Polygonal Quasiconformality and Grunsky’s Operator

Abstract

This paper concerns the old problem of the connection between the dilatations of a given quasisymmetric homeomorphism h of a circle and the associated polygonal quasiconformal maps with a fixed finite number of boundary points, namely whether $k\left(h\right)=sup{k}_n$, where the supremum is taken over all possible n-gons formed by the disk with n distinguished boundary points. A still open question is whether such equality is valid under the additional assumption that the naturally related univalent functions with quasiconformal extensions have equal Grunsky and Teichmüller norms. We solved this problem in the negative for $n\ge 4$.

1. Background and Statement of Problem

It is well known that any quasisymmetric homeomorphism h of $\widehat{\mathbb{R}}=\mathbb{R}\cup \{\infty \}$ onto itself, i.e., such that

$$M^{-1}\le {\displaystyle \frac{h(x+t)-h\left(x\right)}{h\left(x\right)-h(x+t)}}\le M$$

for any $x\in \mathbb{R}$ and $t>0$, with $M=M\left(h\right)<\infty$, admits quasiconformal extensions $f^{\ast }\left(z\right)$ onto the upper half-plane $U=\{z:Imz>0\}$ with dilatations $k\ge k_0\left(M\right)$ (and symmetrical extensions to the lower half-plane $U^{\ast }=\{z:Imz<0\}$). Let $f_0^{\ast }$ be an extremal quasiconformal extension of h, meaning with the minimal dilatation $k_0$.

The Beltrami coefficient ${\mu }_f\left(z\right)={\partial }_{\overline{z}}f/{\partial }_{z}f$ of every symmetric extension f satisfies

$$\mu \left(\overline{z}\right)=\overline{\mu \left(z\right)},z\in \mathbb{C},$$

and f preserves the real axis $\mathbb{R}$. This map naturally generates the corresponding univalent function f on the lower half-plane $U^{\ast }=\{Imz<0\}$ arising as the restriction to $U^{\ast }$ of quasiconformal automorphism of the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$ with Beltrami coefficient $\mu \left(z\right)={\mu }_{f^{\ast }}\left(z\right)$ for $z\in U$ and $\mu \left(z\right)=0$ on $U^{\ast }$. These univalent functions play a crucial role in Teichmüller space theory and in geometric complex analysis (see, e.g., [1]).

The indicated boundary homeomorphisms h are also called M-quasisymmetric. In particular, the 1-quasisymmetric homeomorphisms are reduced to linear polynomials $h\left(x\right)=ax+b$ with real $a\ne 0$ and b.

The notion of quasisymmerty is a specific generalization of the standard geometric symmetry, and one of its important features of is that, in contrast to standard geometric symmetry, M-quasisymmetry is conformally invariant.

One can also use the quasisymmetric homeomorphisms of the unit circle ${\mathbb{S}}^1$, which generate similarly the quasiconformal automorphisms $f^{\mu }\left(z\right)$ conformal on the disk ${\mathbb{D}}^{\ast }=\{z\in \widehat{\mathbb{C}}:\left|z\right|>1\}$ and subject to hydrodynamical normalization $f^{\mu }\left(z\right)=z+b_0+b_1{z}^{-1}+\dots$.

Now consider quasiconformal maps $f:D\to \widehat{\mathbb{C}}$ of a simply connected Jordan domain $D\subset \widehat{\mathbb{C}}$ and regard this domain 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 $\partial D$. Assuming that f has a homeomorphic extension to closure $\overline{D}$, one obtains another polygon $D(f\left(z_1\right),f\left(z_2\right),...,f\left(z_n\right))$, and f moves the vertices into vertices. We shall call such maps polygonal quasiconformal.

In view of the 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 $\mathbb{D}=\left\{\right|z|<1\}$.

For any quasisymmetric map h, we have $k\left(f_0\right)=\underset{n\to \infty }{lim}k\left(f_{\left[n\right]}\right)$, where $f_{\left[n\right]}$ are extremal polygonal quasiconformal maps of $U(x_1,...,x_n)$ onto $U(h\left(x_1\right),...,h\left(x_n\right))$, and the set of vertices {$x_n\}$ becomes dense on $\mathbb{R}$.

