Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 3–10

DISTANCE BETWEEN DOMAINS IN THE SENSE OF LEHTO IS NOT A METRIC

Vladimir Božin and Vladimir Marković

Massachusetts Institute of Technology, Department of Mathematics 77 Massachusetts Ave., Cambridge, MA 02139-4307, U.S.A.; bozin@math.mit.edu University of Minnesota, Department of Mathematics Minneapolis, MN 55455, U.S.A.; markovic@math.umn.edu

Abstract. In this paper we consider the question whether the quotient of the set of domains conformally equivalent to a halfplane by the group of Möbius transformations with distance in the sense of Lehto is a metric space. The answer is shown to be negative in the general case. However, restricted to analytic domains the question has an affirmative answer.

1. Introduction

Let 𝒟 denote the set of complex domains conformally equivalent to the upper half-plane H = {z ∈ C : Im z > 0}. Throughout this paper we shall denote

elements of 𝒟 by D, D̃, ˜ D̃ . The Poincaré density of the hyperbolic metric of D is defined by

η_{D}(z) = \frac{|π′_{D}(z)|}{Im π_{D}(z)},

where π_{D}: D → H is a conformal mapping onto H.

The Schwarzian derivative, or Schwarzian, of a conformal mapping f: D → D̃ is defined in D as the holomorphic function

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

The following two properties of the Schwarzian derivative are well known:

(1) S_{f} ≡ 0 if and only if f is a Möbius transformation.

(2) Cayley’s formula

S(g ∘ f, z) = S(g, f(z))f′(z)^{2} + S(f, z)

holds for conformal mappings f: D → D̃ and g: D̃ → ˜ D̃.

1991 Mathematics Subject Classification: Primary 30C55.

Furthermore, if a weighted sup-norm is defined for functions φ holomorphic in D by

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

we have the identities (see e.g. [3])

(3)  ∥S_{g∘f^{−1}}∥_{D} = ∥S_{g} − S_{f}∥_{D},  ∥S_{f}∥_{D} = ∥S_{f^{−1}}∥_{D}.

Two domains D and D̃ are said to be Möbius equivalent if there is a Möbius transformation of D onto D̃. We denote by Ω the quotient set of the set 𝒟 by the group of Möbius transformations. The Lehto distance between two domains D and D̃ in 𝒟 is defined by

δ(D, D̃) = inf{∥S_{f}∥_{D} : f: D → D̃ conformal}.

The above properties of the Schwarzian derivative imply that

\begin{aligned}δ(D, D̃) & = δ(D̃, D), \\ δ(D, ˜ D̃) & ≤ δ(D, D̃) + δ(D̃, ˜ D̃) \end{aligned}

for all domains D, D̃ and ˜ D̃ in 𝒟. Because δ(D, D̃) = 0 for Möbius equivalent domains D and D̃, the Lehto distance δ defines a pseudometric in the quotient set Ω. We shall see in the sequel that for the subset of Ω consisting of domains with analytic boundaries the Lehto distance does, indeed, define a metric, but that for an arbitrary domain D ∈ 𝒟, the equality δ(D, D̃) = 0 does not imply the Möbius equivalence of D and D̃.

2. Reducing the problem to quadratic differentials

We can consider the space

Q = {φ : φ holomorphic in H, ∥φ∥_{H} < ∞}

as the space of quadratic differentials of bounded norm corresponding to the universal Teichmüller space T (H); cf. [3]. If D is a domain in 𝒟 and f: H → D a conformal mapping of the upper half-plane H onto D, the Kraus–Nehari theorem implies the norm inequality ∥φ∥_{H} ≤ \frac{3}{2} . Thus every conformal mapping f: H → D of a domain D ∈ 𝒟 determines a corresponding quadratic differential φ = S_{f} ∈ Q. For another domain D̃ ∈ 𝒟 and conformal mapping f̃: H → D̃ we denote the respective quadratic differential by φ̃ = Sf̃ ∈ Q. The vanishing δ(D, D̃) = 0 of the Lehto distance is equivalent to the existence of a sequence g_{n}: D → D̃ of conformal mappings so that ∥S_{g_{n}}∥_{D} → 0 when n → ∞. For each g_{n} there is a Möbius transformation A: H → H such that g_{n} = f̃∘ A_{n}^{−1} ∘ f^{−1} . Now by (3) we have the identity ∥S_{g_{n}}∥_{D} = ∥Sf̃ − S_{f∘A_{n}}∥_{H}, and thus we get the following

Lemma 1A. The vanishing δ(D, D̃) = 0 of the Lehto distance between the domains D, D̃ ∈ 𝒟 is equivalent to the existence of a sequence of Möbius transformations A_{n}: H → H such that ∥φ − φ_{n}∥_{H} → 0 when n → ∞, where the sequence of quadratic differentials φ_{n} ∈ Q is defined by φ_{n}(z) = φ(A_{n}z)A′_{n}(z)^{2} = S_{f∘A_{n}}.

