arXiv:math.CV/0312172 v3 21 Jun 2004
WEIL-PETERSSON METRIC ON THE UNIVERSAL TEICHM ¨ULLER SPACE I: CURVATURE PROPERTIES
AND CHERN FORMS
LEON A. TAKHTAJAN AND LEE-PENG TEO
Abstract. We prove that the universal Teichm¨uller space T (1) car-ries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T (1), the Hilbert submanifold T0(1), is a topological group. We define a Weil-Petersson metric on
T (1) by Hilbert space inner products on tangent spaces, compute its Riemann curvature tensor, and show that T (1) is a K¨ahler -Einstein manifold with negative Ricci and sectional curvatures. We introduce and compute Mumford-Miller-Morita characteristic forms for the verti-cal tangent bundle of the universal Teichm¨uller curve fibration over the universal Teichm¨uller space. As an application, we derive Wolpert cur-vature formulas for the finite-dimensional Teichm¨uller spaces from the formulas for the universal Teichm¨uller space.
Contents
1. Introduction 2
2. The universal Teichm¨uller space 5
2.1. Teichm¨uller theory 5
2.2. Homogeneous spaces of Homeoqs(S1) 10
2.3. Teichm¨uller spaces and Teichm¨uller curves of Fuchsian groups 17
2.4. Resolvent kernel 18
2.5. Variational formulas 20
3. T (1) as a Hilbert manifold 23
3.1. Hilbert space structure on tangent spaces 23
3.2. The L2-estimates 28
3.3. The Hilbert manifold structure of T (1) 31 3.4. Integral manifolds of the distribution DT 33
4. Velling-Kirillov and Weil-Petersson metrics 34 4.1. Velling-Kirillov metric on the universal Teichm¨uller curve 34 4.2. Weil-Petersson metric on the universal Teichm¨uller space 35 5. Characteristic forms of the universal Teichm¨uller curve 37 5.1. The form c1(V ) as Velling-Kirillov symplectic form 38
5.2. The Chern form c1(V ) and the resolvent kernel 40
5.3. Mumford-Morita-Miller characteristic forms 42 6. First and second variations of the hyperbolic metric 43
6.1. The first variation 43
6.2. The second variation 44
7. Riemann curvature tensor 46
7.1. The first variation of the Weil-Petersson metric 46 7.2. The second variation of the Weil-Petersson metric 50
7.3. Ricci and sectional curvatures 51
8. Finite-dimensional Teichm¨uller spaces 54
Appendix A. 62
Appendix B. 66
References 69
1. Introduction
The universal Teichm¨uller space T (1) is the simplest Teichm¨uller space that bridges spaces of univalent functions and general Teichm¨uller spaces. Introduced by Bers [Ber65, Ber72, Ber73], the universal Teichm¨uller space is an infinite-dimensional complex manifold modeled on a Banach space. It contains Teichm¨uller spaces of Riemann surfaces as complex submanifolds. The universal Teichm¨uller space T (1) also came to the forefront with the advent of string theory. It contains as a complex submanifold an infinite-dimensional complex Fr´echet manifold M¨ob(S1)\ Diff
+(S1), which plays an
important role in one of the approaches to non-perturbative bosonic closed string field theory based on K¨ahler geometry [BR87a, BR87b]. The manifold M¨ob(S1)\ Diff+(S1) — a homogeneous space of the Lie group Diff+(S1), also
has an interpretation as a coadjoint orbit of the Bott-Virasoro group, and as such carries a natural right-invariant K¨ahler metric [Kir87, KY87].
The complex geometry of the finite-dimensional Teichm¨uller spaces — Teichm¨uller spaces T (Γ) of cofinite Fuchsian groups, has been extensively studied in the context of Ahlfors-Bers deformation theory of complex struc-tures on Riemann surfaces. In particular, A. Weil defined a natural Hermit-ian metric on T (Γ) by the Petersson inner product on the tangent spaces. Called Weil-Petersson metric, it was shown to be a K¨ahler metric by Weil and Ahlfors. In his seminal paper [Ahl62] Ahlfors has studied the curva-ture properties of the Weil-Petersson metric. In particular, he proved that the Bers coordinates on T (Γ) are geodesic at the origin, and computed the Riemann curvature tensor of the Weil-Petersson metric in terms of multiple principal value integrals. Using these formulas, Ahlfors proved that T (Γ) has negative Ricci, holomorphic sectional, and scalar curvatures. Further re-sults have been obtained by Royden [Roy75]. Wolpert re-examined Ahlfors’ approach in [Wol86]. He developed a different method for computing Rie-mann and Ricci curvature tensors, and obtained explicit formulas in terms of the resolvent kernel of the Laplace operator of the hyperbolic metric on the corresponding Riemann surface.
Curvature properties of the infinite-dimensional complex Fr´echet manifold M¨ob(S1)\ Diff+(S1) have been studied by Kirillov and Yuriev [KY87], and
by Bowick and Rajeev [BR87a, BR87b]. In particular, they computed the Riemann curvature tensor of the right-invariant K¨ahler metric and proved that M¨ob(S1)\ Diff+(S1) is a K¨ahler -Einstein manifold.
Since both the finite-dimensional Teichm¨uller spaces T (Γ) and the ho-mogeneous space M¨ob(S1)\ Diff
+(S1) are complex submanifolds of T (1),
it is natural to investigate whether the latter space carries a “universal” K¨ahler metric which can be pulled back to the submanifolds. The imme-diate difficulty is that the universal Teichm¨uller space T (1) is a complex Banach manifold, so that its tangent spaces do not carry Hermitian metric. Nag and Verjovsky [NV90] were the first to address this problem. They have shown that the K¨ahler metric on M¨ob(S1)\ Diff+(S1) is the pull-back
of a certain Hermitian metric defined on a Hilbert subspace of the tan-gent space at the origin of T (1). The latter metric is analogous to the Weil-Petersson metric on finite-dimensional Teichm¨uller spaces. However, finite-dimensional Teichm¨uller spaces T (Γ) embed into T (1) transversally to the Hilbert subspace, so that the Weil-Petersson metric on T (Γ) can not be pulled back from T (1). Nevertheless, following a suggestion by Velling, Nag and Verjovsky [NV90] have shown that the Weil-Petersson metric on T (Γ) can be obtained by a certain “averaging” procedure using Patterson’s uni-form distribution of the “lattice points” of a cofinite Fuchsian group Γ in the hyperbolic plane. The major open problem is to define the Weil-Petersson metric on the whole space T (1), to study its curvature properties, and to find relation between curvatures of this metric and the Weil-Petersson metric on finite-dimensional Teichm¨uller spaces1.
An attempt to define the Weil-Petersson metric on the universal Te-ichm¨uller space based on the completion of diff(S1)/m¨ob(S1)2in the Sobolev’s
3
2-norm was made in [STZ99]. However, the paper [STZ99] does not contain
a rigorous proof that is needed for introducing a Hilbert manifold struc-ture on an infinite-dimensional manifold. Also, the identification between the tangent space diff(S1)/m¨ob(S1) and the space of holomorphic functions
on the unit disk made in [STZ99] is not correct and actually introduces Sobolev’s 92-norm rather than 32-norm. As a result, the corresponding quasi-symmetric homeomorphisms of S1 are of class C3(S1).
Here we introduce Weil-Petersson metric on the universal Teichm¨uller space T (1) and study its curvature properties. We prove that T (1) carries a new structure of a Hilbert manifold such that in the underlying topology T (1) has uncountably many components. We prove that the connected com-ponent of the identity in T (1), the Hilbert submanifold T0(1), is a topological
group. We define the Weil-Petersson metric on T (1) by Hilbert space inner products on tangent spaces. We re-examine the Ahlfors original computation [Ahl62] of the second variation of the hyperbolic metric and of the Riemann tensor for the finite-dimensional Teichm¨uller spaces in terms of the principal
1See the remark on p. 136 in [NV90]. 2Here diff(S1
) and m¨ob(S1
value integrals. We show how to extend the Ahlfors’ method to the case of the universal Teichm¨uller space and how to convert formulas using principal value integrals into closed expressions using resolvent kernel of the Laplace operator on the hyperbolic plane. Our results extend the Wolpert’s formu-las [Wol86] to the infinite-dimensional Hilbert manifold T (1). We also prove that T (1) is a K¨ahler -Einstein manifold with negative Ricci and sectional curvatures. Using the averaging procedure, we derive Wolpert’s curvature formulas [Wol86] for the finite-dimensional Teichm¨uller spaces from the cur-vature formulas for the universal Teichm¨uller space. Finally, we introduce and compute Mumford-Morita-Miller characteristic forms for the vertical tangent bundle associated with the fibration π :T (1) → T (1), where T (1) is the universal Teichm¨uller curve. Here again we consider T (1) and T (1) as Hilbert manifolds and show that the integration over the fibers opera-tion, used in the definition of Mumford-Morita-Miller characteristic forms, is well-defined.
This is the first paper in a series. In the subsequent paper we will con-struct a K¨ahler potential for the Weil-Petersson metric on T (1) and will study the properties of the period mapping.