A long time ago, the question was posed whether for a fixed $n\ge 4$ the equality

$$k\left(h\right)=supk\left(f_{\left[n\right]}\right)$$

(1)

is achieved when one allows the vertices to vary on $\widehat{\mathbb{R}}$ in all possible ways.

In the case $n=4$, when the polygons are quadrilaterals, the distortion of conformal modules $modQ$ of all quadrilaterals $Q\subset \mathbb{D}$ under a quasiconformal homeomorphism $f:D\to \widehat{\mathbb{C}}$ determines its maximal dilatation $K\left(f\right)$, and it was conjectured, accordingly, that in this case the best possible bound in (1) is equal to

$$k_0\left(h\right)=(K_0\left(h\right)-1)/(K_0\left(h\right)+1),$$

where

$$K_0\left(h\right)=\underset{Q}{sup}{\displaystyle \frac{mod{f}_0\left(Q\right)}{modQ}},$$

(2)

and $f_0$ is an extremal extension of h and supremum is taken over all quadrilaterals $Q=U(z_1,z_2,z_3,z_4)$ (i.e., that it suffices to take only quadrilaterals with the vertices on $\widehat{\mathbb{R}}$).

This conjecture was disproved by a counterexample of Anderson and Hinkkanen (see [2]) and by Reich [3]; a complete answer for quadrilaterals is given in [4] (see also [5]). The general problem was solved (also in the negative) by Strebel [6] and the author [7]. Moreover, due to Strebel, if the initial extremal map $f_0$ has no substantional (essential) boundary points, then, for each $n\ge 4$, equality (1) is attained on n-gons only when this map $f_0$ is itself a polygonal map for some n vertices.

Another open problem naturally arises here as follows:

• Question. Is the equality (1) valid (for any $n\ge 4)$ for the boundary values of all univalent functions $f\left(z\right)$ with $\varkappa \left(f\right)=k\left(f\right)$ ?

This problem is connected with numerical applications of quasiconformal analysis. Only some very restricted results are known (see, e.g., [8]).

Recall that the Teichmüller norm $k\left(f\right)$ of a univalent function $f\left(z\right)$ on the disk ${\mathbb{D}}^{\ast }$ equals the minimal dilatation ${\parallel \mu \parallel }_{\infty }$ of its quasiconformal extensions $f^{\mu }\left(z\right)$ to $\mathbb{D}$, and the Grunsky norm is defined by

$$\varkappa \left(f\right)=sup\left|\sum _{m,n=1}^{\infty }\sqrt{mn}{\alpha }_{mn}{x}_m{x}_n\right|:\mathbf{x}=\left(x_n\right)\in S\left(l^2\right)\},$$

where ${\alpha }_{mn}$ are the Grunsky coefficients of f arising from the expansion

$$log{\displaystyle \frac{f\left(z\right)-f\left(\zeta \right)}{z-\zeta }}=-\sum _{m,n=1}^{\infty }{\alpha }_{mn}{z}^{-m}{\zeta }^{-n},(z,\zeta )\in {\mathbb{D}}^{\ast }\times {\mathbb{D}}^{\ast },$$

the principal branch of the logarithmic function is chosen and $\mathbf{x}=\left(x_n\right)$ runs over the unit sphere $S\left(l^2\right)$ of the Hilbert space $l^2$ with norm $\parallel \mathbf{x}\parallel ={\left({\displaystyle \sum _1^{\infty }}{\left|x_n\right|}^2\right)}^{1/2}$.

The Grunsky operator also can be regarded as a matrix operator

$$\mathcal{G}\left(f\right)={\left(\sqrt{mn}{\alpha }_{mn}\left(f\right)\right)}_{m,n=1}^{\infty }$$

acting linearly on the space $l^2$ (with values on $l^2$); it contracts the norms of elements $\mathbf{x}\in l^2$. The norm of this operator equals $\varkappa \left(f\right)$ (cf. [9]).

The inequality $\varkappa \left(f\right)\le 1$ is necessary and sufficient for univalence of f on ${\mathbb{D}}^{\ast }$, while the strengthened inequality $\varkappa \left(f\right)\le k_1\left(k\right)$ provides $k_1\left(k\right)$-quasiconformal extendibility of f across the unit circle ${\mathbb{S}}^1=\left\{\right|z|=1\}$ (see, e.g., [10]; [11], pp. 82–84; [12]; [13]).

