CARDIOIDS AND TEICHMÜLLER SPACES

TOSHIYUKI SUGAWA

Abstract. In this note, we give an expository account on the role played by cardioids in both Teichmüller spaces and in complex dynamics. Especially, we will observe that the shape of the Bers embedding of the Teichmüller space of a once-punctured rectangular torus converges to a cardioid in the sense of Carathéodory when the base surface is pinched along the meridian. A more general and complete result will be included in a forthcoming paper [21] of the author.

1. Introduction

A cardioid is a plane curve similar to {e^{iθ} − e^{2iθ}/2 : θ ∈ ℝ}, see Figure 1 In this note, however, a cardioid means the (open) Jordan region bounded by a cardioid curve. We will call the standard cardioid the domain bounded by the above curve and denote it by C_{0}. Note that the function f_{0}(z) = z − z^{2}/2 maps the unit disk univalently onto C_{0}. Conventionally, we denote by aC_{0}+b the image of C_{0} under the similarity map z ↦ az+b.

It is an accidental coincidence that the cardioid appears both in the Kleinian group theory and in complex dynamics. However, according to Sullivan’s dictionary, the Bers embedding of the Teichmüller space of a Riemann surface (or a Fuchsian group) corresponds to the Mandelbrot set (or its generalization). Therefore, this coincidence might not be a surprizing fact. As is well known, the Mandelbrot set contains a cardioid (the so-called main cardioid) as a connected component of its interior. On the other hand,

Figure 1. The cardioid

Date: August 30, 2006, File: teich HRI.tex.

Key words and phrases. cardioid, Teichmüller space, holomorphic motion, once-punctured torus.

The author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039.

Figure 2. The Mandelbrot set

as Kalme [8] pointed out, the cardioid appears in the universal Teichmüller space in a natural way.

In this note, we point out similarities between the cardioid in the Mandelbrot set and cardioids in the Bers embedded Teichmüller spaces. We emphasis on the role played by holomorphic motions to give a unified aspect on these matters. Moreover, we will observe that the shape of a one-dimensional Teichmüller space tends to that of a cardioid when the base Riemann surface goes to the boundary of the moduli space, at least in a special case. It appears that this is a phenomenon which has not been observed in the literature.

The author would like to thank the organizers of the International Workshop on Teichmüller Theory and Moduli Problems held at the Harish-Chandra Research Institute, Allahabad, India, for giving him a chance to consider this matter seriously. Without their kind invitation to the workshop, this work would not have been carried out due to the author’s busyness and laziness.

2. Cardioid in Mandelbrot set

Even polynomials of the simple form P_{c}(z) = z^{2} + c are not fully understood in the context of iterations, that is, in the theory of complex dynamics. We denote by P_{c}^{n} the n-th iterate of P_{c}, namely, P_{c}^{n} = P_{c}^{n−1} ∘P_{c} for n = 2, 3, …. Here c is a complex parameter. The boundary of the set K_{c} = {z ∈ ℂ : {P_{c}^{n}(z)}_{n=1,2,...} is bounded } is called the Julia set and denoted by J_{c}. The Mandelbrot set M is defined to be the set of c ∈ ℂ for which J_{c} is connected (see Figure 2). It is known that J_{c} is either connected or a Cantor set. Also, it should be noted that c ∉ M iff 0 ∈ K_{c}.

An attracting fixed point of P_{c} is a point α ∈ ℂ such that P_{c}(α) = α and |P′_{c}(α)| < 1. It is easily seen that a neighbourhood of an attracting fixed point of P_{c} is contained in K_{c}. In particular, c ∈ M. A fixed point α of P_{c} must satisfy the relation P_{c}(α)−α = α^{2}−α+c = 0. If we set λ = P′_{c}(α) = 2α, then c = λ/2 − (λ/2)^{2}. We now conclude that P_{c} has an attracting fixed point iff c ∈ D = {λ/2 − (λ/2)^{2} : λ ∈ 𝔻}. Here and hereafter, 𝔻 denotes the unit disk {z ∈ ℂ : |z| < 1}. Therefore, D is contained in the Mandelbrot set. Indeed, it is easy to see that D is a component of the interior of M and D = (1/2)C_{0}. The component D is called the main cardioid of M.