Similarly we have for the Möbius equivalence of domains

Lemma 1B. Two domains D, D̃ ∈ 𝒟 are Möbius equivalent, if and only if there exists a Möbius transformation A: H → H such that φ̃ = φ(Az)A′(z)^{2} .

For any quadratic differential φ ∈ Q we define the subset N(φ) ⊂ Q by

N(φ) = {(φ ∘ A)A′^{2} : A: H → H is a Möbius transformation}.

As an immediate consequence of Lemmas 1A and 1B we have

Lemma 1C. Let f: H → D be a conformal mapping. The set of all D̃ ∈ 𝒟 which satisfy δ(D, D̃) = 0 is equal to the set of domains Möbius equivalent to D if and only if N(S_{f}) is closed in Q.

Finally we prove

Lemma 1D. (Ω, δ) is a metric space, if and only if N(φ) is closed in Q for all φ ∈ Q.

Proof. Should N(φ) be closed in Q for each φ ∈ Q, then δ(D, D̃) = 0 if and only if D and D̃ are Möbius equivalent by Lemma 1C, so that (Ω, δ) would be a metric space. Conversely any φ ∈ Q with ∥φ∥_{H}, < \frac{1}{2} is the Schwarzian of a conformal mapping of the upper half-plane H by the Ahlfors–Weill theorem (cf. [2]). So should δ define a metric in Ω, then N(φ) would by Lemma 1C be closed wherever ∥φ∥_{H} < \frac{1}{2} , and thus actually for all φ ∈ Q.

3. A metric for analytic domains

A domain D ∈ 𝒟 is analytic, if the boundary of D is an analytic curve, i.e. the image of a circle K under a conformal mapping defined in a neighbourhood of K. We denote by C_{0}(H) the subspace of C(H) consisting of continuous functions f vanishing on the boundary of H, and define a subspace Q_{0} of Q by

Q_{0} = {φ ∈ Q : φη_{H}^{−2} ∈ C_{0}(H)}.

Lemma 2. S_{f} ∈ Q_{0} when f: H → D is a conformal mapping onto an analytic domain D.

Proof. Let M: H → U be a Möbius transformation of H onto the unit disc U, and g = f ∘ M^{−1}: U → D. Because D is an analytic domain, the mapping g extends to a conformal mapping defined in a neighbourhood of the closed unit disk \overline{U}. Having a holomorphic extension into a neighbourhood of \overline{U}, the Schwarzian S_{g} remains bounded on \overline{U}, so that lim_{|w|→1} S_{g}(w)η_{U}^{−2}(w) = 0. Thus we also have lim_{z→r} S_{f}(w)η_{H}^{−2}(z) = 0 for every boundary point r ∈ \overline{R} of H, so that the Schwarzian S_{f} belongs to Q_{0} when D is an analytic domain.

Lemma 3. N(φ) is closed in Q for every φ ∈ Q_{0} .

Proof. Let φ ∈ Q_{0} and A_{n} a sequence of Möbius automorphisms of H such that φ_{n} = (φ ∘ A_{n})A′_{n}^{2} → φ̃ in Q when n → ∞. By the compactness principle (e.g. [4]) we may suppose that the sequence A_{n} converges locally uniformly in H either to a Möbius automorphism A: H → H, or to a constant c ∈ \overline{R}. Should the sequence A_{n} converge to a constant function, we would have for all z ∈ H

φ̃(z)η_{H}(z)^{−2} = lim_{n→∞} φ(A_{n}z)η_{H}(A_{n}z)^{−2} = 0

because φ ∈ Q_{0} . Thus φ̃ vanishes identically, and we have ∥φ̃∥_{H} = 0. But φ_{n} → φ̃ in Q, and as ∥φ_{n}∥_{H} = ∥φ∥_{H} for all n, and we have φ = φ̃ = 0 ∈ Q_{0} , so that φ̃ = φ ∈ N(φ). When the sequence A_{n} converges to a Möbius automorphism A, we have φ̃ = (φ ∘ A)A′^{2} , which obviously belongs to N(φ). Thus N(φ) is a closed subset of Q for all φ ∈ Q_{0} .

Denoting by Ω_{A} the quotient of the set of analytic domains by the group of Möbius transformations we get as an immediate consequence of Lemmas 2, 3 and 1D.