These norms are related by $\varkappa \left(f\right)\le k\left(f\right)$ and for most of univalent functions we have the strict inequality $\varkappa \left(f\right)<k\left(f\right)$. On the other hand, the functions with $\varkappa \left(f\right)=k\left(f\right)$ play a crucial role in many applications.

2. Basic Features of Grunsky Norm

This norm measures the deviation of a univalent function from the identity map. Its features underly the main result of this paper and are presented as follows:

Proposition 1.

For any univalent function f on ${\mathbb{D}}^{\ast }$, we have the equality

$$\varkappa \left(f^{t{\mu }_0}\right)=\left|t\right|{\displaystyle \frac{\left|t\right|+\alpha \left(f\right)}{1+\alpha \left(f\right)\left|t\right|}}=\alpha \left(f\right)\left|t\right|+(1-\alpha {\left(f\right)}^2){\left|t\right|}^2+\dots ,$$

(3)

where ${\mu }_0$ is an extremal Beltrami coefficient among quasiconformal extensions $f^{\mu }$ of f onto $\mathbb{D},\left|t\right|<1/\parallel {\mu }_0{\parallel }_{\infty }$, and

$$\alpha \left(f\right)=sup\left\{\right|{\int \int }_D{\mu }_0\left(z\right)\psi \left(z\right)dxdy|:\psi \in A_1^2\left(\mathbb{D}\right),{\parallel \psi \parallel }_{A_1}=1\}(z=x+iy),$$

(4)

$A_1\left(\mathbb{D}\right)$ denotes the subspace in $L_1\left(\mathbb{D}\right)$ formed by integrable holomorphic functions (quadratic differentials) on $\mathbb{D}$, and $A_1^2\left(\mathbb{D}\right)$ is its subset consisting of ψ with zeros even order in D, i.e., of the squares of holomorphic functions.

This proposition is a special case of a general theorem established in [14] for arbitrary quasidisks. Due to [15], every $\psi \in A_1^2\left(\mathbb{D}\right)$ has the following form:

$$\psi \left(z\right)={\displaystyle \frac{1}{\pi }}\sum _{m+n=2}^{\infty }\sqrt{mn}{x}_m{x}_n{z}^{(m+n)}$$

and ${\parallel \psi \parallel }_{A_1\left(\mathbb{D}\right)}={\parallel \mathbf{x}\parallel }_{l^2}=1,\mathbf{x}=\left(x_n\right)$. Hence,

$$\alpha \left(f\right)=\underset{\mathbf{x}=\left(x_n\right)\in S\left(l^2\right)}{sup}{\displaystyle \frac{1}{\pi \parallel {\mu }_0{\parallel }_{\infty }}}\left|\underset{\left|z\right|<1}{\int \int }{\mu }_0\left(z\right)\sum _{m+n\ge 2}^{\infty }\sqrt{mn}{x}_m{x}_n{z}^{m+n-2}dxdy\right|.$$

In contrast, a Beltrami coefficients ${\mu }_0$ is extremal in its equivalence class (has the minimal $L_{\infty }$-norm) if and only if

$$\parallel {\mu }_0{\parallel }_{\infty }=sup\left\{\left|\underset{D^{\ast }}{\int \int }\mu \left(z\right)\psi \left(z\right)dxdy\right|:\psi \in A_1\left(\mathbb{D}\right),{\parallel \psi \parallel }_{A_1}=1\right\}.$$

So, the $L_{\infty }$-norm of any extremal Beltrami coefficient is attained on the (entire) subspace $A_1$ of $L_1$ (this is given by the Hamilton–Krushkal–Reich–Strebel theorem).

Comparison with Proposition implies that the equality $\varkappa \left(f\right)=k\left(f\right)$ is valid only when both quantities $k\left(f\right)$ and $\varkappa \left(f\right)$ coincide with $\alpha \left(f\right)$ given by (4).