We expect that the Julia set J_{c} varies holomorphically (in a sense) on the parameter c ∈ D. Indeed, a repelling periodic point z_{c} of P_{c} is a holomorphic function of c ∈ D (if one takes a suitable branch). In this way, we obtain a function f_{c}(z_{0}) = z_{c} on R_{0} = {e^{2πiθ} : 2^{n}θ ∈ ℤ ∃n ∈ ℕ} for each c ∈ D. Since these repelling periodic points do not collide for c ∈ D, the function f_{c} is injective on R_{0}. Thus the map (c, z) ↦ f_{c}(z) gives a holomorphic motion of R_{0} over D. Since R_{0} is dense in ∂𝔻, the holomorphic motion of R_{0} extends to that of ∂𝔻 (see [11]), which will be still denoted by the same letter f_{c}(z). Since the set of repelling periodic periodic points is dense in the Julia set, one has the relation J_{c} = f_{c}(∂𝔻) for c ∈ D. By a theorem of Mañé-Sad-Sullivan [11], the map f_{c} can be extended to a quasiconformal map of ℂ̂ and thus J_{c} = f_{c}(∂𝔻) is known to be a quasidisk for c ∈ D. For details and interesting figures, see Astala and Martin [2].

It is known that near the cusp point 1/4 of the main cardiod D rather complicated phenomena of phase transition are observed, which are called the parabolic explosion. See [3] for details and references.

3. Bers embedding of Teichmüller space

We give minimal basics in the theory of Teichmüller spaces to state and to prove our result. For details, see [14] or [7].

In the following, we denote by ℍ the upper half-plane {z ∈ ℂ : Im z > 0} and by ℍ^{∗} the lower half-plane {z ∈ ℂ : Im z < 0}. Let Γ be a Fuchsian group acting on ℍ, namely, Γ is a discrete subgroup of PSL(2, ℝ). We denote by Belt(ℍ, Γ) the space of Beltrami coefficient for Γ on ℍ, more precisely,

Belt(ℍ, Γ) = \{μ ∈ L^{∞}(ℍ) : ∥μ∥_{∞} < 1, (μ ∘ γ)\frac{\overline{γ′}}{γ′} = μ ∀γ ∈ Γ\}.

For μ ∈ Belt(ℍ, Γ), we denote by f^{μ} the quasiconformal map f of ℂ̂ = ℂ∪{∞} which is determined by the Beltrami equations fz̄ = μf_{z} in ℍ and fz̄ = 0 in ℍ^{∗} and normalization conditions f(0) = 0, f(1) = 1 and f(∞) = ∞. Note that f^{μ}Γ(f^{μ})^{−1} is a Kleinian group acting on f^{μ}(ℍ) and f^{μ}(ℍ^{∗}) properly discontinuously.

Noting that f^{μ} is conformal on ℍ^{∗}, we define a holomorphic function Φ(μ) on ℍ^{∗} by Φ(μ) = S_{f^{μ}|ℍ^{∗}}, where S_{f} stands for the Schwarzian derivative of f :

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

It is convenient to measure the Schwarzian derivative by the norm

∥φ∥_{ℍ^{∗}} = sup_{z∈ℍ^{∗}}(−2 Im z)^{2}|φ(z)|.

We denote by B_{2}(ℍ^{∗}) the Banach space consisting of analytic functions φ on ℍ^{∗} with ∥φ∥_{ℍ^{∗}} < ∞. Let

B_{2}(ℍ^{∗}, Γ) = {φ ∈ B_{2}(ℍ^{∗}) : (φ ∘ γ)(γ′)^{2} = φ ∀γ ∈ Γ},

which is a closed subspace of B_{2}(ℍ^{∗}).

It is known that Φ(μ) ∈ B_{2}(ℍ^{∗}, Γ) for μ ∈ Belt(ℍ, Γ) and that the image Φ(Belt(ℍ, Γ)) coincides with the Bers embedding of the Teichmüller space of Γ. We set Teich(Γ) = Φ(Belt(ℍ, Γ)) and we identify it with the Teichmüller space of Γ (or equivalently, of the orbifold ℍ/Γ). Note that Teich(Γ) is a bounded domain in B_{2}(ℍ^{∗}, Γ). The map Φ_{Γ} = Φ : Belt(ℍ, Γ) → Teich(Γ) is called the Bers projection and known to be a holomorphic submersion. In the case when Γ is the trivial group 1, the set Teich(1) is called the universal Teichmüller space.

