The complex structure of the Teichm\"uller space of circle diffeomorphisms in the Zygmund smooth class

We provide the complex Banach manifold structure for the Teichmueller space of circle diffeomorphisms whose derivatives are in the Zygmund class. This is done by showing that the Schwarzian derivative map is a holomorphic split submersion. We prove similar results for the pre-Schwarzian derivative map and investigate the fiber space over the Teichmueller space. Especially, we show that this projection is also a holomorphic split submersion and induces the structure of a disk bundle. Finally, we introduce the little Zygmund class and consider the closed subspace given by this family of mappings in the Teichmueller space.


Introduction
The groups of circle diffeomorphisms of certain regularity have been studied in the framework of the theory of the universal Teichmüller space.In the previous papers [10] and [15], we considered the class of circle diffeomorphisms whose derivatives are γ-Hölder continuous for 0 < γ < 1, and gave the foundation of this Teichmüller space.In the limiting case of this γ-Hölder continuity as γ → 1, one can adopt the Zygmund continuous condition.The corresponding Teichmüller space T Z of those circle diffeomorphisms has been defined in Tang and Wu [14].In this present paper, we endow T Z with the complex Banach manifold structure and prove several important properties of T Z regarding this complex structure.
We denote by Bel(D * ) the space of Beltrami coefficients µ, which are measurable functions on the exterior of the unit disk D * = {z | |z| > 1} ∪ {∞} with the L ∞ norm µ ∞ less than 1.To investigate the Zygmund smooth class, we use the space It is said that µ and ν in Bel(D * ) are Teichmüller equivalent if the normalized quasiconformal self-homeomorphisms f µ and f ν of D * having µ and ν as their complex dilatations coincide on the unit circle S = ∂D * .The normilization is given by fixing three distinct points on S. The universal Teichmüller space T is the quotient space of Bel(D * ) by the Teichmüller equivalence.We denote this projection by π : Bel(D * ) → T .Similarly, the Teichmüller space T Z is defined to be the quotient space of Bel Z (D * ) by the Teichmüller equivalence.This can be identified with the set of all normalized orientation-preserving self-diffeomorphisms f of S whose derivatives f ′ are continuous and satisfy the following Zygmund condition: there exists some constant C > 0 such that |f ′ (e i(θ+t) ) − 2f ′ (e iθ ) + f ′ (e i(θ−t) )| ≤ Ct for all θ ∈ [0, 2π) and t > 0. See [14,Theorem 1.1] for these characterizations of T Z as well as those introduced next.
To introduce the complex structure to T , we prepare the complex Banach space A(D) of holomorphic functions ϕ on the unit disk D = {z | |z| < 1} with In the case of T Z , we consider the complex Banach space as the corresponding space.It is obvious that ϕ A ≤ ϕ A Z .
For every µ ∈ Bel(D * ), let f µ be the quasiconformal homeomorphism of the extended complex plane C with complex dilatation 0 on D and µ on D * .Then, the Schwarzian derivative map Φ is defined by the correspondence of µ to S fµ| D , where for a locally univalent meromorphic function f : D → C, the Schwarzian derivative of f is defined by It is known that S fµ| D belongs to A(D) and Φ : Bel(D * ) → A(D) is a holomorphic split submersion (see [12,Section 3.4]).Moreover, Φ projects down to a well-defined injection α : T → A(D) such that α • π = Φ.This is called the Bers embedding of T .By the property of Φ, we see that α is a homeomorphism onto the image.This provides T with the complex Banach structure of A(D), which is the unique complex structure on T such that the Teichmüller projection π : Bel(D * ) → T is a holomorphic map with surjective derivatives at all points of Bel(D * ).
For the Teichmüller space T Z , it was proved in [14, Theorem 2.8] that Moreover, the following theorem was also proved.Indeed, as the Teichmüller projection π : Bel Z (D * ) → T Z is a quotient map satisfying α • π = Φ, the continuity of α is equivalent to that of Φ.
In this paper, as the universal Teichmüller space T and certain other spaces are in the case, we improve these results.
