Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05)

The universal Teichmüller space and related topics

Toshiyuki Sugawa

Abstract. In this survey, we give an expository account of the universal Teichmüller space with emphasis on the connection with univalent functions. In the theory, the Schwarzian derivative plays an important role. We also introduce many interesting results involving Schwarzian derivatives and pre-Schwarzian derivatives, as well. Keywords. universal Teichmüller space, univalent function, Schwarzian derivative, pre-Schwarzian derivative. 2000 MSC. Primary: 30F60; Secondary: 30C55, 30C62.

Contents

1. Preliminary 2 1.1. Quasiconformal mappings 2 1.2. Hyperbolic Riemann surfaces 4 1.3. Quadratic differentials 4 1.4. Univalent functions 5 1.5. Grunsky inequality 6 1.6. Schwarzian derivative 7

2. The universal Teichmüller space 8 2.1. Definition 1: the quotient space of quasiconformal maps 8 2.2. Definition 2: quasisymmetric functions 9 2.3. Definition 3: marked quasidisks 9 2.4. Definition 4: Bers embedding 10 2.5. Equivalence of T_{1} through T_{4} 11

3. Analytic properties of the Bers embedding 12 3.1. The Teichmüller space of a Riemann surface 12 3.2. Relationship with quasi-Teichmüller spaces 13 3.3. The Bers projection 14 3.4. Convexity 14 3.5. Teichmüller distance and other natural distances (metrics) 15 4. Pre-Schwarzian models 16

Version March 9, 2006.

4.1. The models T̂(𝔻) and T̂(ℍ) 16

4.2. The model T̂(𝔻^{∗}) 17 4.3. Loci of typical subclasses of 𝒮 18

5. Univalence criteria 19 5.1. Univalence criteria due to Nehari and Ahlfors-Weill 19 5.2. Inner radius and outer radius 20 5.3. Pre-Schwarzian counterpart 21 5.4. Directions of further investigation 22

References 22

1. Preliminary

In this section, we prepare basic tools to understand the universal Teichmüller space. The material is more or less standard, but for convenience, an expository account will be given without proofs. The most convenient reference for overall topics is perhaps the recently published handbook [62].

1.1. Quasiconformal mappings. A homeomorhism f of a plane domain D onto another domain D′ is called a quasiconformal map if f has locally square integrable partial derivatives (in the sense of distribution) and satisfies the inequality

|fz̄| ≤ k|f_{z}|

almost everywhere in D, where k is a constant with 0 ≤ k < 1,

f_{z} = \frac{1}{2}(f_{x} − if_{y}), fz̄ = \frac{1}{2}(f_{x} + if_{y})

and

f_{x} = \frac{∂f}{∂x}, f_{y} = \frac{∂f}{∂y} .

It turns out that f preserves sets of (2-dimensional) Lebesgue measure zero and, in particular, f_{z} ≠ 0 a.e. Thus the quotient μ = fz̄/f_{z} is well defined as a Borel measurable function on D and satisfies ∥μ∥_{∞} ≤ k < 1. This function is sometimes called the complex dilatation of f and denoted by μ_{f}. More specifically, f is also called a K-quasiconformal map, where K = (1+k)/(1−k). The minimal K = (1+ ∥μ∥_{∞})/(1 − ∥μ∥_{∞}) is called the maximal dilatation of f and denoted by K(f). It is known that a 1-quasiconformal map is conformal (i.e., biholomorphic) and vice versa. The composition of a K_{1}-quasiconformal map and a K_{2}-quasiconformal map is K_{1}K_{2}-quasiconformal map and the inverse map of a K-quasiconformal map is also K-quasiconformal. In particular, K-quasiconformality is preserved under composition with conformal maps. Therefore, K-quasiconformality and, hence, quasiconformality can be defined for homeomorphisms between Riemann surfaces. In particular, we can argue quasiconformality of a homeomorphism of the Riemann sphere ℂ̂ = ℂ ∪ {∞}.

More precise information about compositions of quasiconformal maps will be needed later. Let f : Ω → Ω′ and g : Ω → Ω′′ be quasiconformal maps. Then the complex dilatation of g ∘ f^{−1} is given by

(μ_{g∘f^{−1}} ∘ f)\frac{\overline{f_{z}}}{f_{z}} = \frac{μ_{g} − μ_{f}}{1 − \overline{μ_{f}} · μ_{g}} . (1.1.1)

In particular, we obtain the following lemma.

Lemma 1.1.2. Let f : Ω → Ω′ and g : Ω → Ω′′ be quasiconformal maps. Then g ∘ f^{−1} is conformal on Ω′ if and only if μ_{f} = μ_{g} a.e. in Ω.

Fundamental in the theory of quasiconformal maps is the following exsistence and uniqueness theorem.

Theorem 1.1.3 (The measurable Riemann mapping theorem). For any measurable function μ on ℂ with ∥μ∥_{∞} < 1, there exists a unique quasiconformal map f : ℂ → ℂ such that f(0) = 0, f(1) = 1 and fz̄ = μf_{z} a.e. in ℂ.

For the proof of the theorem and for more information about quasiconformal maps, the reader should consult the books [3] and [70] as well as the paper [4] by Ahlfors and Bers.

We denote by Belt(D) the open unit ball of the space L^{∞}(D) for a domain (or, more generally, a measurable set) D. An element μ of Belt(D) is called a Beltrami coefficient on D. For a Beltrami coefficient μ on ℂ, the function f given in the measurable Riemann mapping theorem will be denoted by f^{μ} throughout the present survey.

Let μ be a Beltrami coefficient on the outside 𝔻^{∗} of the unit disk. We extend μ to μ^{∗} ∈ Belt(ℂ) by setting μ^{∗}(z) = \overline{μ(1/z̄)} for z ∈ 𝔻^{∗}. Let f be a quasiconformal automorphism of ℂ̂ fixing 1, −1, −i with μ_{f} = μ^{∗}. Since f(z) and 1/\overline{f(1/z̄)} both have the same complex dilatation μ^{∗} and satisfy the same normalization condition, they must be equal by uniqueness part of the measurable Riemann mapping theorem. In particular, |f(z)|^{2} = 1 for z ∈ ∂𝔻, and consequently, f maps 𝔻^{∗} onto itself. We define w_{μ} : ℂ̂ → ℂ̂ by w_{μ} = f. Recall that w_{μ} fixes 1, −1 and −i.

The following fact was observed by Ahlfors and Bers [4].

Theorem 1.1.4. Let μ_{t} be a family of Beltrami coefficients on ℂ holomorphically parametrized over a complex manifold X. Then the map t ↦ f^{μ_{t}}(z) is holomorphic on X for a fixed z ∈ ℂ.

We do not explain the meaning of “holomorphically parametrized” here. It is, however, sufficient practically to know that (tμ + ν)/(1 + tν̄μ) is a family of Beltrami coefficients holomorphically parametrized over the unit disk |t| < 1, where μ, ν ∈ Belt(ℂ).

1.2. Hyperbolic Riemann surfaces. A connected complex manifold of complex dimension one is called a Riemann surface. The Poincaré-Koebe uniformization theorem tells us that every Riemann surface R admits an analytic universal covering projection p of the unit disk 𝔻 = {z ∈ ℂ : |z| < 1} onto R except for the case when R is conformally equivalent to the Riemann sphere ℂ̂, the complex plane ℂ, the punctured complex plane ℂ^{∗} = ℂ\{0} or a complex torus (a smooth elliptic curve). Those non-exceptional Riemann surfaces are called hyperbolic.

The group of analytic automorphisms of R is denoted by Aut(R). The group of disk automorphisms Aut(𝔻) is identified with PSU(1, 1) and isomorphic to PSL(2, ℝ) through the Möbius transformation M : ℍ = {z : Im z > 0} → 𝔻 defined by M(z) = (z − i)/(z + i). Thus Aut(𝔻) inherits a structure of real Lie group. A subgroup Γ of Aut(𝔻) is called Fuchsian if Γ is discrete. It is known that Γ is discrete if and only if Γ acts on 𝔻 properly discontinuously. Note also that Γ is torsion-free if and only if Γ acts on 𝔻 without fixed points. The covering transformation group Γ = {γ ∈ Aut(D) : p ∘ γ = p} is a torsion-free Fuchsian group and will be called the Fuchsian model of R. Conversely, for a given torsion-free Fuchsian group Γ the quotient space 𝔻/Γ has natural complex structure so that the projection 𝔻 → 𝔻/Γ becomes an analytic universal covering. In this way, theory of hyperbolic Riemann surfaces can be translated into that of torsion-free Fuchsian groups.

Since the Poincaré metric ρ_{𝔻}(z)|dz| = |dz|/(1−|z|^{2}) is invariant under the pullback by analytic automorphisms of 𝔻, it projects to a smooth metric, denoted by ρ_{R} = ρ_{R}(w)|dw|, on the hyperbolic Riemann surface R via p. The metric ρ_{R} is called the hyperbolic metric of R. Thus ρ_{R} is characterized by the relation ρ_{𝔻} = p^{∗}(ρ_{R}) = ρ_{R}(p(z))|p′(z)||dz|.

Note that ρ_{R} has constant Gaussian curvature −4, in other words, Δ log ρ_{R} = 4ρ_{R}^{2}.

The Schwarz-Pick lemma implies the contraction property f^{∗}ρ_{S} ≤ ρ_{R} for a holomorphic map f : R → S between hyperbolic Riemann surafaces R and S, where equality holds at some (hence every) point in R iff f is a covering projection of R onto S.

1.3. Quadratic differentials. Let H(𝔻) be the set of analytic functions on the unit disk 𝔻 and let n be a non-negative integer. For a Fuchsian group Γ, a function φ ∈ H(𝔻) is said to be automorphic for Γ (with weight −2n) if φ satisfies the functional equation (φ ∘ γ)(γ′)^{n} = φ for every γ ∈ Γ, that is to say, φ(z)dz^{n} is an invariant n-form for Γ. The set of automorphic functions for Γ with weight −2n will be denoted by H_{n}(𝔻, Γ).

An element φ of H_{n}(𝔻, Γ) for a torsion-free Fuchsian group Γ projects to a holomorphic n-form q = q(w)dw^{n} on R = 𝔻/Γ so that p_{n}^{∗}q = φ(z)dz^{n}, where p_{n}^{∗}q means the pull-back q(p(w))(p′(w))^{n} of the n-form q by the canonical projection p : 𝔻 → 𝔻/Γ.