Theorem 1. (Ω_{A}, δ) is a metric space.

4. (Ω, δ) is not a metric space

We are going to give here three slightly different examples of quadratic differentials φ ∈ Q for which N(φ) is not closed in Q. To do this we construct a quadratic differential φ ∈ Q and determine a sequence of Möbius transformations A_{n}: H → H so that the sequence φ_{n} = (φ ∘ A_{n})A′_{n}^{2} converges in Q towards a quadratic differential φ̃ not in N(φ). Particularly, if we choose φ so that ∥φ∥_{H} < \frac{1}{2} , then by the Ahlfors–Weill theorem there are conformal mappings f: H → D and f̃: H → D̃ with S_{f} = φ, Sf̃ = φ̃, so that we have δ(D, D̃) = 0 for the image domains D and D̃, although D and D̃ are not Möbius equivalent.

To begin with let us note that for all a > 0 the function e^{iaz} is in Q with the norm ∥e^{iaz}∥_{H} = (2/ae)^{2} .

Example 1. Let φ(z) = e^{2πiz} + e^{iz}, φ̃(z) = e^{2πiz} − e^{iz} and A_{k}(z) = z + n_{k}, where n_{k} = 2πm_{k} + π + o(1) when k → ∞ (n_{k}, m_{k} ∈ N). To establish the existence of such a sequence n_{k} we notice that the convergents P_{s}/Q_{s}, P_{s}, Q_{s} ∈ N, of the continued fraction expansion of π satisfy

|π − P_{s}/Q_{s}| < \frac{1}{Q_{s}Q_{s+1}} < \frac{1}{Q_{s}^{2}} , P_{s}Q_{s−1} − Q_{s}P_{s−1} = (−1)^{s}

with a strictly increasing sequence of denominators Q_{s} (see e.g. [5]). Thus by choosing an appropriate subsequence we have P_{s_{k}} = n_{k}, Q_{s_{k}} = 2m_{k} + 1 with n_{k}, m_{k} satisfying the above conditions.

It is not difficult to see that (φ ∘ A_{k})A′_{k}^{2} → φ̃ in Q. It remains to be shown that

(4)  φ̃ = (φ ∘ A)A′^{2}

holds for no Möbius transformation A: H → H. Now φ has zeros at z_{k} = (2k + 1)π/(2π − 1) and φ̃ at z̃_{k} = 2kπ/(2π − 1), k ∈ Z. So should (4) hold for a Möbius transformation, then A would be a translation by an odd integral multiple of π/2π − 1. Now for any c ∈ R the coefficients of e^{2πiz} and e^{iz} in the expansion

φ(z + c) = e^{2πic}e^{2πiz} + e^{ic}e^{iz}

are uniquely determined. However, the equations e^{2πic} = 1 and e^{ic} = −1 cannot be simultaneously satisfied for any c, so that φ̃ ≠ (φ ∘ A)A^{2} holds for all Möbius automorphisms A of the upper half-plane H.

Example 2. Suppose that the sequence λ_{k} satisfies ∑_{k=1}^{∞} 3^{2k}e^{−2}π^{−2}|λ_{k}| < +∞ and that λ_{k} ≠ 0 for infinitely many k ∈ N. We define φ and φ̃ by

\begin{aligned}φ(z) & = \frac{e^{iπz}}{(1 − e^{iπz})^{2}} + ∑_{k=1}^{∞} λ_{k}e^{2iπz/3^{k}}, \\ φ̃(z) & = \frac{−e^{iπz}}{(1 + e^{iπz})^{2}} + ∑_{k=1}^{∞} λ_{k}e^{2iπz/3^{k}}. \end{aligned}

Since ∥e^{iπz}/(1 − e^{iπz})^{2}∥_{H} = 1/π^{2} and ∥e^{2iπz/3^{k}}∥_{H} = 3^{2k}/e^{2}π^{2} , we conclude that φ, φ̃ ∈ Q. Setting A_{n}(z) = z + 3^{n} it is easy to see that (φ ∘ A_{n})A′_{n}^{2} → φ̃ in Q. It remains to be shown that φ̃ ∉ N(φ), i.e., that (4) cannot hold for any Möbius automorphism A of H. Now we notice that the limit

lim_{z→r} φ(z)η_{H}^{−2}(z)

is 0 for all r ∈ R \ 2Z and does not exist if r ∈ 2Z. Similarly, the expression φ̃(z)η_{H}^{−2}(z) equals 0 at all boundary points r ∈ R except for the set 2Z+1 of odd integers. Thus a Möbius transformation A satisfying (4) would be a translation by an odd integer. But in the expansion

φ(z + 2l + 1) − φ̃(z) = ∑_{k=1}^{∞} λ_{k}(e^{2iπ(2l+1)/3^{k}} − 1)e^{2iπz/3^{k}}

the coefficients of e^{iπz/3^{k}} are uniquely determined, so that φ(z + 2l + 1) = φ̃(z) cannot hold for any 2l + 1 ∈ Z.