In addition, if $\varkappa \left(f\right)=k\left(f\right)$ and the extremal extension is of Teichmüller type, has Beltrami coefficient ${\mu }_0=k\left|{\psi }_0\right|/{\psi }_0$ with ${\psi }_0\in A_1\left(\mathbb{D}\right)$), then ${\psi }_0={\omega }^2\in A_1^2\left(\mathbb{D}\right)$, and vice versa (in other words, all zeros of ${\psi }_0$ in $\mathbb{D}$ must be of even order).

3. Main Result

The following theorem implies that in the general case the solution of the indicated problem is negative for all approximative polygons.

Theorem 1.

There exist univalent functions $f\left(z\right)=z+b_0+b_1{z}^{-1}+\dots$ on the disk ${\mathbb{D}}^{\ast }$ with equal Grunsky and Teichmüller norms and such that

$$\varkappa \left(f\right)=k\left(f\right)>supk\left(f_{\left[n\right]}\right)$$

(5)

for any $n\ge 4$.

Proof. 

Take an infinite Blashke product

$$b\left(z\right)=-\prod _1^{\infty }{\displaystyle \frac{|z_n|}{z_n}}{\displaystyle \frac{z-z_n}{1-{\overline{z}}_{n}z}}$$

whose zeros satisfy

$$\sum _n(1-|z_n\left|\right)<\infty .$$

It is well known that $b\left(z\right)$ is holomorphic in the unit disk $\mathbb{D}$ and has there only zeros $z_n$ and satisfies: $\left|b\right(z\left)\right|<1$ on $\mathbb{D}$ and $\left|b\right(z\left)\right|=1$ almost everywhere on the unit circle ${\mathbb{S}}^1$. □

It gives rise to the holomorphic quadratic differential ${\psi }_0\left(z\right)d{z}^2$ on $\mathbb{D}$ with

$${\psi }_0\left(z\right)=b{\left(z\right)}^2.$$

This differential has only zeros $z_n$ accumulating to ${\mathbb{S}}^1$, each of even order.

Using this ${\psi }_0$, we define a quasiconformal automorphism $f^{{\widehat{\mu }}_0}$ of the extended plane $\widehat{\mathbb{C}}$ whose Beltrami coefficient ${\widehat{\mu }}_0\left(z\right)$ is equal to

$$k|{\psi }_0\left(z\right)|/{\psi }_0\left(z\right)\mathrm{with}k=k\left(f\right)$$

on the disk $\mathbb{D}$ and equal to zero on the complementary disk ${\mathbb{D}}^{\ast }$. So $f^{{\widehat{\mu }}_0}$ is conformal on ${\mathbb{D}}^{\ast }$, and can normalize it by

$$f^{{\widehat{\mu }}_0}\left(z\right)=z+{\widehat{b}}_0+{\widehat{b}}_1{z}^{-1}+\dots ,\left|z\right|>1;f^{{\widehat{\mu }}_0}\left(0\right)=0.$$

In addition, one can also assume that $f_m\left(0\right)=0$, using the translations $f_m\left(z\right)-f_m\left(0\right)$, which do not change the Beltrami coefficients of $f_m$ (the latter is needed to have compactness of the family $\left\{f_m\right\}$ in spherical metric on $\widehat{\mathbb{C}}$).

On the contrary, now assume that for some $n_0\ge 4$, we have the equality

$$supk\left(f_{\left[n_0\right]}\right)=k\left(f\right)$$

(6)

(or, equivalently, the equality (1) for the corresponding quasisymmetric homeomorphism of ${\mathbb{S}}^1$). Then one can select a sequence of $n_0$-gons

$$P_m=\mathbb{D}(z_{1,m},\dots z_{n_0,m}),m=1,2,...,$$

and the corresponding extremal polygonal maps $f_m$ of $P_m$ onto the domains $f_m\left(P_m\right)$ with vertices $f_m\left(z_{1,m}\right),\dots f_m\left(z_{n_0,m}\right)$, which carry the original vertices $z_j$ to the vertices of $f_m\left(P_m\right)$ and such that

$$\underset{m\to \infty }{lim}k\left(f_m\right)=k\left(h\right)=k\left(f\right).$$

(7)

The Beltrami coefficients of these maps are ${\mu }_{f_m}\left(z\right)=k_m\left|{\phi }_m\left(z\right)\right|/{\phi }_m\left(z\right)$ with $k_m=k\left(f_m\right)$ and ${\phi }_m\in A_1\left(\mathbb{D}\right)$. We normalize ${\phi }_m$ letting $\parallel {\phi }_m{\parallel }_{A_1\left(\mathbb{D}\right)}=1$.