We now define two norms for φ ∈ H_{n}(𝔻, Γ) by

∥φ∥_{A_{n}(𝔻,Γ)} = ∫∫ _{ω} |φ(z)|(1 − |z|^{2})^{n−2}dxdy,
∥φ∥_{B_{n}(𝔻,Γ)} = sup_{z∈𝔻}|φ(z)|(1 − |z|^{2})^{n},

where ω is a fundamental domain for Γ, that is, a subdomain of 𝔻 such that ω ∩ γ(ω) = ∅ for every γ ∈ Γ with γ ≠ id, ⋃_{γ∈Γ} γ(ω̄) = 𝔻 and ∂ω is of zero area. We denote by A_{n}(𝔻, Γ) and B_{n}(𝔻, Γ) the subsets of H_{n}(𝔻, Γ) consisting of φ with finite norm ∥φ∥_{A_{n}(𝔻,Γ)} and ∥φ∥_{B_{n}(𝔻,Γ)}, respectively. It is easy to see that these become complex Banach spaces. When Γ is the trivial group 1, we write A_{n}(𝔻) and B_{n}(𝔻) for A_{n}(𝔻, 1) and B_{n}(𝔻, 1), respectively.

The definition of the spaces A_{n}(𝔻) and B_{n}(𝔻) can be extended for hyperbolic Riemann surfaces R. Let H_{n}(R) denote the set of holomorphic n-forms on R and set

∥φ∥_{A_{n}(R)} = ∫∫ _{R} |φ(w)|ρ_{R}(w)^{2−n}dxdy,
∥φ∥_{B_{n}(R)} = sup_{w∈R} |φ(w)|ρ_{R}(w)^{−n}

for φ = φ(w)dw^{n} in H_{n}(R). Here, we should note that |φ(w)|ρ_{R}(w)^{−n} does not depend on the choice of the local coordinate w, in other words, |φ|ρ_{R}^{−n} can be regarded as a function on R.

The Banach spaces A_{n}(𝔻, Γ) and A_{n}(𝔻/Γ) (resp. B_{n}(𝔻, Γ) and B_{n}(𝔻/Γ)) are isometrically isomorphic through the pull-back p_{n}^{∗} by the projection p : 𝔻 → 𝔻/Γ. Also, the following invariance property is convenient to note.

Lemma 1.3.1. Let R and S be hyperbolic Riemann surfaces and let p : R → S be a conformal homeomorphism. Then the pullback operator p_{n}^{∗} : B_{n}(S) → B_{n}(R) is a linear isometry, in other words,

∥p_{n}^{∗}φ∥_{B_{n}(R)} = ∥φ∥_{B_{n}(S)}, φ ∈ B_{n}(S).

In the theory of Teichmüller spaces, it is important to consider the spaces A_{2} and B_{2} as we shall see later. A 2-form q(w)dw^{2} is traditionally called a quadratic differential.

1.4. Univalent functions. In connection with the universal Teichmüller space, the theory of univalent functions is of particular importance. The best textbook in this direction is [68] by O. Lehto.

We denote by 𝒮 the set of analytic univalent functions f on the unit disk so normalized that f(0) = 0 and f′(0) = 1. An analytic function f around the origin is said to be strongly normalized if f(0) = f′(0) − 1 = f′′(0) = 0. Let 𝒮_{0} be the subset of 𝒮 consisting of strongly normalized functions. For f ∈ 𝒮 , the function g = f/(1 + af), where a = f′′(0)/2, is strongly normalized but not necessarily analytic in 𝔻. It is thus natural to consider the wider class

𝒮̃_{0} = {f : meromorphic and univalent in 𝔻 and strongly normalized}

than 𝒮_{0}.

The following meromorphic counterpart is also useful in the theory of univalent functions. Let Σ be the set of meromorphic univalent functions F on the exterior 𝔻^{∗} = {ζ ∈ ℂ̂ : |ζ| > 1} of the unit disk so normalized that

F (ζ) = ζ + b_{0} + \frac{b_{1}}{ζ} + \frac{b_{2}}{ζ^{2}} + … (1.4.1)

in |ζ| > 1.

For f ∈ 𝒮 , the function F (ζ) = 1/f(1/ζ) belongs to Σ and satisfies the condition 0 ∉ F (𝔻^{∗}), and vice versa. Let Σ′ denote the set of those functions F ∈ Σ that satisfy 0 ∉ F (𝔻^{∗}). Moreover, b_{0} = 0 for a function F (ζ) = ζ+b_{0}+b_{1}/ζ+… in Σ if and only if f ∈ 𝒮̃_{0}, where f(z) = 1/F (1/z). Hence, if we set

Σ_{0} = {F ∈ Σ : F (ζ) − ζ → 0 as ζ → ∞},

the correspondence f(z) ↦ F (ζ) = 1/f(1/ζ) gives bijections of 𝒮̃_{0} onto Σ_{0} and of 𝒮_{0} onto Σ′_{0}, where we define Σ′_{0} = Σ_{0} ∩ Σ′.

1.5. Grunsky inequality. For a meromorphic function F near the point at infinity with an expansion of the form (1.4.1), we take a single-valued branch of log((F (ζ) − F (ω))/(ζ − ω)) in |ζ| > R and |ω| > R for sufficiently large R > 0 and expand it in the form

log \frac{F (ζ) − F (ω)}{ζ − ω} = − ∑_{j=1}^{∞} ∑_{k=1}^{∞} \frac{b_{j,k}}{ζ^{j}ω^{k}}

there. The coefficients b_{j,k} are called the Grunsky coefficients of F. It is easy to see that b_{j,k} = b_{k,j} and b_{1,k} = b_{k} for j, k ≥ 1, where b_{k} is the coefficient in (1.4.1). The last relation is deduced in the following way. If we write F (ζ) = ζ+b_{0}+G(ζ), then G(ζ) = O(|ζ|^{−1}) as ζ → ∞. Fix ω for a moment. Since

log \frac{F (ζ) − F (ω)}{ζ − ω} = log (1 + \frac{G(ζ) − G(ω)}{ζ − ω}) = \frac{G(ζ) − G(ω)}{ζ − ω} + O(|ζ|^{−2}),

we obtain

∑_{k=1}^{∞} \frac{b_{1,k}}{ω^{k}} = − lim_{ζ→∞} ζ log \frac{F (ζ) − F (ω)}{ζ − ω}

= − lim_{ζ→∞} ζ \frac{G(ζ) − G(ω)}{ζ − ω}

= G(ω),

from which the required relation follows.

The following theorem is greatly useful in the theory of Teichmüller spaces as well as the theory of univalent functions. See [42], [29] or [87] for the proof and applications.

Theorem 1.5.1 (Grunsky). A meromorphic function F (ζ) with expansion of the form (1.4.1) around ζ = ∞ is analytically continued to a univalent meromorphic function in |ζ| > 1 if and only if the inequality

∑_{k=1}^{∞} k |∑_{j=1}^{∞} b_{j,k}x_{j}|^{2} ≤ ∑_{j=1}^{∞} \frac{|x_{j}|^{2}}{j} (1.5.2)

holds for an arbitrary sequence of complex numbers x_{1}, x_{2}, ….

The inequality in (1.5.2) is known as the strong Grunsky inequality. Noting b_{1,k} = b_{k}, we take (x_{1}, x_{2}, x_{3}, …) = (1, 0, 0, …) to obtain

∑_{k=1}^{∞} k|b_{k}|^{2} ≤ 1. (1.5.3)

This inequality is known as Gronwall’s area theorem.

It is also known that inequality (1.5.2) can be replaced in the above theorem by the (classical) Grunsky inequality:

|∑_{j=1}^{∞} ∑_{k=1}^{∞} b_{j,k}x_{j}x_{k}| ≤ ∑_{j=1}^{∞} \frac{|x_{j}|^{2}}{j} . (1.5.4)

The symmetric matrix (\sqrt{jk}b_{j,k}) defines a linear operator on ℓ^{2}, where b_{j,k} are the Grunsky coefficient of a meromorphic function F (ζ) around ζ = ∞. This is sometimes called the Grunsky operator and will be denoted by G[F] in the following. The strong Grunsky inequality says that F ∈ Σ if and only if G[F] is a bounded linear operator on ℓ^{2} with operator norm ≤ 1. Here, the operator norm ∥G[F]∥ of G[F] is defined by

∥G[F]∥^{2} = sup_{∥y∥_{2}=1} ∑_{k=1}^{∞} |∑_{j=1}^{∞} \sqrt{jk}b_{j,k}y_{j}|^{2} ,

where ∥y∥_{2} = (∑_{k} |y_{k}|^{2})^{1/2} for y = (y_{1}, y_{2}, …). Thus, F ∈ Σ ⇔ ∥G[F]∥ ≤ 1. It is known (cf. [87]) that F has a quasiconformal extension to ℂ̂ if and only if ∥G[F]∥ < 1.

1.6. Schwarzian derivative. For a non-constant meromorphic function f on a domain, we define T_{f} and S_{f} by

T_{f} = \frac{f′′}{f′} = (log f′)′,
S_{f} = (T_{f})′ − \frac{1}{2}(T_{f})^{2} = \frac{f′′′}{f′} − \frac{3}{2} (\frac{f′}{f} )^{2} .

These are called the pre-Schwarzian derivative and the Schwarzian derivative of f, respectively. Note that T_{f} is analytic at a finite point z_{0} if and only if f is analytic and injective around z_{0}. Similarly, S_{f} is analytic at z_{0} if and only if f is meromorphic and injective around z_{0}. The following two lemmas show usefulness of these operations.

Lemma 1.6.1. Let f be a non-constant meromorphic function on a domain D. The pre-Schwarzian derivative of f vanishes on D if and only if f is (the restriction of) a similarity. The Schwarzian derivative of f vanishes on D if and only if f is (the restriction of) a Möbius transformation.

Lemma 1.6.2. Let f and g be non-constant meromorphic functions for which the composition f ∘ g is defined. Then

T_{f∘g} = (T_{f}) ∘ g · g′ + T_{g} = g_{1}^{∗}(T_{f}) + T_{g},
S_{f∘g} = (S_{f}) ∘ g · (g′)^{2} + S_{g} = g_{2}^{∗}(S_{f}) + S_{g}.

Combining these lemmas, we observe that S_{L∘f∘M} = M_{2}^{∗}(S_{f}) for Möbius transformations L and M. Thus the Schwarzian derivative behaves like a quadratic differential.

2. The universal Teichmüller space

We have two choices to develop the theory of the (universal) Teichmüller space; the unit disk model or the upper half-plane model. Although they can be translated into each other, in principle, via the Möbius transformation z ↦ (z − i)/(z + i), both models have their own advantage and thus can be chosen at will according to the purpose. In the present survey, we will take the unit disk model to connect with the theory of univalent functions in a direct way.