Here is the more detailed content of the paper. In Section 2 we present necessary facts from Teichm¨uller theory, mainly following classical mono-graphs by Ahlfors [Ahl87], Lehto [Leh87] and Nag [Nag88]. Namely, in Section 2.1 we briefly cover: the main definitions, the group structure of the universal Teichm¨uller space T (1), the Bers embedding, structure of T (1) as an infinite-dimensional complex Banach manifold modeled on the com-plex Banach space A∞(D), and the basic properties of the universal
Te-ichm¨uller curve π :T (1) → T (1). In Section 2.2 we realize T (1) and T (1) as homogeneous spaces of the group Homeoqs(S1) of quasi-symmetric
homeo-morphisms of S1, and by using conformal welding we identify T (1) andT (1) with the spaces of univalent functions on the unit disk D. We describe the de-composition of the tangent bundle ofT (1) over the fiber π−1(0) and present
isomorphisms between the tangent spaces. Lemma 2.5 which describes a special property of the quasiconformal mapping with harmonic Beltrami differential seems to be a new result. In Section 2.3 we present, in a suc-cinct form, basic facts about the Teichm¨uller spaces and Teichm¨uller curves of Fuchsian groups, including the definition of the Weil-Petersson metric, and Patterson’s lemma on the uniform distribution of lattice points on the hyperbolic plane. In Section 2.4 we collect necessary properties of the resol-vent kernel G = 12(∆0+12)−1 of the Laplace operator ∆0 on the hyperbolic
plane, and in Section 2.5 we present Ahlfors’ classical variational formulas. In Section 3 we introduce new Hilbert manifold structure on T (1). Namely, in Section 3.1 we define the Hilbert subspaces H−1,1(D∗) and A
2(D) of the
tangent spaces to T (1) and to A∞(D). In Theorem 3.3 we prove that the
differential of the Bers embedding β : T (1)→ A∞(D) is a bounded bijection
between these Hilbert spaces. In Section 3.2 we prepare all L2-estimates used in Section 3.3. The main result there is Theorem 3.10 — the existence
of a Hilbert manifold atlas for T (1). In Theorem 3.13 we prove that the Bers embedding is also a biholomorphic mapping of Hilbert manifolds. In Section 4.1, following [Teo02], we recall the definition of the Velling-Kirillov metric on the universal Teichm¨uller curveT (1) considered as a Banach man-ifold, and in Section 4.2 we define the Weil-Petersson metric on the Hilbert manifold T (1). In Section 5.1 we prove that Velling-Kirillov metric is real-analytic onT (1) by explicitly constructing its real-analytic K¨ahler potential — Theorem 5.3. We introduce Mumford-Miller-Morita characteristic forms by considering π :T (1) → T (1) as a fibration of Hilbert manifolds. The lat-ter property is crucial for the operation “integration over the fibers” (which are non-compact) to be well-defined. In Theorem 5.10 we explicitly com-pute Mumford-Miller-Morita forms in terms of the resolvent G. This is an infinite-dimensional generalization of Wolpert’s result in [Wol86]. In Sec-tion 6 we give a simple derivaSec-tion of the second variaSec-tion of the hyperbolic metric — Proposition 6.3. In Section 7 we prove that the Weil-Petersson metric on T (1) is K¨ahler and explicitly compute its Riemann and Ricci cur-vature tensors, showing that T (1) is a K¨ahler -Einstein manifold. The main results there are Theorem 7.7 and 7.11. They are based on a more technical Proposition 7.2 and Lemma 7.8, and the proof of the latter is presented in Appendix B. In Section 8 we derive Wolpert’s curvature formulas [Wol86] for finite-dimensional Teichm¨uller spaces from the corresponding “universal” curvature formulas for T (1), obtained in Section 7. Finally, in Appendix A we prove that T0(1) and the corresponding Teichm¨uller curve T0(1) — the
inverse image of T0(1) under the projection π, are topological groups in
Hilbert manifold topology. Moreover, we show that T0(1) is the closure of
M¨ob(S1)\ Diff
+(S1) in T (1) with respect to the Hilbert manifold topology,
and prove that T0(1) is the inverse image of β(T (1))∩ A2(D) under the Bers
embedding.
Acknowledgments. We appreciate useful discussions with C. Bishop. The work of the first author was partially supported by the NSF grant DMS-0204628. The work of the second author was partially supported by the grant NSC 91-2115-M-009-017. The second author also thanks CTS for the fellowship to visit Stony Brook University in the Summer of 2003, where part of this work was done.
2. The universal Teichm¨uller space
2.1. Teichm¨uller theory. Here we present, in a succinct form, necessary facts from Teichm¨uller theory (for more details, see monographs [Ahl87, Leh87, Nag88] and the exposition in [Teo02]).
2.1.1. Main definitions. Let D = {z ∈ C : |z| < 1} be the open unit disk and let D∗ = {z ∈ C : |z| > 1} be its exterior. Denote by L∞(D∗) and
and D respectively, and let L∞(D∗)1 be the open unit ball in L∞(D∗). Two
classical models of the universal Teichm¨uller space T (1) are the following. Model A.Extend every µ∈ L∞(D∗)
1 to D by the reflection (2.1) µ(z) = µ 1 ¯ z z2 ¯ z2 , z∈ D,
and consider the unique quasiconformal (q.c.) mapping wµ: C→ C, which
fixes −1, −i and 1 (i.e., is normalized) and satisfies the Beltrami equation (wµ)z¯= µ(wµ)z .
Here and in what follows subscripts z and ¯z always stand for the partial derivatives ∂z∂ and ∂ ¯∂z, unless it is explicitly stated otherwise. Due to the reflection symmetry (2.1) the q.c. mapping wµ satisfies
1 wµ(z) = wµ 1 ¯ z (2.2)
and fixes domains D, D∗, and the unit circle S1. For µ, ν ∈ L∞(D∗) 1, set
µ∼ ν if wµ|S1 = wν|S1. The universal Teichm¨uller space T (1) is defined as
a set of equivalence classes of normalized q.c. mappings wµ,
T (1) = L∞(D∗)1/∼ .
Model B.Extend every µ∈ L∞(D∗)
1 to be zero outside D∗, and consider
the unique q.c. mapping wµ which satisfies the Beltrami equation
wµz¯ = µwzµ,
and is normalized by the conditions f (0) = 0, f′(0) = 1 and f′′(0) = 0. Here f = wµ|
Dis holomorphic on D and prime stands for the derivative. For
µ, ν ∈ L∞(D∗)
1, set µ∼ ν if wµ|D= wν|D. The universal Teichm¨uller space
T (1) is defined as a set of equivalence classes of normalized q.c. mappings wµ,
T (1) = L∞(D∗)1/∼ .
Since wµ|S1 = wν|S1 if and only if wµ|D = wν|D, these two definitions of
the universal Teichm¨uller space are equivalent. The set T (1) is a topological space with the quotient topology induced from L∞(D∗)1. Denote byL∞(D∗)
the subspace of L∞(D∗) consisting of real-analytic Beltrami differentials.
Every point in T (1) can be represented by µ∈ L∞(D∗) [Leh87, Sect. III.1.1].
The space T (1) has a unique structure of a complex Banach manifold, such that the projection map
Φ : L∞(D∗)1 → T (1)
is a holomorphic submersion. The differential of Φ at the origin D0Φ : L∞(D∗)→ T0T (1)
is a complex linear surjection of holomorphic tangent spaces. The kernel of D0Φ is the subspaceN (D∗) of infinitesimally trivial Beltrami differentials.
Explicitly, N (D∗) = µ∈ L ∞(D∗) :ZZ D∗ µ φ d2z = 0 for all φ∈ A1(D∗) , where d2z = dx∧ dy, z = x + iy, and
A1(D∗) = φ holomorphic on D ∗: Z Z D∗ |φ|d2z <∞ .
The Banach space of bounded harmonic Beltrami differentials on D∗ is de-fined by Ω−1,1(D∗) =nµ∈ L∞(D∗) : µ(z) = (1− |z|2)2φ(z), φ∈ A∞(D∗) o , where A∞(D∗) = φ holomorphic on D∗: kφk∞= sup z∈D∗ (1 − |z|2)2φ(z) < ∞ . The Banach space Ω−1,1(D∗) is not separable. The decomposition
(2.3) L∞(D∗) =N (D∗)⊕ Ω−1,1(D∗)
identifies the holomorphic tangent space T0T (1) = L∞(D∗)/N (D∗) at the
origin of T (1) with the Banach space Ω−1,1(D∗). The universal Teichm¨uller space T (1) is a complex Banach manifold modeled on Ω−1,1(D∗).
Remark 2.1. Traditionally, the universal Teichm¨uller space is defined using the complex Banach space L∞(D)
1. The reflection (2.1) establishes natural
complex anti-linear isomorphism between L∞(D∗)1 and L∞(D)1, and the
universal Teichm¨uller space in the traditional definition is complex conjugate to the space T (1) defined above.
2.1.2. The group structure. The unit ball L∞(D∗)1carries a group structure
induced by the composition of q.c. mappings. The group law λ = ν∗ µ−1
is defined through wλ = wν◦ wµ−1, where µ−1 stands for the inverse element
to µ, i.e., µ∗ µ−1 = 0. The group law is given explicitly by λ = ν− µ 1− ¯µν (wµ)z (wµ)z¯ ◦ w−1µ .
It follows from this formula that L∞(D∗)
1 is a subgroup of L∞(D∗)1.
For every λ ∈ L∞(D∗)
1 set [λ] = Φ(λ) ∈ T (1). The group structure on
L∞(D∗)
1 projects to T (1) by [λ]∗ [µ] = [λ ∗ µ]. For every µ ∈ L∞(D∗)1 the
right translations
are biholomorphic automorphisms of T (1). The left translations, in general, are not even continuous mappings (see, e.g., [Leh87, Sect. III.3.4]). For every µ∈ L∞(D∗)
1 the kernel of DµΦ is the subspace D0Rµ(N (D∗)) of L∞(D∗)
and
T[µ]T (1) = D0R[µ](T0T (1))≃ D0Rµ Ω−1,1(D∗).
2.1.3. The Bers embedding. Let A∞(D) be the complex Banach space
A∞(D) = φ holomorphic on D : kφk∞= sup z∈D (1 − |z|2)2φ(z) < ∞ . The Bers embedding β : T (1) ֒→ A∞(D) is defined as follows. Denote by
S(f) the Schwarzian derivative of a conformal map f, S(f) = fzzz fz − 3 2 fzz fz 2 . For every µ ∈ L∞(D∗)
1 the holomorphic function S(wµ|D) ∈ A∞(D) (by
Kraus-Nehari inequality it lies in the ball of radius 6 in A∞(D)). Set
β([µ]) =S(wµ|D).
