ON THE BERS CONJECTURE FOR

FUCHSIAN GROUPS OF THE SECOND KIND

DEDICATED TO PROFESSOR TATSUO FUJI’I’E ON HIS SIXTIETH BIRTHDAY

Toshiyuki Sugawa

§1. Introduction. Suppose that D is a simply connected domain of hyperbolic type in the Riemann sphere Ĉ = C ∪ {∞}. Then the Poincaré metric ρ_{D} in D is defined by

ρ_{D}(z) = \frac{|g′(z)|}{1 − |g(z)|^{2}} , z ∈ D,

where g is any conformal mapping of D onto the unit disk Δ = {z : |z| < 1}. B_{2}(D) will denote the Banach space consisting of all holomorphic functions φ in D such that the norm

∥φ∥_{D} = sup_{z∈D} |φ(z)|ρ_{D}(z)^{−2}

is finite.

If f is a locally univalent meromorphic function, the Schwarzian derivative of f is given by

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

We set after Flinn [6]

S = {S_{f} : f is conformal in Δ},
J = {S_{f} ∈ S : f(Δ) is a Jordan domain },
T = {S_{f} ∈ S : f(Δ) is a quasidisk }.

T is called the universal Teichmüller space. It is known that T ⊂ J ⊂ S ⊂ B_{2}(Δ), T is open, S is closed and T = IntS (see [7], [9]). Let Γ be a Fuchsian group acting on Δ and B_{2}(Δ, Γ) denote the closed subspace of B_{2}(Δ) :

{φ ∈ B_{2}(Δ) : (φ ∘ γ) · (γ′)^{2} = φ for all γ ∈ Γ }.

Further we set S(Γ) = S ∩ B_{2}(Δ, Γ), J(Γ) = J ∩ B_{2}(Δ, Γ), T (Γ) = T ∩ B_{2}(Δ, Γ). Then T (Γ) coincides with the Bers embedding of the Teichmüller space of Γ (see [3]).

Bers conjectured that S = \overline{T} i.e., S(1) = \overline{T (1)} in [2]. Generalizing this conjecture, by the Bers conjecture for Γ we shall mean that S(Γ) = \overline{T (Γ)}.

In [8], Gehring showed that the Bers conjecture is false i.e., S \ \overline{T} ≠ ϕ, in fact essentially he showed that S \ \overline{J} ≠ ϕ. Moreover Flinn proved that J \ \overline{T} ≠ ϕ in [6] (see the next section for the details). Our main purpose in this paper is to show the following

Typeset by AMS-TEX

Theorem 1. Suppose that Γ is a Fuchsian group of the second kind. Then S(Γ) \ \overline{J} ≠ ϕ and J(Γ) \ \overline{T} ≠ ϕ.

Corollary. When Γ is of the second kind, the Bers conjecture for Γ is false. In other words, S(Γ) ⫌ \overline{T (Γ)}.

On the other hand, the author does not know whether the Bers conjecture for Fuchsian groups of the first kind is true or not.

This paper is organized as follows. In §2 we introduce simply connected domains which are constructed by Gehring and Flinn and quote two results from Flinn [6] for later use. Gehring’s (or Flinn’s) domain is essentially obtained by removing a spiral (respectively, countable sequence of closed Jordan regions approximating a spiral) from a disk so that the boundary of the domain is adequately irregular. For a given Fuchsian group Γ of the second kind, in §3 we construct a Γ -invariant simply connected domain Δ′ in Δ which contains the Gehring’s or Flinn’s domain adjacent to a free side of the Dirichlet fundamental region of Γ. Let F : Δ → Δ′ be a Riemann mapping function of Δ′, then S_{F} ∈ S(G) where G = F^{−1}Γ F is a Fuchsian group. While it turns out later that S_{F} ∉ \overline{J} (respectively, S_{F} ∉ \overline{T}) and that G is qc(=quasiconformally) equivalent to Γ (Lemma 4, Corollary), these facts need not imply that S(Γ) \ \overline{J} ≠ ϕ (respectively, J(Γ) \ \overline{T} ≠ ϕ). Now we consider to deform Δ′ by an appropriate qc mapping so that the Schwarzian derivative as above belongs to S(Γ). In §4, such a deformation is presented and we state a slightly general result (Theorem 2) which contains Theorem 1. §5 is devoted to the proof of Theorem 2.