2.1. Definition 1: the quotient space of quasiconformal maps. We denote by QC(𝔻) the set of quasiconformal automorphisms of the unit disk 𝔻. As we will observe later, every function in QC(𝔻) extends to a unique homeomorphism of the closed unit disk \overline{𝔻}. Thus, we may think that every f ∈ QC(𝔻) is a self-homeomorphism of the closed unit disk \overline{𝔻}. Two functions f and g in QC(𝔻) are said to be Teichmüller equivalent and denoted by f∼^{T}g if there exists a disk automorphism L ∈ Aut(𝔻) such that g = L ∘ f on ∂𝔻. The quotient space QC(𝔻)/∼^{T} is a model of the universal Teichmüller space and will be denoted by T_{1} for a moment. The equivalence class represented by f ∈ QC(𝔻) will be denoted by [f] below.

Let f, g ∈ QC(𝔻). The Teichmüller distance between p = [f] and q = [g] is defined by

d_{1}(p, q) = inf_{f_{1}∼^{T}f,g_{1}∼^{T}g} \frac{1}{2} log K(g_{1} ∘ f_{1}^{−1}).

Recall here that K(f) denotes the maximal dilatation of f. By a compactness property of quasiconformal maps, one can check that d_{1}(p, q) is indeed a distance on T_{1}. In this way, T_{1} becomes a metric space. It can also be shown that T_{1} is a complete metric space with metric d_{1} by a normality property of the set of normalized K-quasiconformal automorphisms of ℂ (see [70]).

2.2. Definition 2: quasisymmetric functions. The notion of quasisymmetric functions was created by Beurling and Ahlfors [18] for functions on the real line. We give here a corresponding definition of quasisymmetric functions on the unit circle. A sense-preserving homeomorphism h of the unit circle ∂𝔻 is called quasisymmetric if

\frac{1}{M} ≤ \frac{|h(e^{i(s+t)}) − h(e^{is})|}{|h(e^{is}) − h(e^{i(s−t)})|} ≤ M, s ∈ ℝ, 0 < t < \frac{π}{2}

for a constant M ≥ 1. The set of all quasisymmetric funtions on the unit circle will be denoted by QS(∂𝔻). The main result in [18] can be stated as follows.

Theorem 2.2.1 (Beurling-Ahlfors). The restriction of a quasiconformal automorphism of the unit disk to the unit circle is quasisymmetric. Conversely, a quasisymmetric function on the unit circle can be extended to a quasiconformal automorphism of the unit disk.

Two functions h_{1} and h_{2} on the unit circle are called Möbius equivalent if there exists a disk automorphism L ∈ Aut(𝔻) such that h_{2} = L ∘ h_{1}. Let T_{2} denote the quotient space of QS(∂𝔻) by the Möbius equivalence. By the above theorem of Beurling and Ahlfors, one readily sees that T_{1} can be identified with T_{2} in a natural manner.

In order to get rid of taking quotient, we can define T_{2} as follows. A (sensepreserving) homeomorphism h of ∂𝔻 is said to be normalized if h fixes the points 1, −1 and −i. Since every Möbius equivalence class of quasisymmetric functions is represented by a unique normalized one, one can identify T_{2} with the set of normalized quasisymmetric functions on the unit circle.

See [37] for a modern treatment of quasisymmetric functions.

2.3. Definition 3: marked quasidisks. A simply connected domain D in ℂ̂ is called a quasidisk if D is the image of the unit disk under a quasiconformal automorphism of ℂ̂. If D is the image under a K-quasiconformal automorphism, then D is called a K-quasidisk. Many characteristic properties of quasidisks are known. See, for instance, [40].

Let D be a quasidisk (or a Jordan domain more generally) and x_{1}, x_{2}, x_{3} are positively ordered (distinct) points on ∂D. The quadruple (D, x_{1}, x_{2}, x_{3}) will be called a marked quasidisk. By the Riemann mapping theorem and the Carathéodory extension theorem, there exists a unique conformal homeomorphism g : ℍ → D with g(0) = x_{1}, g(1) = x_{2} and g(∞) = x_{3}.

We denote by Q the set of all marked quasidisks in ℂ̂. Two marked quasidisks (D, x_{j}) and (D′, x′_{j}) are said to be Möbius equivalent if D′ = L(D) and x′_{j} = L(x_{j}), j = 1, 2, 3, for some Möbius transformation L ∈ Möb = Aut(ℂ̂). We can define a pseudo-metric on Q by

d((D, x_{j}), (D′, x′_{j})) = ∥S_{f}∥_{B_{2}(D)},

where f is a conformal homeomorphisms of D onto D′ with f(x_{j}) = x′_{j}. It is easy to see that d(D, D′) = 0 if and only if D and D′ are Möbius equivalent.

The set T_{3} of Möbius equivalence classes of all marked quasidisks constitutes another model of the universal Teichmüller space and the above-defined pseudometric gives a metric on T_{3}, which will be denoted by d_{3}, i.e.,

d_{3}(p, q) = inf_{(D,x_{j})∈p,(D′,x′_{j})∈q} d((D, x_{j}), (D′, x′_{j}))

for p, q ∈ T_{3}.

We can again take a suitable normalization to avoid the process of quotient and even marking. For instance, we may say that a quasidisk D is normalized if its boundary contains the points 0, 1 and ∞ in positive order along the boundary curve. If we denote by Q_{0} the set of normalized quasidisks in ℂ̂, then T_{3} can be identified with Q_{0} naturally, and the restriction of the distance d on Q_{0} corresponds to the distance d_{3} on T_{3}.

In the above, the marking is important. For two simply connected hyperbolic domains D_{1} and D_{2}, we set

d(D, D′) = inf_{f:D→D′ conformal} ∥S_{f}∥_{B_{2}(D)}.

It is easy to see that d is a pseudo-distance. Lehto [68] posed a question whether or not d(D, D′) = 0 implies that D and D′ are Möbius equivalent. Osgood and Stowe [83] answered to this question in the negative (see also [19]).

2.4. Definition 4: Bers embedding. Let D be a hyperbolic domain in ℂ̂. We define a subset T (D) of B_{2}(D) to consists of those holomorphic quadratic differentials φ(z)dz^{2} on D such that φ = S_{f} for some univalent meromorphic function f on D which extends to a quasiconformal automorphism of the Riemann sphere. Note that ∥S_{f}∥_{B_{2}(D)} ≤ 12 for every univalent meromorphic function f on D (see §5.2 and [10]). By Lemmas 1.6.1 and 1.6.2, for a Möbius transformation L, the pull-back L_{2}^{∗} gives an isometoric isomorphism of B_{2}(L(D)) onto B_{2}(D). In particular, for a circle domain Δ, that is, the interior or the exterior of a circle, or a half-plane, the space B_{2}(Δ) is isomorphic, say, to B_{2}(𝔻^{∗}). The space T_{4} = T (𝔻^{∗}) (or its equivalent) is a model of the universal Teichmüller space and known as the Bers embedding of the universal Teichmüller space.

Ahlfors [2] showed the following.

Theorem 2.4.1. T (𝔻^{∗}) is a bounded, connected and open subset of B_{2}(𝔻^{∗}).

Thus T_{4} = T (𝔻^{∗}) inherits a complex structure and a metric from B_{2}(𝔻^{∗}). We denote by d_{4} the distance, namely, d_{4}(φ, ψ) = ∥φ − ψ∥_{B_{2}(𝔻^{∗})} for φ, ψ ∈ T_{4}. Since T (𝔻^{∗}) is bounded, the distance d_{4} is not complete.

2.5. Equivalence of T_{1} through T_{4}. We see now that the above definitions of the universal Teichmüller space are all equivalent. Firstly, consider the restriction map QC(𝔻) → QS(∂𝔻) defined by f ↦ f|_{∂𝔻}. Then this map yields a bijection of T_{1} onto T_{2}.

Secondly, we see the equivalence of T_{3} and T_{4}. For φ ∈ T_{4} = T (𝔻^{∗}), by definition, there exists a quasiconformal map f of ℂ̂ fixing 0, 1, ∞ such that f is conformal on 𝔻^{∗} and satisfies S_{f} = φ. Then the image D = f(𝔻^{∗}) is a normalized quasidisk. Therefore, the correspondence φ ↦ D gives a map T_{4} → T_{3}. We next show that this map is bijective. Suppose that a normalized quasidisk D is given. By definition, D = h(𝔻^{∗}) for some quasiconformal map h of ℂ̂ with h(1) = 0, h(−1) = 1 and h(−i) = ∞. Let μ = μ_{h}|_{𝔻^{∗}} and set f = h ∘ (w_{μ})^{−1} : ℂ̂ → ℂ̂. Then f is quasiconformal map of ℂ̂, is conformal on w_{μ}(𝔻^{∗}) = 𝔻^{∗} and satifies f(1) = 0, f(−1) = 1 and f(−i) = ∞. Therefore, φ = S_{f} belongs to T_{4} = T (𝔻^{∗}). In this way, we obtain the map of T_{3} into T_{4}, which is obviuosly the inverse map of the previously defined map of T_{4} to T_{3}. We have now concluded that T_{3} and T_{4} are equivalent by those maps.

Finally, we connect T_{1} with T_{4}. Let h ∈ QC(𝔻). We define μ ∈ Belt(ℂ) by

μ = \begin{cases}μ_{h} & on 𝔻, \\ 0 & on 𝔻^{∗} \end{cases}

and define a quasiconformal map f : ℂ̂ → ℂ̂ by f = f^{μ}, where f^{μ} was defined in §1.1. Since f is conformal in 𝔻^{∗}, the Schwarzian derivative S_{f} belongs to T_{4} = T (𝔻^{∗}). Note that f ∘h^{−1} is conformal in 𝔻 by construction. Let h_{1} ∈ QC(𝔻) be Teichmüller equivalent to h and define f_{1} in the same way as above. We claim now that S_{f_{1}} = S_{f}. By assumption, h_{1} = L ∘ h on ∂𝔻 for an L ∈ Aut(𝔻). Define a map g : ℂ̂ → ℂ̂ by

g = \begin{cases}f_{1} ∘ f^{−1} & on ℂ̂ \ f(𝔻), \\ f_{1} ∘ h_{1}^{−1} ∘ L ∘ h ∘ f^{−1} & on f(𝔻). \end{cases}

It is clear that g is conformal on f(𝔻) and f(𝔻^{∗}). Furthermore, since h_{1}^{−1}∘L∘h = id on ∂𝔻, the map g is continuous on ℂ̂. Since C = f(∂𝔻) and g(C) = f_{1}(∂𝔻) are both quasicircles, it turns out that g is quasiconformal in ℂ̂. Since μ_{g} = 0 a.e., we conclude that g is conformal, hence, a Möbius map. Because of the relation f_{1} = g ∘ f on f(𝔻^{∗}), S_{f} = S_{f_{1}} follows as required.

