Georgian Mathematical Journal Volume 8 (2001), Number 4, 815–822

A REMARK ON POLYGONAL QUASICONFORMAL MAPS

S. L. KRUSHKAL

To Reiner Kühnau on his 65th birthday

Abstract. Given a quasisymmetric map h : R̂ → R̂, let f_{0} be an extremal quasiconformal extension of h onto the upper half-plane U = {z ∈ ℂ : ℑz > 0} whose dilatation k(f_{0}) = inf{k(f) : f|∂U = h_{0}} =: k(h). Let k_{n} be the minimal dilatation of polygonal quasiconformal maps f : U → U satisfying f(x_{j}) = h(x_{j}), j = 1, 2, …, n, for any n points of R̂ (vertices of n-gons). Already a long time ago, the question was posed whether k(h) = sup k_{n}, where the supremum is taken over all possible n-gons of such kind. The answer is obtained (in the negative) for the case of quadrilaterals (n = 4).

We show that in the case of pentagons (n = 5) the answer is also negative, i.e., there are quasisymmetric h with k(h) > sup k_{5}.

2000 Mathematics Subject Classification: Primary: 30C62, 30C75, 30F60. Secondary: 30C35, 30F45, 31A15. Key words and phrases: Extremal quasiconformal maps, polygon, Grunsky coefficients, Fredholm eigenvalues.

1. Introduction. Any quasisymmetric homeomorphism h of R̂ = R ∪ {∞} onto itself, i.e., such that

M^{−1} ≤ \frac{h(x + t) − h(x)}{h(x) − h(x + t)} ≤ M

for any x ∈ R and t > 0, with M = M(h) < ∞, admits quasiconformal extensions onto the upper half-plane U = {z : ℑz > 0} (and symmetrical extensions to the lower half-plane U^{∗} = {z : ℑz < 0}).

Let f_{0} be an extremal quasiconformal extension of h whose dilatation

k(f_{0}) = inf{k(f) = ∥∂z̄f/∂_{z}f∥_{∞} : f|∂U = h} =: k(h).

Recall that quasiconformal maps are (generalized) homeomorphic solutions of the Beltrami equation ∂z̄w = μ(z) ∂_{z}w in a domain D ⊂ Ĉ = ℂ ∪ {∞} with ∥μ∥_{∞} < 1, where the derivatives ∂z̄ and ∂_{z} are distributional and belong locally to L_{2}. The function μ is called the Beltrami coefficient of the map w, and the quantity K(f) = (1 + k(w))(1 − k(w)) is its maximal dilatation. The maps with k(w) ≤ k_{0} are called k_{0}-quasiconformal.

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Given a simply connected Jordan domain D ⊂ Ĉ, one can regard it as a (topological) polygon D(z_{1}, …, z_{n}) whose vertices are n distinguished boundary points z_{1}, …, z_{n} ordered in accordance with the orientation of ∂D. Quasiconformal maps of polygons moving the vertices into vertices will be called polygonal quasiconformal.

In view of conformal invariance of q-quasiconformality, it will suffice for us to deal with the maps of polygons whose domains are either the upper half-plane U or the unit disk Δ = {z : |z| < 1}.

For any quasisymmetric map h, we have k(f_{0})= lim_{n→∞} k(f_{n}), where f_{n} are extremal polygonal quasiconformal maps of U(x_{1}, …, x_{n}) onto U(h(x_{1}), …, h(x_{n})), and the set of vertices {x_{n}} becomes dense on R.

Already a long time ago, the question was posed whether for a fixed n ≥ 4 (or even for n ≤ N) the equality

k(h) = sup k(f_{n})  (1)

is achieved when one allows the vertices to vary on R̂ in all possible ways.

The answer is known only in the case of quadrilaterals (n = 4). The distortion of conformal modules mod Q of all quadrilaterals D ⊂ Δ under a quasiconformal homeomorphism f : D → Ĉ determines its maximal dilatation K(f), and it was conjectured that the bound

K_{0}(h) = sup_{Q} \frac{mod f_{0}(Q)}{mod Q},

where f_{0} is an extremal extension of h and the supremum is taken over all quadrilaterals Q = U(z_{1}, z_{2}, z_{3}, z_{4}), is the best possible.

This conjecture was disproved by a counterexample of Anderson and Hinkkanen (see [1]) and by Reich [15]; a complete answer for quadrilaterals has been given in [21] (see also [12]).

