Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 191–202

DEGENERATION OF QUASICIRCLES: INNER AND OUTER RADII OF TEICHMÜLLER SPACES

John A. Velling

Brooklyn College, Department of Mathematics Brooklyn, NY 11210, U.S.A.; jvelling@bklyn.bitnet

Abstract. A univalent function f: D → Ĉ with Schwarzian derivative having sup norm 2 can always be normalized to be arbitrarily close to log(1 + z)/(1 − z) on a given compact subset of D. Using this, necessary and sufficient conditions for the Bers embedding of a Teichmüller space (centered at a given surface) to have minimum possible inner radius are established in terms of hyperbolic geometry of the given surface. These conditions are the existence of points with arbitrarily large injectivity radius or simple closed geodesics with arbitrarily wide geodesic annular neighborhoods.

1. Statements of the main results

Let Ω denote a domain in the complex plane. If f: Ω → Ĉ is a nonconstant meromorphic function we let

(1.1)  𝒮_{f}(z) = \frac{f′′′}{f′} − \frac{3}{2}(\frac{f′′}{f′} )^{2}

denote the Schwarzian derivative of f. This is analytic on Ω if and only if f is locally univalent. It satisfies the transformation law

(1.2)  𝒮_{ϕ∘f∘ψ}(z) = 𝒮_{f}(ψ(z)) · ψ′(z)^{2} + 𝒮_{ψ}(z)

for ϕ ∈ Möb, the group of orientation preserving Möbius transformations of Ĉ. Furthermore, 𝒮_{f} ≡ 0 if and only if f ∈ Möb.

Now let D denote the unit disk in C. Nehari [19] showed that if f: D → Ĉ satisfies

(1.3)  ∥𝒮_{f}∥_{∞} = sup_{z∈D} \frac{(1 − |z|^{2})^{2}}{4} |𝒮_{f}(z)| ≤ \frac{1}{2}

1991 Mathematics Subject Classification: Primary 30C75; Secondary 32E35.

Partially supported by PSC-CUNY grant #661365.

then f is univalent. We let Möb(D) denote the subgroup of Möb preserving D, and observe, from (1.2), that if ψ ∈ Möb(D), then ∥𝒮_{f∘ψ}∥_{∞} = ∥𝒮_{f}∥_{∞}. Also note from (1.2) that with respect to the Möb(D) changes of coordinates on D, 𝒮_{f} is a quadratic differential on D. We denote by Q^{∞}(D) the space of holomorphic quadratic differentials on D which are bounded in the sense of (1.3). This is an infinite dimensional complex Banach space.

Ahlfors and Weill [2] strengthened Nehari’s result by showing that if ∥𝒮_{f}∥_{∞} < \frac{1}{2} then f(D) is a quasidisk, i.e. f extends continuously to S^{1} = ∂D, and f(S^{1}) is a quasicircle. In [6], Gehring and Pommerenke showed that if ∥𝒮_{f}∥_{∞} = \frac{1}{2} then f extends continuously to S^{1} and either f(D) is a Jordan domain (f(S^{1}) is a Jordan curve) or else f(D) is a region in Ĉ bounded between two circles tangent at a point. Herein the first order of business is to identify the complete obstruction to f(S^{1}) being a quasicircle under the hypothesis that ∥𝒮_{f}∥_{∞} = \frac{1}{2} .

If z ∈ C, let Re (z) and Im (z) denote the real and imaginary parts of z. The univalent function

(1.4)  f^{∗}(z) = log \frac{1 + z}{1 − z} = 2 tanh^{−1} z  (z ∈ D)

satisfies (1 − z^{2})^{2}𝒮_{f^{∗}}(z) ≡ 2 and maps D onto the parallel strip

(1.5)  𝒯 = \{t : −\frac{π}{2} < Im (t) < \frac{π}{2} \}.