In this way, we obtain the mapping of T_{1} to T_{4} : [h] ↦ S_{f|_{𝔻}^{∗}}. It is not difficult to see that this mapping is bijective. This map is called the Bers embedding.

3. Analytic properties of the Bers embedding

3.1. The Teichmüller space of a Riemann surface. It is beyond the scope of the present survey to develop the theory of Teichmüller spaces of Riemann surfaces in full generality. Here, our forcus will be on the Bers embedding of the Teichmüller space of a Riemann surface. See [76], [45], [35], [36] for general properties of Teichmüller spaces. See also [1], [115] for a differential geometric approach, [95] for an algebraic approach.

For simplicity, we assume a Riemann surface R to be hyperbolic, in other words, there exists a torsion-free Fuchsian group Γ acting on 𝔻 such that 𝔻/Γ is conformally equivalent to R. Thus, we can identify R with 𝔻/Γ. We denote by p : 𝔻 → 𝔻/Γ = R the canonical projection. Two quasiconformal maps f_{j} : R → S_{j}, j = 1, 2, are called Teichmüller equivalent if there exists a conformal homeomorphism g : S_{1} → S_{2} such that f_{2}^{−1} ∘ g ∘ f_{1} : R → R is homotopic to the identity relative to the ideal boundary. We omit the explanation of the term “relative to the ideal boundary”. See the references given above for details. Also,

[33] gives several useful equivalent conditions for that.

The Teichmüller space Teich(R) of R is defined as the set of all the Teichmüller equivalence classes of such quasiconformal maps of R onto another surface.

Suppose that f_{1} : R → S_{1} and f_{2} : R → S_{2} are quasiconformal maps. Let Γ_{j} be a Fuchsian model of S_{j} acting on 𝔻 and h_{j} : 𝔻 → 𝔻 be a lift of f_{j}, namely, p_{j} ∘ h_{j} = f_{j} ∘ p, where p_{j} : 𝔻 → 𝔻/Γ_{j} = S_{j} is the canonical projection. Then, it is known that f_{1} and f_{2} are Teichmüller equivalent if and only if h_{1} and h_{2} are Teichmüller equivalent in the sense of §2.1. Note that h_{j} ∘ γ ∘ h_{j}^{−1} ∈ Γ_{j} for each γ ∈ Γ, namely, h_{j}Γh_{j}^{−1} = Γ_{j}.

Set

QC(𝔻, Γ) = {h ∈ QC(𝔻) : hΓh^{−1} is Fuchsian}

and denote by Teich(Γ) the quotient space QC(𝔻, Γ)/∼^{T}. As we have seen, Teich(R) and Teich(Γ) are canonically isomorphic through the universal covering projection p : 𝔻 → 𝔻/Γ = R. Also, Teich(Γ) is naturally contained in Teich(1) = T_{1}. In this sense, the universal Teichmüller space T (𝔻^{∗}) contains all the Teichmüller space of an arbitrary hyperbolic Riemann surface.

By using (1.1.1), the complex dilatation of f ∈ QC(𝔻, Γ) is seen to be contained in

Belt(𝔻, Γ) = \{μ ∈ Belt(𝔻) : (μ ∘ γ)\overline{γ′}/γ′ = μ ∀γ ∈ Γ\}.

Furthermore, for h ∈ QC(𝔻, Γ), let f be the function constructed in §2.5 and let γ ∈ Γ. Since f and γ ∘f ∘γ^{−1} has the same complex dilatation, γ ∘f ∘γ^{−1} = L∘f for an L ∈ Aut(ℂ̂) = Möb by Lemma 1.1.2. Lemma 1.6.2 now implies that γ_{2}^{∗}(S_{f}) = S_{f}. Therefore, S_{f} is contained in the closed subspace B_{2}(𝔻^{∗}, Γ) of B_{2}(𝔻^{∗}) defined in §1.3. As in the previous section, we see that S_{f} depends only on the Teichmüller equivalence class of h in QC(𝔻, Γ) and the corresponding h ↦ S_{f} is one-to-one, we obtain an embedding β_{Γ} : Teich(𝔻, Γ) → B_{2}(𝔻^{∗}, Γ), which is called the Bers embedding of Teich(𝔻, Γ). We set T (𝔻^{∗}, Γ) = β_{Γ}(Teich(𝔻, Γ)).

Bers [15] showed that T (𝔻^{∗}, Γ) is a bounded domain in B_{2}(𝔻^{∗}, Γ). It is obvious that T (𝔻^{∗}, Γ) is contained in T (𝔻^{∗}) by definition. Indeed, by using the Douady-Earle extension [28], it can be seen that

T (𝔻^{∗}, Γ) = T (𝔻^{∗}) ∩ B_{2}(𝔻^{∗}, Γ)

and that T (𝔻^{∗}, Γ) is contractible.

3.2. Relationship with quasi-Teichmüller spaces. In view of the description of the set T (𝔻^{∗}, Γ), it may be natural to consider the following sets more generally. Let D be a hyperbolic domain in ℂ̂ and let G be a subgroup of Aut(D). Typically, G is a Kleinian group and D is a connected component of its region of discontinuity. Then we set (cf. [100])

S(D, G) = {φ ∈ B_{2}(D, G) : ∃f : D → ℂ̂ s.t. φ = S_{f} and f is univalent in D},
T (D, G) = {φ ∈ B_{2}(D, G) : ∃f : D → ℂ̂ s.t. φ = S_{f} and f extends to a qc map of ℂ̂},

For a circle domain Δ and a Fuchsian group Γ acting on Δ, the set S(Δ, Γ) sometimes called the quasi-Teichmüller space of Γ. (But, note that this terminology is not popular.) Clearly, T (D, G) ⊂ S(D, G). It is easy to see that S(𝔻^{∗}) is closed while, as Ahlfors showed, T (𝔻^{∗}) is open in B_{2}(𝔻^{∗}). The boundary of T (Δ, Γ) in B_{2}(Δ, Γ) is called the Bers boundary and is important in relation with the deformation theory of Kleinian groups (see [16]).

When G is the trivial group 1, we write S(D), T (D) for S(D, 1), T (D, 1), respectively. Note that under the mapping f ↦ S_{f}, the sets 𝒮̃_{0} and Σ_{0} correspond to S(𝔻) and S(𝔻^{∗}), respectively, in one-to-one fashion. It is a challenging problem to characterize those functions f in 𝒮 whose Schwarzian derivatives lie on ∂T (𝔻). See [7] and [43] for some attempts.

In 1970’s, it had been a conjecture of Bers [16] that the closure of T (𝔻^{∗}) in B_{2}(𝔻^{∗}) is S(𝔻^{∗}). In 1978, Gehring [39] disproved it. Prior to it, Gehring [38] proved the weaker assertion that the interior of S(𝔻^{∗}) in B_{2}(𝔻^{∗}) coincides with T (𝔻^{∗}). See [34] for a relevant result. Thurston [112] proved the more striking result that S(𝔻^{∗}) even has an isolated point in B_{2}(𝔻^{∗}) (see also [5]). After that, the Bers conjecture was reformulated in the form that the closure of T (𝔻^{∗}, Γ) is equal to S(𝔻^{∗}, Γ) for a cofinite Fuchain group Γ, that is, a finitely generated Fuchsian group of the first kind. (This is nowadays generalized to the Bers-Thurston density conjecture.) Shiga [97] proved a weaker version of it: the interior of S(𝔻^{∗}, Γ) in B_{2}(𝔻^{∗}, Γ) coincides with T (𝔻^{∗}, Γ) for a cofinite Γ. In the line of these studies, the author showed that S(𝔻^{∗}, Γ) \ \overline{T (𝔻^{∗})} ≠ ∅ for a Fuchsian group Γ of the second kind ([101]) and that the interior of S(𝔻^{∗}, Γ) in B_{2}(𝔻^{∗}, Γ) coincides with T (𝔻^{∗}, Γ) for a finitely generated, purely hyperbolic Fuchsian group Γ of the second kind ([102]). Matsuzaki [72] generalized the former to the case of a certain kind of inifinitely generated Fuchsian groups of the first kind. In recent years, a huge amount of progress has been made in the theory of Kleinian groups, which enabled to prove the Bers-Thurston conjecture. See, for instance,

[20] and [81] for partial solutions to the conjecture.

We end the subsection with the following conjecture.

Conjecture 3.2.1. The interior of quasi-Teichmüller space S(𝔻^{∗}, Γ) in B_{2}(𝔻^{∗}, Γ) is equal to the Bers embedding T (𝔻^{∗}, Γ) of the Teichmüller space of a Fuchsian group Γ acting on 𝔻^{∗}.

Note that Zhuravlev (Žuravlev) [119] proved that T (D^{∗}, Γ) is the connected component of the interior of S(𝔻^{∗}, Γ) which contains the origin for an arbitrary Fuchsian group Γ (see also [100]). Thus the conjecture is equivalent to connectedness of the interior of S(𝔻^{∗}, Γ).

3.3. The Bers projection. Let D be a hyperbolic domain in ℂ̂ and denote by E its complement in ℂ̂. We define the map Φ : Belt(E) → B_{2}(D) by Φ(μ) = S_{f^{μ}|_{D}}, where f^{μ} is defined as in §1.1 for μ which is extended to ℂ by setting μ = 0 on D. Recall here that Belt(E) is the open unit ball of the complex Banach space L^{∞}(E) with norm ∥ · ∥_{∞}. It is clear by definition that Φ(Belt(E)) = T (D). The map Φ : Belt(E) → T (D) is called the (generalized) Bers projection. It is known that Φ : Belt(E) → B_{2}(D) is holomorphic (cf. [100]) and that the Fréchet derivative d_{0}Φ : L^{∞}(E) → B_{2}(D) of Φ at the origin is described by

d_{0}Φ[ν](z) = −\frac{6}{π} ∫∫ _{E} \frac{ν(ζ)}{(ζ − z)^{4}} dξdη (ζ = ξ + iη)

for ν ∈ L^{∞}(E). Bers [15] strengthened Ahlfors’ theorem (Theorem 2.4.1) to the following form.

Theorem 3.3.1. The Bers projection Φ : Belt(𝔻) → T (𝔻^{∗}) is a holomorphic split submersion, in other words, the Fréchet derivative of Φ at every point exists and has a (bounded) left inverse.