Theorem 1.2.The following are hold true: (1) The Schwarzian derivative map Φ : The Teichmüller space T Z is endowed with the unique complex Banach manifold structure such that the Teichmüller projection π : Bel Z (D * ) → T Z is a holomorphic map with surjective derivatives at all points of Bel Z (D * ).
These claims are shown in Corollaries 2.5, 2.6, and 2.7, respectively, as the consequences from Theorem 2.4 in Section 2, which supplies the property of split submersion to the holomorphic map Φ given in Theorem 1.1.
Next, we consider a certain fiber space T Z over the Teichmüller space α(T Z ) ∼ = T Z , which is realized in the space of pre-Schwarzian derivatives of f µ | D .Here, for a locally univalent holomorphic function f : D → C, the pre-Schwarzian derivative of f is defined by In Section 3, we are concerned with this fiber space T Z and the structure of the projection Λ : For µ ∈ Bel(D * ), we normalize f µ so that it fixes ∞.Then, ψ = log(f µ | D ) ′ belongs to the space B(D) of all Bloch functions on D, and the pre-Schwarzian derivative map Ψ : Bel(D * ) → B(D) is defined by µ → log(f µ | D ) ′ , which is known to be holomorphic.The corresponding space to Bel Z (D * ) is given as It was proved in [14,Theorem 1.3] that the pre-Schwarzian derivative map Ψ : Bel Z (D * ) → B Z (D) is holomorphic.The fiber space T Z is defined as the image of Ψ in B Z (D), and the projection Λ onto α(T Z ) ⊂ A Z (D) is given by Λ(ψ) = ψ ′ − ψ 2 /2.Regarding the projection Λ, what we obtain in Section 3 are summarized as follows.
Theorem 1.3.Λ : T Z → α(T Z ) is a holomorphic split submersion, and T Z is a topological disk bundle over the Teichmüller space T Z with the projection Λ.
These results are proved in Theorems 3.3 and 3.4 separately.The first one is a consequence of Theorem 1.2.The second one is a basic claim on this fiber space but it has not been considered properly in the literature even for other Teichmüller spaces.
Finally in Section 4, we define a closed subspace of T Z in a canonical way.The universal Teichmüller space T includes the little subspace T 0 consisting of all asymptotically conformal classes, which means that T 0 is defined by the Beltrami coefficients µ tending to 0 near the boundary.Similarly, the little subspace T Z 0 of T Z can be defined by the Beltrami coefficients in Bel Z 0 (D * ), which consists of µ ∈ Bel Z (D * ) with the condition that (|z| 2 − 1) −1 µ(z) tends to 0 near the boundary.Namely, T Z 0 = π(Bel Z 0 (D * )).This corresponds to the subspace consisting of all normalized self-diffeomorphisms f of S whose continuous derivative f ′ satisfies the Zygmund condition in addition with uniformly as t → 0. In Section 4, we state some of our results also for the little subspace T Z 0 and α(T Z 0 ) = Φ(Bel Z 0 (D * )), and for the fiber space T Z 0 = Ψ(Bel Z 0 (D * )).We do not deal with all results mentioned in Sections 2 and 3 in this modification; only basic claims are proved in Theorems 4.1 and 4.3.However, because the arguments are almost the same, we can easily obtain other results as well.For instance, the version of Theorem 1.3 can be read as follows and we mention it here without proof.
) is a holomorphic split submersion, and T Z 0 is a topological disk bundle over the Teichmüller space T Z 0 with the projection Λ.

The complex Banach structure
In this section, we prove Theorem 1.2.This will be done by the combination of Theorem 1.1 and Theorem 2.4, which is the main achievement in this paper.In the first half of this section, we show necessary claims towards this goal.
For µ and ν in Bel(D * ), we denote by µ * ν the complex dilatation of f µ • f ν , and by Proof.By the formula of the complex dilatation of the composition, we have (the symbol ≍ stands for the equality modulo a uniform positive constant multiple), where the multiple constant depends only on ν.Hence, Similarly, for any µ 1 , µ 2 ∈ Bel Z (D * ), we have This shows that the right translation r ν is continuous for every ν ∈ Bel Z (D * ).
The continuity is usually promoted to the holomorphy in the arguments of subspaces of the universal Teichmüller space.