§2. Gehring’s and Flinn’s construction. Fix a ∈ (0, \frac{1}{8π}) and set a closed Jordan arc

γ_{a} = {±ie^{(−a+i)t} : t ∈ [0, ∞)} ∪ {0}.

Theorem A (Gehring [8]). Let F : Δ → Ĉ \ γ_{a} be a Riemann mapping function of Ĉ \ γ_{a} , then S_{F} ∈ S \ \overline{J}.

We set

A = {x + iy : y > 1} ∪ {x + iy : x > −4, y > −1}

and D_{1} = A \ γ_{a}. As we shall find later, the following theorem also holds.

Theorem A’. Let F : Δ → D_{1} be a Riemann mapping function of D_{1}, then S_{F} ∈ S \ \overline{J}.

For each positive integer m∈ N, we set σ_{m} = (\frac{π}{8} )^{m}, τ_{m} = e^{−2πam}, E_{m} = R_{m}∪P_{m} where

P_{m} = {e^{iσ}z : z ∈ γ_{a}, −σ_{m} ≤ σ ≤ σ_{m}} ∪ {z : |z| ≤ τ_{m}},
R_{m} = {x + iy : |x| ≤ sin σ_{m}, −1 ≤ y ≤ − cos σ_{m}} \ Δ.

Then each E_{m} is a closed Jordan region, E_{1} ⊃ E_{2} ⊃ ⋯ and ⋂_{m=1}^{∞} E_{m} = γ_{a}. Let V denote the translation V (z) = z + 8 and set D_{2} = A \ ⋃_{m=1}^{∞} V^{m}(E_{m}). One can easily see that D_{2} is a Jordan domain in Ĉ. The next theorem is essentially due to Flinn.

Theorem B (Flinn [6, Theorem 2]). Let F : Δ → D_{2} be a Riemann mapping function of D_{2} , then S_{F} ∈ J \ \overline{T} .

Theorems A’ and B are direct conclusions of the following Lemmas 1 and 2, respectively. Let α_{1} = {z = e^{(−a+i)t} : t ∈ (0, ∞)}, α_{2} = {z : −z ∈ α_{1}}. Then α_{1} and α_{2} are logarithmic spirals in D_{2} which converge onto the point 0 from opposite sides of γ_{a}.

Lemma 1 (cf. Flinn [6, Lemma 2]). There exists a constant δ_{1} > 0 with the following property. If f is conformal in D_{1} with ∥S_{f}∥_{D_{1}} ≤ δ_{1}, then

lim_{α_{1}∋z→0} f(z) = lim_{α_{2}∋z→0} f(z).

In particular, f(D_{1}) is not a Jordan domain.

Let β be the subarc {x + iy ∈ ∂D_{1} : −4 < x < ∞} of ∂D_{1}. We note that if f : R_{1} → R_{2} is a conformal mapping of a Jordan domain R_{1} onto another Jordan domain R_{2}, then f is uniquely extended to a homeomorphism f̃: \overline{R_{1}} → \overline{R_{2}}.

Lemma 2 (cf. Flinn [6, Proof of Theorem 2]). There exists a constant δ_{2} > 0 with the following property. If f is conformal in D_{2} with ∥S_{f}∥_{D_{2}} ≤ δ_{2} and if f(D_{2}) is a Jordan domain, then f̃(β) is not a quasiarc.

Remark. In Thorems A’ and B, we can replace the domain A by a half plane {x + iy : y > −1}.

§3. Construction of group invariant domains. Let Γ be an arbitrary Fuchsian group of the second kind acting on the unit disk Δ. In this section we construct Γ - invariant simply connected domains which have the same property as the Gehring’s or Flinn’s domain. Since Γ is of the second kind, Ω(Γ) ∩ ∂Δ ≠ ϕ where Ω(Γ) is the region of discontinuity of Γ in Ĉ. Now we pick a sufficiently small disk Y in Ω(Γ) whose boundary is orthogonal to ∂Δ so that no two distinct points of Y are Γ -equivalent. Let σ be a Möbius transformation such that σ(Y) = Δ and that σ(Y^{+}) = Δ^{+} where Y^{+} = Y ∩ Δ and Δ^{+} = {z ∈ Δ : Imz > 0}. Fix r_{0}, r_{1} ∈ (0, 1) such that r_{1} < r_{0}. Let Δ_{r} = {z ∈ Δ : |z| < r}, Δ_{r}^{+} = Δ_{r} ∩ Δ^{+} and Y_{r}^{+} = σ^{−1}(Δ_{r}^{+}) for r ∈ (0, 1).

