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Bers isomorphism on the universal Teichm\"uller curve

We study the Bers isomorphism between the Teichm\"uller space of the parabolic cyclic group and the universal Teichm\"uller curve. We prove that this is a group isomorphism and its derivative map gives a remarkable relation between Fourier coefficients of cusp forms and Fourier coefficients of vector fields on the unit circle. We generalize the Takhtajan-Zograf metric to the Teichm\"uller space of the parabolic cyclic group, and prove that up to a constant, it coincides with the pull back of the Velling-Kirillov metric defined on the universal Teichm\"uller curve via the Bers isomorphism.


Introduction
Let U be the upper half plane and G a Fuchsian group. We denote by T (G) the Teichmüller space of G and F (G) the Bers fiber space of G. In [Ber73], Bers proved the following result. In case G = {id} is the trivial group, Bers theorem implies a biholomorphic bijection T (Γ 0 ) → F (1), where Γ 0 is the parabolic subgroup {z → z + n | n ∈ Z} of PSL(2, R), and F (1) is the universal Teichmüller curve. Both the universal Teichmüller space T (1) and the universal Teichmüller curve F (1) have group structures. We are going to show in Section 2 that as a subspace of T (1), T (Γ 0 ) is also a subgroup. Moreover, we prove in Section 3 that the Bers isomorphism on T (Γ 0 ) is a group isomorphism.
In [Teo04], we proved that the universal Teichmüller curve F (1) is isomorphic to the homogenuous space S 1 \Homeo qs (S 1 ), where Homeo qs (S 1 ) is the space of quasi-symmetric homeomorphisms of the unit circle. Using this isomorphism, we prove in Section 4 that the derivative of the Bers isomorphism at the origin of T (Γ 0 ) gives a remarkable relation between the Fourier coefficients of cusp forms on the upper half plane and Fourier coefficients of vector fields on the unit circle.
In [Teo04], we defined the Velling-Kirillov metric on the universal Teichmüller curve F (1), which we proved is the unique right invariant Kähler metric on F (1). In [TZ91], Takhtajan and Zograf defined a new Kähler metric on T g,n , the Teichmüller space of Riemann surfaces of genus g with n punctures, which plays an important role in the local index theorem. In Section 2, we generalize the definition of this metric to T (Γ 0 ) and still call it the Takhtajan-Zograf metric. By definition, this metric is right-invariant. Hence it is not surprising that we have the following theorem (see Theorem 5.1).