Any region between two circles tangent at a point can be mapped by a Möbius transformation to 𝒯 . If f: D → Ĉ has ∥𝒮_{f}∥_{∞} = \frac{1}{2} we will show that this map f^{∗} embodies the complete obstruction to f(S^{1}) being a quasicircle.

Theorem 1. If f: D → Ĉ has ∥𝒮_{f}∥_{∞} = \frac{1}{2} and f(S^{1}) is not a quasicircle, then there exist sequences of Möbius transformations {ψ_{n} ∈ Möb(D)}_{n=1}^{∞} and {ϕ_{n} ∈ Möb}_{n=1}^{∞} such that the maps

(1.6)  f_{n} = ϕ_{n} ∘ f ∘ ψ_{n}: D → Ĉ

converge to f^{∗} uniformly on compacta.

This is proved in Section 2. In a similar vein we establish in Section 3

Theorem 2. If {f_{n}: D → Ĉ}_{n=1}^{∞} is a sequence of locally univalent functions such that no f_{n} is univalent, ∥𝒮_{f_{n}}∥_{∞} < ∞, and ∥𝒮_{f_{n}}∥_{∞} ↘ \frac{1}{2} as n → ∞, then there exist sequences of Möbius transformations, {ψ_{n} ∈ Möb(D)}_{n=1}^{∞} and {ϕ_{n} ∈ Möb}_{n=1}^{∞}, and a sequence {r_{n} ∈ (0, 1)}_{n=1}^{∞} with r_{n} → 1 as n → ∞, such that the maps given by

(1.7)  g_{n}(z) = ϕ_{n} ∘ f_{n}(r_{n}ψ_{n}(z))

converge to f^{∗} uniformly on compacta.

The author does not know whether the r_{n} are actually needed in this theorem. We apply Theorem 1 to a study of Teichmüller spaces to obtain several rather

immediate results. Theorems 3 and 4 below are essentially corollaries of Theorem 1 and known results. As such they are presented without formal proof.

Let ℛ be a hyperbolic Riemann orbifold, i.e. if H^{2} denotes two dimensional hyperbolic space with curvature ≡ −1 then ℛ = H^{2}/Γ where Γ is a discrete group of isometries of H^{2} . Giving D the metric ds = 2 |dz|/(1 − |z|^{2}) makes D a conformal model of H^{2} . In this case the group of orientation perserving isometries of H^{2} is precisely Möb(D). Discrete subgroups of Möb(D) are called Fuchsian groups. Let Q^{∞}(D; Γ) denote the subspace of Q^{∞}(D) which is Γ-invariant, i.e. A ∈ Q^{∞}(D; Γ) if and only if A ∈ Q^{∞}(D) and

(1.8)  A(γ(z))(γ′(z))^{2} = A(z)  for all z ∈ D, γ ∈ Γ.

We will need the following equivalence, established but not stated in this form by Nakanishi and Yamamoto in [18].

Theorem. [18] Let {Γ_{n}}_{n=1}^{∞} be a sequence of Fuchsian groups. Then for some α ∈ C (and hence for any α ∈ C) there exists a sequence {𝒮_{n} ∈ Q^{∞}(D; Γ_{n})}_{n=1}^{∞} converging uniformly to α dz^{2}/(1 − z^{2})^{2} on compacta in D if and only if the sequence of Riemann surfaces {ℛ_{n} = D/Γ_{n}}_{n=1}^{∞} has the property that, for given r > 0, eventually any ℛ_{n} contains either an isometrically embedded geodesic ball of radius r or a collar of width r about a simple closed geodesic.

In this case we say that the ℛ_{n} have either larger and larger balls or longer and longer tubes.

Let Q^{∞}(ℛ) denote the complex Banach space of bounded quadratic differentials on ℛ. If ℛ = D/Γ then this is canonically isomorphic to Q^{∞}(D; Γ). The Bers embedding B_{ℛ} : T(ℛ) → Q^{∞}(ℛ) maps the Teichmüller space T(ℛ) of equivalence classes of marked Riemann surfaces quasiconformally equivalent to ℛ into Q^{∞}(ℛ). This is an injection, realizing T(ℛ) as a bounded domain in Q^{∞}(ℛ). For background on Bers’ embeddings of Teichmüller spaces, the reader is referred to [3], a beautiful expository paper on this and related topics.