The Bers embedding is a holomorphic map of complex Banach manifolds, and its differential at the origin is
D0β(µ)(z) =− 6 π Z Z D∗ µ(ζ) (ζ− z)4d 2ζ. (2.4)
The complex-linear mapping D0β induces the isomorphism Ω−1,1(D∗) −→∼
A∞(D) of the holomorphic tangent spaces to T (1) and A∞(D) at the origin.
The mapping Λ : A∞(D)→ Ω−1,1(D∗), inverse to D0β, is given by
µ(z) = Λ(φ)(z) =−1 2(1− |z| 2)2φ 1 ¯ z 1 ¯ z4.
According to the Ahlfors-Weill theorem, over the ball of radius 2 in A∞(D)
the map φ7→ [Λ(φ)] is the right inverse to β, β ◦ Λ = id.
2.1.4. The complex structure. For every µ∈ L∞(D∗)1 let Uµ⊂ T (1) be the
image of the ball of radius 2 in A∞(D) under the map h−1µ = Φ◦ Rµ◦ Λ.
The maps hµν = hµ◦ h−1ν : hν(Uµ∩ Uν)→ hµ(Uµ∩ Uν) are biholomorphic as
functions on the Banach space A∞(D). The structure of T (1) as a complex
Banach manifold modeled on the Banach space A∞(D) is explicitly described
by the complex-analytic atlas given by the open covering T (1) = [
µ∈L∞(D∗)
1
Uµ
with coordinate maps hµand transition maps hµν. The canonical projection
Φ : L∞(D∗)1 → T (1) is a holomorphic submersion and the Bers embedding
β : T (1) → A∞(D) is a biholomorphic map with respect to this complex
Remark 2.2. Since every point T (1) can be represented by a real-analytic Beltrami differential, it is sufficient to consider the atlas formed by the charts (Uµ, hµ) with µ∈ L∞(D∗)1.
Complex coordinates on T (1) defined by the coordinate charts (Uµ, hµ)
are called Bers coordinates. For every ν ∈ Ω−1,1(D∗) set φ = D
0β(ν) and
define a holomorphic vector field ∂ε∂
ν on U0 by setting Dh0 ∂ ∂εν = φ
at all points in U0 3. At every point [µ] ∈ U0, identified with the
corre-sponding harmonic Beltrami differential µ, the vector field ∂ε∂
ν in terms of
the Bers coordinates on Uµ corresponds to
˜ φ = Dµhµ ∂ ∂εν =Dµhµ(Dµh0)−1 (φ) = D0(β◦ Φ) DµR−1µ (Λ(φ)) . Using identification Ω−1,1(D∗)≃ A∞(D), provided by the mapping D0β, we
get (2.5) ∂ ∂εν µ = P DµR−1µ (ν) = P (R(ν, µ)) , where (2.6) R(ν, µ) = ν 1− |µ|2 (wµ)z (wµ)z¯ ◦ w−1µ ,
and P : L∞(D∗) → Ω−1,1(D∗) is the projection onto the subspace of
har-monic Beltrami differentials, defined by the decomposition (2.3). Explicitly,
(2.7) (P µ)(z) = 3(1− |z| 2)2 π ZZ D∗ µ(ζ) (1− ζ ¯z)4d 2ζ.
Remark 2.3. Right translating ν ∈ T0T (1) defines a holomorphic tangent
vector
D0R[µ](ν) = (1− |µ|2) ν◦ wµ
(wµ)z¯
(wµ)z ∈ T[µ]
T (1)
at every [µ]∈ T (1). In Bers coordinates on Uµ this tangent vector is
repre-sented by ν ∈ Ω−1,1(D∗). However, the family {D0R[µ](ν)}[µ]∈T (1) of
holo-morphic tangent vectors does not form a smooth vector field on T (1) since the left translations are not continuous on T (1).
2.1.5. The universal Teichm¨uller curve. The universal Teichm¨uller curveT (1) is a natural complex fiber space over T (1) with a holomorphic projection map π : T (1) → T (1). The fiber over each point [µ] is a quasi-disk wµ(D∗)⊂ ˆC
with complex structure induced from ˆCand
T (1) = {([µ], z) : [µ] ∈ T (1), z ∈ wµ(D∗)} . (2.8)
The fibration π : T (1) → T (1) has a natural holomorphic section given by T (1) ∋ [µ] 7→ ([µ], ∞) ∈ T (1) — the “zero section”, which defines the embedding T (1) ֒→ T (1). The universal Teichm¨uller curve is a complex Banach manifold modeled on A∞(D)⊕ C4, and the mapping
T (1)× D∗ ∋ ([µ], z) 7→ ([µ], wµ(z))∈ T (1) is a real-analytic isomorphism.
2.2. Homogeneous spaces of Homeoqs(S1). Let Homeoqs(S1) be the
group of orientation preserving quasi-symmetric homeomorphisms of the unit circle S1(see, e.g., [Leh87] for the definition), and let Diff
+(S1), M¨ob(S1),
and S1 be the subgroups of Homeoqs(S1) consisting, respectively, of smooth
orientation preserving diffeomorphisms of S1, of M¨obius transformations of S1, and of rotations of S1.
Denote by U the set of univalent functions on D and let
D =f ∈ U : f(0) = 0, f′(0) = 1, f′′(0) = 0, f admits a q.c. extension to C , e
D =f ∈ U : f(0) = 0, f′(0) = 1, f admits a q.c. extension to C . According to the Beurling-Ahlfors extension theorem, the maps
T (1)∋ [µ] 7→ wµ|D∈ D
and
T (1)∋ [µ] 7→ wµ|S1 ∈ M¨ob(S1)\Homeoqs(S1)
define bijections
(2.9) D←− T (1)∼ −→ M¨ob(S∼ 1)\Homeoqs(S1),
which endow the spaces D and M¨ob(S1)\Homeo
qs(S1) with the structure
of complex Banach manifolds modeled on the Banach space A∞(D). In
what follows, we will always identify the coset space M¨ob(S1)\Homeo
qs(S1)
with the subgroup of Homeoqs(S1) fixing−1, −i and 1, so that the bijection
T (1)−→ M¨ob(S∼ 1)\Homeo
qs(S1) is a group isomorphism.
Remark 2.4. It is a non-trivial problem to describe the complex Banach manifold structure of the spacesD and M¨ob(S1)\Homeoqs(S1) intrinsically,
without using the bijection (2.9).
2.2.1. Conformal welding. According to Beurling-Ahlfors extension theo-rem, for every γ∈ M¨ob(S1)\Homeo
qs(S1) there exists a unique α∈ M¨ob(S1)
which fixes 1, and univalent functions f and g on D and D∗, satisfying the following properties.
CW1. f and g admit q.c. extensions to C. CW2. α◦ γ = (g−1◦ f)|
S1.
CW3. f (0) = 0, f′(0) = 1, f′′(0) = 0. CW4. g(∞) = ∞.
The factorization CW2 is known as conformal welding. For γ = wµ|S1,
[µ]∈ T (1), f = wµ|D and g = (wµ◦ w−1µ ◦ α−1)|D∗, so that g(D∗) = wµ(D∗).
Here wµis normalized so that f satisfies CW3 and α∈ M¨ob(S1) is uniquely
determined by the conditions α(1) = 1 and CW4. For [µ] ∈ T (1) we will always denote γµ= (α◦ wµ)|S1, fµ= f and gµ= g, so that
γµ= (gµ−1◦ fµ)|S1.
Slightly abusing notations, we will denote by γµ a q.c. extension of γµ =
(α◦ wµ)|S1 ∈ M¨ob(S1)\Homeoqs(S1) given by α◦ wµ. Since α ∈ M¨ob(S1)
fixes 1, the q.c. mapping γµ satisfies the reflection property (2.2) and the
factorization
(2.10) γµ= gµ−1◦ fµ,
where fµ = wµ and gµ = wµ◦ w−1µ ◦ α−1. We will distinguish between
γµ ∈ M¨ob(S1)\Homeoqs(S1) and its q.c. extension by explicitly specifying
either the property CW2 or the factorization (2.10). The following result will be used in Section 3.
Lemma 2.5. Let µ∈ Ω−1,1(D∗)1 = Ω−1,1(D∗)∩ L∞(D∗)1 andγµ= α◦ wµ
the q.c. mapping introduced above. Then the mapping γµ fixes 0 and ∞.
Proof. By the reflection property (2.2) and the factorization (2.10), it is sufficient to prove that fµ= wµ fixes ∞. Denote
γ = ı◦ γµ◦ ı, g = ı◦ gµ◦ ı, f = ı◦ fµ◦ ı, ı∗(µ) = µ◦ ı
ız
ız
, where ı(z) = z−1. The factorization (2.10) for γ
µ gives γ = g−1◦ f, and the
property CW3 for fµ yields the following Laurent expansion of f at∞, f (z) = z +a1
z + a2
z2 +· · · .
(2.11)
We will prove that f (0) = 0 for µ∈ Ω−1,1(D∗)
1by exploiting the argument in
Royden-Earle’s proof of the Ahlfors-Weill theorem, as presented in [Nag88, Sect. 3.8.5].