Teichmüller spaces and Kähler metrics
In this section, we review the results we need from Teichmüller theory and our previous paper [Teo04]. The readers can consult standard texts in Teichmüller theory [Leh87,Nag88] or our papers [Teo04,TT03] for further details.
Let U, L, D and D * be respectively the upper half plane, the lower half plane, the unit disc and the exterior of unit disc. In this paper, we are going to use both the upper half plane model and the unit disc model for the universal Teichmüller space T (1). In case of the disc model, it is described as follows. Let L ∞ (D) be the complex Banach space of bounded Beltrami differentials on D. We denote by L ∞ (D) 1 the unit ball of L ∞ (D). For µ ∈ L ∞ (D) 1 , we extend µ to D * by reflection There is a unique quasi-conformal mapping w µ fixing −1, −i, 1 which solves the Beltrami equation It fixes the unit circle S 1 , D and D * . The universal Teichmüller space is defined as a set of equivalence classes of normalized quasi-conformal mappings where µ ∼ ν if and only if w µ = w ν on the unit circle. The tangent space at identity is identified with Ω −1,1 (D), the space of harmonic Beltrami differentials on D. More explicitly, let be the space of bounded holomorphic quadratic differentials on D. Then The upper half plane model for the Teichmüller space T (1) ≃ L ∞ (U) 1 / ∼ is defined analogously where we replace D and D * in the definition above by U and L respectively; the reflection on µ is defined by µ(z) = µ(z); the quasi-conformal mapping w µ is normalized to fix the points 0, 1, ∞ and two quasi-conformal mappings w µ and w ν are equivalent if and only if they agree on the extended real lineR = R ∪ {∞}. The universal Teichmüller curve F (1) is defined as Here f µ is the quasiconformal map that is holomorphic on D * , normalized such that f (∞) = ∞, f ′ (∞) = 1, (f − z)(∞) = 0 and satisfies the Beltrami equation fz = µf z on D. The tangent space at identity is identified with Ω −1,1 (D) ⊕ C. Let Homeo qs (S 1 ) be the space of quasi-symmetric homeomorphisms on the unit circle, i.e. it consists of orientation preserving homeomorphism of the unit circle ω : S 1 → S 1 that satisfies for some constant M . We identify S 1 \Homeo qs (S 1 ) as the subspace of Homeo qs (S 1 ) consisting of those ω that fix the point 1. Every ω ∈ S 1 \Homeo qs (S 1 ) can be extended to be a symmetric (with respect to S 1 ) quasi-conformal mapping which we still denote by ω. It has a unique conformal welding ω = g −1 • f satisfying the conditions 1) f and g are quasi-conformal mappings. 2) f | D * and g| D are conformal and depend only on We call ( f | D * , g| D ) the pair of univalent functions associated to the point ω ∈ S 1 \Homeo qs (S 1 ), which we also write as (f, g). We identify T (1) as the subspace of S 1 \Homeo qs (S 1 ) consisting of those ω whose corresponding f satisfies the additional condition (f − z)(∞)(0), i.e. the power series expansion of f has the form In fact, for every µ ∈ L ∞ (D) 1 extended by symmetry to D * , there exists a unique symmetric quasi-conformal mapping ω that satisfies the Beltrami equation ωz = µω z and whose corresponding f in the conformal welding of ω has the form (2.2). We denote this ω by ω µ and the corresponding conformal welding by (f µ , g µ ). The isomorphism T : For the purpose of next section, we introduce the space From our reasoning above, it is isomorphic to L ∞ (D) 1 via the canonical correspondence µ → ω µ . Moreover, there is a real analytic isomorphism J : Q/ ∼ ×D → F (1) given by where the equivalence relation ∼ on Q is defined in the obvious way: ω µ ∼ ω ν if and only if ω µ and ω ν agree on S 1 . We also define K : The tangent space at identity of S 1 \Homeo qs (S 1 ) is identified with the Zygmund class vector fields 1 on S 1 . Given a one parameter flow ω t ∈ S 1 \Homeo qs (S 1 ) where ω 0 = id, it defines the tangent vector Since ω t (S 1 ) = S 1 and ω t (1) = 1 for all t, we find that c −n =c n and n∈Z c n = 0.
The Velling-Kirillov metric is a right invariant metric on S 1 \Homeo qs (S 1 ). At the origin, it is given by The group structure on whereas the group structure on F (1) is induced from the group structure on S 1 \Homeo qs (S 1 ), which is defined by composition of quasi-symmetric homeomorphisms.
Let Γ 0 be the parabolic cyclic group Γ 0 = {z → z + n n ∈ Z}. The Teichmüller space T (Γ 0 ) is defined as and ∼ is the same equivalence relation defined on L ∞ (U) 1 .
The isomorphism is given by [µ] → w µ | R . The subspace T (Γ 0 ) is then identified with the space of quasi-symmetric homeomorphisms on R that fix all the integers and satisfy the condition u(x + 1) = u(x) + 1 for all x.
The tangent space at the origin of T (Γ 0 ) is identified with In [TZ91], Takhtajan and Zograf defined a new Kähler metric on T g,n , the Teichmüller space of Riemann surfaces of genus g with n punctures. We generalize this metric to be a right invariant metric on T (Γ 0 ). At the origin, it is defined by A direct computation gives |α n | 2 n 5 .
In this section, we are going to give a separate proof of the Bers isomorphism B : T (Γ 0 ) → F (1). The basic idea follows the same line as in [Ber73]. But we use a different model for F (1). Besides, the proof is much simpler in this case. On the other hand, we are also going to show that B is in fact a group isomorphism.
From this lemma, we find that the map P descends to the map B : From (3.1) and the holomorphic dependance of f ν (0) on ν, we conclude immediately that B is a holomorphic mapping.
On the other hand, by Remark 2.1, definition (2.1), the identification F (1) ≃ S 1 \Homeo qs (S 1 ) and the commutative diagram (3.5) restricted to maps between R and S 1 , we see that Bers isomorphism is nothing but a correspondence between quasi-symmetric homeomorphisms on the unit circle that fix the point 1 and the lifting of these homeomorphisms to the universal cover of S 1 , i.e. R, under the covering map p : R → S 1 . It immediately follows that Lemma 3.2. The mapping B : T (Γ 0 ) → S 1 \Homeo qs (S 1 ) is a bijection.