The inner radius i(T(ℛ)) of T(ℛ) is the supremum of radii of balls in Q^{∞}(ℛ) centered at the origin which are contained in B_{ℛ} (T(ℛ)). A consequence of the Ahlfors–Weill paper, [2], is that if ℛ carries a hyperbolic metric of curvature ≡ −1 then

(1.9)  i(T(ℛ)) ≥ \frac{1}{2}  whenever  B_{ℛ} (T(ℛ)) ≠ \{0\}.

Throughout the paper we will assume this curvature condition.

If Γ is an elementary group then we have equality in (1.9), [8]. On the other hand it can be shown, following Gehring and Pommerenke [6], that the inequality in (1.9) is strict for cofinite Γ ([12], [15]). A rather direct consequence of Theorem 1 and the result of [18] above is

Theorem 3. Let ℛ be a hyperbolic surface such that i(T(ℛ)) = \frac{1}{2} . Then one of the following two conditions holds: (O_{1}) for any r > 0, a hyperbolic geodesic ball of radius r embeds isometrically in

ℛ, or (O_{2}) for any r > 0, a collar of width r exists about some simple closed geodesic of

ℛ.

If ℛ satisfies either (O_{1} ) or (O_{2} ) of Theorem 3, we will say that ℛ has either big balls—(O_{1} )—or long tubes—(O_{2} ). Now if Γ is dissipative on S^{1} , ℛ has big balls. The following proposition is thus almost immediate, given information available in [21].

Proposition 1. If ℛ = D/Γ, where the action of Γ on S^{1} is dissipative, then i(T(ℛ)) = \frac{1}{2} and o(T(ℛ)) = \frac{3}{2} .

As mentioned above, for any hyperbolic ℛ, B_{ℛ} (T(ℛ)) is a bounded domain in Q^{∞}(ℛ). We denote by o(T(ℛ)) the outer radius of T(ℛ), i.e. the infimum of radii of balls in Q^{∞}(ℛ) centered at the origin which contain B_{ℛ} (T(ℛ)). A well-known theorem of Kraus [11] implies that

(1.10)  o(T(ℛ)) ≤ \frac{3}{2}.

In [18] it was shown that o(T(ℛ)) = \frac{3}{2} if and only if either (O_{1} ) or (O_{2} ) of Theorem 3 hold. In [16] and [17] it was shown that if either (O_{1} ) or (O_{2} ) hold on ℛ, then i(T(ℛ)) = \frac{1}{2} . Thus we have

Theorem 4. For a given hyperbolic Riemann surface ℛ, the following are equivalent:

1. i(T(ℛ)) = \frac{1}{2} ;

2. o(T(ℛ)) = \frac{3}{2} ;

3. ℛ has either big balls or long tubes.

Some consequences of these results will be discussed in Section 4.

It follows from [18] that for univalent f: D → Ĉ with either ∥𝒮_{f}∥_{∞} = \frac{1}{2} for which f(D) is not a quasidisk, or ∥𝒮_{f}∥_{∞} = \frac{3}{2} , whence f(D) is necessarily not a quasidisk, we have a sequence {ψ_{n} ∈ Möb(D)}_{n=1}^{∞} such that for some α ∈ C, 𝒮_{f∘ψ_{n}} → α dz^{2}/(1 − z^{2})^{2} uniformly on compacta. We ask if the family of quadratic differentials α dz^{2}/(1 − z^{2})^{2} on D contain the complete obstruction to any univalent map f: D → Ĉ having a quasidisk as its image. This is answered in the negative in Section 5 by giving explicit examples. We have

Theorem 5. There exist univalent maps f: D → Ĉ, with f(D) not a quasidisk, such that for no sequence {ψ_{n} ∈ Möb(D)}_{n=1}^{∞} does 𝒮_{f∘ψ_{n}} → α dz^{2}/(1−z^{2})^{2} uniformly on compacta for any α ∈ C. Thus the special collection of quadratic differentials α dz^{2}/(1 − z^{2})^{2} does not contain the full obstruction for a simply connected domain uniformized by D to be a quasidisk.

