15] who defined a whole procession of higher Selberg zeta functions. For t ∈ C and ℜ(s) > 1, let

First variation
In this section we compute the first variation of log Z(s). We start by computing the variation of the length of an individual closed geodesic, ∂ µ l(γ).

3.1.
Variation of the length of a closed geodesic. We shall demonstrate a variational formula for the length of a closed geodesic in Proposition 2 using [4, Theorem 1.1]. There, Axelsson & Schumacher work in the more general context of families of Kähler-Einstein manifolds. Here, we note that by [2,Theorem 5], for a neighborhood (with respect to the Weil-Petersson metric) in Teichmüller space, the corresponding family of Riemann surfaces, each equipped with the unique hyperbolic Riemannian metric of constant curvature −1, form a holomorphic family. Thus, we are indeed in the setting of [4]. Related works include [3,34] and [14].
Here we shall prove our variational formula, Proposition 4 using Gardiner's formula, [18,Theorem 8.3]. This formula states that the variation ∂ µ l(γ) is given by 2 1 The term with 1 12 − log(A) comes from the derivative of the Riemann zeta function at −1. 2 We note that in [18,Theorem 8.3], they are considering the real variation, so they have ℜ µ, 2 π Θγ . We are taking the complex variation, and so we have the formula above for the variation.
, where a, b ∈ ∂H are the fixed points of γ, γ is the cyclic subgroup of Γ generated by γ, and κ ′ denotes the derivative of κ. We also recall (c.f. [14]) that for any γ ∈ Γ, the vector field is independent of both the path and the starting point z 0 for any weight 4 holomorphic modular form, ϕ for Γ. We may therefore use this integral to define the period integral as in [14, Definition 1.1.1].
Proposition 2. The variation of the length l(γ) is given by Above, z = z(t) is a parametrization of the geodesic representing γ with ż(t) = 1, and the integration is over the footprint of γ in X. We note that the variation is independent of the choice of parametrization, and φ is defined via µ as in (2.3).
Lemma 2. For any γ ∈ Prim(Γ) the differential ∂l(γ) is nowhere vanishing, namely for all t ∈ T , the linear map µ ∈ T Proof. We recall [17,Chapter 8] that for any fixed γ ∈ Prim(Γ) there is a predual ∂l(γ) ♭ , the Fenchel-Nielsen infinitesimal deformation, of the holomorphic differential ∂l(γ) with respect to the Weil-Petterson inner product, so that The Beltrami differential ∂l(γ) ♭ corresponds to a quadratic holomorphic form θ γ . Its norm is computed in [23, Theorem 2] and is non-zero. Thus ∂l(γ) is nonvanishing at any point t ∈ T . Since it is a linear map whose image has complex dimension one, its kernel, Ker ∂l(γ) ⊂ T (1,0) t (T ), at t ∈ T is of co-dimension one, that is dim Ker ∂l(γ) = 3g − 4. The statement for Ker ∂l 0 follows immediately.

3.2.
Convergence. In order to rigorously justify our variational formulas we must demonstrate the convergence of a certain sum. Proof. Let ε > 0 satisfy (3.8) k n < e εk , k ∈ N\{0}.
The last step is to consequently take L larger and larger, allowing us to make ε 1 (L) arbitrary small. This shows that the series converges absolutely and uniformly on compact subsets of ℜ(s) > 1.

3.3.
First variational formula of Z(s) for ℜs > 1. We introduce the local zeta function as in [14], We also recall the local Selberg zeta function for convenience here, We define further the higher local zeta function We now have requisite tools to prove the first variational formula.

Second variation
We prove here the second variational formula for the log of the Selberg zeta function as well as first and second variational formulas for: the squared resolvent, the Ruelle zeta function, the zeta-regularized determinant of the Laplacian, and the higher zeta functions, 4.1. Second variation of Selberg zeta function. In preparation for the proof, we note first the following differentiation formulas Hence, we see that Proof of Theorem 1. We perform the differentiation∂ µ on the result obtained in Proposition 4, By Leibniz's rule, this is Thus, it only remains to consider the second term. For this, we computē Thus, we have for this second term, Hence the expression becomes The second term above is where we have used the simple fact ∂ µ log ℓ(γ)∂ µ ℓ(γ) = |∂ µ log ℓ(γ)| 2 ℓ(γ). By our preliminary calculation (4.1), Thus, the total expression is Finally, we note that convergence follows from the estimates in Corollary 3 and Lemma 3.

4.2.
Applications to further variational formulas. The following formula for Tr(∆ + s(s − 1)) −2 may be proven as a consequence of the Selberg trace formula for the divided resolvents [16], [24, p. 118]. We have a somewhat different proof which may be of independent interest; this proof comprises §A.