We denote by o(Γ) and i(Γ) the outer and inner radii of the Teichmüller space Teich(Γ), in other words, o(Γ) is the smallest number r with Teich(Γ) ⊂ {φ ∈ B_{2}(ℍ^{∗}, Γ) : ∥φ∥_{ℍ^{∗}} < r} and i(Γ) is the largest number r with Teich(Γ) ⊃ {φ ∈ B_{2}(ℍ^{∗}, Γ) : ∥φ∥_{ℍ^{∗}} < r}.

It is well known that i(1) = 2 and o(1) = 6 for trivial group 1, and hence, 2 ≤ i(Γ) ≤ o(Γ) ≤ 6 for an arbitrary Fuchsian group Γ unless Teich(Γ) is a singleton. For a cofinite Fuchsian group Γ, it is also known that i(Γ) > 2 (cf. [16]) and o(Γ) < 6 (due to Sekigawa [20]). Furthermore, Nakanishi and Yamamoto [18] and Nakanishi and Velling [17] gave characterizing conditions for Γ to satisfy o(Γ) = 6 and i(Γ) = 2, respectively. It is remarkable that their conditions are coincident. In particular, o(Γ) = 6 iff i(Γ) = 2.

4. Kalme’s observation

Kalme [8] made an interesting observation on a special holomorphic family of quasiconformal maps of the Riemann sphere. Following [8], we present some facts related to our investigation.

For a complex number α ∈ ℂ, consider the function

F_{α}(z) = z^{α} = e^{α log z}

on ℍ^{∗}, where the branch of log z is taken so that −π < arg z = Im log z < 0 for z ∈ ℍ^{∗}. Since F′_{α}(z) = αz^{α−1} ≠ 0, the map F_{α} is locally univalent for α ≠ 0. A characterization of univalence of F_{α} is known.

Lemma 1 (Royster [19]). Let α ∈ ℂ \ {0}. Then, F_{α} is univalent in ℍ^{∗} iff |α − 1| ≤ 1 or |α + 1| ≤ 1.

An easy computation gives

S_{F_{α}}(z) = \frac{1 − α^{2}}{2z^{2}} = 2(1 − α^{2})φ_{0}^{∗}(z),

where φ_{0}^{∗} ∈ B_{2}(ℍ^{∗}) is given by φ_{0}^{∗}(z) = 1/(4z^{2}). Note that ∥φ_{0}^{∗}∥_{ℍ^{∗}} = 1.

An interesting fact is that the holomorphic motion f_{t} = F_{1−t} (|t| < 1) of ℍ^{∗} can be extended to that of the Riemann sphere in an explicit way:

f_{t}(z) = \begin{cases}zz̄^{−t} & z ∈ \overline{ℍ} \\ z^{1−t} & z ∈ ℍ^{∗}. \end{cases}

Note that f_{t} is normalized and has the Beltrami coefficient

μ_{f_{t}}(z) = −t\frac{z}{z̄} = −t\frac{|φ_{0}(z)|}{φ_{0}(z)}

for z ∈ ℍ, where φ_{0}(z) = \overline{φ_{0}^{∗}(z̄)} = 1/(4z^{2}) for z ∈ ℍ. This is of the form of a Teichmüller differential on ℍ and, by definition of Φ, one has the relation

(4.1)  Φ(−tμ_{0}) = 2(1 − (1 − t)^{2})φ_{0}^{∗} = g_{0}(t)φ_{0}^{∗},

where

μ_{0} = \frac{|φ_{0}|}{φ_{0}} and g_{0}(t) = 4 (t − \frac{t^{2}}{2} ) .

We can now see that the intersection of Teich(1) with the linear span of φ_{0}^{∗} is precisely the cardioid 4C_{0} (times φ_{0}^{∗}).

Hille [6] observed that for ε ∈ ℝ \ {0}, F_{iε} is a universal covering projection onto an annulus, and hence, is never univalent whereas S_{F_{iε}}(z) = (1+ε^{2})/(2z^{2}) and thus the norm ∥S_{F_{iε}}∥_{ℍ^{∗}} tends to 2 as ε → 0. We point out that the point S_{F_{iε}} approaches to the inward cusp of the above cardioid 4C_{0}φ_{0}^{∗} from the outside.