It is with pleasure that Toshihiro Nakanishi is acknowledged for his sustained and motivating interest in these topics, and Richard Schwartz for his lively participation in related conversations. The referee is thanked for having helped make this paper more readable.

2. Proof of Theorem 1

A domain Ω ⊂ C is said to have a c-accessible boundary if each w_{1} , w_{2} ∈ ∂Ω can be joined by an open arc A_{w_{1},w_{2}} ⊂ Ω such that

(2.1)  min_{j=1,2} |w − w_{j}| ≤ c dist(w, ∂Ω)  for  w ∈ A_{w_{1},w_{2}}.

It follows that c ≥ 1.

Gehring and Pommerenke [5, Theorem III.2.3], [6] characterize quasicircles via domains with c-accessible boundaries. In particular, we will need their

Lemma 1 [6]. Let Ω be a Jordan domain in Ĉ. That Ω is a quasidisk is equivalent to the existence of a constant c such that, for all ϕ ∈ Möb with ϕ(Ω) ⊂ C, the domains ϕ(Ω) have c-accessible boundaries. If Ω satisfies this, then ∂Ω is a quasicircle with constant M ≤ 2c.

It follows that if Ω ⊂ Ĉ is a Jordan domain but not a quasidisk, then for any c ≥ 1 there is some ϕ ∈ Möb (with ϕ(Ω) ⊂ C) such that for some w_{1} , w_{2} ∈ ∂ϕ(Ω), any curve A_{w_{1},w_{2}} ⊂ ϕ(Ω) joining w_{1} to w_{2} has some w_{0} on it satisfying

(2.2)  min_{j=1,2} |w_{0} − w_{j}| > c dist(w_{0}, ∂ϕ(Ω)).

Assume now that we are given f: D → Ĉ with ∥𝒮_{f}∥_{∞} = \frac{1}{2} and f(D) = Ω not a quasidisk. Let c = n ∈ Z_{+} , and ϕ_{n} ∈ Möb take Ω into C such that the conditions of the last paragraph hold. Giving ϕ_{n}(Ω) hyperbolic geometry, we take w_{1,n}, w_{2,n} ∈ ∂ϕ_{n}(Ω), let A_{w_{1,n},w_{2,n}} be the hyperbolic geodesic joining w_{1,n} and w_{2,n} in ϕ_{n}(Ω), and w_{0,n} ∈ A_{w_{1,n},w_{2,n}} be such that (2.2) is satisfied. Then as (2.2) is invariant under postcomposition by a complex affine transformation we may assume ϕ_{n} is chosen so that w_{0,n} = 0. Precomposing by a Möbius transformation ψ_{n}, we let f_{n} = ϕ_{n} ∘ f ∘ ψ_{n} and may assume that f_{n}(−1) = w_{1,n}, f_{n}(1) = w_{2,n}, and f_{n}(0) = 0. As we still have freedom left in our choice of ϕ_{n}, we may also assume that f′_{n}(0) = 1. It follows that in this case we have \frac{1}{4} ≤ dist(w_{0}, ∂Ω) ≤ 1, [13, Section 28, Satz 6], so that

(2.3)  min_{j=1,2} |w_{j,n}| > \frac{n}{4}.

We now establish the following

Lemma 2. If {f_{n}: D → C}_{n=1}^{∞}, normalized by f_{n}(z) = z + a_{n}z^{2} + O(z^{3}), is a sequence of maps satisfying ∥𝒮_{f_{n}}∥_{∞} ≤ \frac{1}{2} , and |f_{n}(1)|, |f_{n}(−1)| > n, then

(2.4)  f_{n}(z) → tanh^{−1} (z)

uniformly on compacta in D.