Indeed, Bers showed the above theorem for the projection Φ : Belt(𝔻, Γ) → T (𝔻^{∗}, Γ) for an arbitrary Fuchsian group Γ. In particular, T (𝔻^{∗}, Γ) is shown to be an open subset of B_{2}(𝔻^{∗}, Γ).

3.4. Convexity. Krushkal [57] proved that the Bers embedding T (𝔻^{∗}) of the universal Teichmüller space is not starlike with respect to any point, and hence, not convex in B_{2}(𝔻^{∗}). For non-starlikeness of general Teichmüller spaces, see Krushkal [60] and Toki [113].

In spite of the above fact, the (Bers embededing of the) Teichmüller spaces enjoy many kinds of convexity properties. We briefly list some of them in this subsection.

The most useful is perhaps the following “disk convexity” due to Zhuravlev [119], which is shown as an application of the Grunsky inequality. A weaker version can be proved also by the λ-lemma (see [104]).

Theorem 3.4.1 (Zhuravlev). Let Γ be a Fuchsian group acting on 𝔻^{∗}. Suppose that a continuous map α : \overline{𝔻} → B_{2}(𝔻^{∗}, Γ) is holomorphic in 𝔻 and satisfies α(∂𝔻) ⊂ S(𝔻^{∗}). Then α(\overline{𝔻}) ⊂ S(𝔻^{∗}, Γ). Furthermore, if α(\overline{𝔻}) ∩ T (𝔻^{∗}) is nonempty, then α(𝔻) ⊂ T (𝔻^{∗}, Γ).

Outline of the proof. For each z ∈ \overline{𝔻}, there exists a unique F_{z} ∈ Σ_{0} such that S_{F_{z}} = α(z). Let ℬ(ℓ^{2}) denote the complex Hilbert space consisting of bounded linear operators on ℓ^{2}. Then the map β : 𝔻 → ℬ(ℓ^{2}) defined by z ↦ G[F_{z}] turns to be holomorphic. Then the (generalized) maximum principle implies that

sup_{z∈𝔻}∥β(z)∥ = sup_{z∈∂𝔻}∥β(z)∥ ≤ 1

and that either ∥β(z)∥ < 1 for all z ∈ 𝔻 or else ∥β(z)∥ = 1 for all z ∈ \overline{𝔻}. Theorem 1.5.1 now yields that α(𝔻) ⊂ S(𝔻^{∗}). If we assume that α(z_{0}) ∈ T (𝔻^{∗}) for some point z_{0} ∈ \overline{𝔻} in addition, then ∥β(z_{0})∥ < 1 and thus ∥β(z)∥ < 1 for all z ∈ 𝔻. This means that α(𝔻) ⊂ T (𝔻^{∗}) ∩ B_{2}(𝔻^{∗}, Γ) = T (𝔻^{∗}, Γ).

We remark that the above argument is a variant of Lehto’s principle (see [13] or [68]).

A more sophisticated application of Grunsky inequality to Teichmüller spaces can be found in [98].

Bers and Ehrenpreis [17] proved that finite dimensional Teichmüller spaces are holomorphically convex. After improvements by Kra [55] and Shiga [96] and others, Krushkal [58] strengthened it by showing that the Teichmüller space of an arbitrary Riemann surface R is complex hyperconvex, that is to say, there exists a negative plurisubharmonic function u(x) on Teich(R) such that u(x) → 0 when x tends to ∞. He proved it by pointing out that the function log tanh(d(x, y)) gives the Green function on Teich(R), where d(x, y) denotes the Teichmüller distance of Teich(R). Krushkal [59] also proved that finite dimensional Teichmüller spaces are polynomially convex. See also [61] for a survey in this direction.

3.5. Teichmüller distance and other natural distances (metrics). In §2, we defined two kinds of distances on the universal Teichmüller space; the Teichmüller distance and the distance induced by the Bers embedding. These distances can be defined for the Teichmüller space of an arbitrary Riemann surface. On the other hand, since Teichmüller spaces have complex structure, it carries natural invariant distances for holomorphic maps (see [46] as a general reference).

Let X be a complex (Banach) manifold. The Kobayashi pseudo-distance d_{K}(x, y) is defined as

inf ∑_{j=1}^{N} d_{𝔻}(z_{j−1}, z_{j}),

where the infimum is taken over all finitely many holomorphic maps f_{j} : 𝔻 → X (j = 1, …, N) which satisfy f_{j}(z_{j}) = f_{j+1}(z_{j})(1 < j < N), f_{1}(z_{0}) = x, and f_{N}(z_{N}) = y. Here, d_{𝔻}(z, w) denotes the Poincaré distance of 𝔻:

d_{𝔻}(z, w) = arctanh |\frac{w − z}{1 − z̄w}| .

The following theorem was proved by Royden [92] for finite dimensional case and by Gardiner (see [35] or [36]) for general case. (For a simple proof using the λ-lemma, see [32].)

Theorem 3.5.1. The Kobayashi pseudo-distance of the Teichmüller space of a Riemann surface is equal to the Teichmüller distance.

For other invariant metrics on Teichmüller spaces, see [76, Appendix 6].

Earle [31] proved that the Carathéodory (pseudo)distance of the Teichmüller space of an arbitrary Fuchsian group is complete.

The Weil-Petersson metric is another important (Riemannian) metric on finite dimensional Teichmüller spaces. Since the complex structure of the Teichmüller space of a general Riemann surface is modelled on a complex Banach space which may not be reflexive, this metric cannot be defined on general Teichmüller spaces unless the structure of the space is changed. However, some attempts were made to construct analogs of the Weil-Petersson metric on the universal Teichmüller space, see [77], [78], [109], [110].

4. Pre-Schwarzian models

The Schwarzian derivative plays an important role in the definition of the Teichmüller space. But, it is not easy to treat with Schwarzian derivative, in general, because of its complicated form. Therefore, some attempts of replacing Schwarzian by pre-Schwarzian have been made. See [118] and [6]. Though the pre-Schwarzian model is sometimes called “poor man’s model” (cf. [43]) since it does not have much invariance, this model is interesting in connection with geometric function theory.

When dealing with pre-Schwarzian derivative, the point at infinity plays a special role. Therefore, we have to consider the case ∞ ∈ D separately.

4.1. The models T̂(𝔻) and T̂(ℍ). Let Δ be a disk or a half-plane in ℂ. Set

Ŝ(Δ) = {T_{f} : f : Δ → ℂ is holomorphic and univalent}

and

T̂(Δ) = {T_{f} : f : Δ → ℂ is holomorphic and extends to a qc map of ℂ̂}.

Here, T_{f} denotes the pre-Schwarzian derivative of f (see §1.6). By definition, T̂(Δ) ⊂ Ŝ(Δ).

We recall that the pre-Schwarzian derivative vanishes only when the function is affine. Since each circle domain in ℂ is similar (affinely equivalent) to either the unit disk 𝔻 or the half-plane ℍ = {z ∈ ℂ : Im z > 0}. Therefore, we may restrict ourselves on the two cases Δ = 𝔻 and ℍ. First let f ∈ 𝒮 . By the well-known inequality (cf. [29])

|(1 − |z|^{2})T_{f}(z) − 2z̄| ≤ 4, (4.1.1)

we obtain ∥T_{f}∥_{B_{1}(𝔻)} ≤ 6. In particular, Ŝ(𝔻) ⊂ B_{1}(𝔻). Note also that the constant 6 is sharp as the Koebe function K(z) = z/(1 − z)^{2} shows. It is easy to see that Ŝ(𝔻) is closed in B_{1}(𝔻).

Let L(z) = (z − i)/(z + i). Note that ∥T_{L}∥_{B_{1}(ℍ)} = 4 and hence T_{L} ∈ T̂(ℍ). Since L_{1}^{∗} : B_{1}(𝔻) → B_{1}(ℍ) is a linear isometry and T_{f∘L} = L_{1}^{∗}(T_{f}) + T_{L}, the space T̂(ℍ) is contained in B_{1}(ℍ) and it is isometrically equivalent to T̂(𝔻). In this sense, it is enough to consider only T̂(𝔻).

We define the map π : B_{1}(𝔻) → B_{2}(𝔻) by π(ψ) = ψ′ − ψ^{2}/2. By definition, π(Ŝ(𝔻)) = S(𝔻) and π(T̂(𝔻)) = T (𝔻). Duren, Shapiro and Shields [30] noticed that this map is continuous (see also §5.3).

Astala and Gehring [6] proved an analogous result to the case of Schwarzian derivative.

Theorem 4.1.2. The interior of Ŝ(𝔻) in B_{1}(𝔻) is equal to T̂(𝔻), while the closure of T̂(𝔻) in B_{1}(𝔻) is not equal to Ŝ(𝔻). Moreover, ∂T (𝔻) \ π(∂T̂(𝔻)) is not empty.

Zhuravlev [118] revealed the following remarkable property of T̂(𝔻).

Theorem 4.1.3 (Zhuravlev). The space T̂(𝔻) decomposes into the uncountably many connected components T̂_{0} and T̂_{ω}, ω ∈ ∂𝔻, where

T̂_{0} = {T_{f} ∈ T̂(𝔻) : f(𝔻) is bounded } and T̂_{ω} = {T_{f} ∈ T̂(𝔻) : f(z) → ∞ as z → ω}.

Moreover, {ψ ∈ B_{1}(𝔻) : ∥ψ − ψ_{ω}∥_{B_{1}(𝔻)} < 1} ⊂ T̂_{ω} holds for each ω ∈ ∂𝔻, where ψ_{ω}(z) = 2ω̄/(1 − ω̄z) is the pre-Schwarzian derivative of the function z/(1 − ω̄z).

Note that the map π is not injective even in each connected component of T̂(𝔻). Therefore, we should note that this model of the universal Teichmüller space has some redundancy.

4.2. The model T̂(𝔻^{∗}). There is some subtlety in consideration of the pre-Schwarzian model of the universal Teichmüller space T̂(𝔻^{∗}) on the exterior 𝔻^{∗} of the unit circle. The first thing to note is the fact that the Banach space B_{1}(𝔻^{∗}) is not the right space on which T̂(𝔻^{∗}) is modeled. We define

Ŝ(𝔻^{∗}) = {T_{F} : F ∈ Σ}

and

T̂(𝔻^{∗}) = {T_{F} : F ∈ Σ extends to a quasiconformal map of ℂ̂}.