Namely, f satisfies the Beltrami equation with the Beltrami differential ν = ı∗(µ)|D, which is supported on D. The fundamental theorem from the
theory of q.c. mappings (see, e.g. [Ahl87]) asserts that f admits the series representation
f (z) = z + P (ν)(z) + P (νH(ν))(z) + P (νH(νH(ν)))(z) +· · · , (2.12)
which is uniformly and absolutely convergent on C. Here for h∈ C2(D) we
denote P (h)(z) =−1 π Z Z D h(ζ) ζ− zd 2ζ, H(h)(z) =−1 π Z Z D h(ζ) (ζ− z)2 d 2ζ,
where the latter integral — the Hilbert transform, is understood in the principal value sense. Since ν has compact support, it immediately follows from the definition of the operators P and H that the series (2.12) has the Laurent expansion (2.11) at ∞. We will prove that for ν ∈ Ω−1,1(D)
each term of this series vanishes at z = 0. Representing ν(z) = −12(1− |z|2)2P∞n=0anz¯nand using polar coordinates, we get for any (n−1) – iterate
of the operator νH, n > 1, P (νH(νH(ν . . . H(ν))))(0) = 1 2π nZZ D . . . ZZ D P m1,...mnam1. . . amnr m1+1 1 . . . rnmn+1e−im1θ1. . . e−imnθn r1eiθ1(r 2eiθ2 − r 1eiθ1)2. . . (r neiθn− rn−1eiθn−1)2 (1− r21)2. . . (1− rn2)2dr1dθ1. . . drndθn = ∞ X m1,...,mn=0 am1. . . amnIm1,...,mn,
where each integral in the definition of H is understood in the principal value sense. The interchange of the orders of summation and integration can be easily justified. For fixed r1 6= 0, r1 6= r2, r2 6= r3, . . . , rn−1 6= rn, let
Im1,...,mn(r1, . . . , rn) = Z 2π 0 . . . Z 2π 0 e−im1θ1. . . e−imnθndθ 1· · · dθn r1eiθ1(r 2eiθ2 − r 1eiθ1)2. . . (r neiθn − rn−1eiθn−1)2.
A change of variables θk7→ θk+ θ, k = 1, . . . , n gives
Im1,...,mn(r1, . . . , rn) = e
−i(m1+...+mn+(2n−1))θI
m1,...,mn(r1, . . . , rn).
Since all mk≥ 0 and 2n−1 > 0 for n ≥ 1, we have e−i(m1+...+mn+(2n−1))θ 6= 1
and hence
Im1,...,mn(r1, . . . , rn) = 0.
Remark 2.6. Since P (f )z = H(f ), it also follows from the proof that fz(0) =
1.
Similar to (2.9) , there are bijections e
D←− T (1)∼ −→ S∼ 1\Homeoqs(S1),
where we always identify the coset space S1\Homeo
qs(S1) with the stabilizer
of 1 in Homeoqs(S1) (see, e.g., [Teo02]). For every γ ∈ S1\Homeoqs(S1)
there exist unique univalent functions f and g on D and D∗, satisfying the properties CW1, CW4 and
CW2′. γ = (g−1◦ f)|
S1;
CW3′. f (0) = 0, f′(0) = 1.
Namely, the fibration π : T (1) −→ T (1) corresponds to the fiber space S1\Homeo
qs(S1) over M¨ob(S1)\Homeoqs(S1) with the fibers isomorphic to
S1\ M¨ob(S1)≃ D∗. The points in the fiber at [µ]∈ T (1) correspond to the
points σw◦ γµ∈ S1\Homeoqs(S1), w∈ D∗ with5
σw(z) = 1− w 1− ¯w 1− z ¯w z− w ∈ S 1\ M¨ob(S1).
Using the properties CW1 and CW2 for γµ, we get the factorization CW2′
for γ is γ = σw◦ γµ= (g−1◦ f)|S1, where f = λw◦ fµ, g = λw◦ gµ◦ σw−1, and λw(z) = z cwz + 1 , cw =− 1 gµ(w) =−1 2 f′′(0) f′(0),
so that (gµ◦ σw−1)(∞) = gµ(w), and the functions f and g satisfy the
prop-erties CW3′ and CW4 respectively. The mapping
T (1) ∋ ([µ], gµ(w))7→ γ = σw◦ γµ∈ S1\Homeoqs(S1)
establishes the isomorphismT (1)−→ S∼ 1\Homeo
qs(S1).
As before, we will also denote by γ a q.c. extension of γ∈ S1\Homeo
qs(S1)
which satisfies the reflection property (2.2) and admits the factorization γ = g−1◦ f.
Remark 2.7. It is known [Kir87] that Diff+(S1) is an infinite-dimensional
Lie group and homogeneous spaces M¨ob(S1)\ Diff
+(S1) and S1\ Diff+(S1)
are infinite-dimensional complex Fr´echet manifolds. In this case conformal welding readily follows from the Riemann mapping theorem without using q.c. mappings [Kir87]. Note that our convention for the conformal welding is different from that in [Kir87]: we are using right cosets instead of left cosets.
The bijectionT (1)−→ S∼ 1\Homeo
qs(S1) endows the universal Teichm¨uller
curve T (1) with the group structure. Explicitly, ([λ], z) = ([ν], ζ)∗ ([µ], w)−1, where (2.13) λ = ν− µ 1− ¯µν γz γz¯ ◦ γ−1 and (2.14) z =wλ◦ γ ◦ (wν)−1(ζ).
Here γ is a q.c. extension of σu◦γµ, u = gµ−1(w), and the point ([λ], z)∈ T (1)
does not depend on the choice of the extension γ. The mapping
T (1)∋ [µ] 7→ γµ∈ S1\Homeoqs(S1)≃ T (1)
is a complex–analytic embedding of T (1) into T (1) which is not a group homomorphism. On the other hand, the natural embedding
M¨ob(S1)\Homeoqs(S1)≃ T (1) ∋ [µ] 7→ wµ∈ S1\Homeoqs(S1)≃ T (1)
is a group homomorphism, though not a complex–analytic mapping. Con-sidering wµ as an element of S1\Homeoqs(S1) via this embedding, it admits
a conformal welding
wµ= g−1µ ◦ fµ
(2.15)
that satisfies the properties CW1, CW2′, CW3′, CW4. In terms of the
functions fµand gµ satisfying properties CW1–CW4, we have
fµ= λµ◦ fµ and gµ= λµ◦ gµ◦ α−1µ ,
where αµ ∈ PSU(1, 1) is such that wµ = αµ◦ γµ, and λµ ∈ PSL(2, C) is
uniquely determined by the conditions fµ(0) = 0, (fµ)′(0) = 1 and g
µ(∞) =
∞.
2.2.2. The horizontal and vertical subspaces. The right translations R([µ],z):
T (1) → T (1) are biholomorphic automorphisms of T (1) [Ber73]. The holo-morphic tangent space toT (1) at ([µ], z) is identified with the holomorphic tangent space at (0,∞) — the origin of T (1) by
T([µ],z)T (1) = D(0,∞)R([µ],z)(T(0,∞)T (1)) ≃ T(0,∞)T (1).
The holomorphic tangent space at the origin naturally splits into the direct sum of horizontal and vertical subspaces,
T(0,∞)T (1) = Ω−1,1(D∗)⊕ C.
The identification of holomorphic tangent spaces provides a natural splitting of the tangent space at every point inT (1) into the direct sum of horizontal and vertical subspaces. Lifts of horizontal and vertical tangent vectors at
the origin of T (1) to every point in the fiber at the origin are explicitly described as follows.
TV1. Let µ ∈ Ω−1,1(D∗) ⊂ T
(0,∞)T (1) be a horizontal tangent vector to
T (1) at the origin. A curve ([tµ], z(t)), z(0) = z, which defines the horizontal lift of µ to the point (0, z) ∈ T (1) in the fiber π−1(0) at
the origin, for small t is given by the equation ([µ(t)],∞) ∗ (0, z) = ([tµ], z(t)). Using (2.13), (2.14) and Lemma 2.5, we get
µ(t) = (σz−1)∗(tµ) = tµ◦ σz−1(σ
−1
z )′
(σz−1)′
and z(t) = wtµ(z).
Thus the horizontal lift of µ∈ T(0,∞)T (1) to every point in the fiber
(0, z)∈ π−1(0) is the vector field
τµ= ∂ε∂µ 0+ ˙w µ(z)∂ ∂z, where w˙ µ(z) = dz dt(0)
(cf. [Wol86]). At the point (0, z) ∈ π−1(0) the vector field τ
µ is
identified with the horizontal tangent vector (σz−1)∗µ∈ T(0,∞)T (1).
TV2. Let 1 ∈ C ⊂ T(0,∞)T (1) be the vertical tangent vector to T (1) at
the origin, given by the value of the vector field ∂z = ∂z∂ at z =∞.
A curve defining the right translate of 1 to the point (0, z) ∈ T (1) in the fiber π−1(0) at the origin for small t is given by the equation
(0, t−1)∗ (0, z) = (0, z(t)), and it follows from (2.14) that
dz dt(0) =
(1− z)(1 − |z|2)
(1− ¯z) .
Thus the right translate of 1 ∈ T(0,∞)T (1) to the point (0, z) ∈
π−1(0) is the vector (1−z)(1−|z|2
)
(1−¯z) ∂z at (0, z). As a result, the vector
field ∂z at every point (0, z) ∈ π−1(0) is identified with the vertical
tangent vector
(1− ¯z)
(1− z)(1 − |z|2)1∈ T(0,∞)T (1).
2.2.3. The isomorphisms of the tangent spaces. The real tangent vector space TR
0 S1\Homeoqs(S1) to S1\Homeoqs(S1) at the origin is identified
with the subspace of Zygmund class continuous real-valued vector fields u = u(θ)dθd on S1 (see, e.g., [Teo02] for the definition), satisfying
Z 2π 0
In particular, the Fourier series u(θ) =Pn∈Zcneinθ is absolutely convergent. For|z| = 1 set ˜ u(z) = i X n∈Z\{0} cnzn+1.
The function ˜u on S1 admits the decomposition ˜
u = u++ u−,
where u+ and u− are boundary values of functions holomorphic on D and
D∗ respectively and u+(0) = 0. Explicitly, u+(z) =i ∞ X n=1 cnzn+1, u−(z) =i ∞ X n=1 c−nz1−n.
It is a difficult problem to characterize the Zygmund class in terms of the Fourier series (cf. Remark 2.4). On the other side, in terms of the Fourier series the almost complex structure J on TR
0 S1\Homeo(S1) is explicitly
given by the classical conjugation operator J u = i X n∈Z\{0} sgn(n)cneinθ d dθ for u = X n∈Z\{0} cneinθ d dθ.