5. One-dimensional Teichmüller spaces.

From now on, we restrict ourselves on one-dimensional Teichmüller spaces. For simplicity, we assume Fuchsian groups to be torsion-free. Then dim Teich(Γ) = 1 iff the signature of Γ is (0,4) or (1,1). Since to each Fuchsian group Γ of signature (0,4), there corresponds a Fuchsian group Γ′ of signature (1,1) such that Teich(Γ) = Teich(Γ′) (cf. [9]), without loss of generality we may further assume that Γ is of signature (1,1). Then the quotient Riemann surface ℍ/Γ is represented by the once-punctured torus of the form T = (ℂ \ Ω)/Ω, where

Ω = ⟨ω_{1}, ω_{2}⟩_{ℤ} = {mω_{1} + nω_{2} : m, n ∈ ℤ}

and ω_{1}, ω_{2} ∈ ℂ with Im (ω_{2}/ω_{1}) > 0. We also write Teich(T) for Teich(Γ) when T = ℍ/Γ.

In recent years, much progress has been made in the study of one-dimensional Teichmüller spaces. For instance, Minsky [12] proved that the Bers embedding of a onedimensional Teichmüller space is a Jordan domain. By using Minsky’s method, Miyachi [13] showed that the one-dimensional Teichmüller space is “cusp-shaped” at every boundary point corresponding to a cusp. Computer graphics of the Bers embeddings are presented by [9] and [10]. Also, the inner and outer radii of the Teichmüller space of a once-punctured squre torus are numerically computed in [22].

In [21], the following result is proved.

Theorem 2. The Bers embedding Teich(T) of the Teichmüller space of a once-punctured torus T converges to a cardioid in the sense of Carathéodory (under suitable indentification of B_{2}(T) = B_{2}(𝔻, Γ) with ℂ), when T goes to infinity in the moduli space

There is some technicality with the suitable choice of the basis of the vector space B_{2}(T) so that one can identify B_{2}(T) with ℂ. Therefore, in this note, we restrict ourselves to the case when T is a rectangular torus for the sake of simplicity.

Let Ω_{λ} be the lattice generated by λ > 0 and i = \sqrt{−1} over ℤ, namely, Ω_{λ} = \{m + ni : m, n ∈ ℤ}. We set T_{λ} = (ℂ \ Ω_{λ})/Ω_{λ}. We denote by π_{λ}(ζ) = [ζ] the canonical projection ℂ \ Ω_{λ} → T_{λ}.

Take a holomorphic universal covering projection p_{λ} of the upper half-plane ℍ onto ℂ \ Ω_{λ} so that p_{λ}(i) = a_{λ} and p′_{λ}(i)/i > 0. Then q_{λ} = π_{λ} ∘ p_{λ} is a holomorphic universal covering projection of ℍ onto T_{λ}. Let Γ_{λ} be the covering transformation group of q_{λ} : ℍ → T_{λ}. Note that ℍ/Γ_{λ} = T_{λ}.

Let φ̃_{λ} be the pullback of the quadratic differential dζ^{2} on ℂ \ Ω_{λ} under the map p_{λ}, namely, φ̃_{λ} = (p′_{λ})^{2}. Further let

φ_{λ} = (∥φ̃_{λ}∥_{ℍ})^{−1}φ̃_{λ}.

Then, by definition, φ_{λ} ∈ B_{2}(ℍ, Γ_{λ}) and ∥φ_{λ}∥_{ℍ} = 1.

Set also

φ_{λ}^{∗}(z) = \overline{φ_{λ}(z̄)}.

Then φ_{λ}^{∗} ∈ B_{2}(ℍ^{∗}, Γ_{λ}) and ∥φ_{λ}^{∗}∥_{ℍ^{∗}} = 1.

Since dζ^{2} on ℂ \ Ω_{λ} projects to a nontrivial holomorphic quadratic differential on T_{λ} via π_{λ}, the vector space B_{2}(ℍ^{∗}, Γ_{λ}) is spanned by φ_{λ}^{∗}. Let

U_{λ} = {w ∈ ℂ : wφ_{λ}^{∗} ∈ Teich(Γ_{λ})}.