For an arbitrary µ ∈ Bel(D * ), by choosing satisfied.We will prove the desired claim by induction.Suppose that we have obtained , and the argument in the first part of this proof implies that σ(Φ(µ k+1 * ν −1 k )) ∈ Bel Z (D * ) and this yields a bi-Lipschitz self-diffeomorphism of D * .Then, its composition with f ν k is also a bi-Lipschitz diffeomorphism whose complex dilatation is defined to be ν k+1 .Namely, . This is Teichmüller equivalent to µ k+1 and belongs to Bel Z (D * ) by Proposition 2.1.Thus, the induction step proceeds.We obtain ν = ν n as a required replacement of µ.
Remark 2. For circle diffeomorphisms whose derivatives are γ-Hölder continuous for 0 < γ < 1, the conformally barycentric extension (the Douady-Earle extension) yields an appropriate bi-Lipschitz diffeomorphism (see [10,Theorem 6.10]), but we do not know whether this works also for the Zygmund smooth class.The Beurling-Ahlfors extension of self-diffeomorphisms of R whose derivatives are in the Zygmund smooth class is investigated in [7].
We are ready to state the main claim in our arguments.
Theorem 2.4.For every µ ∈ Bel Z (D * ), there exists a holomorphic map s µ : Proof.The existence of a local holomorphic right inverse s ν of Φ such that s ν • Φ(ν) = ν can be proved by the standard argument if f ν is bi-Lipschitz in the hyperbolic metric for ν ∈ Bel Z (D * ).This is obtained as the generalized Ahlfors-Weill section by using the bi-Lipschitz quasiconformal reflection with respect to the quasicircle f ν (S).See [5,11] for a general argument, and [10,16] for its application to particular Teichmüller spaces.In our present case, the same proof can be applied.
For an arbitrary µ ∈ Bel Z (D * ), Lemma 2.3 shows that there exists ν ∈ Bel Z (D * ) such that ν is Teichmüller equivalent to µ and f ν is bi-Lipschitz in the hyperbolic metric.Then, we can take a local holomorphic right inverse s ν defined on some neighborhood By using the right translations r µ and r ν which are biholomorphic automorphisms of Bel Z (D * ) as in Lemma 2.2, we set The statements of Theorem 1.2 correspond to the following corollaries to Theorem 2.4.Remark 3.For a surjective holomorphic map Φ : B → A from a domain B ⊂ Y to a domain A ⊂ X of Banach spaces X and Y in general, Φ is a holomorphic split submersion if and only if both of the following conditions are satisfied (see [12, p.89]): (1) For every ϕ ∈ A, there exists a holomorphic map s defined on a neighborhood V ⊂ A of ϕ such that Φ • s = id V ; (2) Every µ ∈ B is contained in the image of some local holomorphic right inverse s as given in (1).For certain Teichmüller spaces defined by the supremum norm such as the universal Teichmüller space and Teichmüller spaces of circle diffeomorphisms, it has been proved that the Schwarzian derivative map Φ is a holomorphic split submersion.However, for Teichmüller spaces defined by the integrable norm such as the Weil-Petersson Teichmüller space and the BMO Teichmüller space, only condition (1) has been verified.
Corollary 2.6.The Bers embedding α : Corollary 2.7.The Teichmüller space T Z is endowed with the unique complex Banach manifold structure such that the Teichmüller projection π : Bel Z (D * ) → T Z is a holomorphic map with surjective derivatives at all points of Bel Z (D * ).
Proof.As the Bers embedding α : T Z → A Z (D) is a homeomorphism onto the image by Corollary 2.6, T Z is endowed with the complex Banach structure as the domain α(T Z ) in A Z (D).The uniqueness follows from [4, Proposition 8].

Pre-Schwarzian derivatives
Let ψ = log f ′ for a locally univalent holomorphic function f on D. Its derivative P f = ψ ′ is called the pre-Schwarzian derivative of f .Let B(D) be the space of all holomorphic functions ψ on D such that Such a ψ is called a Bloch function.Then, ignoring the difference of complex constant functions, we can regard B(D) as the complex Banach space with norm • B .We note that ψ ∈ B(D) if and only if ψ ′ ∈ A Z (D).