As for n ≥ 5, it has been showed by Strebel [19] that equality (1) holds for extremal quasiconformal maps f_{0} without essential boundary points.

The purpose of this note is to show that in the case of pentagons (n = 5) the answer is also negative:

Theorem. For each k ∈ (0, 1), there exist quasisymmetric maps h with

k(h) = k > sup k(f_{5}),  (2)

where the supremum is taken over the extremal polygonal maps of all possible pentagons U(x_{1}, …, x_{5}).

The proof goes along the lines of [15] and [5]; it involves the fundamental Reich–Strebel inequality for polygonal maps and the Grunsky coefficient inequality for holomorphic functions with quasiconformal extension.

2. The Grunsky coefficients. For convenience, we collect here the main facts concerning the properties of the Grunsky matrix of univalent functions.

We shall use the following notations. Let D be a hyperbolic simply-connected domain in Ĉ. We denote by A_{1}(D) the subspace of L_{1}(D) formed by holomorphic functions in D , and put

< μ, φ >_{D}= ∫ ∫_{D} μ(z)φ(z)dxdy, μ ∈ L_{∞}(D), φ ∈ L_{1}(D) (z = x + iy)

and

A_{1}^{2}(D) = {φ ∈ A_{1}(D) : φ = ω^{2}};

the last set consists of the integrable holomorphic functions in D with zeros of even order.

The Grunsky univalence criterion [3] says that a Ĉ-holomorphic function

f(z) = z + a_{0} + a_{1}z^{−1} + ⋯  (3)

in a neighborhood of the infinite point z = ∞ extends to an injective holomorphic function in the disk

D^{∗} = {z ∈ Ĉ : |z| > 1}

(with the same expansion (2)) if and only if its Grunsky coefficients, defined from the expansion

log \frac{f(z) − f(ζ)}{z − ζ} = − ∑_{m,n=1}^{∞} c_{mn}z^{−m}ζ^{−n}, (z, ζ) ∈ (Δ^{∗})^{2},

satisfy for any x = (x_{1}, x_{2}, …) ∈ l^{2} the inequality

| ∑_{m,n=1}^{∞} \sqrt{mn} c_{mn}x_{m}x_{n}| ≤ ∥x∥^{2} (∥x∥^{2} = (| ∑_{1}^{∞} |x_{n}|^{2})^{1/2}).  (4)

It is also known that, for functions (3) having k-quasiconformal extensions to Ĉ, inequality (4) is sharpened as follows:

κ(f) = sup \{| ∑_{m,n=1}^{∞} \sqrt{mn} c_{mn}x_{m}x_{n}| : x ∈ l^{2}, ∥x∥ = 1\} ≤ k∥x∥^{2};  (5)

and on the other hand, any holomorphic function f in Δ^{∗} satisfying (5) is univalent and quasiconformally extendible to Ĉ with a dilatation k′ ≥ k (see, e.g., [9] [13]; [8], pp. 82–84).

We shall denote the class of univalent functions in Δ^{∗} of form (3) by Σ, and let Σ(k) consist of f ∈ Σ having k-quasiconformal extensions to Ĉ.

The following statement provides a complete description of the class of functions for which inequality (5) is both necessary and sufficient to belong to Σ(k), i.e., the value κ(f) coincides with the least (extremal) dilatation k(f) among possible extensions of f.

Proposition ([5], [6]). The equality

κ(f) = inf{∥μ∥_{∞} : w^{μ}|Δ^{∗} = f}  (6)

holds if and only if the function f is the restriction to Δ^{∗} of a quasiconformal self-map w^{μ_{0}} of ℂ with the Beltrami coefficient μ_{0} satisfying the condition

sup | < μ_{0}, φ >_{Δ} | = ∥μ_{0}∥_{∞},  (7)

where the supremum is taken over holomorphic functions φ ∈ A_{1}^{2}(Δ) with ∥φ∥_{A_{1}(Δ)} = 1.

Geometrically, this means that the Carathéodory metric on the immersion of the holomorphic disk {tμ_{0} : t ∈ Δ} ⊂ L_{∞}(Δ) into the universal Teichmüller space T equals the intrinsic Teichmüller–Kobayashi metric of this space.