Immediate consequences of this are

1. Re (𝒮_{f_{n}}(0)) → 2,

2. a_{n} → 0, and

3. the verity of Theorem 1. (The f_{n} given preceding Lemma 2 satisfy the criteria of Lemma 2.) To establish Lemma 2, we use a technique of Hawley and Schiffer [7] (see

also [6]). We precompose with h: T = \{t ∈ C : | Im (t)| < \frac{1}{2}π\} → D given by h(t) = tanh (\frac{1}{2}t). Let g_{n} = f_{n} ∘ h, so that

(2.5)  g_{n}(t) = t + a_{n}t^{2} + O(t^{3})

and

(2.6)  𝒮_{g_{n}}(t) = −\frac{1}{2} + \frac{1}{4}(1 − h(t)^{2})^{2}𝒮_{f_{n}}(h(t)).

Evidently Re (𝒮_{g_{n}}(t)) ≤ 0 with equality if and only if 𝒮_{f_{n}}(h(t)) = 2/(1−h(t)^{2})^{2} . It suffices to show

(2.7)  g_{n} → id

uniformly on compacta in T.

Defining

(2.8)  v_{n}(t) = |g′_{n}(t)|^{−1/2}  for  t ∈ R,

we note that since g_{n}: T → C is univalent in T, v_{n} never vanishes. Also

(2.9)  \frac{v′_{n}}{v_{n}} = −\frac{1}{2} Re (\frac{g′′_{n}}{g′_{n}} ),  \frac{v′′_{n}}{v_{n}} − (\frac{v′_{n}}{v_{n}} )^{2} = \frac{1}{2} Re (\frac{d}{dt} \frac{g′′_{n}}{g′_{n}} )

so that

(2.10)  v′′_{n}(t) = p_{n}(t) · v_{n}(t)

where

(2.11)  p_{n}(t) = −\frac{1}{2} Re (𝒮_{g_{n}}(t)) + \frac{1}{2} Im (\frac{g′′_{n}(t)}{g′_{n}(t)})^{2} ≥ 0.

We now verify that (2.7) does indeed hold.

Suppose that v_{n}(1) ≥ 1 + ε. Assume for the moment that v_{n}(t) ≥ 1 for t ∈ (0, 1). The mean value theorem tells us that for some τ_{n} ∈ (0, 1), v′_{n}(τ_{n}) = ε, so that v′_{n}(1) ≥ ε. As v_{n} is convex, v_{n}(t) ≥ 1 + εt for t ≥ 1. Thus |g′_{n}(t)| ≤ (1 + εt)^{−2} , so that

(2.12)  |w_{2,n}| ≤ ∫_{0}^{∞} |g′_{n}(t)| dt ≤ 1 + ∫_{1}^{∞} \frac{1}{(1 + εt)^{2}} dt ≤ 1 + \frac{1}{ε}.

We see that \frac{1}{4}n ≤ (1+(1/ε)), or that n ≤ 4(1+(1/ε)). Similarly if v_{n}(−1) ≥ 1+ε and v_{n}(t) ≥ 1 for t ∈ (−1, 0), then n ≤ 4(1+(1/ε)). The convexity of v_{n} implies that indeed v_{n}(t) ≥ 1 on either t ∈ (−1, 0) or (0, 1), so that as n → ∞ we see that v_{n}(±1) → 1.

As we may, for any t ∈ R rather than just t = ±1, study in the same fashion the behavior of v_{n}(t), this means that as n → ∞, v_{n}(t) → 1 for all t ∈ R. Note that if {f_{n}}_{n=1}^{∞} is a sequence of positive convex functions on R such that f_{n}(0) = 1 for all n, and, for all t ∈ R, f_{n}(t) → 1, then f_{n} → 1 uniformly on compact sets in R. We conclude that v_{n} → 1 uniformly on compact sets in R.