Moreover, from the commutative diagram
we conclude immediately that Lemma 3.3. The mapping B : T (Γ 0 ) → S 1 \Homeo qs (S 1 ) is a group homomorphism.

Grouping together, we have shown that
Theorem 3.4. The Bers isomorphism B : T (Γ 0 ) → F (1) is a biholomorphism between complex manifolds and a group isomorphism.

The derivative mapping
In this section, we study the derivative of the Bers isomorphism at the origin. The following theorem gives a significance relation between the Fourier coefficients of cusp forms and Fourier coefficients of vector fields on S 1 .
Theorem 4.1. The derivative of the mapping P : T (Γ 0 ) → Homeo qs (S 1 )/S 1 at the origin is given by the following linear isomorphism Proof. Given µ ∈ Ω −1,1 (U, Γ 0 ), for t in a neighborhood of 0, we consider the one parameter family w tµ . Let v tµ = σ ωtν (0) • ω tν , where µ and ν are related by (3.1) and by definition P( It is well known that (see e.g. [Ahl87,Nag88]) w tµ is real analytic in U and therefore v tµ is real analytic in D. Taking derivative with respect to t and putting t = 0, we havė . By a well-known theorem of Ahlfors [Ahl61], we havė where p(z) is a polynomial of degree two. Since w tµ fixes 0, 1, ∞, we havė w µ vanishes at 0, 1, ∞, and we find that p(z) is a constant and is equal to Restricted to R, equation (4.1) and (4.2) give us This implies the assertion of the theorem.
Remark 4.4. T (Γ 0 ) can be thought of as the parameter space of all the onepunctured surfaces. This theorem implies that the tangent vector on T (Γ 0 ) that purely 'moves the puncture' corresponds to the automorphic form which has only the first coefficient nonzero. On the other hand, the automorphic form with vanishing first coefficient does not 'move the puncture'.
Since we have proved in [Teo04] that the Velling-Kirillov metric is Kähler, we obtain immediately Corollary 5.2. The Takhtajan-Zograf metric on T (Γ 0 ) is a right-invariant Kähler metric.
Remark 5.3. Let G andĠ be Fuchsian groups as in Theorem 1.1. Although T (Ġ) and F (G) are submanifolds of T (Γ 0 ) and F (1) respectively, the Bers isomorphism B : T (Γ 0 ) → F (1) we study in Section 3 does not induce the Bers isomorphism B G : T (Ġ) → F (G). In fact, when G = {id}, we do not have an explicit expression of the biholomorphism h : U/Ġ → (U/G)\{â} of Theorem 1.1, and thus we cannot write down the isomorphism B G : T (Ġ) → F (G) and its derivative mapping explicitly as in Section 4. Nevertheless, it will be interesting to compare the Velling-Kirillov metric on F (G) to the Takhtajan-Zograf metric on T (Ġ) under Bers isomorphism B G .
Bowick and Rajeev [BR87] and Kirillov and Yuriev [KY87] have calculated the curvature tensor of the Velling-Kirillov metric on the homogenous space S 1 \ Diff + (S 1 ), a natural submanifold of S 1 \Homeo qs (S 1 ) ≃ F (1). Via the Bers isomorphism B : T (Γ 0 ) → F (1), we can use their result to find the curvature tensor of the Takhtajan-Zograf metric on T (Γ 0 ) and study the properties of the curvature. This will in turn give the curvature properties of the Takhtajan-Zograf metric on the Teichmüller spaces of cofinite punctured Riemann surfaces. We will consider this question in a subsequent paper.