Remark 2.8. Note that our definition of the operator J differs by a nega-tive sign from the definition in [Kir87, NV90] for the homogeneous space S1\ Diff
+(S1).
The holomorphic and anti-holomorphic tangent vectors at the origin are v = u− iJ u 2 = ∞ X n=1 cneinθ d dθ and ¯v = u + iJ u 2 = −1 X n=−∞ cneinθ d dθ. For every smooth functionF in a neighborhood of the origin in T (1) and u∈ TR 0 S1\Homeoqs(S1) set ˙ F[u] = d dt t=0 F(γt),
where γt is a curve in S1\Homeoqs(S1) with the tangent vector u at the
origin. Corresponding directional derivatives of F at the origin in T (1) in the holomorphic and anti-holomorphic directions v and ¯v are defined by (2.16) ∂F(v) = 1 2 ˙ F[u] − i ˙F[J u], and ∂¯F(¯v) = 1 2 ˙ F[u] + i ˙F[J u]. For s∈ R let Hs(S1) = ( u = ∞ X n=−∞ aneinθ d dθ : ∞ X n=−∞ |n|2s|an|2 <∞ )
be the Sobolev space of complex-valued vector fields on S1. The properties of the tangent spaces T0S1\Homeoqs(S1), T0D and Te 0M¨ob(S1)\Homeoqs(S1),
which will be used in Section 5, can be succinctly summarized as follows (see [Teo02] for details).
TS1. Under the R-linear isomorphism TR
0 S1\Homeoqs(S1)−→ T∼ 0De u(θ) = X n∈Z\{0} cneinθ 7→ u+(z) = i ∞ X n=1 cnzn+1,
and ˙f|D= u+, ˙g0|D∗ =−u−, where
˙ f = d dtf t t=0 , ˙g = d dtgt t=0 ,
γt = g−1t ◦ ft is a smooth curve in T (1) tangent to u at the origin,
and ˙g0(z) = ˙g(z)− ˙g′(∞)z.
TS2. Under the R-linear isomorphism
T0R M¨ob(S1)\Homeoqs(S1)−→ T∼ 0T (1) D0β −−→ A∞(D) u(θ) = X n∈Z\{−1,0,1} cneinθ 7→ d3u+ dz3 (z) = i ∞ X n=2 (n3− n)cnzn−2. TS3. If φ(z) = ∞ X n=2 (n3− n)anzn−2∈ A∞(D) then ∞ X n=2 n2s|an|2 <∞ for all s < 1. TS4. TR
0 S1\Homeoqs(S1)⊂ Hs(S1) for all s < 1.
2.3. Teichm¨uller spaces and Teichm¨uller curves of Fuchsian groups. Let Γ be a Fuchsian group, i.e., a discrete subgroup of PSU(1, 1). Let
L∞(D∗, Γ) =
µ∈ L∞(D∗) : µ◦ γγ′
γ′ = µ for all γ ∈ Γ
be the space of bounded Beltrami differentials for Γ and L∞(D∗, Γ)1 = L∞(D∗)1∩ L∞(D∗, Γ)
be the open unit ball in L∞(D∗, Γ). The Teichm¨uller space of the Fuchsian
group Γ is defined by
T (Γ) = L∞(D∗, Γ)1/∼ ,
where the equivalence relation is the same as the one used to define the universal Teichm¨uller space T (1) in Section 2.1.1. The Teichm¨uller space T (Γ) has a natural structure of a complex Banach manifold such that the
tangent space at the origin of T (Γ) is identified with the Banach space Ω−1,1(D∗, Γ) of bounded harmonic Beltrami differentials for Γ,
Ω−1,1(D∗, Γ) = Ω−1,1(D∗)∩ L∞(D∗, Γ).
For every Fuchsian group Γ the canonical embedding T (Γ) ֒→ T (1) is holo-morphic, so that the universal Teichm¨uller space T (1) contains all the Te-ichm¨uller spaces T (Γ) as complex submanifolds. The universal Teichm¨uller space T (1) is the Teichm¨uller space for the trivial Fuchsian group Γ ={1}.
The inverse image of T (Γ) under the projection mapT (1) → T (Γ) is called the Bers fiber spaceBF(Γ). The quasi-Fuchsian group Γµ= wµ◦ Γ ◦ (wµ)−1 acts on the fiber wµ(D∗) at the point [µ] ∈ T (Γ). The Teichm¨uller curve
T (Γ) of the Fuchsian group Γ is a fiber space over T (Γ) with the fiber Γµ\wµ(D∗) at the point [µ]∈ T (Γ).
The domain D∗ is a model of the hyperbolic plane H2. The hyperbolic (Poincar´e) metric on D∗ — a Hermitian metric of constant Gaussian curva-ture −1, is
(2.17) ds2= ρ(z)|dz|2 = 4|dz|
2
(1− |z|2)2 ,
and the hyperbolic area 2-form is ρ(z) d2z. The Fuchsian group Γ is of finite type (cofinite) if the corresponding Riemann surface — the orbifold Γ\D∗, has a finite hyperbolic area. In this case, the Teichm¨uller space T (Γ)
is a finite-dimensional complex manifold with a natural Hermitian metric, called the Weil-Petersson metric. It is defined as Petersson’s inner product on tangent spaces T[µ]T (Γ) ≃ Ω−1,1(D∗, Γµ), where [µ] ∈ T (Γ) and Γµ =
wµ◦ Γ ◦ w−1µ . For µ, ν ∈ T0T (Γ),
hµ, νiW P =
Z Z
Γ\D∗
µ¯νρ(z)d2z.
The Weil-Petersson metric on T (Γ) is a K¨ahler metric.
The following result, due to Patterson [Pat75], will be used in Section 8. Here we present it in a convenient form as in [Teo02].
Lemma 2.9. Let Γ be a cofinite Fuchsian group and h ∈ L∞(D∗, ρ(z)d2z) be Γ–automorphic, i.e., h◦ γ = h for all γ ∈ Γ. Then
Z Z Γ\D∗ h(z)ρ(z)d2z = lim r→1+ A(Γ\D∗) A(D∗ r) Z Z D∗ r h(z)ρ(z)d2z,
where D∗r = {z ∈ D∗ : |z| ≥ r}, A(Γ\D∗) is the hyperbolic area of the
Riemann surface Γ\D∗, and A(D∗
r) is the hyperbolic area of D∗r.
2.4. Resolvent kernel. Let
be the Laplace-Beltrami operator of the hyperbolic metric on D, acting on functions. It is well-known (see, e.g., [Hej76, Lan85]) that the differential ex-pression (2.18) defines a unique positive, self-adjoint operator on the Hilbert space L2(D, ρ(z)d2z), which we still denote by ∆0. Let
G = 12 ∆0+12
−1
be (a one-half of) the resolvent of ∆0 at the regular point λ =−12.
Remark 2.10. Note that the Laplace-Beltrami operator in [Hej76, Lan85] is 4∆0, so that the regular point λ =−12 for the operator ∆0 corresponds to
λ =−2 for the Laplace-Beltrami operator in [Hej76, Lan85].
The resolvent G is a bounded integral operator on L2(D, ρ(z)d2z) with
kernel (2.19) G(z, w) = 2u + 1 2π log u + 1 u − 1 π, where u(z, w) is a point-pair invariant on D,
u(z, w) = |z − w|
2
(1− |z|2)(1− |w|2).
The resolvent kernel G(z, w) has the following properties (see, e.g., [Hej76] and [Lan85, Sect. XIV.3]).
RK1. G is symmetric, G(z, w) = G(w, z), and is a point-pair invariant, G(γz, γw) = G(z, w) for all γ ∈ PSU(1, 1).
RK2. G(z, w) is positive for all z, w∈ D.
RK3. If g ∈ BC∞(D) — the space of smooth bounded functions on D,
then the integral f (z) =
ZZ
D
G(z, w)g(w)ρ(w)d2w
is absolutely convergent for all z ∈ D and f = G(g) ∈ BC∞(D)
satisfies the differential equation
2 ∆0+ 12(f ) = g. Conversely, if f ∈ BC∞(D) and g = 2 ∆ 0+12(f ) ∈ BC∞(D), then f = G(g). RK4. For all z∈ D, ZZ D G(z, w)ρ(w)d2w = 1.
The last property immediately follows from RK3 since 2 ∆0+ 12
(1) = 1, where 1 is the constant function equal to 1 on D.
The resolvent kernel G of the Laplace-Beltrami operator on D∗ is given by the same formula (2.19) and satisfies the properties RK1 – RK4.
When Γ is a cofinite Fuchsian group, we denote by GΓ the one-half of the
resolvent of the Laplace-Beltrami operator on the Riemann surface Γ\D at λ =−1
2. It is a bounded integral operator on L2(Γ\D, ρ(z)d2z) with kernel
(2.20) GΓ(z, w) =
X
γ∈Γ
G(z, γw), z, w∈ D,
and it enjoys all the properties RK1-RK4. The corresponding resolvent kernel on Γ\D∗ is given by the same formula with z, w∈ D∗.
Remark 2.11. The operator GΓ plays a prominent role in the Weil-Petersson
geometry of the finite-dimensional Teichm¨uller space T (Γ) [Wol86].
2.5. Variational formulas. Here we collect necessary variational formulas. To simplify the computations in the following sections, we will use different realizations of the hyperbolic plane H2, given either by the unit disk D or its exterior D∗, or by the upper half-plane U.
Let l and m be integers and Γ a Fuchsian group (we will be primarily interested in the cases when Γ ={1}, i.e., is a trivial group, and when Γ is a cofinite Fuchsian group). Using the model H2 ≃ D, tensor of type (l, m) for Γ is a C∞-function ω on D satisfying
ω(γz)γ′(z)lγ′(z)m= γ(z) for all γ ∈ Γ.