If F (ζ) = ζ + b_{0} + b_{1}/ζ + b_{2}/ζ^{2} + …, then T_{F} (ζ) = 2b_{1}/ζ^{3} + ⋯ = O(ζ^{−3}) as ζ → ∞. Therefore, the norm

B(ψ) = sup_{ζ∈𝔻^{∗}}(|ζ|^{2} − 1)|ζψ(ζ)| (4.2.1)

is more natural. Indeed, Becker’s univalence criterion [12] and Avhadiev’s inequality [8]

(|ζ|^{2} − 1) |\frac{ζF′′(ζ)}{F′(ζ)}| ≤ 6 (4.2.2)

imply the following result.

Theorem 4.2.3. If a meromorphic function F (ζ) = ζ + b_{0} + b_{1}/ζ + … in |ζ| > 1 satisfies B(T_{F}) ≤ 1, then F ∈ Σ. Conversely, every function F in Σ satisfies B(T_{F}) ≤ 6.

We set

B′_{1}(𝔻^{∗}) = {ψ ∈ B_{1}(𝔻^{∗}) : lim_{ζ→∞} ζ^{2}ψ(ζ) = 0}.

Then, it is easy to see that B′_{1}(𝔻^{∗}) = {ψ : 𝔻^{∗} → ℂ holomorphic and B(ψ) < ∞}. The above theorem now yields that Ŝ(𝔻^{∗}) is a bounded subset of B′_{1}(𝔻^{∗}).

We define π : B′_{1}(𝔻^{∗}) → B_{2}(𝔻^{∗}) as before by π(ψ) = ψ′ − ψ^{2}/2. Then π is continuous [13, Lemma 6.1]. By definition, π(Ŝ(𝔻^{∗})) = S(𝔻^{∗}) and π(T̂(𝔻^{∗})) = T (𝔻^{∗}). Since T (𝔻^{∗}) is an open set and T̂(𝔻^{∗}) = π^{−1}(T (𝔻^{∗})), the set T̂(𝔻^{∗}) is also open in B′_{1}(𝔻^{∗}). In this way, we see that the space T̂(𝔻^{∗}) is a complex Banach manifold modeled on B′_{1}(𝔻^{∗}). We remark that π does not map B_{1}(𝔻^{∗}) into B_{2}(𝔻^{∗}).

The set T̂(𝔻^{∗}) seems to be less investigated, but could be more useful. For instance, the mapping F ↦ T_{F} sends Σ_{0} to Ŝ(𝔻^{∗}) bijectively. Recall that the mapping F ↦ S_{F} sends Σ_{0} to S(𝔻^{∗}) bijectively. Therefore, the mapping π sends Ŝ(𝔻^{∗}) to S(𝔻^{∗}) bijectively.

4.3. Loci of typical subclasses of 𝒮. Since the differential operator T_{f} is closely related with geometric function theory, many classical subclasses of univalent functions correspond to sets with nice properties in Ŝ(𝔻).

We recall several fundamental classes in univalent function theory. We denote by 𝒜 the set of analytic functions f in the unit disk 𝔻 so normalized that f(0) = 0 and f′(0) = 1. A function f ∈ 𝒜 is called starlike (convex) if f is univalent and if f(𝔻) is starlike with respect to the origin (convex). We denote by 𝒮^{∗} and 𝒦 the sets of starlike and convex functions in 𝒜 , respectively. A function f ∈ 𝒜 is called close-to-convex if e^{iα}f′/g′ has positive real part in 𝔻 for a convex function g and for a real constant α. Denote by 𝒞 the set of close-to-convex functions in 𝒜 . It is known that 𝒞 ⊂ 𝒮 (cf. [29]).

It is interesting to see how pre-Schwarzians of those functions are located in the space Ŝ(𝔻). The following result gives an answer to this question.

Theorem 4.3.1 ([27], [51]). {T_{f} : f ∈ 𝒦 } and {T_{f} : f ∈ 𝒞 } are both convex subsets of Ŝ(𝔻).

It may be natural to conjecture the following.

Conjecture 4.3.2 ([48]). The subset {T_{f} : f ∈ 𝒮^{∗}} of Ŝ(𝔻) is starlike with respect to the origin.

Note that the vector operations in B_{1}(𝔻) is translated to the Hornich operations in the space of uniformly locally univalent functions (see, for example, [48]). Also, see Casey [22] for relations between subclasses of 𝒮 and (the closure) of T̂(𝔻).

5. Univalence criteria

As is well developed in Lehto’s textbook [68], univalence criteria are closely connected with the universal Teichmüller space. The present section will be devoted to this topic.

5.1. Univalence criteria due to Nehari and Ahlfors-Weill. Nehari [79] proved the following result, which is fundamental in the Teichmüller spaces.

Theorem 5.1.1. Every meromorphic univalent function f on the unit disk satisfies the inequality ∥S_{f}∥_{B_{2}(𝔻)} ≤ 6. Conversely, if a meromorphic function f on the unit disk satisfies the inequality ∥S_{f}∥_{B_{2}(𝔻)} ≤ 2, then f must be univalent.

The constants 6 and 2 are sharp since the Koebe function K(z) = z/(1 − z)^{2} satisfies ∥S_{K}∥_{B_{2}(𝔻)} = 6 and since the function f(z) = ((1 + z)/(1 − z))^{iϵ}, ϵ > 0, is never univalent but ∥S_{f}∥_{B_{2}(𝔻)} = 2(1 + ε^{2}) can approach 2 (Hille [44]). The former assertion was first proved by Kraus [56] and reproved by Nehari. Therefore, it is called nowadays the Kraus-Nehari theorem. The Kraus-Nehari theorem is a consequence of the Bieberbach theorem. By the Möbius invariance of (1−|z|^{2})^{2}|S_{f}(z)|, it is enough to show the inequality only at the origin, namely, |S_{f}(0)| ≤ 6 for f ∈ 𝒮 . A straightforward computation gives S_{f}(0) = 6(a_{3} − a_{2}^{2}) for f(z) = z + a_{2}z^{2} + a_{3}z^{3} + …. If we set F (ζ) = 1/f(1/ζ) = ζ + b_{0} + b_{1}/ζ + …, then b_{1} = a_{2}^{2} − a_{3}, and thus the inequality |b_{1}| ≤ 1 (see (1.5.3)) implies the required one.

The class N = {f ∈ 𝒜 : ∥S_{f}∥_{B_{2}(𝔻)} ≤ 2} is sometimes called the Nehari class. Gehring and Pommerenke [41] showed that f ∈ N maps the unit disk conformally onto a Jordan domain unless f(𝔻) is Möbius equivalent to the parallel strip {z : |Im z| < π/4}. For further development, see [24], [25] and [26].

In connection with Nehari’s theorem, Ahlfors and Weill established the following quasiconformal extension criterion. For φ ∈ B_{2}(𝔻^{∗}) with ∥φ∥_{B_{2}(𝔻^{∗})} < 2, we set α[φ] ∈ Belt(𝔻) by α[φ](z) = −ρ_{𝔻}(z)^{−2}φ(1/z̄)z̄^{−4}/2. Note that the map α is the restriction of a bounded linear operator which maps B_{2}(𝔻^{∗}, Γ)_{2} = {φ ∈ B_{2}(𝔻^{∗}, Γ) : ∥φ∥_{B_{2}(𝔻)} < 2} into Belt(𝔻^{∗}, Γ) for every Fuchsian group Γ.

Theorem 5.1.2 (Ahlfors-Weill). The map α : B_{2}(𝔻^{∗})_{2} → Belt(𝔻) is the local inverse of the Bers projection Φ : Belt(𝔻) → T (𝔻^{∗}), in other words, Φ(α[φ]) = φ for φ ∈ B_{2}(𝔻^{∗}) with ∥φ∥_{B_{2}(𝔻^{∗})} < 2.

Corollary 5.1.3. The universal Teichmüller space T(𝔻^{∗}) contains the open ball centered at the origin with radius 2 in B_{2}(𝔻^{∗}).

The map α : B_{2}(𝔻^{∗})_{2} → Belt(𝔻) is sometimes called the Ahlfors-Weill section.

5.2. Inner radius and outer radius. Let D be a hyperbolic domain in ℂ̂. The inner radius σ_{I}(D) and the outer radius σ_{O}(D) of univalence is defined respectively by

σ_{I}(D) = sup{σ ≥ 0 : ∥S_{f}∥_{B_{2}(D)} ≤ σ ⇒ f is univalent in D},
σ_{O}(D) = sup{∥S_{f}∥_{B_{2}(D)} : f : D → ℂ̂ is univalent}.

We also define the number τ(D) ∈ [0, +∞] as ∥S_{p}∥_{B_{2}(𝔻)}, where p is a holomorphic universal covering projection of 𝔻 onto D. The quantity τ(D) is independent of the choice of p and thus well defined. Note that τ(D) < ∞ if and only if ∂D is uniformly perfect (cf. [88] or [105]).

Summarizing theorems of Ahlfors [2], Gehring [38], Nehari [79], we obtain the following.

Theorem 5.2.1. σ_{I}(Δ) = 2, σ_{O}(Δ) = 6, τ(Δ) = 0 hold for a circle domain Δ. Let D be a simply connected hyperbolic domain. Then σ_{O}(Δ) ≤ 12 and τ(D) ≤ 6. Moreover, D is a quasidisk if and only if σ_{I}(D) > 0.

The inequality σ_{O}(Δ) ≤ 12 is shown as follows. Let f : D → ℂ̂ be univalent and set Ω = f(D). Take a conformal map g : 𝔻^{∗} → D and set h = f ∘ g. Then, by Lemmas 1.3.1, 1.6.2 and the Kraus-Nehari theorem, we obtain

∥S_{f}∥_{B_{2}(D)} = ∥g_{2}^{∗}(S_{f})∥_{B_{2}(𝔻^{∗})} = ∥S_{h} − S_{g}∥_{B_{2}(𝔻^{∗})} ≤ ∥S_{h}∥_{B_{2}(𝔻^{∗})} + ∥S_{g}∥_{B_{2}(𝔻^{∗})} ≤ 12.

It is a remarkable fact due to Beardon and Gehring [10] that σ_{O}(D) ≤ 12 holds even for an arbitrary hyperbolic domain D.

The inner and outer radii of univalence are better understood in the context of (quasi-) Teichmüller space.

Lemma 5.2.2. Let g : 𝔻^{∗} → D be a conformal homeomorphism of 𝔻^{∗} onto a simply connected hyperbolic domain D. Then {φ ∈ S(𝔻^{∗}) : ∥φ − S_{g}∥ < σ_{I}(D)} is the maximal open ball centered at S_{g} contained in T (𝔻^{∗}). On the other hand, σ_{O}(D) = max{∥φ − S_{g}∥_{B_{2}(𝔻^{∗})} : φ ∈ S(𝔻^{∗})}.