4.2.2.
Variation of the Ruelle zeta function.
For the case s = 1, we have Proof. The proof follows immediately from (2.10).
As a corollary, we obtain the variation of the holomorphic analytic torsion.
Corollary 5. The first and second variations of the logarithm of the holomorphic analytic torsion, T 0 (X), defined in (2.4), are respectively

4.2.4.
Variation of the higher zeta functions. We conclude this section by computing the variation of the higher zeta functions.
Lemma 5. For a fixed point s ∈ C with ℜ(s) > 1, we have The sum in the right hand side of the equation above is absolutely convergent.
Proof. First compute Third, To finish the lemma, we need to prove that we can change the order of differentiation with respect to µ (or s) and the summation over the set of primitive closed geodesics, that is Lemma 6. For any fixed point s ∈ C with ℜ(s) > 1 and t > 1, we have The sum in the right hand side of the equation above is absolutely convergent in ℜ(s) > 1.
Proof. The proof follows the term-wise differentiation of ∂ ∂µ log z(s, t). First we compute The differentiation is justified because t+k−1 k (s + k) is a polynomial in k and hence the right hand side of the the previous equation converges for ℜ(s) > 1 by and especially for j = 0, d ds (log z γ (s, t)) = k≥0 t+k−1 k · l(γ) e l(γ)·(s+k) − 1 .
As before, the differentiation is justified by Lemma 3. Note that By the Christmas stocking identity, It is straightforward to repeat the calculations for the Selberg zeta function to compute the second variation of these zeta functions as well; this is left as an exercise for the reader.

5.
The asymptotics of∂ µ ∂ µ log Z(s) for large s We begin by demonstrating the asymptotics of each of the terms A γ (s) and B γ (s) as s → ∞.
in the precise sense that Proof. We write Since we are interested in the behavior as s → ∞, we may assume that s > ln(2) l 0 .
Then we have More generally, by the absolute convergence of the series, the function is continuous for |t| < 1. In particular, it is continuous at t = 0. Hence, using we have Next, we estimate the II a term, Consequently, it is plain to see that In a similar way, we compute the asymptotic behavior of B γ (s) as s → ∞.
in the precise sense that Proof. The proof is quite similar to that of the preceding proposition. We write We therefore define three terms, We begin by computing that .
We note that lim , for all k ≥ 0, when s > ln(2) l 0 .
We therefore see that the function is continuous for all |t| < 1. In particular, this function is continuous at t = 0. We therefore have, with t = e −sl(γ) , Hence, we see that We shall estimate the other two terms, We therefore see that Next, We compute the sum Hence, In particular, Recalling that We are now poised to complete the proof of Theorem 2.
Proof of Theorem 2. We first assume, recalling (1.6), This immediately implies For all γ with l(γ) > l 0 , we note that by Propositions 7 and 8, Hence we see that by Theorem 1, This together with Proposition 8 for the asymptotics of B γ0 (s) as s → ∞ completes the proof of the theorem in this case. Next, we assume that ∂ µ l 0 = 0.
Hence, the term with the slowest exponential decay as s → ∞ is A γ0 (s). We similarly see that for all γ with l(γ) > l 0 = l(γ 0 ) that We therefore have by Theorem 1, The proof is then completed by the asymptotics of A γ0 (s) as s → ∞ given in Proposition 7. The statements for the Ruelle zeta function follow immediately upon noting that log R(s) is simply given by the k = 0 term in log Z(s).
Proof of Corollary 1. By Lemma 2 there is an orthogonal decomposition with respect to the Weil-Petersson metric of the tangent space T (1,0) t (T ) at each point t ∈ T corresponding to the surface, X = X t . This decomposition is Hence, we may write Then, the leading terms in e sl0∂ µ ∂ µ log Z(s) are Recalling the estimates on the terms A γ and B γ , we see that the remainder term satisfies the estimate given in Corollary 1.
for sufficiently large s. Therefore, − log Z(s) is plurisubharmonic in a relatively compact open set for large s. This is in a way similar to the genus one case. Write a genus one Riemann surface as X = C/Z 2 τ , Z 2 τ = Z + Zτ , ℑτ > 0. The analytic torsion for the trivial bundle is given by (see [22] and [7]) Now, e −πℑτ /6 = e πiτ /12 2 , and e πiτ /12 is a holomorphic function of τ . Thus, ∞ k=0 (1 − e 2πikτ ) is also a holomorphic function of τ , and we compute where the norm above is with respect to the hyperbolic metric. In particular, − log det(∂ * ∂ ) is plurisubharmonic on the whole moduli space. The systole l 0 in this case is represented (up to SL(2, Z)-action, i.e., up by biholomorphic mappings) by τ , |τ | ≤ 1, and its differential ∂l 0 = ∂|τ | has rank one. However, for genus g ≥ 2, it is not clear to us whether there exists such t ∈ T such that ∂l 0 is of full rank. Some questions related to the rank of of ∂l 0 have been studied in [25].
5.1. Curvature asymptotics. We now consider the special case when s = m ∈ N, beginning by recalling Proposition 9. [Teorema 2 [40]] The second variation Let m ≥ 2. Then, the second variation The curvature Ric (m) (µ, µ) has been known to admit an expansion of the form c 2 m 2 + c 1 m + c 0 + · · · for large m. 3 In [21] the first two coefficients have been found explicitly for general fiberation of Kähler manifolds. We determine c 0 and the remainder in Corollary 2.
Proof of Corollary 2. We use (5.5) to writē In case ∂ µ l 0 = 0, we have proven that This shows that which shows that , m → ∞. 3 We began by computing the coefficients of m −k for k ∈ N using the expression of the curvature given in Proposition 1. These coefficients become increasingly complicated expressions at an exponential rate, and in the end, they vanished in each increasingly lengthy calculation. Indeed, our results show that there are no further non-zero coefficients c k m k for k ∈ Z with k ≤ −1, because the remainder is O(k −N ) for any N ∈ N as k → ∞.
In case ∂µl 0 = 0, the remainder is even smaller, because we have proven that The same argument then shows that Remark 4. In a recent preprint [30] we have been able to prove a general result in the setting of families of Kähler manifolds. Our result shows that the leading three terms of the curvature Ric (m) (µ, µ) and the Quillen curvature agree. In the case of Riemann surfaces this can be proved independently using the Bergman kernel expansion [20] and Berndtsson's curvature formula in Proposition 1.