Let ωε be a smooth family of tensors of type (l, m) for Γ
εµ= wεµ◦ Γ ◦ w−1εµ,
where µ∈ Ω−1,1(D, Γ) and ε∈ C is sufficiently small. Set
(wεµ)∗(ωε) = ωε◦ wεµ((wεµ)z)l((wεµ)¯z)m,
which is a tensor of type (l, m) for Γ — a pull-back of the tensor ωε by wεµ. Lie derivatives of the family ωε along vector fields ∂/∂εµ and ∂/∂ ¯εµ
are defined in the standard way, Lµω = ∂ ∂ε ε=0(wεµ) ∗(ωε) and L ¯ µω = ∂ ∂ ¯ε ε=0(wεµ) ∗(ωε).
When ω is a function on T (Γ) — a tensor of type (0, 0), the Lie derivatives reduce to directional derivatives
Lµω = ∂ω(µ) and Lµ¯ω = ¯∂ω(¯µ)
— the evaluation of the 1-forms ∂ω and ¯∂ω on the holomorphic and anti-holomorphic tangent vectors µ and ¯µ to T (Γ) at the origin. Corresponding real vector fields ∂t∂
µ are defined by ∂ ∂tµ = ∂ ∂εµ + ∂ ∂ ¯εµ , so that ∂ ∂εµ = 1 2 ∂ ∂tµ − i ∂ ∂tiµ and ∂ ∂ ¯εµ = 1 2 ∂ ∂tµ + i ∂ ∂tiµ .
For the model H2 ≃ U we have ∂ ∂εwεµ(z) =− 1 π ZZ U
R (wεµ(z), wεµ(u)) µ(u)(wεµ)2u(u)d2u,
(2.21) ∂ ∂ ¯εwεµ(z) =− 1 π ZZ U Rwεµ(z), wεµ(u) µ(u)(wεµ)2u(u)d2u,
where the q.c. mapping wεµ is normalized by fixing 0, 1,∞ and the kernel
R is R(z, u) = z(z− 1) (u− z)u(u − 1) = 1 u− z + z− 1 u − z u− 1. Setting F [µ] = ∂ ∂ε ε=0 wεµ and Φ[µ] = ∂ ∂ ¯ε ε=0 wεµ, we get from (2.21) F [µ](z) =− 1 π Z Z U R(z, u)µ(u) d2u, (2.22) Φ[µ](z) =− 1 π Z Z U R(z, ¯u)µ(u) d2u.
The function Φ[µ](z) is holomorphic on U and satisfies Φ[µ]zzz(z) =− 6 π ZZ U µ(u) (¯u− z)4d 2u.
As it follows from (2.7), the projection P : L∞(U)→ Ω−1,1(U) is given by
(2.23) (P µ)(z) =−3(z− ¯z) 2 π ZZ U µ(u) (u− ¯z)4d 2u.
Equivalently, for µ(z) = (z−¯2z)2φ(z) with φ ∈ A∞(U), Φ[µ]zzz = φ on U.
The function F [µ] satisfies F [µ]z¯= µ on U, and is holomorphic on the lower
half-plane U .
Lemma 2.12. Forµ∈ Ω−1,1(U) and z∈ U, lim ε→0 Z Z U(z,ε) µ(u) (u− z)4d 2u = lim ε→0 Z Z U(z,ε) µ(u) (u− z)5d 2u = 0, where U(z, ε) = U\ {u ∈ U : |u − z| < ε}.
Proof. The proof of the first formula essentially follows the classical Ahlfors’ proof in [Ahl87, Lemma 2 in Sect. VI D] by using µ(u) = (u−¯2u)2φ(u) with φ∈ A∞(U), the identity
(u− ¯u)2 (u− z)4 = ∂ ∂u − 1 u− z + ¯ u− z (u− z)2 − 1 3 (¯u− z)2 (u− z)3 ,
and Stokes’ theorem. The second formula is proved similarly. Another classical result of Ahlfors [Ahl61] is the following.
Lemma 2.13. Forµ∈ Ω−1,1(U) and z∈ U,
F [µ](z) = (z− ¯z)
2
2 Φ
′′(z) + (z− ¯z)Φ′(z) + Φ(z),
where Φ(z) = Φ[µ](z).
Remark 2.14. It follows from Lemma 2.13 that F [µ]zzz= 0 for µ∈ Ω−1,1(U),
in agreement with Lemma 2.12.
Corollary 2.15. For µ∈ Ω−1,1(U) and z∈ U,
Z Z
U
µ(u) (u− z)(u − ¯z)3d
2u = 0.
Proof. Using (2.22), we have F [µ](z)− (z− ¯z) 2 2 Φ ′′(z)− (z − ¯z)Φ′(z)− Φ(z) =− 1 π Z Z U µ(u) (z− ¯z) 3 (u− z)(u − ¯z)3d 2w. For µ∈ L∞(U)1 set
Kµ(u, v) = (wµ)u(u)(wµ)v(v) (wµ(u)− wµ(v))2 and Kµ(u, ¯v) = (wµ)u(u)(wµ)v(v) (wµ(u)− wµ(v))2 . We have from (2.21) the following formulas [Ahl62]
∂ ∂εKεµ(z, u) =− 1 π Z Z U µ(v)Kεµ(z, v)Kεµ(v, u) d2v, (2.24) ∂ ∂ ¯εKεµ(z, u) =− 1 π Z Z U µ(v)Kεµ(z, ¯v)Kεµ(¯v, u) d2v, and ∂ ∂εKεµ(z, ¯u) =− 1 π Z Z U µ(v)Kεµ(z, v)Kεµ(v, ¯u) d2v, (2.25) ∂ ∂ ¯εKεµ(z, ¯u) =− 1 π Z Z U µ(v)Kεµ(z, ¯v)Kεµ(¯v, ¯u) d2v,
For the model H2 ≃ D the q.c. mapping wµ is normalized by fixing
−1, −i, 1. The kernel R is given by
R(z, u) = (z + 1)(z + i)(z− 1) (u− z)(u + 1)(u + i)(u − 1),
and formulas similar to (2.22) hold for F and Φ. In particular, let f be a q.c. mapping such that f|D ∈ D, and let µ be a Beltrami differential
supported on the quasi-disk Ω∗ = f (D∗). Let vtµ be the solution on C of
the Beltrami equation
(vtµ)¯z= tµ(vtµ)z,
satisfying vtµ(0) = 0, v′tµ(0) = 1 and vtµ′′(0) = 0. Then
˙v = d dt t=0 vtµ
is a holomorphic function on Ω = f (D) and
(2.26) ˙vzzz(z) =− 6 π Z Z Ω∗ µ(u) (u− z)4d 2u. 3. T (1) as a Hilbert manifold
In this section we are going to endow T (1) with a structure of a complex manifold modeled on the separable Hilbert space
A2(D) = φ holomorphic on D :kφk 2 2 = ZZ D |φ|2ρ−1(z)d2z <∞ of holomorphic functions on D. In the corresponding topology, the universal Teichm¨uller space T (1) is a disjoint union of uncountably many components on which the right translations act transitively.
3.1. Hilbert space structure on tangent spaces. Let A2(D∗) = φ holomorphic on D ∗ :kφk2 2 = Z Z D∗ |φ|2ρ−1(z)d2z <∞ be the Hilbert space of holomorphic functions on D∗.
Lemma 3.1. The vector spaces A2(D) and A2(D∗) are subspaces of A∞(D)
andA∞(D∗) respectively. The natural inclusion maps A2(D) ֒→ A∞(D) and
A2(D∗) ֒→ A∞(D∗) are bounded linear mappings of Banach spaces.
Proof. It is sufficient to consider only the spaces of holomorphic functions on D. For every φ∈ A2(D), let φ =P∞n=2(n3−n)anzn−2be the power series
expansion. Then kφk22 = ZZ D |φ|2ρ−1d2z = π 2 ∞ X n=2 (n3− n)|an|2,
and by Cauchy-Schwarz inequality, |φ(z)| = ∞ X n=2 (n3− n)anzn−2 ≤ ∞ X n=2 (n3− n)|an|2 !1/2 ∞ X n=2 (n3− n)|z|2n−4 !1/2
for every z∈ D. Since
∞ X n=2 (n3− n)|z|2n−4= 6 (1− |z|2)4, we have kφk∞= sup z∈D (1 − |z|2)2φ(z) ≤ r 12 π kφk2. Let H−1,1(D) = µ = ρ −1φ, φ holomorphic on D :¯ kµk2 2 = Z Z D |µ|2ρ(z)d2z <∞ and H−1,1(D∗) = µ = ρ −1φ, φ holomorphic on D¯ ∗:kµk2 2 = ZZ D∗ |µ|2ρ(z)d2z <∞ be the Hilbert spaces of harmonic Beltrami differentials on D and D∗ re-spectively. It follows from Lemma 3.1 that the natural inclusion maps H−1,1(D) ֒→ Ω−1,1(D) and H−1,1(D∗) ֒→ Ω−1,1(D∗) are bounded and un-der the linear mapping D0β, H−1,1(D∗)−→ A∼ 2(D).
Remark 3.2. It follows from the proof of Lemma 3.1 that every µ∈ H−1,1(D)
(respectively in H−1,1(D∗)) satisfies lim
|z|→1µ(z) = 0.
Indeed, for given ε > 0 let N be such that
∞
X
n=N
Then |µ(z)| ≤(1− |z| 2)2 4 N −1X n=2 (n3− n)anzn−2 +(1− |z| 2)2 4 ∞ X n=N (n3− n)|an|2 !1/2 ∞ X n=2 (n3− n)|z|2n−4 !1/2 ≤(1− |z| 2)2 4 N −1X n=2 (n3− n)anzn−2 + √ 6ε 4 , so that lim sup |z|→1 |µ(z)| ≤ √ 6ε 4 . Since ε is arbitrary this proves the assertion.