In particular, for any function f ∈ Σ(k) having k-quasiconformal extension onto \overline{Δ} with the Beltrami coefficient μ of the form

μ(z) = k|φ(z)|/φ(z), φ ∈ A_{1}^{2}(Δ) \ {0},  (8)

we have the equality

κ(f) = k.  (9)

If the boundary quasicircle L = f(∂Δ) is asymptotically conformal, i.e., satisfies

max_{z∈L(a,b)} \frac{|a − z| + |z − b|}{|a − b|} → 1 as |a − b| → 0,

where L(a, b) is the smaller subarc of L between the points a, b ∈ L, then equality (8) is both necessary and sufficient for f ∈ Σ(k) to have (9) ([7]; see also [6], [11]). In particular, this holds for C^{1}-smooth curves.

The properties of asymptotically conformal curves and of related conformal maps are described, for example, in [14].

Using Strebel’s frame mapping criterion (see [18], [2]), one can establish that each f ∈ Σ(k) mapping the unit circle S^{1} = ∂Δ onto an asymptotically conformal curve admits a unique extremal quasiconformal extension to \overline{Δ} of Teichmüller’s type, i.e., whose Beltrami coefficient μ_{0} = k|φ|/φ with φ ∈ A_{1}(Δ) \ {0} (see [7]). This yields (together with (8) and (9)) that for any f ∈ Σ(k) which maps Δ^{∗} onto a domain with asymptotically conformal boundary and has extremal extension with the Beltrami coefficient is of the form

μ_{0} = k|φ|/φ, φ ∈ A_{1}(Δ) \ A_{1}^{2}(Δ),  (10)

(i.e., with φ having at least one zero in Δ of odd order), one obtains the strong inequality κ(f) < k.

The simplest example of coefficients (10) is given by μ_{0}(z) = |z|^{m}/z^{m} with odd m ≥ 1.

3. The proof of the Theorem consists of four stages.

(a) For a given quasisymmetric map h, let f_{0} be an extremal quasiconformal extension of h to U. Fix a pentagon U(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}) with the vertices a_{1}, …, a_{5} ∈ R̂. We can assume that these vertices are ordered by

−∞ < a_{1} < a_{2} < ⋯ < a_{5} = ∞.

By the general Teichmüller theorem on extremal quasiconformal maps (see, e.g., [20], [4]), the Beltrami coefficient of the extremal map

f : U(a_{1}, …, a_{5}) → U(h(a_{1}), …, h(a_{5}))

is of the form

μ_{f}(z) = k \frac{|φ_{0}(z)|}{φ_{0}(z)},

where k = k(f) and φ_{0} is a rational function of the form

φ_{0}(z) = ∑_{k=1}^{4} \frac{c_{k}}{z − a_{k}} .  (11)

Here the numbers c_{k} are real because the quadratic differential φ_{0}(z)dz^{2} takes the real values on R; hence

φ_{0}(z̄) = \overline{φ_{0}(z)}.

First observe that after composing h with the elements of P SL(2, R) (fractional linear automorphisms of U), the initial pentagon is transformed conformally to a pentagon U(0, 1, a, b, ∞) for which representation (11) takes the most simple form

φ_{0}(z) = \frac{(c_{1} + c_{2})z − (c_{1}b + c_{2}a)}{z(z − 1)(z − a)(z − b)},

which shows that φ_{0} has only one real zero. This yields that the extremal quasiconformal maps of any pentagon D(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}), whose domain D is bounded by a C^{1+α} smooth curve onto other ones, are determind by holomorphic quadratic differentials

ψ_{0} = (φ_{0} ∘ g)(g′)^{2},  (12)

where g maps conformally D(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}) onto U(0, 1, a, b, ∞), and each ψ_{0} has only one zero; moreover, this zero is located on the boundary of D. Consequently, ψ_{0} does not vanish in D, and it is possible to define a singlevalued branch of \sqrt{ψ_{0}} in the whole domain D.

When at least two vertices of a pentagon tend to the same limit point, it degenerates into an n-gon with n ≤ 4 vertices.

(b) Now, assuming that equality (1) holds for h with n = 5, one can select a sequence of pentagons

P_{m} = U(a_{1,m}, a_{2,m}, a_{3,m}, a_{4,m}, a_{5,m}), m = 1, 2, …,

and the corresponding extremal quasiconformal maps

f_{m} : P_{m} → U(h(a_{1,m}), …, h(a_{5,m}))

which carry the vertices a_{j} to h(a_{j}), j = 1, …, 5, so that

lim_{m→∞} k(f_{m}) = k(h) = k(f_{0}).  (13)

The Beltrami coefficients of these maps are

μ_{f_{m}}(z) = k_{m}|φ_{m}(z)|/φ_{m}(z), k_{m} = k(f_{m}),

where each φ_{m}(z) is a rational function of form (11).