We now apply the fundamental Reich–Strebel inequality for polygons (see [16], Theorem 7), which involves an extremal Beltrami coefficient ${\mu }_0$ and the related quadratic differentials ${\phi }_m\left(z\right)$ on $\mathbb{D}$. This inequality implies

$$Re{\int \int }_{\mathbb{D}}{\displaystyle \frac{{\mu }_0\left(z\right){\phi }_m\left(z\right)}{1-|{\mu }_0{\left(z\right)|}^2}}dxdy\ge {\displaystyle \frac{k_m}{1-k_m}}-{\int \int }_{\mathbb{D}}\left|{\phi }_m\left(z\right)\right|{\displaystyle \frac{|{\mu }_0{\left(z\right)|}^2}{1-|{\mu }_0{\left(z\right)|}^2}}dxdy.$$

For the maps f with constant $|{\mu }_0\left(z\right)|$ in U (equal to $k\left(h\right)$), the last inequality provides the relation

$${\displaystyle \frac{1}{1-k{\left(h\right)}^2}}Re{\int \int }_{\mathbb{D}}{\mu }_0\left(z\right){\phi }_m\left(z\right)dxdy\ge {\displaystyle \frac{k_m}{1-k_m}}-{\displaystyle \frac{k{\left(h\right)}^2}{1-k{\left(h\right)}^2}}.$$

Combining this with Equation (7), one obtains

$$\underset{m\to \infty }{lim\; inf}Re{\int \int }_{\mathbb{D}}{\mu }_0\left(z\right){\phi }_m\left(z\right)dxdy\ge k\left(h\right),$$

which is possible only if

$$\underset{m\to \infty }{lim}Re{\int \int }_{\mathbb{D}}{\mu }_0\left(z\right){\phi }_m\left(z\right)dxdy=k\left(h\right).$$

In our case,

$${\mu }_0\left(z\right)={\widehat{\mu }}_0\left(z\right):=k\left(f\right)\left|{\psi }_0\left(z\right)\right|/{\psi }_0\left(z\right),$$

and since the functions ${\phi }_m\left(z\right)$ do not vanish identically on the unit disk, the last equality implies

$$\underset{\phi \in A_1^2\left(\mathbb{D}\right):{\parallel \phi \parallel }_{A_1\left(\mathbb{D}\right)}=1}{sup}\left|{\int \int }_{\mathbb{D}}{\widehat{\mu }}_0\left(z\right)\phi \left(z\right)dxdy\right|=k\left(f\right).$$

(8)

On the other hand, by applying the mean value inequality

$$|{\phi }_m\left(z_0\right)|\le {\displaystyle \frac{1}{\pi {\delta }^2}}\underset{|z-{\zeta }_0|}{\int \int }{\phi }_m\left(z\right)dxdy(z=x+iy)$$

with $\delta <1-|z_0|$ (and passing, if needed, to a subsequence), one obtains that the functions ${\phi }_m$ converge locally uniformly on $\mathbb{D}$ to an integrable holomorphic function ${\phi }_0$.

If ${\phi }_0\left(z\right)\equiv 0$ on $\mathbb{D}$, then $\left\{{\phi }_m\right\}$ is so-called degenerated sequence for the extremal Beltrami coefficient ${\widehat{\mu }}_0$ (equivalently, the boundary limit function $f^{{\widehat{\mu }}_0}\left(e^{i\theta }\right)$ has a substantial point on ${\mathbb{S}}^1$), and in this case the extremal extension of $f\left(z\right)$ onto $\mathbb{D}$ cannot be unique.

But such a situation is impossible for Beltrami extremal coefficients of Teichmüller type (in particular, for ${\widehat{\mu }}_0$). This contradiction yields that the limit function ${\psi }_0\left(z\right)$ must necessarily have the isolated zeros in $\mathbb{D}$.