It follows from the identity theorem that g_{n} → id uniformly on compacta, as desired.

3. Proof of Theorem 2

We pick r_{n} so that F_{n}: D → Ĉ given by F_{n}(z) = f_{n}(r_{n}z) is univalent on D but not on S^{1} . Such clearly exist, and that r_{n} → 1 as n → ∞ follows from an estimate of Kra and Maskit [10, Lemma 5.1]. Pick z_{z,1} , z_{n,2} ∈ S^{1} such that F_{n}(z_{n,1}) = F_{n}(z_{n,2}). Take ψ_{n} ∈ Möb(D) such that ψ_{n}(z_{n,1}) = −1 and ψ_{n}(z_{n,2}) = 1. By pre- and post-composition with Möbius transformations we may assume G_{n}(z) = ϕ_{n} ∘ F_{n} ∘ ψ_{n}(z) is univalent on D, extends holomorphically to S^{1} , and G_{n}(−1) = G_{n}(1) = ∞. Note that we still have two degrees of freedom left in the choice of ϕ_{n}, one in the choice of ψ_{n}.

With h: 𝒯 → D as in Section 2, let g_{n} = G_{n} ∘ h. We wish to choose ϕ_{n}, ψ_{n} so that g_{n} → id uniformly on compacta. As in Section 2, let v_{n} = |g′_{n}|^{−1/2} .

Lemma 3. Let t ∈ R. As t → ±∞, v_{n}(t) = O( exp (−|t/2|)).

Proof of Lemma 3. With G_{n} as in the preceding paragraph, we let V_{n}(z) = |G′_{n}(z)|^{−1/2} . As G_{n} has a pole at both ±1, G′_{n} has a double pole at both. Thus V_{n} vanishes at ±1. One readily checks that

(3.1)  v_{n}(t) = V_{n}(h(t)) · |\frac{dh}{dt}|^{−1/2}  with  z = h(t) = tanh (\frac{t}{2}).

Thus, as V_{n}(z) = O(|z − 1|) as z → 1,

(3.2)  v_{n}(t) = O( tanh (\frac{1}{2}t) − 1) · O( cosh (\frac{1}{2}t)) = O(e^{−t}) · O(e^{t/2}) = O(e^{−t/2})  as t → ∞.

Since the same argument works as t → −∞, the lemma is established.

Lemma 4. For any fixed ϕ_{n} such that our normalization holds,

(3.3)  max_{t∈R} v_{n}(t)

is defined and ≠ ∞, independent of the freedom of choice of ψ_{n}.

Proof of Lemma 4. By Lemma 3, there exists a maximum over R for v_{n}(t). Our choice of ψ_{n} is determined up to post-composition by an element of Möb(D) fixing 1 and −1. Since if ψ ∈ Möb(D) fixes both 1 and −1, ψ ∘ ψ_{n} ∘ h(t) = ψ_{n} ∘ h(t + τ) for some fixed τ ∈ R, the proof of our lemma reduces down to showing that max_{t∈R} v_{n}(t) = max_{t∈R} v_{n}(t + τ). But this is immediate, proving the lemma.

We now choose ψ_{n} so that the maximum in (3.3) occurs at t = 0, and choose ϕ_{n} so that g_{n}(0) = 0, g′_{n}(0) = 1.

Since the g_{n} are univalent on 𝒯 and normalized as in the last paragraph, we consider any convergent subsequence (still called {g_{n}}_{n=1}^{∞} for convenience) which converges uniformly on compacta in 𝒯 . Then v_{n} also converges uniformly on compacta in 𝒯 . Let g^{∗} be the limit of the g_{n}, and v^{∗} the limit of the v_{n}. As in (2.8), v^{∗} satisfies an ODE

(3.4)  (v^{∗})′′(t) = p^{∗}(t) · v^{∗}(t),

where p^{∗} ≥ 0. Thus v^{∗} is convex for t ∈ R.

