Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 237–247

EXTREMAL QUASICONFORMAL POLYGON MAPPINGS FOR ARBITRARY SUBDOMAINS OF COMPACT RIEMANN SURFACES

Kurt Strebel

Freiestraße 14, CH-8032 Zürich Switzerland; kstrebel@math.unizh.ch

Abstract. By a polygon we originally mean a plane disk Δ with a finite number P of distinguished boundary points ω_{i}. Let (Δ, P), P = {ω_{i}} and (Δ′, P′), P′ = {ω′_{i}} be two polygons. The extremal quasiconformal mapping f_{n} of (Δ, P) onto (Δ′, P′), f_{n}(ω_{i}) = ω′_{i} for all i, is a Teichmüller mapping, associated with a quadratic differential φ_{n}, called polygon differential. To a given extremal qc mapping f: Δ → Δ′ we can associate extremal polygon mappings f_{n}, f_{n}(ω_{i}) = f(ω_{i}) for all vertices ω_{i}. This construction has applications in the theory of extremal qc mappings. It is the purpose of this article to generalize the notion of polygon and polygon mappings and prove their properties for arbitrary subdomains of compact Riemann surfaces.

I. Introduction: the disk case

Let Δ : |z| < 1 and Δ′ : |w| < 1 be unit disks in the z - and w-plane respectively and let w = f(z) be a quasiconformal mapping of Δ onto Δ′ with maximal dilatation K and complex dilatation μ, ∥μ∥_{∞} = (K − 1)/(K + 1). Let P be a finite set of points ω_{i}, i = 1, …, n, n ≥ 4 on ∂Δ. The disk Δ together with the set P = {ω_{i}} is called a polygon, the points ω_{i} are its vertices. The mapping f takes the polygon (Δ, P) with vertices ω_{i} into a polygon (Δ′, P′) with vertices ω′_{i} = f(ω_{i}), i = 1, …, n. Let f_{n} be the extremal qc mapping of Δ onto Δ′ which takes the vertices ω_{i} into the vertices ω′_{i}. This mapping determines a pair of quadratic differentials φ_{n} on Δ and ψ_{n} on Δ′. The trajectory structures of φ_{n} and ψ_{n} partition the disks into finitely many horizontal strips R_{jn} in Δ and R′_{jn} in Δ′. They are mapped by the integrals Φ_{n}(z) = ∫ \sqrt{φ_{n}(z)} dz and Ψ_{n}(w) = ∫ \sqrt{ψ_{n}(w)} dw onto Euclidean horizontal rectangles

\begin{matrix}Φ_{n}(R_{jn}) : 0 < ξ_{n} < a_{jn}, 0 < η_{n} < b_{jn}, \\ Ψ_{n}(R′_{jn}) : 0 < ξ′_{n} < K_{n}a_{jn}, 0 < η′_{n} < b_{jn}. \end{matrix}

The extremal mapping f_{n} satisfies the relation

Ψ_{n} ∘ f_{n} ∘ Φ_{n}^{−1}(ζ_{n}) = K_{n}ξ_{n} + iη_{n}, ζ_{n} = ξ_{n} + iη_{n},

2000 Mathematics Subject Classification: Primary 30C62, 30C75.

with K_{n} ≥ 1 the constant dilatation of f_{n}. This is a consequence of Teichmüller’s theorem; the extremal f_{n} is uniquely determined.

Next we introduce the metrics induced by the two quadratic differentials. They are nothing else than the Euclidean length elements

|dΦ_{n}(z)| = |φ_{n}(z)|^{1/2}|dz|, |dΨ_{n}(w)| = |ψ_{n}(w)|^{1/2}|dw|

in the Φ_{n} - and Ψ_{n} -plane respectively. The length inequality is usually expressed for closed trajectories. In our case the trajectories are cross cuts of Δ and Δ′, but they become closed curves by reflection on ∂Δ and ∂Δ′ respectively. The length inequality says that every closed trajectory is shortest in its free homotopy class with respect to the planes punctured at the points P_{n} and P′_{n} respectively. Here we have actually trajectory intervals connecting vertical sides of the disks, with lengths one half of the lengths of the closed trajectories.