Then, our result can be stated as in the following.

Theorem 3. The domain U_{λ} converges to the cardioid 4C_{0} in the sense of Carathéodory as λ → 0 + .

Since U_{λ} is a simply connected bounded domain (indeed, a Jordan domain by Minsky’s theorem), there exists a conformal homeomorphism g_{λ} of the unit disk 𝔻 onto U_{λ} such that g_{λ}(0) = 0 and g′_{λ}(0) > 0. Recall that g_{0}(t) = 2(1 − (t − 1)^{2}) is a conformal map of 𝔻 onto the cardioid 4C_{0}. Then, the last theorem can be restated in the following way.

Theorem 4. The conformal map g_{λ} converges to the map g_{0} locally uniformly on 𝔻.

We illustrate the shape of the boundary of Teich(Γ_{λ}) for several values of λ. We remark that the boundary curve is drawn very roughly near the “main” cusp when λ is small. This is simply because of our algorithm of computation.

6. Proof of Theorem 4

Let ρ_{ℂ\Ω_{λ}}(ζ)|dζ| and ρ_{T_{λ}}([ζ])|dζ| be the hyperbolic metrics on ℂ \ Ω_{λ} and T_{λ}, respectively. The density functions (the hyperbolic densities) ρ_{ℂ\Ω_{λ}}(ζ)|dζ| and ρ_{T_{λ}}([ζ]) are characterized by the relations ρ_{T_{λ}}([ζ]) = ρ_{ℂ\Ω_{λ}}(ζ) and ρ_{ℂ\Ω_{λ}}(p_{λ}(z))|p′_{λ}(z)| = ρ_{ℍ}(z) = 1/(2 Im z). The following assertion can be found (in a more general form) in [22].

Figure 3. Bers embedding for the punctured square torus (λ = 1)

CARDIOIDS AND TEICHMÜLLER SPACES 7

Figure 4. Bers embedding for small λ Figure 5. Bers embedding for very small λ

Figure 6. Bers embedding for very very small λ Figure 7. The cardioid 4C_{0} = {2(1 − α^{2}) : |α − 1| < 1}.

Lemma 5. The hyperbolic density ρ_{T_{λ}}([ζ]) of T_{λ} takes its minimum at the point [a_{λ}] = [(λ + i)/2].

Corollary 6. The supremum of (2 Im z)^{2}|φ_{λ}(z)| over z ∈ ℍ is attained at z = i.

Proof. Letting ζ = p_{λ}(z), we have the relation

(2 Im z)^{2}|φ_{λ}(z)| = ρ_{ℂ\Ω_{λ}}(ζ)^{−2},

which is known to take its maximum at ζ = a_{λ} = p_{λ}(i) by the last lemma. □

By the normalization of φ_{λ}, we have thus the relation 4|φ_{λ}(i)| = ∥φ_{λ}∥_{ℍ} = 1. On the other hand, φ̃_{λ}(i) = p′_{λ}(i)^{2} < 0. Therefore, φ_{λ}(i) = φ̃_{λ}(i)/∥φ̃_{λ}∥_{ℍ} < 0 and thus

(6.1)  φ_{λ}(i) = −\frac{1}{4}.

We also note the following fact, which will be used later.

Lemma 7. The hyperbolic density ρ_{ℂ\Ω_{λ}}(ξ + i/2) converges to π/2 uniformly in ξ ∈ ℝ as λ → 0 + .

Proof. Observe that ℂ \ Ω_{λ} converges to the parallel strip S = {ζ : 0 < Im ζ < 1} with respect to the point i/2 in the sense of Carathéodory when λ → 0+. Thus, the hyperbolic density ρ_{ℂ\Ω_{λ}} converges to ρ_{S} locally uniformly on S (cf. [5]). Since ρ_{ℂ\Ω_{λ}}(ξ + i/2) is a periodic function of ξ with period λ and since ρ_{S}(ξ +i/2) = π/2, the assertion follows. □

We now define the curves

α_{λ}(s) = [a_{λ} + λs] (0 ≤ s ≤ 1)
β_{λ}(s) = [a_{λ} + is] (0 ≤ s ≤ 1)

on T_{λ}, where we recall that a_{λ} = (λ + i)/2. Then, by the obvious symmetry of T_{λ}, these are simple hyperbolic geodesics of T_{λ} (see Figure 8).