Let M(Δ) be the space of Beltrami coefficients supported in Δ : {μ ∈ L^{∞}(Δ) : ∥μ∥_{∞} < 1}. For μ ∈ M(Δ), w^{μ} will denote the normalized μ-conformal self-mapping of Δ which fixes three points 1, i, −1 ∈ ∂Δ. We set

M_{Y_{r_{0}}^{+}} (Δ)_{k} = {μ ∈ L^{∞}(Δ) : μ = 0 on Y_{r_{0}}^{+} and ∥μ∥_{∞} < k},

for k ∈ (0, 1]. Notice that w^{μ} is conformal in Y_{r_{0}}^{+} for μ ∈ M_{Y_{r_{0}}^{+}} (Δ)_{k}.

Lemma 3. Let δ_{0} = min{δ_{1}, δ_{2}} where δ_{1} and δ_{2} are as in Lemmas 1 and 2, respectively, then there exists a constant k ∈ (0, 1] such that for any μ ∈ M_{Y_{r_{0}}^{+}} (Δ)_{k} the following is valid:

∥S_{w^{μ}}∥_{Y_{r_{1}}^{+}} ≤ \frac{δ_{0}}{2} .

Proof. We can extend w^{μ} to a qc homeomorphism of Ĉ by the rule

w^{μ}(z) = 1/\overline{w^{μ}(1/z̄)}.

It is well known that w^{μ}(z) converges to z uniformly on each compact set in C as ∥μ∥_{∞} → 0 (see [1], for example). Since w^{μ} is conformal in Y_{r_{0}} = σ^{−1}(Δ_{r_{0}}) for μ ∈ M_{Y_{r_{0}}^{+}} (Δ)_{1} and Y_{r_{1}}^{+} is relatively compact in Y_{r_{0}}, S_{w^{μ}} converges to 0 uniformly on Y_{r_{1}}^{+} as ∥μ∥_{∞} → 0, in particular ∥S_{w^{μ}}∥_{Y_{r_{1}}^{+}} converges to 0 as ∥μ∥_{∞} → 0. □

For each ε ∈ (0, r_{1}) we choose a τ = τ_{ε} ∈Möb such that τ(R̂) = {x − i : x ∈ R} ∪ {∞} and τ(Δ_{ε}^{+}) ⊃ A.

Construction 1 (Gehring type).

We set Δ′_{1} = Δ′_{1}(ε) = Δ \ ⋃_{γ∈Γ} γ((τ ∘ σ)^{−1}(γ_{a})).

Construction 2 (Flinn type).

We set Δ′_{2} = Δ′_{2}(ε) = Δ \ ⋃_{γ∈Γ} γ((τ ∘ σ)^{−1}(⋃_{m=1}^{∞} V^{m}(E_{m}))).

We note that (τ ∘σ)^{−1}(γ_{a}) ⊂ Y^{+}, that (τ ∘σ)^{−1}(⋃_{m=1}^{∞} V^{m}(E_{m})) ⊂ Y^{+} and that (γ(Y^{+}))_{γ∈Γ} is a disjoint family. Therefore Δ′_{j} is a Γ -invariant simply connected domain contained in Δ for j = 1, 2; furthermore Δ′_{2} is a Jordan domain. If we let F_{j} : Δ → Δ′_{j} be a Riemann mapping function and set G_{j} = F_{j}^{−1}Γ F_{j} which is a subgroup of Möb acting discontinuously on Δ i.e., a Fuchsian group acting on Δ, then S_{F_{1}} ∈ S(G_{1}) and S_{F_{2}} ∈ J(G_{2}).

Now we state a lemma which guarantees that G_{j} is qc equivalent to Γ.

Lemma 4. Let k ∈ (0, 1] be as in Lemma 3. For sufficiently small ε ∈ (0, r_{1}), there exists a qc mapping h_{j} of Δ′_{j} = Δ′_{j}(ε) onto Δ with the following properties for j = 1, 2.

(1) ∥μ(h_{j})∥_{∞} < k where μ(h_{j}) is the Beltrami coefficient of h_{j},

(2) h_{j} is conformal in Δ′_{j} ∩ Y_{r_{0}}^{+},

(3) h_{j} ∘ γ = γ ∘ h_{j} for all γ ∈ Γ.

Because the qc mapping f_{j} = h_{j} ∘ F_{j} : Δ → Δ deforms G_{j} into Γ, we have the following

Corollary. G_{j} is qc equivalent to Γ.