Noting that if {f_{n}: R → [0, 1], f_{n}(0) = 1}_{n=1}^{∞} is a sequence of continuous functions which converge pointwise to a convex function f^{∗}, then f^{∗} ≡ 1, we deduce that v^{∗} ≡ 1. As this limit is independent of the subsequence chosen in the last paragraph from the original sequence {g_{n}}_{n=1}^{∞}, this completes the proof of the theorem.

4. Some consequences

Let ℛ_{1} and ℛ_{2} be two quasiconformally equivalent hyperbolic Riemann surfaces. If ℛ_{1} satisfies either condition (O_{1} ) or (O_{2} ) then ℛ_{2} does also: if ℛ_{1} has big balls then the fact that K -quasiconformal maps are bi-Hölder of exponent 1/K (see [1, Chapter 1]) implies that ℛ_{2} does as well, and ℛ_{1} having long tubes is equivalent to annuli of arbitrarily large moduli being embedded in ℛ_{1} —a property which is preserved by quasiconformal deformation (see [4, p. 159]). We may thus say that the satisfaction of (O_{1} ) or (O_{2} ) is determined by the underlying quasiconformal structure, and have

Corollary 1. If ℛ is a quasiconformal hyperbolic surface and ℛ_{1} , ℛ_{2} are two conformal structures on ℛ compatible with its quasiconformal structure, then the following conditions are equivalent:

1. i(T(ℛ_{1})) = \frac{1}{2} ;

2. o(T(ℛ_{1})) = \frac{3}{2} ;

3. i(T(ℛ_{2})) = \frac{1}{2} ;

4. o(T(ℛ_{2})) = \frac{3}{2} ;

5. ℛ has big balls or long tubes (in the sense of Theorem 3), independent of the compatible complete hyperbolic structure.

Following this line of thought, we let K_{1/3} denote the middle thirds Cantor set, and Ĉ \ K_{1/3} its complement in the number sphere. As this hyperbolic Riemann surface is quasiconformally equivalent to a normal cover of a genus two surface, with Z ∗ Z acting as the group of deck transformations (see [22] for the construction), we have that hyperbolically Ĉ \ K_{1/3} satisfies neither (O_{1} ) nor (O_{2} ). This yields

Corollary 2. For the Riemann surface Ĉ\K_{1/3} one has i(T(Ĉ\K_{1/3})) > \frac{1}{2} and o(T(Ĉ \ K_{1/3})) < \frac{3}{2} .

Theorems 3 and 4 may be seen to hold in the more general case of sequences of Riemann surfaces. One has, in particular,

Corollary 3. Let {ℛ_{n}}_{n=1}^{∞} be a sequence of hyperbolic Riemann surfaces. The following are equivalent:

1. i(T(ℛ_{n})) → \frac{1}{2} ;

2. o(T(ℛ_{n})) → \frac{3}{2} ;

3. as n → ∞ the ℛ_{n} have either larger and larger balls or longer and longer tubes (in the sense of the result of [18] mentioned in Section 1).

For a hyperbolic Riemann surface ℛ, T(ℛ) depends only on the conformal type of ℛ, not on any marking. Thus we may consider i(T(ℛ)) and o(T(ℛ)) as functions on the moduli space of unmarked Riemann surfaces quasiconformally equivalent to ℛ. In the case where ℛ has finite hyperbolic area, is of genus g, and has n punctures, we denote by ℳ^{g,n} this moduli space. Mumford [14] showed that if {ℛ_{k} ∈ ℳ^{g,n}}_{k=1}^{∞} is a sequence of Riemann surfaces leaving every compact set in ℳ^{g,n}, then the ℛ_{n} are developing short geodesics. By the collar lemma (see [9]) this implies that the ℛ_{n} are developing long tubes. In this finite volume case, the converses also clearly hold. Thus one has

Corollary 4. i(T(ℛ_{n})) → \frac{1}{2} and o(T(ℛ_{n})) → \frac{3}{2} if and only if the \{ℛ_{n}\}_{k=1}^{∞}

eventually leave every compacta in ℳ^{g,n}.