Let T = Ψ(Bel(D * )), which consists of those ψ = log f ′ for a conformal homeomorphism f of D that is quasiconformally extendable to D * with f (D) bounded.It is known that this is an open subset of B(D) but there are other uncountably many connected components of the open subset in B(D) consisting of those ψ = log f ′ with f (D) unbounded.See [17].
In the case of the Teichmüller space T Z , we define the corresponding space which we regard as the complex Banach space by ignoring the difference of complex constant functions.We note that ψ ∈ B Z (D) if and only if ψ ′ ∈ B(D).Moreover, we see that ψ B ψ B Z (see [19,Theorem 5.4]).Hereafter, the symbol stands for the inequality modulo a uniform positive constant multiple.The space B Z (D) was defined in [14] by using sup z =w∈D |ψ ′ (z) − ψ ′ (w)|/d(z, w) for the hyperbolic distance d on D, but they are equivalent (see [19,Theorem 5.5]).
Concerning the pre-Schwarzian derivative model of T Z , the following theorem has been obtained: • B Z .Moreover, unlike in the case of the universal Teichmüller space T , it is proved in [14,Theorem 1.3] that T Z is exactly the set of those ψ = log f ′ in B Z (D) for a conformal homeomorphism f of D that is quasiconformally extendable to D * with its complex dilatation in Bel Z (D * ).In other words, for every We examine the structure of T Z and the pre-Schwarzian derivative map Ψ : Bel Z (D * ) → T Z .The strategy is to factorize the Schwarzian derivative map Φ by Ψ and to bring the properties of Φ to Ψ. Due to the relation Proof.It is easy to see that Λ is Gâteaux holomorphic.Hence, to prove that Λ is holomorphic, it suffices to show that Λ is locally bounded.See [1, p.28] and [2,Theorem 14.9].The local boundedness can be verified as follows.
and, by [18,Proposition 8], we see that Therefore, We can improve Theorem 3.1 immediately in virtue of Theorem 2.4.It also asserts the same property for Λ = Λ| T Z .
Proof.By Theorem 2.4, Φ = Λ • Ψ is a holomorphic split submersion.Combined with Theorem 3.1 and Lemma 3.2, this implies the statement.Remark 4. For the Teichmüller space T γ of circle diffeomorphisms of γ-Hölder continuous derivatives, the fact that the pre-Schwarzian derivative map Ψ is a holomorphic split submersion was asserted in [15,Theorem 1.2].Once we know that Ψ is holomorphic (in fact, it suffices to prove its continuity to see this), the fact that the Schwarzian derivative map Φ is a holomorphic split submersion shown in [10,Theorem 7.6] implies that so is Ψ by similar arguments as above.
We can examine the structure of T Z more closely.In fact, T Z is a topological disk bundle over T Z though we do not know whether or not this is biholomorphically equivalent to the Bers fiber space over T Z .To see this, we use another pre-Schwarzian derivative map given by a different normalization.For any ζ ∈ D * , we impose the condition f µ (ζ) = ∞ on the quasiconformal homeomorphism f µ conformal on D with complex dilatation µ on D * .Then, P fµ| D is well defined for every µ ∈ Bel(D * ) by this normalization and ψ = log(f µ | D ) ′ belongs to B(D).We also assume f µ (0) = 0 and (f µ ) ′ (0) = 1 hereafter, and denote such a map f µ by Theorem 3.4.T Z is a topological bundle over T Z ∼ = α(T Z ) with the projection Λ : T Z → α(T Z ) and the fiber homeomorphic to D * .Proof.For every ϕ 0 ∈ α(T Z ) = Φ(Bel Z (D * )), we take some µ 0 ∈ Bel Z (D * ) such that Φ(µ 0 ) = ϕ 0 .By Theorem 2.4, there is a holomorphic map s µ 0 : V ϕ 0 → Bel Z (D * ) defined on some neighborhood V ϕ 0 ⊂ α(T Z ) of ϕ 0 such that s µ 0 (ϕ 0 ) = µ 0 .Then, we define a local trivialization L : From this, we see that L is injective.For any ϕ ∈ V ϕ 0 , we define the restriction of L onto the fiber Λ −1 (ϕ) as We can find this fact in [14,Theorem 1.3] and also see it by the argument below.Then, for Hence, ℓ ϕ is surjective for every ϕ ∈ V ϕ 0 , and so is L. A simple computation yields , Thus, the continuity of ℓ ϕ can be verified.In fact, ℓ ϕ is equi-continuous when ϕ moves locally and so does µ.Since ℓ ϕ is a continuous bijection, the continuity of ℓ −1 ϕ follows if we prove that ℓ ϕ (ζ) diverges as ζ ∈ D * tends to the boundary ∂D * = S.We apply the above computation for Here, We can also see that as ζ ∈ D * tends to S because ℓ ϕ extends to S continuously by allowing the value ∞.