We now apply the Reich–Strebel inequality for polygons to the extremal Beltrami coefficient μ_{0} = μ_{f_{0}} of h and to quadratic differentials φ_{m} (see [16], Theorem 7) which yields

ℜ ∫ ∫_{U} \frac{μ_{0}(z)φ_{m}(z)}{1 − |μ_{0}(z)|^{2}} dxdy ≥ \frac{k_{m}}{1 − k_{m}} − ∫ ∫_{U} |φ_{m}(z)| \frac{|μ_{0}(z)|^{2}}{1 − |μ_{0}(z)|^{2}} dxdy. (14)

For the maps f_{0} with constant |μ_{0}(z)| in U (equal to k(h)), the last inequality implies

\frac{1}{1 − k(h)^{2}} ℜ < μ_{0}, φ_{m} >_{U} ≥ \frac{k_{m}}{1 − k_{m}} − \frac{k(h)}{1 − k(h)^{2}} .

Combining this with (13), one obtains

lim inf_{m→∞} ℜ < μ_{0}, φ_{m} >_{U} ≥ k(h),

which is possible only if

lim_{m→∞} ℜ < μ_{0}, φ_{m} >_{U}= k(h).  (15)

Since φ_{m}(z) ≠ 0 in U, it follows from (15) that

sup_{φ∈A_{1}^{2}(U): ∥φ∥_{A_{1}(U)}=1} | < μ_{0}, φ >_{U} | = k(h).  (16)

(c) Conjugating h and its extensions by

g : Δ → U, ζ ↦ z = i(1 + ζ)/(1 − ζ),

we pass to pentagons Δ(z_{1}, …, z_{5}) with the vertices on S^{1} and to quasiconformal automorphisms f of the disk Δ, and extend their Beltrami coefficients onto the whole complex plane ℂ, setting μ_{f}(z) = 0 for |z| > 1. Then the corresponding maps are conformal on Δ^{∗}, and we normalize them by means of (3). All this does not change the dilatations.

Denote by h̃ and f̃_{0} the homeomorphisms corresponding to h and f_{0} via such transforms, respectively. Then

μf̃_{0}(ζ) = −(μ ∘ g) \frac{(ζ̄ − 1)^{2}}{(ζ − 1)^{2}} =: μ̃_{0}(ζ),

and, similar to (16),

sup_{φ̃∈A_{1}^{2}(Δ): ∥φ̃∥_{A_{1}(Δ)}=1} | < μ̃_{0}, φ̃ >_{Δ} | = k(h̃).

This means, due to Proposition, that

κ(f̃_{0}) = k(f̃_{0}) = k(h).

(d). Now take quasisymmetric homeomorphisms

h = f^{k|φ|/φ}|R

with arbitrary k ∈ (0, 1) and φ satisfying (10) such that quasicircles h̃(S^{1}), determined by h following (c), are asymptotically conformal. Then it follows from the above that all such homeomorphisms satisfy (2). This completes the proof of the Theorem.

4. Remark on the Fredholm eigenvalues. Due to Kühnau–Schiffer’s theorem (see [10], [17]), the reciprocal value to the Grunsky constant κ(f̃_{0})) is equal to the least positive Fredholm eigenvalue λ_{L} of the quasicircle L = f̃_{0}(S^{1}) defined by

\frac{1}{λ_{L}} := sup \frac{|D_{f(Δ)}(u) − D_{f(Δ^{∗})}(u)|}{D_{f(Δ)}(u) + D_{f(Δ^{∗})}(u)}, D_{G}(u) = ∫ ∫_{G} (∇u)^{2}dxdy,

where the supremum is taken over functions u continuous on Ĉ and harmonic on Ĉ \ L. In the case of a smooth curve, λ_{L} coincides with the least positive eigenvalue of the double-layer potential over L.

These eigenvalues produce another obstruction to equality (1) for quadrilaterals and pentagons.

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(Received 02.03.2001)

Author’s address:

Research Institute for Mathematical Sciences Department of Mathematics and Computer Science Bar-Ilan University, 52900 Ramat Gan Israel