Example 3. We now define φ and φ̃ by

\begin{aligned}φ(z) & = (e^{2πiz} − e^{−π})(∑_{k=1}^{∞} \frac{1}{10^{k}} e^{πiz/2^{k}}), \\ φ̃(z) & = (e^{2πiz} − e^{−π})(∑_{k=1}^{∞} \frac{1}{10^{k}} e^{πi/2^{k}(z+(4^{k}−1)/3)}), \end{aligned}

and set A_{n}(z) = z + \frac{1}{3}(4^{n} − 1). Since ∥e^{πiz/2^{k}}∥_{H} = 4^{k+1}/e^{2}π^{2} , the series

s(z) = ∑_{k=1}^{∞} \frac{1}{10^{k}} e^{πiz/2^{k}}

converges in Q. Because e^{2πiz} − e^{−π} is bounded in H, both functions φ and φ̃ are in Q, and it is not difficult to see that (φ ∘ A_{n})A′_{n}^{2} → φ̃ in Q.

For Im(z) ≤ 1 we have

|\frac{1}{10^{k}} e^{πiz/2^{k}} | > 2|\frac{1}{10^{k+1}} e^{πiz/2^{k+1}} |,

so that the series s(z) has no zeros with Im(z) < 1. Since all zeros of e^{2πiz} − e^{−π} are lying on the line Im(z) = \frac{1}{2} , we see that the functions φ and φ̃ do not have any zeros in the horizontal strip \{z ∈ H : Im(z) < \frac{1}{2}\}. On the other hand, the functions φ and φ̃ have zeros on the line Im(z) = \frac{1}{2} at the points \frac{1}{2}i + l with l ∈ Z. We conclude that a Möbius automorphism of H satisfying φ̃ = (φ ∘ A)A′^{2} would be a translation by an integer. Now for any l ∈ Z we have an expansion

φ(z + l) − φ̃(z) = (e^{2πiz} − e^{−π}) ∑_{k=1}^{∞} c_{n}e^{πiz/2^{k}}

with uniquely determined coefficients c_{n}. Hence, for φ(z + l) − φ̃(z) to vanish identically we should have

l ≡ \frac{1}{3}(4^{k} − 1) = 1 + 4 + ⋯ + 4^{k−1} (mod 2^{k+1})

for every k = 1, 2, …. This is obviously impossible for any fixed l ∈ Z, so that φ̃ ≠ (φ ∘ A)A′^{2} for all Möbius transformations A.

By Lemma 1D and the above three counterexamples we get thus the following theorem.

Theorem 2. (Ω, δ) is not a metric space.

5. Further results and discussion

Having shown in the previous section that N(φ) is not closed for some φ ∈ Q we now ask how large the set of such φ actually is.

Theorem 3. The set of quadratic differentials φ ∈ Q with N(φ) not closed is nowhere dense in Q.

Proof. We proved above that N(φ) is not closed in Q if and only if there is a sequence of Möbius transformations A_{n} of H such that (φ ∘ A_{n})A′_{n}^{2} → φ̃ and A_{n} → c ∈ \overline{R}. We shall show that this is possible only for quadratic differentials φ in a nowhere dense subset of Q.

Let Ψ_{0} ∈ Q be an arbitrary quadratic differential. We shall show that within any ball B ⊂ Q of radius ε > 0 centered at Ψ_{0} there is a ball B′ of radius \frac{1}{4}ε such that N(φ) is closed for every φ ∈ B′. First choose a quadratic differential Φ_{0} ∈ Q_{0} with ∥Φ_{0}∥_{H} = 1 and a point z_{0} ∈ H such that |Ψ_{0}(z_{0})|η_{H}(z_{0})^{−2} > ∥Ψ_{0}∥_{H} − \frac{1}{4}ε, and further a point z_{1} ∈ H with |Φ_{0}(z_{1})|η_{H}(z_{1})^{−2} sufficiently close to 1 and a Möbius transformation A: H → H mapping z_{0} to z_{1} so that

|\frac{3}{4}εΦ_{0}(Az_{0})A′(z_{0})^{2}e^{iθ} + Ψ_{0}(z_{0})|η_{H}(z_{0})^{−2} > ∥Ψ_{0}∥_{H} + \frac{1}{2}ε.