We have shown that ℓ ϕ = L| {ϕ}×D * is a homeomorphism for every ϕ ∈ V ϕ 0 .Finally, we check that the bijection L is a homeomorphism.For the continuity of L, we use the triangle inequality Then, for the first term in the right side, we apply the equi-continuity of ℓ ϕ to show that the norm converges to 0 as ζ → ζ * .For the second term, we apply the continuity of Ψ ζ * • s µ 0 to show that the norm converges to 0 as ϕ → ϕ * .
For the continuity of L −1 , we consider any ψ = L(ϕ, ζ) and ψ * = L(ϕ * , ζ * ) in Λ −1 (V ϕ 0 ), and choose ψ = L(ϕ * , ζ).Then, we have The definition of the norm • A Z ×D * on the product space is obvious and |ζ − ζ * | should be interpreted appropriately if either ζ or ζ * is ∞.The convergence ψ → ψ * implies that ϕ → ϕ * , and hence ψ − ψ → 0 by the continuity of L. This also implies that By the continuity of ℓ −1 ϕ * , we have that ζ → ζ * .Thus, we obtain that L −1 is continuous, and L is a homeomorphism.Remark 5.For the Teichmüller space T γ of circle diffeomorphisms of γ-Hölder continuous derivatives, the same result holds true for the fiber space T γ defined in the same manner.

The little Zygmund smooth class
In this section, we supply additional results concerning the little class of Zygmund smoothness.For the space Bel Z (D * ) of the Beltrami coefficients, its little subspace is defined as The Teichmüller space of the little Zygmund smooth class is defined by T Z 0 = π(Bel Z 0 (D * )).We remark that the usage of naught 0 in this section is different from [10,15].
We see that Bel Z 0 (D * ) is a closed subspace of Bel Z (D * ).Indeed, assuming that a sequence {µ k } k∈N in Bel Z 0 (D * ) and µ ∈ Bel Z (D * ) are given so that µ k − µ Z → 0 as k → ∞, we can show that µ ∈ Bel Z 0 (D * ).For every ε > 0, we choose some which implies that µ ∈ Bel Z 0 (D * ).For a normalized self-diffeomorphisms f of S whose derivative f ′ satisfies the Zygmund condition, we say that f ′ satisfies the little Zygmund condition if uniformly as t → 0. We can show that (f µ ) ′ satisfies the little Zygmund condition if and only if µ belongs to Bel Z 0 (D * ).Indeed, this is verified by examining the proof of [14, Theorem 1.1] with the fact that (f µ ) ′ | S satisfies the little Zygmund condition defined in a similar way to the case of self-diffeomorphisms of S if and only if [3,Theorem 5.3] and [20]).
For the spaces A Z (D) and B Z (D) of holomorphic functions, the little subspaces are similarly defined:
depends only on the Teichmüller equivalence class [ν] ∈ T Z .After obtaining Theorem 2.4 and Corollary 2.7 below, we see that R [ν] is a biholomorphic automorphism of T Z .

Corollary 2 . 5 .
The Schwarzian derivative map Φ : Bel Z (D * ) → A Z (D) is a holomorphic split submersion onto the image.Proof.By Theorem 1.1, Φ is holomorphic, and by Theorem 2.4, it is a holomorphic split submersion.