Finally we note that these results hold, with little modification, for hyperbolic orbifolds, i.e. ℛ = D/Γ where Γ may have elliptic elements.

5. Obstructions to quasidisk-hood may be quite complicated

We now examine in a somewhat more general setting the obstruction to a simply connected domain being a quasidisk. First note that the set of α ∈ C such that α dz^{2}/(1 − z^{2})^{2} is the Schwarzian derivative of a map from D to a quasidisk forms the interior of the bounded set bordered by the cardioid

Λ = {α = 2(1 − re^{2iθ}) : r = 4 cos^{2} θ, 0 ≤ θ ≤ π},

in C. For α exterior to this cardioid, any map from D to Ĉ having α dz^{2}/(1−z^{2})^{2} as its Schwarzian is not even univalent (see [8]). We saw in Theorem 1 above that for univalent maps f: D → Ĉ with ∥𝒮_{f}∥_{∞} ≤ \frac{1}{2} the point α = 2 on Λ embodies the entire obstruction to f(D) being a quasidisk. (Never mind the fact that all such maps are a priori univalent.) It was seen in [18] that if f: D → Ĉ is univalent with ∥𝒮_{f}∥_{∞} = \frac{3}{2} then the point α = −6 embodies the obstruction to f(D) being a quasidisk.

Now consider the space of oriented simply connected domains Ω in Ĉ which are uniformized by D. We say two such domains are equivalent, denoted by Ω_{1} ∼ Ω_{2} , precisely when there is some ϕ ∈ Möb such that Ω_{2} = ϕ(Ω_{1}). The space of such domains mod ∼ is naturally parametrized by the Schwarzian derivatives of Riemann mappings to the Ω, mod precomposition by elements of Möb(D). We have the following straightforward consequence of the Nakanishi–Yamamoto theorem

Corollary 5. There exists a quasidisk Ω such that no sequence of uniformizing maps f_{n}: D → Ω has the property that for some α ∈ C, 𝒮_{f_{n}} → α dz^{2}/(1−z^{2})^{2} uniformly on compacta in D.

Proof. Such a quasidisk may be taken to be one of the components of the domain of discontinuity of a quasifuchsian group uniformizing two conformally distinct, quasiconformally equivalent marked finite hyperbolic area Riemann surfaces. As the geometries of the two surfaces are finite and fixed, neither big balls nor long tubes exist.

This allows us to show the following, which includes Theorem 5 of the introduction.

Theorem 5′. There exists a simply connected non-quasidisk domain Ω uniformized by D such that

1. no sequence of uniformizing maps f_{n}: D → Ω has the property that for some α ∈ C, 𝒮_{f_{n}} → α dz^{2}/(1 − z^{2})^{2} uniformly on compacta in D,

2. for any uniformizing map f: D → Ω, there exists a sequence of quasidisks Ω_{k} and uniformizing maps g_{k}: D → Ω_{k} such that g_{k} → f uniformly on compacta in D (and, moreover, 𝒮_{g_{k}} → 𝒮_{f} in the topology of (1.3)).

Degeneration of quasicircles 201

Thus the special collection of quadratic differentials α dz^{2}/(1 − z^{2})^{2} does not contain the full obstruction for a simply connected domain uniformized by D to be a quasidisk.

Proof. For an example here, take a totally degenerate group on the boundary of Bers’ embedding of Teichmüller space for a finite hyperbolic area Riemann surface. Once again, the finite geometry of the surface does not allow for big balls or long tubes. In fact, one may use Corollary 4 to show that ∥𝒮_{f}∥_{∞} may be taken arbitrarily close to either \frac{1}{2} or \frac{3}{2} .

Addendum. Since the writing of this article, Ohtake’s [20] has appeared, containing related results.

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202 John A. Velling

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Received 5 June 1992