For every [µ]∈ T (1) let D0R[µ] H−1,1(D∗)
be the subspace of the tan-gent space T[µ]T (1) = D0R[µ] Ω−1,1(D∗)
with a Hilbert space structure isomorphic to H−1,1(D∗). Let DT be the distribution on T (1), defined by
the assignment
T (1)∋ [µ] 7→ D0R[µ] H−1,1(D∗)
⊂ T[µ]T (1).
Similarly, let DA be the distribution on A∞(D), defined by
A∞(D)∋ φ 7→ A2(D)⊂ TφA∞(D)≃ A∞(D).
The next statement asserts that under the Bers embedding β : T (1) → A∞(D) the distribution DT is isomorphic to the restriction of the
distribu-tion DA to β(T (1)).
Theorem 3.3. For every [µ]∈ T (1) the linear mapping D0 β◦ R[µ]
: H−1,1(D∗)→ A2(D)
is a topological isomorphism.
Proof. Let ν∈ H−1,1(D∗). Set w
t= wtν∗µ = wtν◦ wµ and let wt = g−1t ◦ ft
be the conformal welding associated with the q.c. mapping wt by (2.15).
Let vt = ft◦ f−1, where wµ = g−1 ◦ f is the factorization for wµ, and set
Ω = f(D) = g(D), Ω∗= f(D∗) = g(D∗). Since β([tν∗ µ]) = S(ft) =S(vt)◦ f fz2+S(f), we have D0 β◦ R[µ] (ν) = d dt t=0S(f t) = ˙v zzz◦ f fz2, where ˙v = d dt t=0vt.
The q.c. mapping vtis holomorphic on Ω and satisfies vt◦g = gt◦wtν. Since
gt and g are holomorphic on D∗, the Beltrami differential of vt is given by
t˜ν(z) = 0, z∈ Ω, t(ν◦ g−1)(z)g−1z (z) g−1z (z) , z ∈ Ω∗.
It follows from (2.26) that
D0 β◦ R[µ] (ν) f−1(z) fz−1(z)2= ˙vzzz(z) =− 6 π Z Z Ω∗ ˜ ν(u) (u− z)4d 2u. (3.1)
Let ρ1(z) = (ρ◦ f−1)(z)|fz−1(z)|2 and ρ2(z) = (ρ◦ g−1)(z)|g−1z (z)|2 be the
hyperbolic metric densities on the domains Ω and Ω∗ respectively. Classical inequalities (see e.g., [Leh87, Nag88])
1 4 ≤ η
2
i(z)ρi(z) ≤ 4, i = 1, 2,
where η1(z) and η2(z) stand, respectively, for the distances of z ∈ Ω and
z ∈ Ω∗ to the quasi-circle f(S1), yield the following estimates (cf. [Nag88, Sect. 3.4.5]) ZZ Ω d2z |u − z|4 ≤ Z Z |z−u|≥η2(u) d2z |u − z|4 = π η2(u)2 ≤ 4πρ2 (u), u∈ Ω∗, and ZZ Ω∗ d2u |u − z|4 ≤ 4πρ1(z), z ∈ Ω.
From here it follows kD0 β◦ R[µ] (ν)k22 = Z Z D D0 β◦ R[µ] (ν) 2ρ−1d2z = ZZ Ω ˙vzzz 2ρ−11 d2z ≤6 2 π2 Z Z Ω ρ1(z)−1 Z Z Ω∗ d2v |v − z|4 Z Z Ω∗ |˜ν(u)|2d2u |u − z|4 d 2z ≤6 2· 4 π Z Z Ω ZZ Ω∗ |˜ν(u)|2d2u |u − z|4 d 2z≤ 62· 42Z Z Ω∗ |˜ν(u)|2ρ2(u)d2u =62· 42 Z Z D∗ |ν|2ρ(u)d2u = 576kνk22.
To prove that the mapping D0 β◦ R[µ]
is onto, we adapt to our case Bers’ arguments, as presented in [Nag88, Sect. 3.5]. For φ ∈ A2(D) set
q = (φ◦ f−1)(f−1
z )2, and choose µ in the equivalence class of [µ]∈ T (1) to be
the conformally natural extension of (g−1◦f)|
S1, constructed by Douady and
on C which fixes the quasi-circle f(S1). According to the Bers reproducing formula [Ber66], q(z) =−3 π Z Z Ω∗ q(h(u))(u− h(u))2h ¯ u(u) (u− z)4 d 2u. (3.2)
Analogous to L∞(D∗) and Ω−1,1(D∗), consider the Banach spaces L∞(Ω∗)
and
Ω−1,1(Ω∗) =µ∈ L∞(Ω∗) : µ = ρ−12 q, q is holomorphic on Ω¯ ∗ . Denote by ˜P the corresponding projection ˜P : L∞(Ω∗) → Ω−1,1(Ω∗). The mapping
µ7→ (g∗)−1(µ) = µ◦ g−1(g−1)′ (g−1)′
establishes the isomorphisms L∞(D∗)≃ L∞(Ω∗) and Ω−1,1(D∗)≃ Ω−1,1(Ω∗),
and ˜P = (g∗)−1◦ P ◦ g∗. Define ν∈ Ω−1,1(D∗) by (g∗)−1(ν)(z) = ˜P 1 2(q(h(z))(z− h(z)) 2h ¯ z(z) ∈ Ω−1,1(Ω∗).
The comparison between (3.2) and (3.1) shows that D0 β◦ R[µ]
(ν) = φ.
To prove that ν∈ H−1,1(D∗) we use the Earle-Nag [EN88] estimate, 1
C ≤ |z − h(z)|
4ρ
1(h(z))ρ2(z)≤ C, z ∈ Ω∗,
(3.3)
where the constant C depends only on kµk∞. Since the operator ˜P gives
the orthogonal projection of L2(Ω∗, ρ2(z) d2z) onto H−1,1(Ω∗), we get by the
Earle-Nag inequality Z Z D∗ |ν|2ρ(z)d2z = ZZ Ω∗ |(g∗)−1(ν)(z)|2ρ2(z)d2z ≤1 4 Z Z Ω∗ |q(h(z))(z − h(z))2hz¯(z)|2ρ2(z)d2z ≤C 4 Z Z Ω∗ |q(h(z))h¯z(z)|2ρ−11 (h(z))d2z.
Since h is sense reversing, for
κ = h
−1 z
we have kκk∞ < 1. Now ZZ Ω∗ |q(h(z))hz¯(z)|2ρ1(h(z))−1d2z = ZZ Ω |q(z)|2ρ1(z)−1|(h¯z◦ h−1)(z)hz−1¯ (z)|2(1− |κ(z)|2)d2z = ZZ Ω |q(z)|2 1− |κ(z)|2ρ1(z) −1d2z ≤C1 Z Z Ω |q(z)|2ρ1(z)−1d2z = C1kφk22 <∞,
so thatkνk2 ≤ C2kφk2. This also proves that the inverse map to D0 β◦ R[µ]
is bounded, so that D0 β◦ R[µ]
is a topological isomorphism. Remark 3.4. It follows from the proof of the first part of Theorem 3.3 that D0 β◦ R[µ]
= D0(β◦ Φ ◦ Rµ) extends to be a bounded linear operator on
L2(D∗, ρ(z)d2z) and the estimate
kD0(β◦ Φ ◦ Rµ) (ν)k2=kDµ(β◦ Φ) (D0Rµ(ν))k2 ≤ 24kνk2
holds for all ν∈ L2(D∗, ρ(z)d2z) and µ∈ L∞(D∗)
1.
3.2. The L2-estimates. The lemmas below are needed for the rigorous definition of a complex Hilbert manifold structure on T (1).
Lemma 3.5. For every ε > 0 there exists 0 < δ < 1 such that for all µ∈ Ω−1,1(D∗) with kµk ∞< δ, |(wµ )z(z)|2 (1− |wµ(z)|2)2 − 1 (1− |z|2)2 < ε (1− |z|2)2
for all z∈ D ∪ D∗. The same inequality holds for w
µ−1 = w−1µ .
Proof. Using the isomorphism Ω−1,1(D∗) −→ Ω∼ −1,1(D) given by reflection
(2.1) and property (2.2), it is sufficient to prove the estimate for z ∈ D. Since γµ = α◦ wµ, where α ∈ PSU(1, 1), the estimate holds for wµ if and
only if it holds for γµ. By Lemma 2.5 γµ fixes 0 and ∞, and by the result
of Ahlfors and Bers in [AB60] (see also the remark of Bers in [Ber73]) the functional
L∞(D∗)1 ∋ µ 7→ (γµ)z(0)∈ C
is real-analytic at µ = 0. In particular, for every ε > 0 there exists 0 < δ < 1 such that for all µ∈ Ω−1,1(D∗)
1 withkµk∞< δ, |(γµ)z(0)|2− 1 < ε.
For z ∈ D, let ˜µ = µ ◦ σz, where σz(w) = 1+zww+z . Then γµ˜ = ˜σz◦ γµ◦ σz for
some ˜σz ∈ PSU(1, 1). Since ˜µ ∈ Ω−1,1(D∗)1, it follows from Lemma 2.5 that
γµ˜(0) = 0. Therefore from ˜σz(γµ(z)) = 0, one obtains
|(γµ)z(z)|2
(1− |γµ(z)|2)2
(1− |z|2)2 =|(γµ˜)z(0)|2.
Since k˜µk∞ = kµk∞, the assertion follows. On the other hand, if µ ∈
Ω−1,1(D∗)
1, then µ−1 ∈ L∞(D∗)1 andkµ−1k∞=kµk∞. Hence the assertion
also holds for wµ−1.