5.2.
The cases m = 1, 2. We have seen that for large m, the sign of the second variation can be either positive or negative, depending upon the variation of the length of the shortest closed hyperbolic geodesic. Here we consider the special cases m = 1, 2. For m = 1 we show that there are surfaces and harmonic Beltrami differentials for which the second variation is strictly positive, as well as surfaces and harmonic Beltrami differentials for which the second variation is strictly negative.
Proof. If X is hyperelliptic then there exist harmonic Beltrami differentials µ such that R(µ, µ)u, u = 0 for all u ∈ H 0 (K). This follows for example from [6, Lemma 3.3 & Proposition 3.4]. Thus by Propositions 9 and 1, we havē To prove the second statement, for any Riemann surface, X, let ω be any non-zero abelian differential of norm 1. Then ω 2 is a holomorphic quadratic differential. Let µ = ρ −1 ω 2 be the corresponding harmonic Beltrami differential. We compute the curvature Ric (1) (µ, µ) using Proposition 1. To do this, let {ω j } be an orthonormal basis of H 0 (K), and fix The case m = 2 is of special interest because H 0 (K 2 ) can be viewed as the dual of the tangent space of Teichmüller space. Moreover, the negative of the Ricci curvature, −Ric (2) is therefore the Ricci curvature of the Weil-Pettersson metric. This object has been studied intensively for quite some time; see for example [36]. We demonstrate elementary upper and lower estimates of the variation in this case.
Proposition 11. Let m = 2 and B = B (2) be the Bergman kernel in this case. We have∂ Proof. In this case we havē We shall prove the estimates by demonstrating upper and lower estimates for the Ricci curvature in this case. In this way, we are also simultaneously demonstrating lower and upper estimates for the Ricci curvature of the Weil-Petersson metric.
By Proposition 1 We have the estimates ( 1,1 + 1) −1 ≤ 1, and ( 0 + 1) −1 ≤ 1 as operators. In this case, we actually have the unitary equivalence of 1,1 and 0 via the natural identification of f (z)dz ⊗ dz with f (z)y 2 , as the metric tensor y −2 dz ⊗ dz is globally defined . Hence, Applying the Cauchy-Schwarz inequality to the first term we obtain As for the second term, this is just X |µ| 2 B, so we have the lower bound . This implies the lower estimate.
To prove the upper estimate, we select one of the basis elements, u j , to be the corresponding quadratic form of µ. So, for example, we set We therefore have It follows from the unitary equivalence of the operators 1,1 and 0 that We proceed to estimate using [31]. On [31, p. 30], we see that in Wolf's notation, our µ = Φ g0 . We also note that the operator denote ∆ there is equal to −∆ 0 = −2 0 in our notation. Hence, the statement of [31, Lemma 5.1] is in our context Of course, this is equivalent to We use this estimate in (5.6) to obtain So, recalling the fact that the volume of our surface is 4π(g − 1), we have the lower estimate X |µ| 4 ≥ µ 4 W P 1 4π(g − 1) .
It would be interesting to find some more effective estimates for the variations in terms of the eigenvalues of m−1,1 and the geometry of the Riemann surface. The related questions of estimating the Weil-Petersson sectional curvature has been studied extensively; see [39] and references therein. For non-prime γ, with γ = kγ p , γ p ∈ Prim(Γ), N p (γ) := N (γ p ).
If |z| → ∞, |f (z)| = O(|z| −3 ), thus we can compute the integral over R using a large half disk contour together with the residue theorem. The estimate shows that the integral on the curved arc of the half disk is vanishing as the radius of the disk tends to infinity. Hence Res(f ; z).