Lemma 4 is obtained in an obvious way by the following

Lemma 5. For sufficiently small ε ∈ (0, r_{1}), there exists a qc mapping H_{j} : σ(Y′_{j}) → Δ^{+} with the following properties for j = 1, 2.

(1) ∥μ(H_{j})∥_{∞} < k where μ(H_{j}) is the Beltrami coefficient of H_{j},

(2) H_{j} is conformal in σ(Y′_{j}) ∩ Δ_{r_{0}}^{+},

(3) H_{j} = identity on ∂Δ^{+} \ [−1, 1], where Y′_{j} = Y′_{j}(ε) = Y ∩ Δ′_{j}(ε).

Proof of Lemma 5. Let H_{(ε)} : σ(Y′_{j}(ε)) ∩ Δ_{r_{0}}^{+} → Δ_{r_{0}}^{+} be the conformal mapping which fixes three points r_{0}, r_{0}i, −r_{0}. Noting that Δ_{r_{0}}^{+} \ \overline{Δ_{ε}^{+}} ⊂ σ(Y′_{j}(ε)), we find that H_{(ε)}|_{Δ_{r_{0}}^{+} \\overline{Δ_{ε}^{+}}} can be extended to a conformal mapping H̃_{(ε)} in \{z : ε < |z| < r_{0}^{2}/ε\}

by the reflection principle. Let Θ_{ε} : (0, π) → (0, π) be the mapping defined by the rule r_{0}e^{iΘ_{ε}(θ)} = H̃_{(ε)}(r_{0}e^{iθ}) for all θ ∈ (0, π), then Θ_{ε} is a smooth mapping such that

Θ′_{ε}(θ) = |H′_{(ε)}(r_{0}e^{iθ})|.

Now we set

H_{(ε)}(re^{iθ}) = re^{i(tθ+(1−t)Θ_{ε}(θ))} for r ∈ [r_{0}, 1], θ ∈ (0, π),

where r = t + (1 − t)r_{0}, then this extended H_{(ε)} has the properties (2) and (3). On the other hand, clearly H̃_{(ε)}(z) converges to z uniformly on each compact set in C^{∗} = C \ {0} as ε → 0, hence Θ_{ε}(θ), Θ′_{ε}(θ) uniformly converges to θ, 1 as ε → 0, respectively. Therefore the explicit expression

|μ(H_{(ε)})(re^{iθ})| = \{ \begin{matrix}|\frac{(r−r_{0})(Θ′_{ε}(θ)−1)−(Θ_{ε}(θ)−θ)ri}{2(1−r_{0})+(r−r_{0})(Θ′_{ε}(θ)−1)+(Θ_{ε}(θ)−θ)ri}|, & r ∈ (r_{0}, 1], \\ 0, & r ∈ (0, r_{0}) \end{matrix}

shows that ∥μ(H_{(ε)})∥_{∞} → 0 as ε → 0, so for sufficiently small ε ∈ (0, r_{1}), H_{j} = H_{(ε)} : σ(Y′_{j}) → Δ^{+} has the properties (1), (2) and (3). □

Henceforth we fix an ε ∈ (0, r_{1}) for which Lemma 4 holds.

§4. Deformations by partially conformal qc mappings. In this section we present a method to construct the family of group-invariant domains which includes desired one.

Let M(Δ, Γ) be the space of Beltrami coefficients for Γ with support in Δ i.e., the subset of L^{∞}(Δ) consisting of all μ ∈ L^{∞}(Δ) with ∥μ∥_{∞} < 1 and

(μ ∘ γ) · \overline{γ′}/γ′ = μ for all γ ∈ Γ.

Set D_{j}^{μ} = w^{μ}(Δ′_{j}) for μ ∈ M(Δ, Γ). If Γ^{μ} denotes the Fuchsian group w^{μ}Γ (w^{μ})^{−1} acting on Δ, then D_{j}^{μ} is a Γ^{μ}-invariant simply connected domain whose boundary is homeomophic to the Δ′_{j}’s. We take a Riemann mapping function F_{j}^{μ} : Δ → D_{j}^{μ} and set G_{j}^{μ} = (F_{j}^{μ})^{−1}Γ^{μ}F_{j}^{μ} and φ_{j}^{μ} = S_{F_{j}^{μ}}. Since Γ^{μ} acts discontinuously on D_{j}^{μ} , G_{j}^{μ} acts also discontinuously on Δ, hence G_{j}^{μ} is a Fuchsian group. And clearly φ_{1}^{μ} ∈ S(G_{1}^{μ}) and φ_{2}^{μ} ∈ J(G_{1}^{μ}). Because the logarithmic spiral γ_{a} is a quasiarc, the general qc mapping w^{μ} may unfasten the spirals removed, thus we must restrict Beltrami coefficients μ to be considered on a certain class of M(Δ, Γ). In this article we only consider