Lehto [66] proved the following relations.

Theorem 5.2.3. The relation σ_{O}(D) = τ(D) + 6 holds for a simply connected hyperbolic domain D. Furthermore, 2 − τ(D) ≤ σ_{I}(R) ≤ min{2, 6 − τ(D)}.

As for the quantity τ(D), the following are known. For a convex domain D, we have τ(D) ≤ 2. This result is repeatedly re-discovered by many mathematicians; [86], [91], [114], [80], [66]. Suita [108] refined this result by showing the sharp inequality

τ(f(𝔻)) ≤ \begin{cases}2, & 0 ≤ α ≤ 1/2, \\ 8α(1 − α), & 1/2 ≤ α ≤ 1 \end{cases}

for a convex function f ∈ 𝒦 of order α, namely, when Re (1+zf′′(z)/f′(z)) > α.

It is known that τ(D) ≤ 6(K^{2} − 1)/(K^{2} + 1) for a K-quasidisk D (see [68]). See also [69], [52], [23], [73], [9] for other classes of domains.

It is not easy to determine, or even to estimate from below, the value of σ_{I}(D), in general. Known examples are sectors [63], triangles [65], the interiors and the exteriors of regular polygons [21], [65], some other polygonal domains [74], [75], the exteriors of hyperbolas [64].

For a general method of estimating σ_{I}(D) from below, see [67], [68] and [106]. See also [107].

5.3. Pre-Schwarzian counterpart. One can define quantities similarly as in the previous section with respect to pre-Schwarzian derivative. We add the symbol ˆ to indicate it. For instance,

σ̂_{I}(D) = sup{σ ≥ 0 : ∥T_{f}∥_{B_{1}(D)} ≤ σ ⇒ f is univalent in D}

for a hyperbolic domain D in ℂ. In the case when D = 𝔻^{∗}, we adopt the norm B(ψ) :

σ̂_{I}(𝔻^{∗}) = sup{σ ≥ 0 : B(T_{F}) ≤ σ ⇒ f is univalent in 𝔻^{∗}}.

Duren, Shapiro and Shields [30] proved that σ̂_{I}(𝔻) ≥ 2(\sqrt{5}−2) = 0.472 ⋯ by observing that ∥ψ′∥_{B_{2}(𝔻)} ≤ 4∥ψ∥_{B_{1}(𝔻)} and thus π(ψ) = ψ′ − ψ^{2}/2 is a continuous map of B_{1}(𝔻) into B_{2}(𝔻). Note that Wirths [116] found the sharp constant C = (13\sqrt{3} + 55\sqrt{11})/64 = 3.20204 … for the estimate ∥ψ′∥_{B_{2}(𝔻)} ≤ C∥ψ∥_{B_{1}(𝔻)}. Nowadays, the best value for this univalence criterion is known.

Theorem 5.3.1. σ̂_{I}(Δ) = 1 and σ̂_{O}(Δ) = 6 for Δ = 𝔻, ℍ and 𝔻^{∗}.

Becker [11], [12] showed that σ̂_{I}(𝔻) ≥ 1 and σ̂_{I}(𝔻^{∗}) ≥ 1 and Becker-Pommerenke [14] showed that equality hold for Δ = 𝔻 and that σ̂_{I}(ℍ) = 1. Pommerenke [89] showed the sharpness for Δ = 𝔻^{∗}.

By (4.1.1) and the fact that the Koebe function K satisfies ∥T_{K}∥_{B_{1}(𝔻)} = 6, we see that σ̂_{O}(𝔻) = 6. σ̂_{O}(ℍ) = 6 can be seen by noting the relation

∥ψ∥_{B_{1}(ℍ)} = lim_{r→1−} ∥ψ∥_{B_{1}(Δ_{r})}

for ψ ∈ B_{1}(ℍ), where Δ_{r} = {z : |z − i(1 + r^{2})/(1 − r^{2})| < 2r/(1 − r^{2})}. The formula σ̂_{O}(𝔻^{∗}) = 6 follows from the fact that the inequality in (4.2.2) is sharp for each ζ.

For concrete estimates of τ̂(D) for several geometric classes of domains, see [117], [103], [82], [49], [50].

In spite of relative simplicity of the operation T_{f}, very little is known for quantities σ̂_{I}(D) and σ̂_{O}(D). Stowe [99] gave non-trivial examples of domains D for which σ̂_{I}(D) ≥ 1.

5.4. Directions of further investigation. The Bers embedding of Teichmüller spaces is still mysterious. We know very little about the shape of it. Pictures of one-dimensional Teichmüller spaces were recently given in [53] and [54]. Note that the first attempt towards it was done by Porter [90] as early as in 1970’s.

It is an interesting and important problem to describe the intersection of T (Δ) or T̂(Δ) with a (complex) one-dimensional vector subspace of B_{2}(Δ) or B_{1}(Δ) for a circle domain. Completely known examples are essentially, as far as the author knows, the linear hull of 1/(1 − z) in B_{1}(Δ) [93] and the linear hull of z^{−2} in B_{2}(ℍ) in [47], only.

The results presented above could be generalized to various directions. We end this survey with remarks on possible ways to study furthermore.

In this section, we considered mainly the case when the domain is simply connected. When the domain is multiply connected, the problem will become much more difficult. See [84] and [85] for fundamental information.

We were concerned here with only pre-Schwarzian and Schwarzian derivatives. On the other hand, several definitions of higher-order Schwarzian derivatives have been proposed (e.g., [111], [94]). Thus, we may develop the theory for those higher-order Schwarzian derivatives.

Of course, we may consider domains in ℂ^{n} or ℝ^{n} but with great difficulty caused by the lack of canonical metrics such as hyperbolic metric, the lack of Riemann mapping theorem and so on. Note that Martio and Sarvas [71] gave some injectivity conditions even in higher dimensions.

References

1. W. Abikoff, The Real Analytic Theory of Teichmüller Space, Lecture Notes in Mathematics, vol. 820, Springer-Verlag, 1976.

2. L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291–301.

3. , Lectures on Quasiconformal Mappings, van Nostrand, 1966.

4. L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404.

5. K. Astala, Selfsimilar zippers, Holomorphic Functions and Moduli I (D. Drasin, C. J. Earle, F. W. Gehring, I. Kra, and A. Marden, eds.), Springer-Verlag, 1988, pp. 61–73.

6. K. Astala and F. W. Gehring, Injectivity, the BMO norm and the universal Teichmüller space, J. Analyse Math. 46 (1986), 16–57.

7. , Crickets, zippers, and the Bers universal Teichmüller space, Proc. Amer. Math. Soc. 110 (1990), 675–687.

8. F. G. Avhadiev, Conditions for the univalence of analytic functions (Russian), Izv. Vysš. Učebn. Zaved. Matematika 1970 (1970), no. 11 (102), 3–13.

9. R. W. Barnard, L. Cole, K. Pearce, and G. B. Williams, The sharp bound for deformation of a disk under a hyperbolically convex map, to appear in Proc. London Math. Soc.

10. A. F. Beardon and F. W. Gehring, Schwarzian derivatives, the Poincaré metric and the kernel function, Comment. Math. Helv. 55 (1980), 50–64.

11. J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23–43.

12. , Löwnersche Differentialgleichung und Schlichtheitskriterien, Math. Ann. 202 (1973), 321–335.

13. , Conformal mappings with quasiconformal extensions, Aspects of Contemporary Complex Analysis, Proc. Conf. Durham, 1979, Academic Press, 1980, pp. 37–77.

14. J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math. 354 (1984), 74–94.

15. L. Bers, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113–134.

16. , On boundaries of Teichmüller spaces and on Kleinian groups: I, Ann. of Math. (2) 91 (1970), 570–600.

17. L. Bers and L. Ehrenpreis, Holomorphic convexity of Teichmüller spaces, Bull. Amer. Math. Soc. 70 (1964), 761–764.

18. A. Beurling and L. V. Ahlfors, The boundary correspondence for quasiconformal mappings, Acta Math. 96 (1956), 125–142.

19. V. Božin and V. Marković, Distance between domains in the sense of Lehto is not a metric, Ann. Acad. Sci. Fenn. Math. 24 (1999), 3–10.

20. J. Brock and K. Bromberg, On the density of geometrically finite Kleinian groups, Acta Math. 192 (2004), 33–93.

21. D. Calvis, The inner radius of univalence of normal circular triangles and regular polygons, Complex Variables Theory Appl. 4 (1985), 295–304.

22. S. D. Casey, The inclusion of classical families in the closure of the universal Teichmüller space, Michigan Math. J. 39 (1992), 189–199.

23. Y. M. Chiang, Some remarks on Lehto’s domain constant, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), 285–293.

24. M. Chuaqui and B. Osgood, Sharp distortion theorems associated with the Schwarzian derivative, J. London Math. Soc. 48 (1993), 289–298.

25. M. Chuaqui, B. Osgood, and Ch. Pommerenke, John domains, quasidisks, and the Nehari class, J. Reine Angew. Math. 471 (1996), 77–114.

26. M. Chuaqui and Ch. Pommerenke, Characteristic properties of Nehari functions, Pacific J. Math. 188 (1999), 83–94.

27. J. A. Cima and J. A. Pfaltzgraff, A Banach space of locally univalent functions, Michigan Math. J. 17 (1970), 321–334.

28. A. Douady and C. J. Earle, Conformally natural extensions of homeomorphisms of the circle, Acta Math. 157 (1986), 23–48.

29. P. L. Duren, Univalent Functions, Springer-Verlag, 1983.

30. P. L. Duren, H. S. Shapiro, and A. L. Shields, Singular measures and domains not of Smirnov type, Duke Math. J. 33 (1966), 247–254.

31. C. J. Earle, On the Carathéodory metric in Teichmüller spaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Princeton Univ. Press, 1974, pp. 99–103. Ann. of Math. Studies, No. 79.

32. C. J. Earle, I. Kra, and S. L. Krushkal’, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), 927–948.

33. C. J. Earle and C. McMullen, Quasiconformal isotopies, Holomorphic Functions and Moduli I (D. Drasin, C. J. Earle, F. W. Gehring, I. Kra, and A. Marden, eds.), Springer-Verlag, 1988, pp. 143–154.