On the other hand, for any $(n_0+3)$-gon $U(0,1,x_1,x_2,\dots ,x_{n_0},\infty )$, the extremal map

$$f_0:U(0,x_1,x_2,\dots ,x_{n_0},\infty ))\to U(0,1,h\left(x_1\right),h\left(x_2\right),\dots ,h\left(x_{n_0}\right),\infty )$$

has the Beltrami coefficient of the form ${\mu }_0\left(z\right)=k\left|{\phi }_0\left(z\right)\right|/{\phi }_0\left(z\right)$, where

$${\phi }_0\left(z\right)=\sum _1^{n_0}{\displaystyle \frac{c_j}{z(z-1)(z-a_j)}}$$

where the numbers $c_j$ are real, because the quadratic differential ${\phi }_0\left(z\right)d{z}^2$ takes the real values on $\mathbb{R}$.

This yields that the extremal quasiconformal maps of any $n_0$-gon $D(a_1,\dots ,a_{n_0})$, whose domain D is bounded by a $C^{1+\alpha }$ smooth curve onto other ones, are determined by holomorphic quadratic differentials $\psi =({\phi }_0\circ g){\left(g^{\prime }\right)}^2$, where g maps conformally $D(a_1,\dots ,a_{n_0})$ onto $U(0,1,x_1,x_2,\dots ,x_{n_0-3},\infty )$ (or, equivalently, onto the unit disk with the distinguished boundary points). Therefore, every such function $\psi$ can have in the disk $\mathbb{D}$ at most $n_0-4$ zeros.

The same remains valid for the limit function

$${\widehat{\psi }}_0\left(z\right)=\underset{m\to \infty }{lim}{\phi }_m\left(z\right),\left|z\right|<1,$$

and $\parallel {\widehat{\psi }}_0{\parallel }_{A_1\left(\mathbb{D}\right)}\le 1$.

Put ${\widehat{\mu }}_0=k\left|{\widehat{\psi }}_0\right|/{\widehat{\psi }}_0$. The corresponding quasiconformal is also polygonal, and the above construction yields

$$f^{{\widehat{\mu }}_0}\left(z_j\right)=f\left(z_j\right),j=1,\dots ,n_0.$$

In view of the uniqueness of the extremal quasiconformal map between the polygons $P_m$ and $f^{{\widehat{\mu }}_0}\left(P_m\right)$, the function ${\widehat{\psi }}_0$ must coincide with the initial function ${\psi }_0\left(z\right)=b{\left(z\right)}^2$ up to a positive factor.

We again reached a contradiction, because the chosen Blashke function $b\left(z\right)$ has infinite number of zeros in $\mathbb{D}$. This contradiction completes the proof of the theorem.

4. Remarks

In the simplest cases of quadrilaterals and pentagons ($n=4,5$), the approximating polygonal maps also are defined by nonvanishing holomorphic quadratic differentials in $\mathbb{D}$ and hence also have equal Grunsky and Teichmüller norms. Nevertheless, in the general case, the dilatations of these maps do not approach the dilatation of the limit function.

We also mention a case of arbitrary n, when the equality (1) is valid. It arises when the extrema1 quasiconformal extension of f is of Teichmüller type and its defining holomorphic quadratic differential $\phi$ is zero free in $\mathbb{D}$.

Indeed, using the arguments applied in the proof of the theorem, one obtains that in this case the approximating polygonal maps $f_n$ also must have nonvanishing quadratic differentials ${\psi }_n$ on $\mathbb{D}$ (at least for large n). Hence, $\varkappa \left(f_n\right)=k\left(f_n\right)=k_{\left[n\right]}$, and the subharmonicity of these functions leads to the equality (1).

Comparing with Strebel’s result that then the original map must be polygonal itself, one derives that this is possible only when all zeros of quadratic differentials ${\psi }_0$ defining the extremal extension are real.

5. Open Question

The above theorem solves the problem in the case when the initial quasisymmetric function h and the corresponding univalent function $f\left(z\right)$ on ${\mathbb{D}}^{\ast }$ have extremal Teichmüller extensions defined by quadratic differential ${\psi }_0$ with a infinite number of zeros.

The case of a finite number of zeros (all of even order) remains open. In this case, the above proposition yields that the equality (8) is compatible with the assumption that f has equal Grunsky and Teichmüller norms.

An unsolved question is whether this assumption provides, in turn, the relation (6) for appropriate $n_0$ (depending on the number of zeros of ${\psi }_0$).