M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k} = M_{Y_{r_{0}}^{+}} (Δ)_{k} ∩ M(Δ, Γ)
= {μ ∈ M(Δ, Γ) : μ = 0 on Y_{r_{0}}^{+} and ∥μ∥_{∞} < k}.

Since w^{μ} is conformal in Y_{r_{0}}^{+} for μ ∈ M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k}, it is expected that the spirals are but slightly deformed by w^{μ}. In fact, we have the following result for this class which is proved in the rest of this paper.

Theorem 2. Let k be as in Lemma 3. For μ ∈ M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k}, we have

φ_{1}^{μ} ∈ S(G_{1}^{μ}) \ \overline{J},
φ_{2}^{μ} ∈ J(G_{2}^{μ}) \ \overline{T} .

Theorem 2 and Lemma 4 prove Theorem 1.

Proof of Theorem 1. Let h_{j} be as in Lemma 4. We set

μ_{j} = \{ \begin{matrix}μ(h_{j}), & on Δ′_{j} , \\ 0, & on Δ \ Δ′_{j} , \end{matrix}

where μ(h_{j}) is the Beltrami coefficient of h_{j} and set D′_{j} = w^{μ_{j}}(Δ′_{j}). Notice that μ_{j} ∈ M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k}. Since h_{j} and w^{μ_{j}}|_{Δ′_{j}} have the same Beltrami coefficient, w^{μ_{j}} ∘ h_{j}^{−1} : Δ → D′_{j} is a Riemann mapping function of D′_{j}. Therefore we can take w^{μ_{j}} ∘h_{j}^{−1} as F_{j}^{μ_{j}}. By definition, G_{j}^{μ_{j}} = h_{j} ∘ (w^{μ_{j}})^{−1}Γ^{μ_{j}}w^{μ_{j}} ∘ h_{j}^{−1} = h_{j}Γ h_{j}^{−1} = Γ. By virture of Theorem 2, φ_{1}^{μ_{1}} ∈ S(Γ) \ \overline{J}, φ_{2}^{μ_{2}} ∈ J(Γ) \ \overline{T} , hence Theorem 1 is proved. □

Remark. The family (φ_{j}^{μ})_{μ∈M_{Y{+ \atop r_{0}}} (Δ,Γ)_{k}} is ample in a certain sense. We now explain

this in the following. We recall that F_{j} : Δ → Δ′_{j} is a Riemann mapping function of Δ′_{j} and G_{j} = F_{j}^{−1}Γ F_{j} is a Fuchsian group acting on Δ. In this paragraph we assume that Γ is non-elementary and choose F_{j} so that 1, i, −1 ∈ Λ(G_{j}) = Ĉ\Ω(G_{j}). Let F_{j}^{∗} : M(Δ, Γ) → M(Δ, G_{j}) be the pullback of Beltrami coefficients by F_{j} , namely F_{j}^{∗}(μ) is the Beltrami coefficient of the qc mapping w^{μ} ∘ F_{j} for μ ∈ M(Δ, Γ). Since w^{F_{j}^{∗}(μ)} and w^{μ} ∘ F_{j} have the same Beltrami coefficient, we can choose w^{μ} ∘ F_{j} ∘ (w^{F_{j}^{∗}(μ)})^{−1} : Δ → D_{j}^{μ} as the Riemann mapping function F_{j}^{μ} , then we have G_{j}^{μ} = w^{F_{j}^{∗}(μ)}G_{j}(w^{F_{j}^{∗}(μ)})^{−1}.

Generally, for ν ∈ M(Δ, G_{j}) the group isomorphism g ↦ w^{ν}g(w^{ν})^{−1} (g ∈ G_{j}) determines an element of the reduced Teichmüller space T^{#}(G_{j}) of G_{j} (see, for example, Earle [4], [5], Nag [10]). Let this point in T^{#}(G_{j}) be denoted by Φ^{#}(ν). It turns out that Φ^{#}(M_{K}(Δ, G_{j})_{k}) is a neighborhood of Φ^{#}(0) in T^{#}(G_{j}) for any k ∈ (0, 1] and any measurable set K ⊂ Δ such that p(K) is relatively compact in the double Ω(G_{j})/G_{j} where p : Ω(G_{j}) → Ω(G_{j})/G_{j} is the canonical projection. (For example, combine [11, Corollay 2] with [4]. This fact was pointed out to the author by H.Ohtake.)

On the other hand F_{j}^{∗}(M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k}) = M_{F_{j}^{−1}(Y_{r_{0}}^{+})}(Δ, G_{j})_{k} and p(F_{j}^{−1}(Y_{r_{0}})) is

relatively compact in Ω(G_{j})/G_{j} , hence Φ^{#}(F_{j}^{∗}(M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k})) is a neighborhood

of Φ^{#}(0). In other words, qc deformations G_{j} → G_{j}^{μ} (g ↦ w^{F_{j}^{∗}(μ)}g(w^{F_{j}^{∗}(μ)})^{−1}) for μ ∈ M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k} cover a neighborhood of the identity mapping G_{j} → G_{j} in T^{#}(G_{j}).

The above proof of Theorem 1 shows virtually that there exists an isomorphism G_{j} → Γ which belongs to Φ^{#}(F_{j}^{∗}(M_{Y_{r_{0}}^{+}} (Δ, Γ)_{k})).

§5. Proof of Theorem 2.

Proof of the first part. φ_{1}^{μ} ∉ \overline{J}.

By the same argument in [6] or [8], it is sufficient to prove the following

Claim 1. There exists a constant δ > 0 with the following property. If f is conformal in D_{1}^{μ} with ∥S_{f}∥_{D_{1}^{μ}} ≤ δ, then f(D_{1}^{μ}) is not a Jordan domain.

Proof of Claim 1. We set δ = δ_{1}/2 where δ_{1} is as in Lemma 1. Suppose that f is conformal in D_{1}^{μ} with ∥S_{f}∥_{D_{1}^{μ}} ≤ δ. Set g = f ∘ w^{μ} ∘ (τ ∘ σ)^{−1}|_{D_{1}}, β_{j} = w^{μ} ∘ (τ ∘ σ)^{−1}(α_{j}) for j = 1, 2 and w_{0} = w^{μ} ∘ (τ ∘ σ)^{−1}(0). Since

∥S_{g}∥_{D_{1}} = ∥S_{f∘w^{μ}}∥_{(τ∘σ)^{−1}(D_{1})} ≤ ∥S_{f}∥_{D_{1}^{μ}} + ∥S_{w^{μ}}∥_{Y_{r_{1}}^{+}} ≤ δ_{1} ,

Lemma 1 implies that

lim_{β_{1}∋w→w_{0}} f(w) = lim_{β_{2}∋w→w_{0}} f(w).

Thus f(D_{1}^{μ}) is not a Jordan domain. □

Proof of the second part. φ_{2}^{μ} ∉ \overline{T} .

Similary it is sufficient to prove the following

Claim 2. There exists a constant δ > 0 with the following property. If f is conformal in D_{2}^{μ} with ∥S_{f}∥_{D_{2}^{μ}} ≤ δ, then f(D_{2}^{μ}) is not a quasidisk.

Proof of Claim 2. We set δ = δ_{2}/2 where δ_{2} is as in Lemma 2. Suppose that f is conformal in D_{2}^{μ} with ∥S_{f}∥_{D_{2}^{μ}} ≤ δ. Further suppose that f(D_{2}^{μ}) is a Jordan domain. We shall show that ∂f(D_{2}^{μ}) is not a quasicircle. Set g = f ∘ w^{μ} ∘ (τ ∘ σ)^{−1}|_{D_{2}}. Since ∥S_{g}∥_{D_{2}} ≤ ∥S_{f}∥_{D_{2}^{μ}} + ∥S_{w^{μ}}∥_{Y_{r_{1}}^{+}} ≤ δ_{2} and g(D_{2}) is a Jordan domain, Lemma 2 produces that g̃(β) is not a quasiarc. Hence ∂f(D_{2}^{μ}) is not a quasicircle because g̃(β) ⊂ ∂f(D_{2}^{μ}). □

Acknowledgement. The author would like to thank Prof. Masahiko Taniguchi for giving helpful suggestions, which greatly simplified his previous argument. He also thanks Prof. Hiromi Ohtake for valuable informations.

Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan

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