We take the lifts α̂_{λ} and β̂_{λ} of them starting at i via the covering projection q_{λ} : ℍ → T_{λ}. We denote by A_{λ} and B_{λ} the elements of Γ_{λ} ⊂ PSL(2, ℝ) corresponding to α̂_{λ} and β̂_{λ}, respectively. In other words, α̂_{λ}(1) = A_{λ}(α̂_{λ}(0)) = A_{λ}(i) and β̂_{λ}(1) = B_{λ}(β̂_{λ}(0)) = B_{λ}(i). See Figure 9.

Let σ and τ be the hyperbolic lengths of curves α_{λ} and β_{λ}, respectively. Then A_{λ} and B_{λ} can be represented by

A_{λ} = ± (\begin{matrix}e^{σ} & 0 \\ 0 & e^{−σ} \end{matrix}) and B_{λ} = ± (\begin{matrix}cosh τ & sinh τ \\ sinh τ & cosh τ \end{matrix}) .

Since the commutator [A_{λ}, B_{λ}] must be parabolic, the following relation is required:

sinh σ sinh τ = 1.

We remark that this sort of relation was already found by Hayman [4, §7].

We note the following fact.

Lemma 8. The hyperbolic length σ of α_{λ} tends to 0 as λ → 0 + .

CARDIOIDS AND TEICHMÜLLER SPACES 9

Figure 8. Geodesics on T_{λ} Figure 9. Lifts of curves on ℍ

Proof. By definition, σ can be computed by

σ = ∫_{0}^{λ} ρ_{T_{λ}}([ξ + i/2])dξ = ∫_{0}^{λ} ρ_{ℂ\Ω_{λ}}(ξ + i/2)dξ.

By Lemma 7, we see that σ ∼ πλ/2 as λ → 0 + . In particular, the assertion follows. □

A crucial result is the following variant of a proposition due to T. Nakanishi [15, Proposition 3.1].

Lemma 9 (Nakanishi’s lemma). Let G_{n} be a sequence of Fuchsian groups acting on ℍ each of which contains a hyperbolic element of the form z ↦ M_{n}z such that M_{n} → 1 as n → ∞. Further let φ_{n} be an element of B_{2}(ℍ, G_{n}) such that φ_{n} converges locally uniformly to a holomorphic function φ_{∞} on ℍ. Then φ_{∞}(z) = c/z^{2} for some constant c.

Proof. Write φ_{n}(z) = P_{n}(z)/z^{2} and φ_{∞}(z) = P_{∞}(z)/z^{2}. Then P_{n} → P_{∞} locally uniformly. Since φ_{n}(M_{n}z)M_{n}^{2} = φ_{n}(z), we have P_{n}(M_{n}^{k}z) = P_{n}(z) for any k ∈ ℤ. Letting n → ∞, we can see that P_{∞} is constant along the positive imaginary axis, which implies that P_{∞} is constant. □

Lemma 10. The quadratic differential φ_{λ}(z) converges to φ_{0}(z) = 1/(4z^{2}) locally uniformly on ℍ as λ → 0 + .

Proof. Since ∥φ_{λ}∥_{ℍ} = 1, the family {φ_{λ}} is locally bounded on ℍ and thus normal. Let λ_{n} be any sequence of positive numbers tending to 0 such that φ_{λ_{n}} converges locally uniformly on ℍ. Then Lemmas 8 and 9 imply that the limit function has the form c/z^{2}. On the other hand, by (6.1), the constant c must be 1/4, which is independent of the sequence λ_{n}. Thus a standard argument gives us the local uniform convergence of φ_{λ} to 1/(4z^{2}). □

Let

μ_{λ} = \frac{|φ_{λ}|}{φ_{λ}}.

Then tμ_{λ} ∈ Belt(ℍ, Γ_{λ}) for every t ∈ 𝔻. Since Φ(−tμ_{λ}) ∈ B_{2}(ℍ^{∗}, Γ_{λ}), we can write Φ(−tμ_{λ}) = cφ_{λ}^{∗} for some c ∈ ℂ for each t ∈ 𝔻. We define the function h_{λ} : 𝔻 → ℂ by the relation

Φ(−tμ_{λ}) = h_{λ}(t)φ_{λ}^{∗}.