Corollary 3.6. Letµ∈ Ω−1,1(D∗),kµk∞ < δ, where δ corresponds to ε = 1
in the previous lemma. Then for every λ ∈ L∞(D∗)
1 the linear mapping
DλRµ extends to an invertible bounded linear operator on the Hilbert space
L2(D∗, ρ(z)d2z). Moreover, DλRµ(ν) 2≤ √ 2 (1− kµk∞)2kνk2 , for allν ∈ L2(D∗, ρ(z)d2z) and λ∈ L∞(D∗)
1. The same inequality holds for
DλRµ−1. Proof. Since DλRµ(ν) = (1− |µ|2) ν◦ w µ 1 + µλ◦ wµ(w(wµµ))zz 2 (wµ)z (wµ)z , and 1− |µ|2 1 + µλ◦ wµ(w(wµµ))zz 2 ∞ ≤ 1 (1− kµk∞)2 for all λ∈ L∞(D∗)
1, we have by using Lemma 3.5 andkµk∞=kµ−1k∞,
Z Z D∗ DλRµ(ν) 2ρ(z)d2z≤ (1 − kµk∞)−4 ZZ D∗ ν◦ wµ (wµ)z (wµ)z 2 ρ(z)d2z =(1− kµk∞)−4 Z Z D∗ |ν|2 4|(wµ−1)z| 2 (1− |wµ−1|2)2 (1− |µ−1|2)d2z ≤2(1 − kµk∞)−4 ZZ D∗ |ν|2ρ(z)d2z = 2(1− kµ|∞)−4kνk22.
Replacing everywhere µ by µ−1 we get the same estimate for D
λRµ−1.
Denote byO(D∗)
1the subgroup of L∞(D∗)1generated by µ ∈ Ω−1,1(D∗),kµk∞<
Lemma 3.7. For every µ∈ O(D∗)
1 there exists C > 0 such that
kRµ(λ1)− Rµ(λ2)k2 < Ckλ1− λ2k2
for all λ1, λ2 ∈ L∞(D∗)1 satisfying λ1− λ2 ∈ L2(D∗, ρ(z)d2z).
Proof. Suppose first thatkµk∞ < δ. Set λ(t) = λ1+ tν, where ν = λ2− λ1,
so that λ(t)∈ L∞(D∗)
1, 0≤ t ≤ 1. By fundamental theorem of calculus,
Rµ(λ1)− Rµ(λ2) = Z 1 0 d dtRµ(λ(t))dt = Z 1 0 Dλ(t)Rµ(ν)dt. Using Corollary 3.6, kRµ(λ1)− Rµ(λ2)k22= Z Z D∗ Z 1 0 Dλ(t)Rµ(ν)(z) 2 ρ(z)d2z ≤ Z 1 0 Z Z D∗ Dλ(t)Rµ(ν)(z) 2 ρ(z)d2z dt ≤C2kνk22= C2kλ1− λ2k22.
The same estimate also holds for Rµ−1.
Since every µ ∈ O(D∗)1 can be written as µεnn ∗ · · · ∗ µε11, where µi ∈
Ω−1,1(D∗),kµik∞< δ, and εi =±1, i = 1, . . . , n, we have
Rµ= Rεµ11 ◦ · · · ◦ R
εn
µn,
and the assertion of the lemma follows.
Remark 3.8. Applying the same argument, we get from Corollary 3.6 that for every µ∈ O(D∗)
1 there exists C > 0, depending only onkµk∞ such that
DλRµ(ν) 2 ≤ Ckνk2,
for all ν∈ L2(D∗, ρ(z)d2z) and λ∈ L∞(D∗)
1.
Lemma 3.9. For every µ∈ O(D∗)
1 there exists C > 0 such that
k(β ◦ Φ)(λ ∗ µ) − (β ◦ Φ)(µ)k2 ≤ Ckλk2
for all λ∈ L2(D∗, ρ(z)d2z)∩ L∞(D∗)
1.
Proof. Set φ(t) = (β◦ Φ)(tλ ∗ µ). By fundamental theorem of calculus, (β◦ Φ)(λ ∗ µ) − (β ◦ Φ)(µ) = Z 1 0 dφ dt(t)dt, where dφ dt(t) = Dtλ(β◦ Φ ◦ Rµ) (λ) = (Dtλ∗µ(β◦ Φ) ◦ DtλRµ) (λ)
by chain rule. Since (D0Rµ)−1= DµRµ−1, it follows from Remarks 3.4 and
3.8 that
kDtλ∗µ(β◦ Φ)(ν)k2 ≤ 24kDtλ∗µR(tλ∗µ)−1(ν)k2≤ C1kνk2.
Using Remark 3.8 again, we get dφ dt(t) 2 =k (Dtλ∗µ(β◦ Φ) ◦ DtλRµ) (λ)k2≤ C2kλk2, 0≤ t ≤ 1. Therefore, (β ◦ Φ)(λ ∗ µ) − (β ◦ Φ)(µ) 2 2 = ZZ D Z 1 0 dφ dt(t, z)dt 2 ρ(z)−1d2z ≤ Z 1 0 ZZ D dφ dt(t, z) 2ρ(z)−1d2z dt ≤C22 λ 2 2,
which concludes the proof.
3.3. The Hilbert manifold structure of T (1). For every µ∈ O(D∗)1 let
Vµ⊂ Uµ⊂ T (1) be the image under the map h−1µ = Φ◦Rµ◦Λ of the open ball
of radiuspπ/3 about the origin in A2(D), which by Lemma 3.1 is contained
in the ball of radius 2 in A∞(D). Here (Uµ, hµ) is the coordinate chart Uµ
of the complex-analytic atlas for T (1) as a complex Banach manifold (see Section 2.1.4). Let ˜ hµ= hµ Vµ : Vµ→ A2(D).
The main result of this subsection is the following. Theorem 3.10. For everyµ, ν ∈ O(D∗)
1 the sets ˜hµ(Vµ∩ Vν) and ˜hν(Vµ∩
Vν) are open in A2(D) and the map
˜
hµν = ˜hµ◦ ˜h−1ν : ˜hν(Vµ∩ Vν)−→ ˜hµ(Vµ∩ Vν)⊂ A2(D)
is a biholomorphic function on the Hilbert space A2(D).
Proof. First we prove that the sets ˜hµ(Vµ∩ Vν) and ˜hν(Vµ∩ Vν) are open in
A2(D). Since Vµ∩ Vν 6= ∅ (otherwise there is nothing to prove), there exist
φ1 ∈ ˜hµ(Vµ∩ Vν) and φ2 ∈ ˜hν(Vµ∩ Vν), kφ1k2,kφ2k2 < p π/3, such that ˜ h−1 µ (φ1) = ˜h−1ν (φ2), i.e., (Φ◦ Rµ◦ Λ)(φ1) = (Φ◦ Rν◦ Λ)(φ2).
Setting λ1= Λ(φ1), λ2= Λ(φ2) and κ = ν∗ µ−1, we get
Φ(λ1) = Φ(λ2∗ κ).
The sets hµ(Uµ∩ Uν) and hν(Uµ∩ Uν) are open in A∞(D), so that there
exists δ1 > 0 such that hµ(Uµ∩ Uν) contains a ball of radius δ1 about φ1
continuous function in the Banach space A∞(D), so that there exists δ2 > 0
such that the inverse image by hµνof the ball of radius δ1 about φ1in A∞(D)
contains the ball of radius δ2 about φ2 in A∞(D). According to Lemma 3.1,
the latter ball contains any ball of radius δ3<
p
π/12 δ2 about φ2 in A2(D).
Now for every ϕ2 ∈ A2(D) satisfying kϕ2− φ2k2< δ3 set
ϕ1 = hµν(ϕ2) = (β◦ Φ ◦ Rκ◦ Λ)(ϕ2).
We claim that δ3 > 0 can be chosen such that ϕ1 ∈ A2(D) and kϕ1k2 <
p
π/3, which implies that ˜hν(Vµ∩ Vν) contains the ball of radius δ3 about
φ2 in A2(D). Indeed, set λ = Λ(ϕ2), so that ϕ1 = (β ◦ Φ)(λ ∗ κ). Since
λ− λ2 ∈ L2(D∗, ρ(z)d2z), we have by Lemmas 3.9 and 3.7,
kϕ1− φ1k2 =k(β ◦ Φ)(λ ∗ κ) − (β ◦ Φ)(λ2∗ κ)k2
≤ Ckλ ∗ λ−12 k2 ≤ C2kλ − λ2k2
= 2C2kϕ2− φ2k2 < 2C2δ3,
where the constant C > 0 (chosen to be the same for both Lemmas 3.7 and 3.9) depends only on λ2 and κ. Choosing δ3 small enough we have
kϕ1k2 <
p π/3.
The same argument applied to the map ˜hνµ= ˜h−1µν proves that ˜hµ(Vµ∩Vν)
is open in A2(D).
It remains to prove that the map ˜hµν is a holomorphic function in the
Hilbert space A2(D). It is bounded, so according to [Bou67] it is sufficient
to prove that for every ϕ∈ ˜hν(Vµ∩ Vν) and every η ∈ A2(D) the mapping
C ∋ t 7→ φ(t) = ˜hµν(ϕ + tη) ∈ A2(D) is a holomorphic function in some neighborhood of 0 in C. For this purpose we use the standard argument based on the fact that the map hµν is already a holomorphic function in
the Banach space A∞(D) and the mapping C ∋ t 7→ φ(t) ∈ A∞(D) is a
holomorphic function in some neighborhood of 0 in C. Thus there exists δ > 0 such that for every |t0| < δ,
φ(t) − φ(t0)− (t − t0)dφ dt(t0) ∞ = o(|t − t0|) as t → t0.
Moreover, δ can be chosen such that ϕ+tη∈ ˜hν(Vµ∩Vν) for|t| < δ. Then for
every z ∈ D the complex-valued function φ(t)(z) is holomorphic on |t| < δ and φ(t, z)− φ(t0, z)− (t − t0) dφ dt(t0, z) = 1 2πi I |w−t0|=δ1 φ(w, z) 1 w− t − 1 w− t0 − t− t0 (w− t0)2 dw = (t− t0) 2 2πi I |w−t0|=δ1 φ(w, z) (w− t0)2(w− t) dw,
where δ1 > 0 is such that the disk of radius δ1 about t0 is inside the disk of