Let α_{jn} be a trajectory interval of φ_{n} in Δ. It is stretched by K_{n} onto the corresponding trajectory interval α′_{jn} of ψ_{n} in the w-plane. Let

dw = p(z) dz + q(z) dz̄

be the differential of the qc mapping f. Its length in terms of the ψ_{n} -metric is

|dw̃| = |ψ_{n}(w)|^{1/2}|dw| = |ψ_{n}(w)|^{1/2}|p(z) dz + q(z) dz̄|.

Now the length inequality reads

K_{n}a_{jn} ≤ ∫_{f(α_{jn})} |dw̃| = ∫_{f(α_{jn})} |ψ_{n}(w)|^{1/2}|p(z) dz + q(z) dz̄|.

Integration over the disk Δ gives the polygon inequality

(1)  K_{n} ≤ ∫∫_{|z|<1} |φ_{n}(z)|\frac{|1 + μ(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}|^{2}}{1 − |μ(z)|^{2}} dx dy,

z = x + iy, μ(z) = q(z)/p(z) the complex dilatation of f. Developing the integrand gives the equivalent form

(1′)  \begin{aligned}\frac{k_{n}}{1 − k_{n}} & ≤ Re ∫∫_{|z|<1} \frac{μ(z)φ_{n}(z)}{1 − |μ(z)|^{2}} dx dy + ∫∫_{|z|<1} |φ_{n}(z)|\frac{|μ(z)|^{2}}{1 − |μ(z)|^{2}} dx dy, \\ k_{n} & = \frac{K_{n} − 1}{K_{n} + 1}, \end{aligned}

which is easier to explore. This is true for an arbitrary qc self-mapping of Δ and all inscribed polygons (for details, see [5], pp. 383–386).

Next, let f be extremal for its boundary values, with maximal dilatation K. Then, K_{n} ≤ K for all polygons. Let the point sets P_{n} become dense on ∂Δ for n → ∞. By general principles of qc mappings K_{n} → K. An easy consequence is the so called Hamilton–Krushkal relation

(2)  Re ∫∫_{Δ} μ(z)φ_{n}(z) dx dy → ∥μ∥ = ∥μ∥_{∞} = ess sup |μ(z)|

for ∥φ_{n}∥ = ∫∫_{Δ} |φ_{n}(z)| dx dy = 1, n → ∞, as a necessary condition for f to be extremal. (The computations are immediate if |μ(z)| = ∥μ∥ a.e. In the general case we first realize that the integral tends to zero for every subset E ⊂ Δ with |μ(z)| < ∥μ∥ − ε, ε > 0. For a detailed discussion see [5].)

Another application of the polygon inequality is the following. Let K be the maximal dilatation of an extremal mapping f. Can one have

sup K_{n} = K, n ≤ N < ∞?

Can e.g. the above relation hold for quadrilaterals? Unless the boundary mapping has a so called substantial (essential) boundary point, this is only possible if f itself is a polygon mapping (see [7]).

II. The general case

The Hamilton–Krushkal relation (2) is an important ingredient in the theory of extremal qc mappings. The polygon inequality (1) provides us with Hamilton sequences which have an intuitive geometric meaning.

However, the inequality has so far only been proved for the disk (see [5], inequality 3.2.6). It is the purpose of this paper to prove it for arbitrary plane domains Ω and, slightly more generally, for arbitrary subdomains of compact Riemann surfaces.

Let Ω be a subdomain of a compact Riemann surface R the boundary ∂Ω of which contains infinitely many points. Let μ be an extremal Beltrami coefficient in Ω. We set μ_{0} = μ in Ω and μ_{0} = 0 in R \ Ω. (This extension of μ was used in [2] to investigate different kinds of local boundary dilatations.) Then, μ_{0} is a Beltrami differential in R. For the norms we have ∥μ∥ = ∥μ_{0}∥.

Let f_{0} be a μ_{0} -quasiconformal mapping of R onto a compact surface R′, and let Ω′ = f_{0}(Ω). The complex dilatation of f_{0} in Ω is μ. Therefore any qc mapping f of Ω with complex dilatation μ differs from the restriction f_{0} | Ω by a conformal mapping. However, f_{0} is qc in all of R, and in particular homeomorphic on ∂Ω. If μ is extremal on Ω, f_{0} | Ω is an extremal qc mapping of Ω onto f_{0}(Ω).

Choose a finite set P_{n} of points ω_{i} on ∂Ω and let G_{n} be the surface R punctured at the set P_{n}, G_{n} = R \ P_{n}. Let f_{n} be the extremal qc mapping of G_{n} onto G′_{n} = R′ \ P′_{n} which takes ω_{i} into ω′_{i} = f_{0}(ω_{i}) for all indices i and is homotopic to f_{0} modulo the boundary P′_{n} = ∂G′_{n}. It is a Teichmüller mapping associated with a rational quadratic differential φ_{n} of finite norm ∥φ_{n}∥ < ∞, which is holomorphic in R except for possible poles of the first order at the points ω_{i}. Changing the earlier nomenclature we call G_{n} a polygon, f_{n} a polygon mapping and φ_{n} a polygon differential.

We are trying to establish a polygon inequality similar to (1). In the disk case we used reflection on the boundary of the disk and thus had actually to deal with quadratic differentials with closed trajectories. The polygon differentials in this more general case have no symmetry. Therefore we have to make use of the general trajectory structure. This gives rise to a length inequality between homotopic closed curves which can then be integrated over the whole surface R. The fact that μ_{0} = 0 outside Ω finally reduces the domain of integration to Ω, and we arrive at the polygon inequality (1), but now for general domains Ω ⊂ R.

III. Trajectory structure

This chapter is based on reference [6], where the definitions, statements and their proofs can be found.

Let φ be a rational quadratic differential of finite norm on the extended plane or, more generally, on an arbitrary compact Riemann surface R. Its critical points are finitely many zeroes and, possibly, finitely many first order poles. The critical trajectories, i.e. those which tend in at least one direction to a critical point of φ, are finite in number ([6], §7.1). All the other trajectories are either closed Jordan curves or spirals. A spiral is a trajectory which is not closed but both ends of which diverge, i.e. have a limit set which consists of at least two points ([6], §10.2).

Every closed trajectory α of an arbitrary quadratic differential φ on an arbitrary Riemann surface R is embedded in a ring domain swept out by closed trajectories homotopic to α and of the same φ-length ([6], §9.4).

The behaviour of non closed trajectories is more complicated. But on a compact Riemann surface and for a rational quadratic differential of finite norm every divergent trajectory ray α^{+} is recurrent ([6], §11, Theorem 11.1 and Corollary). The limit set A of a recurrent ray α^{+} is the closure of a domain ([6], §11, Theorem 11.2 and Corollaries). The boundary of A consists of critical trajectories of finite length connecting zeroes of φ. Both subrays α^{+} and α^{−} of a spiral have the same limit set, which we call for short a spiral set. There are only finitely many spiral sets, and different ones are non-overlapping. Therefore the entire surface R is decomposed into finitely many ring domains and spiral sets.

Let A be a spiral set and choose a closed vertical interval β in the interior A^{∘} of A. Then, A is covered by a finite set of closed, horizontal, non-overlapping strips S_{i} corresponding to horizontal rectangles in the Φ = ∫ \sqrt{φ} -plane with both their vertical sides on β but no other points in common with β. Strips of the first kind have their vertical intervals on different sides of β, strips of the second kind on the same side, either both on the right-hand side or both on the left-hand side (for the construction see ([6], §11.3).

The finitely many rays α^{+} and α^{−} with initial point on β^{+} or β^{−} which end up in a critical point or an end point of β before they meet β again, together with the endpoints of β, determine the strip decomposition of A.

IV. Length inequality

Let α be a closed trajectory of φ. If γ is any closed curve which is freely homotopic to α, then its φ-length ∫_{γ} |φ(z)|^{1/2} |dz| is at least as big as the φlength of α, with equality only if γ is a closed trajectory in the free homotopy class of α ([6], §17.1). The length inequality, in combination with quasiconformal mappings, will be integrated over the annulus swept out by the closed trajectories of φ parallel to α.

Since we cannot restrict ourselves to quadratic differentials with closed trajectories, we need a length inequality for spiral domains A^{∘}.

Let β be a compact vertical interval in A^{∘} and let S be a horizontal strip of the first kind based on β. Let α be a trajectory interval in S with initial point P^{+} and end point P^{−} on the two different sides of β. Let δ be the subinterval of β with initial point P^{−} and end point P^{+} . The horizontal length of the closed Jordan curve α + δ is equal to a = ∫_{α} |φ(z)|^{1/2} |dz| ([6], §24, Theorem 24.1, where the property is expressed for the vertical trajectories and vertical length = height rather than for the horizontal length).

Let γ be any closed curve in the free homotopy class of α + δ. Then, since its φ-length is at least equal to its horizontal length with respect to φ and this is minimized (see [6], Definition 24.1), in its free homotopy class, by a, we have ∫_{γ} |φ(z)|^{1/2} |dz| ≥ a.

The process is slightly more complicated for strips of the second kind, since in this case the closed curve α + δ evidently would not minimize the horizontal length in its free homotopy class. In order to get a minimizing closed curve we must combine a trajectory interval α_{1} of the second kind on one side of β with a trajectory interval α_{2} of the second kind on the other side of β. The two connecting subintervals δ_{1} and δ_{2} on β can be chosen in such a way that the curve α_{1} + δ_{1} + α_{2} + δ_{2} is a Jordan curve, possibly after a slight shift of δ_{1} , say, away from β, (see [6], Fig. 68, p. 156, where again the reasoning is carried out for heights instead of horizontal lengths). Then, again by ([6], Theorem 24.1), the φ-length of any closed curve γ in the free homotopy class of the above step curve is at least equal to the sum a_{1} + a_{2} of the lengths of α_{1} and α_{2} .

V. Polygon inequality

We are now ready to prove the Polygon Inequality (1) and, equivalently (1′), for the general case of Section II.

First, let α be a closed trajectory of φ_{n}. In the maximal annulus S determined by α on G_{n} = R − {ω_{i}} we introduce the parameter ζ = ξ + iη = Φ_{n}(z) = ∫ \sqrt{φ_{n}(z)} dz, z a local parameter on R. The Φ_{n} -image of S cut along a vertical trajectory β is a rectangle 0 ≤ ξ ≤ a, 0 ≤ η ≤ b. The trajectory α is mapped by f_{n} onto a closed trajectory α′ = f_{n}(α) of the image differential ψ_{n} = f_{n}(φ_{n}) and the ring domain S on G_{n} onto a ring domain S′ = f_{n}(S) on R′ swept out by closed trajectories of ψ_{n}. The mapping f_{0} takes α into a closed curve α̃ = f_{0}(α) of ψ_{n} -length at least equal to K_{n}a, because it is freely homotopic on G′_{n} to the closed trajectory α′ with ψ_{n} -length K_{n}a.

Let us introduce the notations, in terms of local parameters z and w on R and R′ respectively, w = f_{0}(z), dw = p(z) dz + q(z) dz̄,

dz = \frac{dz}{dζ} dζ = \frac{1}{Φ′_{n}(z)} dζ.

This gives, for the ψ_{n} -length of α̃,

(3) K_{n}a ≤ ∫_{α̃} |dΨ_{n}| = ∫_{α̃} |ψ_{n}(w)|^{1/2} |dw| = ∫_{α} |ψ_{n}(w)|^{1/2}|p(z) dz + q(z) dz̄|.

Along the horizontals in the ζ = ξ + iη -plane we have dz = dξ/Φ′_{n}(z), and thus

(4)  K_{n}a ≤ ∫ \frac{|ψ_{n}(w)|^{1/2}}{|φ_{n}(z)|^{1/2}} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dξ,

and hence by integration over the η -variable

(5)  \begin{aligned}K_{n}ab & ≤ ∫∫_{Φ_{n}(S)} \frac{|ψ_{n}(w)|^{1/2}}{|φ_{n}(z)|^{1/2}} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dξ dη \\ & = ∫∫_{S} |ψ_{n}(w)|^{1/2}|φ_{n}(z)|^{1/2} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dx dy. \end{aligned}

There is the variable w = f_{0}(z) which has no place in S but in S̃ = f_{0}(S). By means of the Jacobian du dv = J(w/z) dx dy = (|p(z)|^{2} −|q(z)|^{2}) dx dy we go back to S ⊂ R with the integral

(6)  K_{n}ab ≤ ∫∫_{S} |ψ_{n}(w)|^{1/2}|φ_{n}(z)|^{1/2}J^{1/2} \frac{|p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}|}{J^{1/2}} dx dy.

Secondly, let A be a spiral set of the trajectory structure of φ_{n}. Choose a closed, regular vertical interval β in the interior A^{∘} of A. β is the basis of a finite set of non overlapping horizontal φ_{n} -strips which cover A. They go over into horizontal rectangles in the ζ = ξ + iη -plane by the conformal mapping Φ_{n}. For given ε > 0 we can choose β so short that the ψ_{n} -length of β̃ = f_{0}(β) is smaller than ε.

Let S be a strip of the first kind. It connects an interval on the positive side of β with one on the negative side. Let α be a trajectory interval of φ_{n} in S and δ the interval on β which connects the two end points of α. It can be considered as a shift vector on β pointing from the end point of α on the negative side of β to its initial point on the positive side. It is the same for all trajectory intervals α in S (for details see [6], p. 68).

The curve α + δ is mapped by f_{n} onto a curve α′ + δ′ on R′, composed of a horizontal interval α′ and a vertical interval δ′ of the image differential ψ_{n} = f_{n}(φ_{n}). The curve f_{0}(α + δ) = f_{0}(α) + f_{0}(δ) = α̃ + δ̃ has ψ_{n} -length at least equal to the ψ_{n} -length of α′, which is equal to |α′|_{ψ_{n}} = K_{n} · |α|_{φ_{n}} = K_{n} · a. For the lengths of the vertical intervals δ, δ′, we have |δ|_{φ_{n}} = |δ′|_{ψ_{n}} = d. We use the rectangle 0 ≤ ξ ≤ a, 0 ≤ η ≤ b, b the φ_{n} -height of the strip S, in the ζ = ξ + iη - plane, ζ = Φ_{n}(z), as parameter domain. From |α̃ + δ̃|_{ψ_{n}} = |α̃|_{ψ_{n}} + |δ̃|_{ψ_{n}} ≥ K_{n}a and |δ̃|_{ψ_{n}} < ε we get |α̃|_{ψ_{n}} ≥ K_{n}a − ε, and hence

(7)  \begin{aligned}K_{n}a − ε & ≤ ∫_{α} |ψ_{n}(w)|^{1/2}|p(z) dz + q(z) dz̄| \\ & = ∫_{Φ_{n}(α)} |ψ_{n}(w)|\frac{1}{|φ_{n}(z)|^{1/2}} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dξ. \end{aligned}

Integration over η yields

(8)  \begin{aligned}K_{n}ab − εb & ≤ ∫∫ |ψ_{n}(w)|^{1/2} \frac{1}{|φ_{n}(z)|^{1/2}} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dξ dη \\ & = ∫∫_{S} |ψ_{n}(w)|^{1/2}|φ_{n}(z)|^{1/2} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dx dy. \end{aligned}

Here, z = x + iy is an arbitrary parameter on R. Since ψ_{n}(w) depends on w = f_{0}(z), we introduce the Jacobian J(w/z) = |p(z)|^{2} − |q(z)|^{2} . Then, the integral reads

(9)  K_{n}ab − εb ≤ ∫∫_{S} |ψ_{n}(w)|^{1/2}J^{1/2}|φ_{n}(z)|^{1/2} \frac{|p + q \frac{φ_{n}}{|φ_{n}|}|}{J^{1/2}} dx dy.

We sum it up over all strips S of the first kind based on β. Denoting the area of S in terms of φ_{n} by |S|_{φ_{n}} we have

(10) K_{n} ∑ |S|_{φ_{n}} − ε ∑ b ≤\begin{matrix}∫ & ∫ \\ ∑ S \end{matrix} |ψ_{n}(w)|^{1/2}J^{1/2}|φ_{n}(z)|^{1/2} \frac{|p + q \frac{φ_{n}}{|φ_{n}|}|}{J^{1/2}} dx dy.

We pass now to the strips of the second kind. If we want to have a length inequality, they have to be combined, each strip on the right-hand side with a strip on the left-hand side of the same height. To that end take the highest strip S_{r} on the right-hand side and the highest strip S_{l} on the left-hand side. Let b_{r} and b_{l} be the respective heights and let b_{r} > b_{l}. Then we cut off a horizontal rectangle of height b_{r} − b_{l} from S_{r} to get S̃_{r} with height b_{l}. We can therefore glue S̃_{r} to S_{l}. The total number of strips is reduced by one. Therefore the process comes to an end.

Let α_{1} + δ_{1} + α_{2} + δ_{2} be a closed step curve with respect to φ_{n}. It is mapped by f_{n} onto a similar step curve with respect to ψ_{n}, with horizontal ψ_{n} -length K_{n}a_{1} + K_{n}a_{2} . The f_{0} image α̃_{1} + δ̃_{1} + α̃_{2} + δ̃_{2} has ψ_{n} -length greater or equal to K_{n}(a_{1} + a_{2}). On the other hand, it is |α̃_{1}|_{ψ_{n}} + |α̃_{2}|_{ψ_{n}} + |δ̃_{1}|_{ψ_{n}} + |δ̃_{2}|_{ψ_{n}}. Setting |δ̃_{1}|_{ψ_{n}} = d̃_{1} , |δ̃_{2}|_{ψ_{n}} = d̃_{2} we find

(11)  |α̃_{1}|_{ψ_{n}} + |α̃_{2}|_{ψ_{n}} ≥ K_{n}(a_{1} + a_{2}) − d̃_{1} − d̃_{2} > K_{n}(a_{1} + a_{2}) − 2ε.

With the values |α̃_{1}|_{ψ_{n}} = ∫α̃_{1} |dΨ_{n}|, |α̃_{2}|_{ψ_{n}} = ∫α̃_{2} |dΨ_{n}| we get

(12)  \begin{aligned}K_{n}(a_{1} + a_{2}) − 2ε & < ∫_{α̃_{1}+α̃_{2}}|\frac{dΨ_{n}}{dw} | |dw| = ∫_{α_{1}+α_{2}} |ψ_{n}(w)|^{1/2}|p dz + q dz̄| \\ & = ∫ |ψ_{n}(w)|^{1/2} \frac{1}{|φ_{n}(z)|^{1/2}} |p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dξ. \end{aligned}

The integral with respect to ξ is thought to be extended over the rectangles corresponding to the two strips. We integrate over η and get, with the height of both strips S_{r} and S_{l} being equal to b

(13) \begin{aligned}K_{n}(a_{1} + a_{2})b − 2εb & ≤ ∫∫ |ψ_{n}(w)|^{1/2} \frac{1}{|φ_{n}(z)|^{1/2}} |p + q \frac{φ_{n}}{|φ_{n}|}| dξ dη \\ & = ∫∫_{S_{r}+S_{l}} |ψ_{n}(w)|^{1/2}|φ_{n}(z)|^{1/2} |p + q \frac{φ_{n}}{|φ_{n}|}| dx dy. \end{aligned}

Now we sum up over all strips S of the first kind and all pairs of strips S_{r} + S_{l} of the second kind in A. We generically denote the height of the strips by b. Thus K_{n}ab = K_{n}∥φ_{n}∥_{S}, K_{n}(a_{1} + a_{2})b = K_{n}∥φ_{n}∥_{S_{r}+S_{l}}. This gives

(14)  ∑_{(S)} K_{n}∥φ_{n}∥_{S} − ε|β|_{φ_{n}} ≤\begin{matrix}∫ & ∫ \\ ∑ S \end{matrix} |ψ_{n}(w)|^{1/2}|φ_{n}(z)|^{1/2} |p + q \frac{φ_{n}}{|φ_{n}|}| dx dy

and

(15) ∑_{(S_{r}+S_{l})} K_{n}∥φ_{n}∥_{S_{r}+S_{l}} − 2ε\frac{1}{2}|β|_{φ_{n}} ≤\begin{matrix}∫ & ∫ \\ ∑ S_{r}+S_{l} \end{matrix} |ψ_{n}|^{1/2}|φ_{n}|^{1/2} |p + q \frac{φ_{n}}{|φ_{n}|}| dx dy

and hence

(16)  K_{n}∥φ_{n}∥_{A} − 2ε|β|_{φ_{n}} ≤ ∫∫_{A} |ψ_{n}|^{1/2}|φ_{n}|^{1/2} |p + q \frac{φ_{n}}{|φ_{n}|}| dx dy.

The integral over A does not depend on the length of the vertical interval β, so we can let |β|_{φ_{n}} → 0. Then, ε becomes arbitrarily small too, which leads to

(17)  K_{n}∥φ_{n}∥_{A} ≤ ∫∫_{A} |ψ_{n}(w)|^{1/2}|φ_{n}(z)|^{1/2} |p + q \frac{φ_{n}}{|φ_{n}|}| dx dy.

We add up the inequalities for all ring domains and spiral sets of φ_{n} and get the inequality (17) for R instead of A. Finally, choosing ∥φ_{n}∥ = 1, we have, inserting the Jacobian J(w/z)

(18)  K_{n} ≤ ∫∫_{R} |ψ_{n}(w)|^{1/2}J^{1/2}|φ_{n}(z)|^{1/2} \frac{|p + q \frac{φ_{n}}{|φ_{n}|}|}{J^{1/2}} dx dy.

To this expression we apply Schwarz’s inequality which gives

(19)  K_{n}^{2} ≤ ∫∫_{R} |ψ_{n}(w)|J dx dy · ∫∫_{R} |φ_{n}(z)| \frac{|p + q \frac{φ_{n}}{|φ_{n}|}|^{2}}{|p|^{2} − |q|^{2}} dx dy.

With ∫∫_{R} |ψ_{n}(w)|J(w/z) dx dy = ∥ψ_{n}∥_{R′} = K_{n}∥φ_{n}∥_{R} = K_{n} we finally get

(20)  \begin{aligned}K_{n} & ≤ ∫∫_{R} |φ_{n}(z)| \frac{|p(z) + q(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}|^{2}}{|p(z)|^{2} − |q(z)|^{2}} dx dy \\ & = ∫∫_{R} |φ_{n}(z)| \frac{|1 + μ_{0} \frac{φ_{n}(z)}{|φ_{n}(z)|}|^{2}}{1 − |μ_{0}(z)|^{2}} dx dy. \end{aligned}

Remember that μ_{0}(z) = 0 outside of Ω and μ_{0}(z) = μ(z) in Ω. Then, the above integral over R reduces to one over Ω,

(21)  K_{n} ≤ ∫∫_{Ω} |φ_{n}(z)|\frac{|1 + μ(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}|^{2}}{1 − |μ(z)|^{2}} dx dy.

This is the Polygon Inequality. The polygon mapping f_{n} is the extremal qc mapping on R which takes the points ω_{i} ∈ ∂Ω into the points ω′_{i} = f_{0}(ω_{i}) ∈ ∂Ω′ and is homotopic on G_{n} = R \ {ω_{i}} to f_{0} modulo the points P_{n} = {ω_{i}}. It is a Teichmüller mapping associated with a quadratic differential φ_{n} of finite norm (norm one, if normalized) which is holomorphic on R except for possible first order poles at the vertices ω_{i}.

VI. Homotopy

Let now the set of vertices P_{n} become arbitrarily dense on ∂Ω with n → ∞. By general principles of qc mappings (see e.g. [4], II, §5) there is a subsequence of the sequence (f_{n}) which converges uniformly in R to a qc mapping with maximal dilatation smaller or equal to \overline{lim} K_{n}. Its values on ∂Ω are the same as the values of f_{0} on ∂Ω. We denote its restriction to Ω by f^{∗}, with maximal dilatation K^{∗}. Clearly, the maximal dilatation of f_{0} is the same as the maximal dilatation K of its restriction f to Ω.

An extremal mapping we are looking for has the boundary values of f on ∂Ω and is homotopic to f in Ω modulo the boundary (this is in fact the Teichmüller class of f in Ω).

Let G_{n} = R \ P_{n}, G′_{n} = R \ P′_{n}. In this notation the mappings f_{n}: G_{n} → G′_{n} are homotopic (mod P_{n}) to the mapping f_{0} . This property has an equivalent in terms of covering surfaces. Let Ĝ_{n}, Ĝ′_{n} be the universal covering surfaces of G_{n} and G′_{n} respectively. We lift the qc mappings f_{n} and f_{0} to the universal covering surfaces Ĝ_{n}, Ĝ′_{n}. Lifting is done by corresponding arcs with a fixed initial point z_{0} ∈ R \ P_{n} and its image point by f_{n} or f_{0} in R′ \ P′_{n}, as the case may be. In principle, the point z_{0} can be chosen anywhere in G_{n}. For our purposes it is however important to choose z_{0} ∈ Ω.

Let f̂_{n} be the lift of f_{n}, (f_{0}, P_{n}) the lift of f_{0} to Ĝ_{n} → Ĝ′_{n}.

Besides the multisheeted covering surfaces Ĝ_{n}, Ĝ′_{n} over G_{n}, G′_{n} (which is easier for the lifting process) we represent Ĝ_{n} and Ĝ′_{n} by the disks Δ, Δ′ respectively, without changing the notations of the lifts. The mappings f̂_{n} and (f_{0}, P_{n}) are thus quasiconformal mappings of the disks Δ → Δ′ which agree on the boundary ∂Δ → ∂Δ′. Their limits are quasiconformal mappings g and h of Δ onto Δ′ respectively. Since the projection of (f_{0}, P_{n}) onto G_{n} is always equal to f_{0} , the same is true if we restrict f_{0} to the original mapping f: Ω → Ω′. For f̂_{n} the projection f_{n} maps z_{0} onto f_{n}(z_{0}) = z_{n} → f^{∗}(z_{0}) ∈ Ω′. Therefore g = lim f̂_{n} cannot degerate. On the other hand, since Δ is connected (by arcs), the projection is connected. It cannot contain a point of ∂Ω, but contains all of Ω and is thus equal to Ω. We have: The projection of g is f^{∗} and thus g = f̂^{∗}. The two lifts f̂ and f̂^{∗} to the universal covering surfaces Ω̂ → Ω̂′ (which are represented by Δ and Δ′) have the same boundary values. It follows that f^{∗} = lim f_{n} | Ω and f are homotopic in Ω modulo the boundary ∂Ω.

We conclude: f^{∗} is in the Teichmüller class of f in Ω. Since f is extremal, we must have K ≤ K^{∗}, but evidently only equality is possible. It follows from the Polygon Inequality that lim_{n→∞} K_{n} = K. We finally have the

Theorem. Let Ω be a subdomain of a compact Riemann surface R with infinitely many boundary points. Let μ be an extremal Beltrami coefficient in Ω. Let {ω_{i}} be a sequence of points on ∂Ω which is dense in ∂Ω. Then, there exists a sequence of polygon differentials φ_{n} of norm ∥φ_{n}∥ = 1 with vertices ω_{i}, i = 1, …, n (n → ∞) which is a Hamilton–Krushkal sequence for μ in Ω. The usual computations in fact show that the Teichmüller differentials of the extremal polygon mappings satisfy

lim_{n→∞} Re ∫∫_{Ω} μ(z)φ_{n}(z) dx dy = ∥μ∥_{∞}.

References

[1] Earle, Clifford J., and Curt McMullen: Quasiconformal Isotopies, Holomorphic Functions and Moduli, Vol. I. - Springer-Verlag, 1988, 143–154.

[2] Lakic, Nikola: Substantial boundary points for plane domains and Gardiner’s conjecture. - Ann. Acad. Sci. Fenn. Math. 25, 2000, 285–306.

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

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

[5] Reich, Edgar, and Kurt Strebel: Extremal quasiconformal mappings with given boundary values. - In: Contributions to Analysis, A Collection of Papers Dedicated to Lipman Bers, Academic Press, 1974, 375–391.

[6] Strebel, Kurt: Quadratic Differentials. - Ergeb. Math. Grenzgeb. 3:5, Springer-Verlag, 1984.

[7] Strebel, Kurt: On the dilatation of extremal quasiconformal mappings of polygons. - Comment. Math. Helv. 74, 1999, 143–149.

Received 8 May 2001