We now show the following.

Lemma 11. The function h_{λ} is a conformal homeomorshism of 𝔻 onto U_{λ} satisfying h_{λ}(0) = 0, \overline{h_{λ}(t)} = h_{λ}(t̄), and, in particular, h′_{λ}(0) ∈ ℝ \ \{0\}.

Proof. Since the map t ↦ Φ_{Γ_{λ}}(−tμ_{λ}) is of the form of a Teichmüller map, h_{λ} is known to be a proper holomorphic injection of 𝔻 into U_{λ} (see, for instance, [14]). Since dim Teich(Γ_{λ}) = 1, this map is indeed a conformal homeomorphism.

By the symmetry of Ω_{λ}, we have the relation \overline{p_{λ}(−z̄) − a_{λ}} + a_{λ} = p_{λ}(z). Therefore, φ_{λ} = (p′_{λ})^{2} satisfies \overline{φ_{λ}(−z̄)} = φ_{λ}(z). Hence, \overline{μ_{λ}(−z̄)} = μ_{λ}(z). This implies \overline{Φ(t̄μ_{λ})(−z̄)} = Φ(tμ_{λ})(z), which is equivalent to \overline{h_{λ}(t̄)}\overline{φ_{λ}^{∗}(−z̄)} = h_{λ}(t)φ_{λ}^{∗}(z). Since \overline{φ_{λ}^{∗}(−z̄)} = φ_{λ}^{∗}(z), we have the relation \overline{h_{λ}(t)} = h_{λ}(t̄). □

Remark. The quasiconformal deformation of T_{λ} corresponding to Φ(−tμ_{λ}) can be constructed explicitly. This holomorphic motion of ℍ^{∗} is induced by the holomorphic motion of ℂ \ Ω_{λ} given by

𝔻 × (ℂ \ Ω_{λ}) ∋ (t, ζ) ↦ \frac{ζ − tζ̄}{1 + t}.

See [21] for the details.

By Lemma 10, we see that the Teichmüller differential μ_{λ} = |φ_{λ}|/φ_{λ} coverges pointwise to μ_{0} = |φ_{0}|/φ_{0} = z/z̄. We now use the following basic property of quasiconformal maps. The proof of it can be read from a paper of Ahlfors and Bers [1]. See also [21] for a detailed account.

Proposition 12. Let μ_{n} be a sequence of measurable functions on ℂ such that ∥μ_{n}∥_{∞} ≤ 1 and μ_{n} → μ a.e. Then the normalized tμ_{n}-quasiconformal map f^{tμ_{n}}(z) converges to f^{tμ}(z) locally uniformly in (t, z) ∈ 𝔻 × ℂ.

By the above proposition, f^{−tμ_{λ}}(z) converges to f^{−tμ_{0}}(z) locally uniformly in (t, z) ∈ 𝔻 × ℂ as λ → 0 + . Since f^{−tμ_{λ}} is analytic in ℍ^{∗}, the Weierstrass double series theorem implies that

Φ(−tμ_{λ})(z) = S_{f^{−tμ_{λ}}}(z) → S_{f^{−tμ_{0}}}(z) = Φ(−tμ_{0})(z)

locally uniformly in (t, z) ∈ 𝔻 × ℍ^{∗}. By definition of h_{λ} and (4.1), the above convergence is equivalent to the locally uniform convergence

h_{λ}(t)φ_{λ}^{∗}(z) → g_{0}(t)φ_{0}^{∗}(z).

Since φ_{λ}^{∗} → φ_{0}^{∗} by Lemma 10, we now see that h_{λ} → g_{0} locally uniformly on 𝔻. In particular, h′_{λ}(0) → g′_{0}(0) = 4. Since h′_{λ}(0) is continuous in λ and, by Lemma 11, assumes non-zero real numbers, we also see that h′_{λ}(0) > 0 for all λ > 0. By the uniqueness of the Riemann mapping function and by Lemma 11, we obtain g_{λ} = h_{λ}. Thus the main theorem has been proved.

References

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

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

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