Thus ∥Ψ_{0} + Φ∥_{H} > ∥Ψ_{0}∥_{H} + \frac{1}{2}ε when Φ is defined by

Φ = \frac{3}{4}ε(Φ_{0} ∘ A)A′^{2}e^{iθ}.

For any ∥Φ_{1}∥_{H} < \frac{1}{4}ε, the quadratic differential φ = (Ψ_{0}+Φ)+Φ_{1} is contained in the ball B of radius ε centered at Ψ_{0} with a norm satisfying ∥φ∥_{H} > ∥Ψ_{0}∥_{H} + \frac{1}{4}ε. Because Φ, too, belongs to the subspace Q_{0} , we have (Φ ∘ A_{n})A′_{n}^{2} → 0 in H for any sequence A_{n} of Möbius automorphisms of H converging to a constant. Assuming that (φ ∘ A_{n})A′_{n}^{2} → φ̃ ∈ Q when n → +∞ we would thus also have ((Ψ_{0} + Φ_{1}) ∘ A_{n})A′_{n}^{2} → φ̃ in H as n → +∞. But ∥φ̃∥_{H} = ∥φ∥_{H} > ∥Ψ_{0}∥_{H} + \frac{1}{4}ε, so that for some z ∈ H and n > n_{0} we would have

|(Ψ_{0} + Φ_{1})(A_{n}z)A′_{n}(z)^{2}|η_{H}(z)^{−2} = |(Ψ_{0} + Φ_{1})(A_{n}z)|η_{H}(A_{n}(z))^{−2} > ∥Ψ_{0}∥_{H} + \frac{1}{4}ε,

contradicting the inequality ∥Ψ_{0} + Φ_{1}∥_{H} < ∥Ψ_{0}∥_{H} + \frac{1}{4}ε.

Let us finally discuss the background of our examples, especially Examples 3 and 2.

Let Γ_{j} be a decreasing sequence of Fuchsian groups and denote by Q(Γ_{j}) the increasing sequence of subspaces of Q consisting of all quadratic differentials φ satisfying φ = (φ ∘ A)A′^{2} for all A ∈ Γ_{j}. Let P = \overline{⋃_{j=1}^{∞} Q(Γ_{j})}. Consider now sequences A_{n} of Möbius transformations such that for every j, the transformations A_{n} are eventually in the same right coset of the group Γ_{j}. Thus for every j there is N_{j} such that (φ ∘ A_{m})A′_{m}^{2} = (φ ∘ A_{n})A′_{n}^{2} for all φ ∈ Q(Γ_{j}) whenever n, m > N_{j}. Such a sequence A_{n} obviously induces a mapping F: P → Q when F (φ) is defined by F (φ) = lim_{n→∞}(φ ∘ A_{n})A′_{n}^{2} ∈ Q. Any fixed Möbius transformation A: H → H determines a mapping F_{A}: P → Q, φ̃_{A}(φ) = (φ∘A)A′^{2} corresponding to the constant sequence A_{n} = A.

If P ≠ ⋃_{j=1}^{∞} Q(Γ_{j}), there can be mappings F which are not equal to F_{A} for any Möbius transformation A. This can be clearly seen from Example 3. There Γ_{j} is the group of translations by integral multiples of 2^{j}, and the mapping F corresponding to the sequence A_{n} acts as if it were a “translation” by the 2-adic number 1+4+4^{2} +· · ·. We could choose any other 2-adic number and “translate” any element of P, which is here the smallest closed subspace of Q containing all functions e^{mπiz/2^{n}}, m, n ∈ N.

We end this paper by posing the natural problem of characterizing all φ ∈ Q such that N(φ) is closed in Q.

We would like to thank our adviser M. Mateljević for motivation and many helpful discussions and suggestions.

References

[1] Ahlfors, L.V.: Quasiconformal Mappings. - D. van Nostrand, Princeton, N.J.–Toronto– New York–London, 1966.

[2] Ahlfors, L.V., and G. Weill: A uniqueness theorem for Beltrami equations. - Proc. Amer. Math. Soc. 13, 1962, 976–978.

[3] Lehto, O.: Univalent Functions and Teichmüller Spaces. - Grad. Texts in Math. 109, Springer-Verlag, New York, 1987.

[4] Lehto, O., and K.I. Virtanen: Quasiconformal Mappings in the Plane. - Springer-Verlag, Berlin–Heidelberg–New York, 1973.

[5] Vinogradov, I.M.: Fundamentals of Number Theory. - Nauka, Moscow, 1965.

Received 10 June 1996