34. B. Flinn, Jordan domains and the universal Teichmüller space, Trans. Amer. Math. Soc. 282 (1984), 603–610.

35. F. P. Gardiner, Teichmüller Theory and Quadratic Differentials, Wiley-Intersciences, New York, 1987.

36. F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Amer. Math. Soc., 2000.

37. F. P. Gardiner and D. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), 683–736.

38. F. W. Gehring, Univalent functions and the Schwarzian derivative, Comment. Math. Helv. 52 (1977), 561–572.

39. , Spirals and the universal Teichmüller space, Acta Math. 141 (1978), 99–113.

40. , Characteristic Properties of Quasidisks, Les Presses de l’Université de Montréal, 1982.

41. F. W. Gehring and Ch. Pommerenke, On the Nehari univalence criterion and quasicircles, Comment. Math. Helv. 1984 (59), 226–242.

42. G. M. Goluzin, Geometric theory of functions of a complex variable, American Mathematical Society, Providence, R.I., 1969, Translations of Mathematical Monographs, Vol. 26.

43. D. H. Hamilton, The closure of Teichmüller space, J. Anal. Math. 55 (1990), 40–50.

44. E. Hille, Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc. 55 (1949), 552–553.

45. Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Tokyo, 1992.

46. M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, Berlin-New York, 1993.

47. C. I. Kalme, Remarks on a paper by Lipman Bers, Ann. of Math. (2) 91 (1970), 601–606.

48. Y. C. Kim, S. Ponnusamy, and T. Sugawa, Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives, J. Math. Anal. Appl. 299 (2004), 433–447.

49. Y. C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math. 32 (2002), 179–200.

50. , Norm estimates of the pre-schwarzian derivatives for certain classes of univalent functions, Proc. Edinburgh Math. Soc. 49 (2006), 1–13.

51. Y. J. Kim and E. P. Merkes, On certain convex sets in the space of locally schlicht functions, Trans. Amer. Math. Soc. 196 (1974), 217–224.

52. W. Koepf, Close-to-convex functions, univalence criteria and quasiconformal extensions, Ann. Univ. Marie Curie-Sklodowska 40 (1986), 97–103.

53. Y. Komori and T. Sugawa, Bers embedding of the Teichmüller space of a once-punctured torus, Conform. Geom. Dyn. 8 (2004), 115–142.

54. Y. Komori, T. Sugawa, M. Wada, and Y. Yamashita, Drawing Bers embeddings of the Teichmüller space of once-punctured tori, to appear in Exper. Math.

55. I. Kra, On the Carathéodory metric on the universal Teichmüller space, Complex analysis Joensuu 1978 (Proc. Colloq., Univ. Joensuu, Joensuu, 1978) (Berlin), Lecture Notes in Math., vol. 747, Springer, Berlin, 1979, pp. 230–241.

56. W. Kraus, Über den Zusamenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung, Mitt. Math. Sem. Giessen 21 (1932), 1–28.

57. S. L. Krushkal, On the question of the structure of the universal Teichmüller space, Soviet Math. Dokl. 38 (1989), 435–437.

58. , The Green function of Teichmüller spaces with applications, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 143–147.

59. , Polynomial convexity of Teichmüller spaces, J. London Math. Soc. (2) 52 (1995), 147–156.

60. , Teichmüller spaces are not starlike, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 167–173.

61. S. L. Krushkal′, Complex geometry of the universal Teichmüller space, Sibirsk. Mat. Zh. 45 (2004), 780–808, English translation in Siberian Math. J. 45 (2004), 646–668.

62. R. Kühnau (ed.), Handbook of Complex Analysis: Geometric Function Theory, 2 vols., Elsevier Science B.V., Amsterdam, 2002, 2005.

63. M. Lehtinen, On the inner radius of univalency for non-circular domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 45–47.

64. , Estimates of the inner radius of univalency of domains bounded by conic sections, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 349–353.

65. , Angles and the inner radius of univalency, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), 161–165.

66. O. Lehto, Domain constants associated with Schwarzian derivative, Comment. Math. Helv. 52 (1977), 603–610.

67. , Remarks on Nehari’s theorem about the Schwarzian derivative and schlicht functions, J. Analyse Math. 36 (1979), 184–190.

68. , Univalent Functions and Teichmüller Spaces, Springer-Verlag, 1987.

69. O. Lehto and O. Tammi, Schwarzian derivative in domains of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1978/79), 253–257.

70. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd Ed., Springer-Verlag, 1973.

71. O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1978/79), 383–401.

72. K. Matsuzaki, Simply connected invariant domains of Kleinian groups not in the closures of Teichmüller spaces, Complex Variables Theory Appl. 22 (1993), 92–100.

73. D. Mejía and Ch. Pommerenke, On spherically convex univalent functions, Michigan Math. J. 47 (2000), 163–172.

74. L. Miller-Van Wieren, Univalence criteria for classes of rectangles and equiangular hexagons, Ann. Acad. Sci. Fenn. A I Math. 22 (1997), 407–424.

75. , On Nehari disks and the inner radius, Comment. Math. Helv. 76 (2001), 183–199.

76. S. Nag, The Complex Analytic Theory of Teichmüller Spaces, Wiley, New York, 1988.

77. , Diff(S^{1}), Teichmüller space and period matrices: canonical mappings via string theory, Rev. Roumaine Math. Pures Appl. 39 (1994), 789–827.

78. S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the H^{1/2} space on the circle, Osaka J. Math. 32 (1995), 1–34.

79. Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.

80. , A property of convex conformal maps, J. Analyse Math. 30 (1976), 390–393.

81. K. Ohshika, Realising end invariants by limits of minimally parabolic, geometrically finite groups, arXive:math. GT/0504546 (2005).

82. Y. Okuyama, The norm estimates of pre-Schwarzian derivatives of spiral-like functions, Complex Variables Theory Appl. 42 (2000), 225–239.

83. B. Osgood and D. Stowe, The Schwarzian distance between domains: A question of O. Lehto, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 313–318.

84. B. G. Osgood, Univalence criteria in multiply-connected domains, Trans. Amer. Math. Soc. 260 (1980), 459–473.

85. , Some properties of f′′/f′ and the Poincaré metric, Indiana Univ. Math. J. 31 (1982), 449–461.

86. V. Paatero, Über die konforme Abbildung von Gebieten, deren Ränder von beschränker Drehung sind, Ann. Acad. Sci. Fenn. A I Math.-Phys. XXXIII:9 (1931), 1–79.

87. Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

88. Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (1979), 192–199.

89. , On the Becker univalence criterion, Ann. Univ. Mariae Curie-Skłodowska Sect. A 36/37 (1982/83), 123–124 (1985).

90. R. M. Porter, Computation of a boundary point of Teichmüller space, Bol. Soc. Mat. Mexicana 24 (1979), 15–26.

91. M. S. Robertson, Univalent functions f(z) for which zf′(z) is spirallike, Michigan Math. J. 16 (1969), 97–101.

92. H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud. 66, Princeton Univ. Press, 1971, pp. 369– 383.

93. W. C. Royster, On the univalence of a certain integral, Michigan Math. J. 12 (1965), 385–387.

94. E. Schippers, Distortion theorems for higher order Schwarzian derivatives of univalent functions, Proc. Amer. Math. Soc. 128 (2000), 3241–3249.

95. M. Seppälä and T. Sorvali, Geometry of Riemann Surfaces and Teichmüller Spaces, North-Holland, Amsterdam, 1992.

96. H. Shiga, On analytic and geometric properties of teichmüller spaces, J. Math. Kyoto Univ. 24 (1984), 441–452.

97. , Characterization of quasi-disks and Teichmüller spaces, Tôhoku Math. J. 37 (1985), 541–552.

98. H. Shiga and H. Tanigawa, Grunsky’s inequality and its applications to Teichmüller spaces, Kodai Math. J. 16 (1993), 361–378.

99. D. Stowe, Injectivity and the pre-Schwarzian derivative, Michigan Math. J. 45 (1998), 537–546.

100. T. Sugawa, The Bers projection and the λ-lemma, J. Math. Kyoto Univ. 32 (1992), 701– 713.

101. , On the Bers conjecture for Fuchsian groups of the second kind, J. Math. Kyoto Univ. 32 (1992), 45–52.

102. , On the space of schlicht projective structures on compact Riemann surfaces with boundary, J. Math. Kyoto Univ. 35 (1995), 697–732.

103. , On the norm of the pre-Schwarzian derivatives of strongly starlike functions, Ann. Univ. Mariae Curie-Skłodowska, Sectio A 52 (1998), 149–157.

104. , Holomorphic motions and quasiconformal extensions, Ann. Univ. Mariae Curie-Skłodowska, Sectio A 53 (1999), 239–252.

105. , Uniformly perfect sets: analytic and geometric aspects (Japanese), Sugaku 53 (2001), 387–402, English translation in Sugaku Expo. 16 (2003), 225–242.

106. , A remark on the Ahlfors-Lehto univalence criterion, Ann. Acad. Sci. Fenn. A I Math. 27 (2002), 151–161.

107. , Inner radius of univalence for a strongly starlike domain, Monatsh. Math. (2003), 61–68.

108. N. Suita, Schwarzian derivatives of convex functions, J. Hokkaido Univ. Edu. (Sec. IIA) 46 (1996), 113–117.

109. L. A. Takhtajan and L.-P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239 (2003), 183– 240.

110. , Weil-Petersson geometry of the universal Teichmüller space, Infinite dimensional algebras and quantum integrable systems (Basel), Progr. Math., vol. 237, Birkhäuser, Basel, 2005, pp. 225–233.

111. H. Tamanoi, Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1995), 127–181.

112. W. P. Thurston, Zippers and schlicht functions, The Bieberbach conjecture, Proceedings of the symposium on the occasion of the proof of the Bieberbach conjecture held at Purdue University, West Lafayette, Ind., March 11–14, 1985 (Providence, RI) (D. Drasin, P. Duren, and A. Marden, eds.), Mathematical Surveys and Monographs, vol. 21, American Mathematical Society, 1986, pp. 185–187.

113. M. Toki, On nonstarlikeness of Teichmüller spaces, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 58–60.

114. S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266–267.

115. A. J. Tromba, Teichmüller Theory in Riemannian Geometry, Birkhäuser, Basel, 1992.

116. K.-J. Wirths, Über holomorphe Funktionen, die einer Wachstumsbeschränkung unterliegen, Arch. Math. 30 (1978), 606–612.

117. S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math. J. 28 (1999), 217–230.

118. I. V. Zhuravlev, Model of the universal Teichmüller space, Siberian Math. J. 27 (1986), 691–697.

119. I. V. Žuravlev, Univalent functions and Teichmüller spaces, Soviet Math. Dokl. 21 (1980), 252–255.

Toshiyuki Sugawa  E-mail: sugawa@math.sci.hiroshima-u.ac.jp

Address: Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan