Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 155–190

THE MAPPING BY HEIGHTS FOR QUADRATIC DIFFERENTIALS IN THE DISK

Kurt Strebel

Universität Zürich, Mathematisches Institut Rämistraße 74, CH-8001 Zürich, Schweiz; k610640@czhrzu1a.bitnet

Abstract. The heights of simple closed loops with respect to a holomorphic quadratic differential play an important role on compact Riemann surfaces. Here, the analogue is developed for quadratic differentials of finite norm in the disk. The height of a loop is replaced by the height of a cross cut, which is the same as the vertical distance, with respect to the q.d., of its end points.

1. Introduction

1.1. Let φ ≠ 0 be a holomorphic quadratic differential in the unit disk D := {z; |z| < 1}. It defines invariant length elements |φ(z)|^{1/2}|dz| and area elements |φ(z)| dx dy, z = x + iy.

The φ-length of an arc γ is

|γ|_{φ} := ∫_{γ} |φ(z)|^{1/2}|dz|,

and the φ-distance of a pair of points z_{1} , z_{2} is equal to

d_{φ}[z_{1}, z_{2}] := inf_{{γ}} |γ|_{φ},

where γ varies over all arcs connecting the two points. The φ-area of a point set E ⊂ D is the integral

∫∫_{E} |φ(z)| dx dy,

and the φ-area of D is the L^{1} -norm of φ,

∥φ∥ = ∫∫_{D} |φ(z)| dx dy.

Throughout this paper we will speak about quadratic differentials of finite norm.

1991 Mathematics Subject Classification: Primary 30C62; Secondary 30C75.

Besides that φ also defines an element of height

|Im {φ(z)^{1/2}dz}|

and one of horizontal length

|Re {φ(z)^{1/2}dz}| .

Since the last expression is equal to the first one for the differential −φ, it is enough to look at the elements of height. Similarly as for lengths we can define the φ-height of an arc γ by

h_{φ}(γ) := ∫_{γ} |Im {φ(z)^{1/2}dz}|

and the vertical distance or the φ-height of a pair of points z_{1} , z_{2} by

h_{φ}[z_{1}, z_{2}] := inf_{{γ}} h_{φ}(γ),

with the same meaning of {γ} as before (see Definition 2.1 below).

For a better visualization of the different quantities we introduce, locally and away from the zeroes, the integral of the square root of φ,

w = u + iv = Φ(z) = ∫ ^{z} \sqrt{φ(z)} dz.

The elements of the multivalued function Φ are well defined up to the transformation

Φ_{2}(z) = ±Φ_{1}(z) + const .

The elements of length

|dw| = |φ(z)|^{1/2}|dz|

and of height

|dv| = |Im \{\sqrt{φ(z)} dz\}|

are well defined. The height of an arc γ is nothing but the total variation of the multivalued harmonic function v along γ. If we introduce w as local parameter instead of z, the elements of length and of height become Euclidean, as the expressions show, however with branchings in the large.

1.2. Let \overline{D} be the closed unit disk |z| ≤ 1 and assign a finite number of points ζ_{1}, …, ζ_{N} on ∂D. We call \overline{D} together with the distinguished points ζ_{i} a polygon P, an N -gon in this particular case. The ζ_{i} are its vertices and the intervals on ∂D between the vertices its sides or edges. A quadratic differential φ is said to belong to P, if it is meromorphic in \overline{D}, with at most simple poles at the vertices, and real along the sides of P (i.e. φ(z) dz^{2} real for tangential dz).

It is easy to see (e.g. by means of a conformal mapping of D onto the upper half plane) that φ can be continued to Ĉ by reflection on ∂D. The continuation is a quadratic differential with closed trajectories which sweep out a finite number of disjoint annuli (for details see [1]). Each annulus is split into a symmetric pair of quadrilaterals by two subintervals of different sides of P, separated by at least two vertices. The quadrilaterals in D are called horizontal strips S_{i}. They are mapped onto Euclidean rectangles

0 < u < a_{i}, 0 < v < b_{i}

by a branch of w = u + iv = Φ(z). If we choose a trajectoryα_{i} out of each open strip S_{i}, we get a system of disjoint cross cuts of D. Each cross cut connects two different sides of P, separated by at least two vertices, and different ones connect different pairs of sides.

Conversely, it was shown by H. Renelt and, simultaneously, by J. Hubbard and H. Masur (1976), that one can prescribe a system of Jordan arcs γ_{i} which are cross cuts in the above sense, and the numbers b_{i} > 0. Then, there exists a unique quadratic differential φ, associated with the N -gon P and such that its trajectories are homotopic to the given cross cuts γ_{i} and the heights of its strips (or cylinders) are the given numbers b_{i}. (For a proof and references to the original literature see [1]).

This theorem can be used to set up a bijection of the differentials of two different polygons. Let P and P^{∗} be two N -gons. Given an order preserving correspondence of the vertices (and hence of the sides) of P and P^{∗}, we assign to each polygon differential φ of P a polygon differential φ^{∗} of P^{∗} by the requirement that the strips of φ and φ^{∗} connect corresponding sides of P and P^{∗} respectively and that corresponding strips have the same heights (measured in the φ- and φ^{∗} -metric respectively). This determines the “mapping by heights” for two given polygons with a given correspondence of the vertices.

As an example, let us consider two pentagons P and P^{∗}. The quadratic differentials φ and φ^{∗} associated with them are the squares of the derivatives of conformal maps w = Φ(z), w^{∗} = Φ^{∗}(z^{∗}) respectively, mapping the pentagons onto bus like figures with the same heights of the upper and of the lower parts (Figure 1). The vertices go over into the corners pointing outwards. In terms of the parameters w and w^{∗} we have, by the transformation rule for quadratic differentials, φ ≡ φ^{∗} ≡ 1. The basic theorem says that the numbering of the

Figure 1.

corners and the heights of the two pieces can be given, but then the shape of a “bus” is uniquely determined.

1.3. The purpose of this note is to generalize this mapping for arbitrary quasisymmetric homeomorphisms of the boundary of the unit disk. Rather than heights of strips, which do not exist in the general case, we consider vertical distances (or heights) of pairs of boundary points. It is easy to see (using intersection numbers) that the two polygon differentials φ and φ^{∗} induce the same vertical distances of corresponding pairs of sides. (There is no correspondence of boundary points, except for the vertices.)

Our main result will be, that every quasisymmetric selfmapping of the boundary of the disk induces a selfmapping of the space of holomorphic quadratic differentials of finite norm. Corresponding differentials φ and φ^{∗} determine the same vertical distances of all corresponding pairs of boundary points, which is the characteristic property of the mapping. The proof goes by approximation of differentials by polygon differentials. The main ingredient is the notion of a totally regular trajectory (see [2]). A trajectory α of φ is called regular, if it does not tend, in any of its two directions, to a zero of φ. Otherwise it is called critical. It is known that a regular trajectory of quadratic differential of finite norm has two different end points on ∂D (for a proof see [1, Section 19]). A regular trajectory α is called totally regular, if for any sequence of points {z_{n}} tending to a point z ∈ α and such that the trajectories α_{n} ∋ z_{n} are regular, α_{n} → α in the Euclidean metric of the disk D. It is shown in [2] that there can be at most denumerably many regular trajectories which are not totally regular. Moreover, if {φ_{n}} is a sequence of holomorphic quadratic differentials with uniformly bounded norm which tends locally uniformly to a differential φ not identically equal to zero, then the above statement is true with α_{n} a trajectory of φ_{n} rather than of φ. It will be shown that the vertical distance of two points is equal to the supremum of the vertical distances of pairs of totally regular trajectories separating the two points. This, together with the fact that the totally regular trajectories and their vertical distances are invariant under the constructed mapping gives the result.

For later use we state

Lemma 1.3. Let φ be an arbitrary holomorphic quadratic differential of norm ∥φ∥ ≤ M < ∞ in the unit disk D : |z| < 1. Let ζ be a boundary point of D. Then, for any ε > 0 and ϱ_{2} > 0 there exists a number ϱ_{1} , 0 < ϱ_{1} < ϱ_{2} , such that for some ϱ ∈ [ϱ_{1}, ϱ_{2}]

L(ϱ) = ∫_{σ_{ϱ}} |φ(z)|^{1/2}|dz| < ε,

with σ_{ϱ} = {z; |z − ζ| = ϱ, z ∈ D}. Whereas ϱ depends on φ, ϱ_{1} does not. (For a proof see [2, Lemma 1.1]).

2. Heights (vertical distances)

2.1. Let φ ≠ 0 be a holomorphic quadratic differential of finite norm in the disk D : |z| < 1.

Definition 2.1. Let ζ_{1} and ζ_{2} be boundary points of D. The vertical distance or height of the pair of points ζ_{1} , ζ_{2} with respect to φ is

h_{φ}[ζ_{1}, ζ_{2}] := inf_{{γ}} ∫_{γ} |dv|,

where γ runs over all locally rectifiable open Jordan arcs in D with limit points ζ_{1} and ζ_{2} respectively and v is the imaginary part of w = u + iv = Φ(z) = ∫^{z} \sqrt{φ(z)} dz.

Similarly one defines the vertical distance of two interior points (where the arcs γ are simply rectifiable Jordan arcs in D) or the vertical distance of an interior point and a boundary point.

The vertical distance of two point sets E_{1} and E_{2} is defined as usual:

h_{φ}[E_{1}, E_{2}] := inf {h_{φ}[z_{1}, z_{2}]; z_{1} ∈ E_{1}, z_{2} ∈ E_{2}}.

A special case is the vertical distance of a pair of horizontal geodesics α_{1} and α_{2} . The height of a pair of points z_{1} ∈ α_{1} and z_{2} ∈ α_{2} does not depend on their position, since one can add, to a curve γ connecting z_{1} and z_{2} , arbitrary subintervals of α_{1} and α_{2} ending at z_{1} and z_{2} respectively (because dv = 0 along any horizontal interval). We therefore have, for the vertical distance of α_{1} and α_{2} ,

h_{φ}[α_{1}, α_{2}] = h_{φ}[z_{1}, z_{2}], z_{1} ∈ α_{1}, z_{2} ∈ α_{2}.

It follows from the fact that for any boundary point ζ of D there are circular cross cuts of D, centered at ζ, with arbitrarily short φ-length (Lemma 1.3), that the vertical distance of z_{1} and z_{2} is also equal to the vertical distance of two end points of the horizontal arcs.

It is immediate by computation of the norm using polar coordinates that the distance of a boundary point ζ from the center (and hence from any interior point z) is finite for a.a. ζ ∈ ∂D. Since the vertical distance is smaller or equal to the distance, this is also true for heights.

2.2. Lower semicontinuity of the vertical distance

Lemma 2.2. For any fixed

z_{0} ∈ D, ζ ∈ ∂D

lim inf_{z→ζ} h_{φ}[z_{0}, z] ≥ h_{φ}[z_{0}, ζ].

Proof. It follows from the length area principle (Lemma 1.3), that there exists a sequence of radii ϱ_{n} → 0 such that the φ-length |τ_{n}|_{φ} of the circular cross cuts

τ_{n} := {z ∈ D, |z − ζ| = ϱ_{n}}

tends to zero. Therefore, for every positive ε, there exists a subsequence {τ_{n_{i}}} such that

∑_{i} |τ_{n_{i}}|_{φ} < ε.

To simplify the notation, we call this subsequence {τ_{n}} again (Figure 2).

Figure 2.

Let us first consider the case where

h_{φ}[z_{0}, τ_{0}] + h_{φ}[τ_{0}, τ_{1}] + h_{φ}[τ_{1}, τ_{2}] + ⋯ = A < ∞.

For a given ε > 0 we can find an arc γ_{0} connecting z_{0} with τ_{0} such that

h_{φ}(γ_{0}) < h_{φ}[z_{0}, τ_{0}] + \frac{ε}{2},

The mapping by heights for quadratic differentials 161

and arcs γ_{n} connecting τ_{n−1} with τ_{n} such that

h_{φ}(γ_{n}) < h_{φ}[τ_{n−1}, τ_{n}] + \frac{ε}{2^{(n+1)}} ,

n = 1, 2, . . .. The end points of γ_{n−1} and γ_{n} on τ_{n−1} are connected by a subinterval Δτ_{n−1} of τ_{n−1} . Since

∑_{0}^{∞} |τ_{n}|_{φ} < ε,

we also have

∑_{0}^{∞} |Δτ_{n}| < ε.

We thus get a curve γ, which we write

γ := γ_{0} + Δτ_{0} + γ_{1} + Δτ_{1} + γ_{2} + Δτ_{2} + ⋯

connecting z_{0} with ζ which has height

h_{φ}[z_{0}, ζ] ≤ h_{φ}(γ) < h_{φ}[z_{0}, τ_{0}] + h_{φ}[τ_{0}, τ_{1}] + ⋯ + 2ε = A + 2ε.

On the other hand, there exists an index n such that

h_{φ}[z_{0}, τ_{n}] ≥ h_{φ}[z_{0}, τ_{0}] + h_{φ}[τ_{0}, τ_{1}] + ⋯ + h_{φ}[τ_{n−1}, τ_{n}] > A − ε.

Therefore

h_{φ}[z_{0}, τ_{n}] > h_{φ}[z_{0}, ζ] − 3ε.

If z is separated from z_{0} by τ_{n},

h_{φ}[z_{0}, z] ≥ h_{φ}[z_{0}, τ_{n}],

which proves that

lim inf_{z→ζ} h_{φ}[z_{0}, z] ≥ h_{φ}[z_{0}, ζ].

Let now

h_{φ}[z_{0}, τ_{0}] + ∑_{n=1}^{∞} h_{φ}[τ_{n−1}, τ_{n}] = ∞.

Then, for any M < ∞ there exists an index n such that

h_{φ}[z_{0}, τ_{n}] ≥ h_{φ}[z_{0}, τ_{0}] + ∑_{i=1}^{n} h_{φ}[τ_{i−1}, τ_{i}] > M,

which shows that

lim inf_{z→ζ} h_{φ}[z_{0}, z] = ∞.

2.3. Connections of smallest height; step curves

Lemma 2.3. Let φ be a holomorphic quadratic differential in D. Then, any shortest connection of two interior points of D has minimal height.

Proof. Let z_{0}, z_{1} ∈ D and let γ_{0} be the shortest curve connecting the two points. Then, γ_{0} is a geodesic and hence consists of φ-straight pieces satisfying the angle condition at the zeroes of φ (for details see [1, Theorem 8.1]). If γ_{0} is horizontal, i.e. φ(z) dz^{2} ≥ 0 along γ_{0} , then h_{φ}(γ_{0}) = 0, and thus γ_{0} has minimal height. Otherwise, γ_{0} consists of non horizontal and possibly horizontal straight segments. Let γ be an arbitrary connection of z_{0} and z_{1} . Choose a radius r < 1 such that the disk D_{r} = {z; |z| < r} contains both γ_{0} and γ. Mark the zeroes of φ on γ_{0} and those points z on the non horizontal edges of γ_{0} which lie on a relatively critical trajectory (i.e. one which meets a zero of φ before hitting the circle |z| = r). There can be only finitely many markings on γ_{0} . The trajectories going through non marked points on γ_{0} are cross cuts of D_{r} which separate z_{0} and z_{1} . They sweep out finitely many horizontal strips S_{i} of height b_{i}, say, mapped conformally onto Euclidean horizontal strips by any branch of Φ. Each strip is passed once (of course in general not vertically) by γ_{0} . Since the horizontal pieces of γ_{0} have height zero, h_{φ}(γ_{0}) = ∑ b_{i}.

On the other hand, γ must cross every strip S_{i}. Therefore, by Euclidean geometry in the w-plane, h_{φ}(γ) ≥ ∑ b_{i}.

Of course, there is no uniqueness of connections of minimal height, as there is for curves with minimal length. Since, locally, there always exists a shortest connection (see [1, Theorem 8.1]) we have the following

Corollary 2.3. Every point z has a neighborhood U(z) with the property that any two points z_{0}, z_{1} ∈ U(z) can be joined by an arc of minimal height in U(z).

We will later work with connections of a special type.

Definition 2.3. A step curve (with respect to φ) is a curve which consists of horizontal (φ(z) dz^{2} ≥ 0) and vertical (φ(z) dz^{2} ≤ 0) pieces.

It is easy to see, that, locally, there always exists a connecting step curve of minimal height. It may have to pass through a zero of φ. On the other hand, there always exists a step curve connection avoiding the zero, and of height arbitrarily close to the minimum. The preceding picture shows connections of minimal height in a neighborhood of a third order zero (Figure 3). The arcs consist of two horizontal and one vertical (dotted) or two vertical and one horizontal pieces. If the two points are not in adjacent sectors, the connecting arc of minimal height goes through the zero in the center of the disk.

Of course, a union of step curves is again a step curve. The following is therefore clear.

Theorem 2.3. Let φ ≠ 0 be holomorphic in D, z_{0}, z_{1} ∈ D. Then, every curve γ connecting z_{0} and z_{1} can be replaced by a step curve \overline{γ} which is contained in an arbitrarily small neighborhood of γ and has height

h_{φ}(\overline{γ}) ≤ h_{φ}(γ).

An approximation by a step curve \overline{γ} avoiding the zeroes of φ is possible with height

h_{φ}(\overline{γ}) < h_{φ}(γ) + ε,

for every ε > 0.

3. Heights and separating trajectories

3.1. The vertical distance of two points can be expressed in terms of the totally regular trajectories separating them. We begin with the case where the two points are in D.

Theorem 3.1. The vertical distance of two points z_{0}, z_{1} ∈ D is zero if and only if there is no totally regular trajectory which separates them.

Remark. As the preceding picture (Figure 4) shows, the statement is wrong for regular but not totally regular trajectories. The two points z_{0}, z_{1} have vertical distance zero, although they are separated by the regular trajectory α. The two points lie on regular trajectories α_{0} and α_{1} respectively, which have an end point in common with α. Note that a totally regular trajectory cannot have an end point in common with another trajectory.

Proof. Assume first that α is totally regular and separates the two points. Then, there are totally regular trajectories α̃ in every neighborhood (in the Euclidean metric of D) of α. If we choose α̃ in a sufficiently small neighborhood of α, it also separates the two points. Therefore all curves γ connecting z_{0} and z_{1} have a subinterval that connects α and α̃. Clearly, α and α̃ have positive vertical distance, and thus

h_{φ}[z_{0}, z_{1}] ≥ h_{φ}[α, α̃] > 0.

Addendum. A similar argument works if α passes through z_{0} , but not through z_{1} . Then, α̃ is chosen such that it separates α from z_{1} . We therefore have:

If h_{φ}[z_{0}, z_{1}] = 0, no totally regular trajectory α can separate the two points nor pass through one of the points without passing through the other.

It is evident, that the above proof also works if one of the points or both are boundary points of D. We will therefore not repeat it further down.

The converse is a consequence of the following

Lemma 3.1 (Trimming lemma). Assume that no totally regular trajectory separates z_{0} from z_{1} . Then, for every step curve \overline{γ} connecting the two points and every ε > 0 there exists a variation \overline{γ}_{0} of \overline{γ} with height h_{φ}(\overline{γ}_{0}) < ε.

Proof. If one of the two points lies on a totally regular trajectory σ, the other one necessarily lies on the same (because otherwise we could find separating ones). But then \overline{γ}_{0} is the connecting subinterval of σ. We can therefore exclude this case in the sequel.

We connect the two points by a step curve \overline{γ}, consisting of horizontal and vertical intervals α_{i} and β_{i} respectively and avoiding the zeroes of φ (except for possible zeroes at the end points, of course). To fix the ideas, we write symbolically

\overline{γ} = z_{0} + β_{0} + α_{1} + β_{1} + ⋯ + α_{N} + β_{N} + z_{N}

calling the last point z_{N} rather than z_{1} . We can always start and terminate with a vertical interval, since \overline{γ} is not assumed to have minimal height. Moreover, by arbitrarily small shifts, if necessary, we can achieve that the horizontal intervals α_{i} lie on totally regular trajectories (which we also call α_{i}) none of which passes through z_{0} or z_{N}.

Figure 5.

Choose a totally regular trajectory σ intersecting β_{0} in a point ζ near z_{0} (Figure 5). The curve \overline{γ} has a last intersection ζ^{∗} with σ. The subinterval of \overline{γ} between ζ and ζ^{∗}, which we denote by (\overline{γ}; ζ, ζ^{∗}), clearly increases monotonically, if ζ → z_{0} , and hence the points ζ^{∗} have a limit z_{0}^{∗} ∈ \overline{γ}. This point is necessarily an interior point of some vertical side β_{i} of \overline{γ}. Otherwise it would lie on a totally regular trajectory, α_{i+1} , say, and hence z_{0} would lie on the same, which we have excluded. The two pairs of points ζ, z_{0} and ζ^{∗}, z_{0}^{∗} have the same distance (which is equal to their vertical distance). We choose σ such that it is smaller than ε/2N and we replace the arc (\overline{γ}; z_{0}, z_{0}^{∗}) by (β_{0}; z_{0}, ζ) + (σ; ζ, ζ^{∗}) + (β_{i}; ζ^{∗}, z_{0}^{∗}), which means that we go, except for two short vertical intervals, along σ.

Now we look at the remaining arc (\overline{γ}; z_{0}^{∗}, z_{N}) of \overline{γ}. Its two end points have the property that there is no separating totally regular trajectory (because such a trajectory would also separate z_{0} and z_{N}) and z_{0}^{∗} cannot lie on a totally regular trajectory, unless z_{N} lies on the same. We therefore have the initial situation and we can continue the trimming process. The two intervals on β_{i} add to one of length (height) less than ε/N. Because there are at most N steps necessary, we have a step curve \overline{γ}_{0} of height less than N · ε/N = ε which connects the two points z_{0} and z_{N}.

The theorem follows immediately: because ε > 0 is arbitrary, h_{φ}[z_{0}, z_{N}] = 0.

3.2. We pass to the general case.

Theorem 3.2. Let z_{0}, z_{1} ∈ D be points with the property that there exist totally regular trajectories σ separating them. Then,

h_{φ}[z_{0}, z_{1}] = sup_{σ′,σ′′} h_{φ}[σ′, σ′′],

where σ′ and σ′′ run over all totally regular trajectories which separate z_{0} and z_{1} .

Proof. The inequality sign in one direction is evident. Take two arbitrary totally regular trajectories σ′, σ′′ separating z_{0} and z_{1} . Then, every curve γ connecting z_{0} with z_{1} has a subarc γ′ connecting σ′ and σ′′. Therefore

∫_{γ} |φ(z)|^{1/2}|dz| ≥ ∫_{γ′} |φ(z)|^{1/2}|dz| ≥ h_{φ}[σ′, σ′′],

and thus

h_{φ}[z_{0}, z_{1}] ≥ h_{φ}[σ′, σ′′].

We conclude that

h_{φ}[z_{0}, z_{1}] ≥ sup_{σ′,σ′′} h_{φ}[σ′, σ′′].

To prove the converse inequality, connect z_{0} and z_{1} by a step curve \overline{γ} as in the proof of Lemma 3.1 (z_{1} = z_{N}).

Orient \overline{γ} from z_{0} to z_{N}. Let ζ′ be the infimum and ζ′′ the supremum of the intersections of separating totally regular trajectories σ with \overline{γ}. We claim that both points are interior points of certain vertical intervals β_{i}, except possibly ζ′ = z_{0} , ζ′′ = z_{N}. For, assume that ζ′ is the end point of a β_{i} (Figure 6). Then α_{i+1} is itself a separating totally regular trajectory. All totally regular trajectories σ in a neighborhood of α_{i} + 1 are also separating, and ζ′ cannot be the infimum. If, on the other hand, ζ′ is the initial point of β_{i}, α_{i} being totally regular cannot separate the two points but must pass through z_{0} , which we have excluded. A similar argument works for ζ′′.

Figure 6.

Let ζ′ ∈ β_{i}. For a given ε > 0 we choose a separating totally regular trajectory σ′ intersecting β_{i} in an ε-neighborhood of ζ′. Let ζ′′ ∈ β_{k} and choose a totally regular trajectory σ′′ cutting β_{k} in an ε-neighborhood of ζ′′. Let γ_{0} be a step curve connecting σ′ and σ′′ with a height

h_{φ}(γ_{0}) < h_{φ}[σ′, σ′′] + ε.

The subarc of \overline{γ} connecting z_{0} with ζ′ is not cut by any totally regular trajectory σ separating z_{0} from ζ′, because σ would also separate z_{0} from z_{N}. By Theorem 3.1 one can therefore connect z_{0} and ζ′ by a step curve γ′ of height < ε. Similarly, we can connect ζ′′ and z_{n} by a step curve γ′′ of height < ε. Let τ′ and τ′′ be the two vertical intervals connecting ζ′ with σ′ and ζ′′ with σ′′ respectively. They have length and thus height smaller than ε. The step curve

γ̃ = γ′ + τ′ + γ_{0} + τ′′ + γ′′,

possibly with two subintervals of σ′ and σ′′ respectively, connects z_{0} with z_{N} and has height

h_{φ}(γ̃) < h_{φ}(γ_{0}) + 4ε < h_{φ}[σ′, σ′′] + 5ε.

Since ε > 0 is arbitrary, this proves that

h_{φ}[z_{0}, z_{N}] ≤ sup_{σ′,σ′′} h_{φ}[σ′, σ′′],

and the theorem is proved.

3.3. In this section and the next one the results of Sections 3.1 and 3.2 are generalized to boundary points of D.

Theorem 3.3. The vertical distance of an interior point z from a boundary point r or of two boundary points r and s is positive if and only if there exists a totally regular trajectory which separates them.

Proof. If there exists a totally regular trajectory σ separating the two points or ending in one of them without tending, in the opposite direction, to the other one, then h_{φ} > 0. This is proved as in 3.1.

The converse is first shown for an interior point and a boundary point. Let h_{φ}[z, r] > 0. Choose a sequence of circular cross cuts τ_{n} := {z ∈ D; |z − r| = ϱ_{n}}, with ϱ_{n} → 0 and |τ_{n}|_{φ} → 0. Because of the lower semicontinuity of the heights, we can find, for a fixed 0 < A < h_{φ}[z, r], an index n such that |τ|_{φ} < \frac{1}{2}A and h_{φ}[z, τ_{n}] > A. For any z_{n} ∈ τ_{n} we clearly also have h_{φ}[z, z_{n}] > A. According to Theorem 3.2 there are two totally regular trajectories σ′ and σ′′ separating z and z_{n} such that h_{φ}[σ′, σ′′] > A. If one of these separates z and r, we are done.

Figure 8.

If none of them does, by topological reasons (Figure 7), both pass through τ_{n}. Therefore h_{φ}[σ′, σ′′] ≤ |τ_{n}|_{φ} < \frac{1}{2}A, a contradiction.

In the case of two boundary points r and s, we choose two circular cross cuts

τ′ = {z ∈ D; |z − r| = ϱ′}, τ′′ = {z ∈ D; |z − s| = ϱ′′}

such that h_{φ}[τ′, τ′′] > A and |τ′|_{φ} < \frac{1}{2}A, |τ′′|_{φ} < \frac{1}{2}A, for some fixed 0 < A < h_{φ}[r, s]. The vertical distance of two arbitrary points z′ ∈ τ′, z′′ ∈ τ′′ is h_{φ}[z′, z′′] > A. Then, there exist totally regular trajectories σ′, σ′′ separating the points z′, z′′ and such that h_{φ}[z′, z′′] > A (Figure 8). If one at least of the two separates r and s, we are done. If none does, they must both cut one of the arcs τ′, τ′′. If both cut the same, we use the earlier argument. Otherwise we choose a totally regular trajectory σ separating σ′ and σ′′ and such that both h_{φ}[σ′, σ] > \frac{1}{2}A and h_{φ}[σ′′, σ] > \frac{1}{2}A (see Corollary 4.1). If σ does not separate r and s, it must intersect at least one of the cross cuts τ′ or τ′′. We then find a contradiction as before. Thus, if h_{φ}[r, s] > 0 there exists a totally regular trajectory separating the two points, as claimed.

3.4. The next theorem is the generalization of 3.2 to boundary points.

Theorem 3.4. Let r and s be boundary points of D. If there are totally regular trajectories σ separating r and s, then

h_{φ}[r, s] = sup_{σ′,σ′′} h_{φ}[σ′, σ′′],

the supremum being taken over all separating pairs of totally regular trajectories. The same is true for an interior point z_{0} and a boundary point .

Proof. We start with the second case. Let z_{0} ∈ D, s ∈ ∂D. Let σ be a totally regular trajectory separating z_{0} and s. Then, for all sufficiently small ϱ > 0 it also separates z_{0} from the circular cross cut

τ(ϱ) := {z ∈ D; |z − s| = ϱ}.

From Theorem 3.2 we conclude that for any z ∈ τ(ϱ)

h_{φ}[z_{0}, z] = sup_{σ′,σ′′} h_{φ}[σ′, σ′′],

σ′ and σ′′ running over the totally regular trajectories separating z_{0} and z.

Figure 9.

Let now 0 < A < h_{φ}[z_{0}, s] ≤ ∞ and ε > 0. There exists ϱ > 0 such that |τ(ϱ)|_{φ} < ε and h_{φ}[z_{0}, z] ≥ h_{φ}[z_{0}, τ(ϱ)] > A. Let σ′, σ′′ be totally regular trajectories separating z_{0} and z and such that h_{φ}[σ′, σ′′] > A. If they do not both separate z_{0} and s (Figure 9), choose σ separating σ′ from σ′′ with ε < h_{φ}[σ, σ′′] < 2ε. Then, σ cannot cut τ(ϱ). For, if it does, so does σ′′, and hence h_{φ}[σ, σ′′] ≤ |τ(ϱ)|_{φ} < ε, a contradiction. Therefore σ separates z_{0} and s, and

h_{φ}[σ′, σ] = h_{φ}[σ′, σ′′] − h_{φ}[σ, σ′′] > A − 2ε.

To show the same for two boundary points r and s, choose a separating totally regular trajectory σ. Then, it is easy to see that

h_{φ}[r, s] = h_{φ}[r, σ] + h_{φ}[σ, s].

For, clearly the sign ≥ holds. On the other hand, any pair of arcs γ′, γ′′ connecting r with σ and σ with s respectively can be completed by a subinterval Δσ of σ to a curve γ = γ′ + Δσ + γ′′ connecting r with s. Therefore

h_{φ}[r, s] ≤ h_{φ}(γ) = h_{φ}(γ′) + h_{φ}(γ′′),

which gives the inequality

h_{φ}[r, s] ≤ h_{φ}[r, σ] + h_{φ}[σ, s].

Let now, for any A′ < h_{φ}[r, σ], A′′ < h_{φ}[σ, s], σ′ and σ′′ be totally regular trajectories separating r from σ and σ from s respectively and such that h_{φ}[σ′, σ] > A′ and h_{φ}[σ, σ′′] > A′′. Then,

h_{φ}[σ′, σ′′] = h_{φ}[σ′, σ] + h_{φ}[σ, σ′′] > A′ + A′′.

We end up with

sup_{σ′,σ′′} h_{φ}[σ′, σ′′] ≥ h_{φ}[r, σ] + h_{φ}[σ, s] = h_{φ}[r, s].

Since the converse inequality is evident, the theorem is proved.

4. Convergence of heights

4.1. In order to prove the convergence of heights we have to give Theorem 3.2 a more constructive form.

Definition 4.1. Given a closed, regular (i.e. without zeroes of φ) vertical interval β̃ with the property that the two trajectories σ and σ′ through its end points are totally regular. Then, the domain bounded by σ and σ′ is called an elementary horizontal strip S spanned by β̃. It is denoted by (β̃; σ, σ′).

S is said to separate the two points z, z′ ∈ D, if both σ and σ′ separate the two points.

Let S_{j}, spanned by β̃_{j}, be disjoint elementary strips separating the two points z and z′. Then, clearly

h_{φ}[z, z′] ≥ ∑_{j} |β̃_{j}|.

Theorem 4.1. For every ε > 0 there exists a finite system of non overlapping elementary strips S_{j}, spanned by β̃_{j}, such that

h_{φ}[z, z′] < ∑_{j} |β̃_{j}| + ε.

Proof. Let

\overline{γ} = z_{0} + β_{0} + α_{1} + β_{1} + ⋯ + α_{N} + β_{N} + z_{N}

be a step curve connecting the two points z_{0} and z′ = z_{N}. The β_{i} are the vertical, the α_{i} the horizontal intervals. We may assume that \overline{γ} does not go through a zero of φ, with the possible exception of z_{0} and z_{N}, and that the α_{i} are lying on different totally regular trajectories not going through an end point of \overline{γ}.

(1) Suppose that there are totally regular trajectories σ with intersection ζ = σ ∩ β_{0} arbitrarily close to z_{0} which do not separate the two points. Let ζ^{∗} be the last intersection of \overline{γ}, oriented from z_{0} to z_{N}, with σ. The point z_{0}^{∗} = lim_{ζ→z_{0}} ζ^{∗} is called the conjugate point of z_{0} . It follows from the above assumptions about \overline{γ} that z_{0}^{∗} is, with exception of the trivial case z_{0}^{∗} = z_{N}, an interior point of some β_{i}. We denote the subarc of \overline{γ} bounded by z_{0} and by z_{0}^{∗} by (\overline{γ}; z_{0}, z_{0}^{∗}). As in the proof of Lemma 3.1, this subarc can be replaced by an arc of height < ε/2N. We continue with the arc (\overline{γ}; z_{0}^{∗}, z_{N}), which has at least one vertical side less than \overline{γ}, namely β_{0} . The totally regular trajectories separating z_{0}^{∗} and z_{N} also separate z_{0} and z_{N}.

Figure 10.

(2) Assume now that there is a half neighborhood of z_{0} on β_{0} which is cut only by totally regular trajectories σ separating the two points z_{0} and z_{N}. Let η_{0} ∈ β_{0} be the supremum of all intersections ζ = σ ∩ β_{0} interior to β_{0} . Then, every totally regular trajectory cutting β_{0} between z_{0} and η_{0} separates z_{0} and z_{N}. We get an elementary strip bounded by σ, with ζ = σ ∩ β_{0} near z_{0} and τ, η = τ ∩ β_{0} near η_{0} (Figure 10). Its spanning vertical interval is, in our notation, β̃_{0} = (β_{0}; ζ, η). We choose the two points in such a way that the sum of the lengths of the two vertical intervals [z_{0}, ζ] and [η, η_{0}] is less than ε/2(N + 1).

To find the remaining arc of \overline{γ}, let η^{∗} be the last intersection of \overline{γ} with τ, and let η_{0}^{∗} = lim η^{∗} if η approaches η_{0} . Eventually, all η^{∗} lie on the same β_{i}, and the two vertical intervals [η, η_{0}] and [η^{∗}, η_{0}^{∗}] have the same length.

If η^{∗} ≡ η, i.e. the only intersection of τ with \overline{γ} is the point η on β_{0} , we have η_{0}^{∗} = η_{0} . If η_{0}^{∗} is an interior point of β_{0} , the curve γ̃ is the same as \overline{γ}, subdivided in the following way:

γ̃ = (β_{0}; z_{0}, ζ) + β̃_{0} + (β_{0}; η, η_{0}) + (\overline{γ}; η_{0}, z_{N}).

It easily follows from the requirements on \overline{γ} that the arc (\overline{γ}; η_{0}, z_{N}) satisfies the same conditions.

If η_{0} = (β_{0} ∩ α_{1}), i.e. η_{0} is the end point of β_{0} , we set
γ̃ = (β_{0}; z_{0}, ζ) + β̃_{0} + (β_{0}; η, η_{0}) + α_{1} + (\overline{γ}; α_{1} ∩ β_{1}, z_{N}).

Again it is immediately clear that (\overline{γ}; α_{1} ∩ β_{1}, z_{N}), i.e. the subarc of \overline{γ} starting with the initial point of β_{1} , satisfies the same conditions as \overline{γ} itself.

Assume now that η^{∗} ≠ η, i.e. that the totally regular trajectory τ intersects \overline{γ} again. Thus η_{0}^{∗} ≠ η_{0} . It is easy to see that η_{0}^{∗} must be an interior point of β_{i}. For, if it is the end point, η_{0}^{∗} = β_{i} ∩ α_{i+1} , η_{0}^{∗} lies on a totally regular trajectory, namely α_{i+1} , and η_{0} lies on the same. This trajectory must separate the two points z_{0} and z_{N}, because it cannot go through z_{N}. If η_{0} is an interior point of β_{0} , it therefore can be slightly pushed up, and if it is the end point, α_{1} and α_{i+1} are lying on the same totally regular trajectory, which was excluded from the beginning.

Likewise one shows that η_{0}^{∗} cannot lie on a totally regular trajectory α_{j}, j > i + 1.

The new step curve is

γ = (β_{0}; z_{0}, ζ) + β̃_{0} + (τ; η, η^{∗}) + (β_{i}; η^{∗}, η_{0}^{∗}) + (\overline{γ}; η_{0}^{∗}, z_{N}),

in words: starting at z_{0} we follow β_{0} to the lower side σ of the elementary strip S, cross it along β_{0} , follow the upper side τ of S from η to η^{∗}, go on β_{i} to η_{0}^{∗}, which is the initial point of the remaining arc (\overline{γ}; η_{0}^{∗}, z_{N}). We continue the procedure with this subarc of \overline{γ}.

Every β_{i} of the original step curve \overline{γ} can give rise to at most one separating elementary strip, and there can be at most N trimmings. Therefore the total height of the final step curve γ̃ is

h_{φ}(γ̃) < ∑ |β̃_{j}| + N \frac{ε}{2N} + (N + 1)\frac{ε}{2(N + 1)} = ∑ |β̃_{j}| + ε,

which proves the theorem.

The constructed step curve γ̃ has almost minimal height. It is clear that among all step curves connecting z_{0} and z_{N} there exist such elements. The special feature of γ̃ is that almost its entire height is attained by the crossings β̃_{j} of disjoint elementary strips.

The following is now evident:

Corollary 4.1. Let h_{φ}[z_{0}, z_{1}] > 0. Then, for every number x, 0 ≤ x ≤ h_{φ}[z_{0}, z_{1}], and every ε > 0 there exists a separating totally regular trajectory σ with the property

x − ε < h_{φ}[z_{0}, σ] < x + ε.
Remember that h_{φ}[z_{0}, z_{1}] = h_{φ}[z_{0}, σ] + h_{φ}[σ, z_{1}].

4.2. We now prove the convergence of vertical distances of pairs of interior points of D. An inequality in one direction is easy, even without bounded norm.

Lemma 4.2. Let (φ_{n}) be a sequence of holomorphic quadratic differentials in D which tends locally uniformly to φ ≠ 0. Let z, z′ ∈ D. Then,

lim sup h_{φ_{n}}[z, z′] ≤ h_{φ}[z, z′].

Proof. Choose a rectifiable curve γ connecting the two points. Then,

h_{φ_{n}}[z, z′] ≤ ∫_{γ} |dv_{n}| → ∫_{γ} |dv|,

hence

lim sup_{n→∞} h_{φ_{n}}[z, z′] ≤ ∫_{γ} |dv|.

Since this is true for all γ, the lemma is proved. Evidently, if h_{φ}[z, z′] = 0, the heights converge.

It is easy to see, with practically the same proof, that the result is also true for point sets. Let E, E′ ⊂ D. Then

lim sup_{n→∞} h_{φ_{n}}[E, E′] ≤ h_{φ}[E, E′].

On the other hand, the lemma does not hold for boundary points r, s, even if (φ_{n}) → φ in norm. Counterexamples can readily been given using conformal mappings.

4.3. An inequality in the other direction can be shown for sequences (φ_{n}) which are bounded in norm.

Lemma 4.3. Let (φ_{n}) be a sequence of holomorphic quadratic differentials in D, with uniformly bounded norm ∥φ_{n}∥ ≤ M < ∞, for all n. Let (φ_{n}) → φ ≠ 0 locally uniformly in D. Then, for each elementary strip S : (β; σ, σ′) of φ

lim_{n→∞} h_{φ_{n}}[σ, σ′] = h_{φ}[σ, σ′] = |β|_{φ}.

Proof. After the end of Section 4.2 we only have to show that

lim inf_{n→∞} h_{φ_{n}}[σ, σ′] ≥ h_{φ}[σ, σ′].

Let z and z′ be the end points of β on σ and σ′ respectively. Let β_{n} be the vertical φ_{n} -interval, starting at z and ending at a point z′_{n} ∈ σ′. Then, by

Figure 11.

Lemma 2.3 and because β_{n} is vertical, h_{φ_{n}}[z, z′_{n}] = |β_{n}|_{φ_{n}}. On the other hand, the right hand term clearly tends to |β|_{φ} = h_{φ}[z, z′]. Since z′_{n} → z′, we have

lim_{n→∞} h_{φ_{n}}[z, z′] = lim_{n→∞} h_{φ_{n}}[z, z′_{n}] = h_{φ}[z, z′].

Choose two points ζ, ζ′ on β, near z and z′ respectively (Figure 11) and such that all the trajectories through ζ and ζ′ of φ and φ_{n} are totally regular. Denote these trajectories by α, α′, α_{n}, α′_{n} respectively. From α_{n} → α, α′_{n} → α′ we get for all sufficiently large n,

h_{φ_{n}}[σ, σ′] ≥ h_{φ_{n}}[α_{n}, α′_{n}] = h_{φ_{n}}[ζ, ζ′],

and hence

lim inf h_{φ_{n}}[σ, σ′] ≥ h_{φ}[ζ, ζ′].

Since this is true for all ζ, ζ′ the result follows.

We now proceed to the general case.

Theorem 4.3. Let φ, φ_{n} be as in Lemma 4.3. Let z, z′ ∈ D and z_{n} → z, z′_{n} → z′. Then

(1)  lim_{n→∞} h_{φ_{n}}[z_{n}, z′_{n}] = h_{φ}[z, z′].

Similarly, for two totally regular trajectories σ, σ′

(2)  lim_{n→∞} h_{φ_{n}}[σ, σ′] = h_{φ}[σ, σ′].

Proof. We first prove equation (1). Evidently Lemma 4.2 also holds for z_{n} → z, z′_{n} → z′ instead of z_{n} ≡ z and z′_{n} ≡ z′. We therefore have again

lim sup_{n→∞} h_{φ_{n}}[z_{n}, z′_{n}] ≤ h_{φ}[z, z′].

To show the reversed inequality, let, for any given ε > 0, γ̃ be a step curve connecting z and z′ as in Lemma 4.1. Let S_{j} : (β_{j}, σ_{j}, σ′_{j}) be a system of disjoint elementary strips separating z and z′ and such that

∑ |β_{j}| > h_{φ}[z, z′] − ε.

Then,

h_{φ_{n}}[z, z′] ≥ ∑_{j} h_{φ_{n}}[σ_{j}, σ′_{j}] → ∑_{j} |β_{j}| > h_{φ}[z, z′] − ε

and hence

lim inf_{n→∞} h_{φ_{n}}[z, z′] ≥ h_{φ}[z, z′].

Because of the locally uniform convergence φ_{n} → φ and the triangle inequality for heights we can replace z and z′ by z_{n} and z′_{n} respectively, which proves part one of the theorem.

To prove (2), we choose two arbitrary points z ∈ σ and z′ ∈ σ′. We have h_{φ_{n}}[σ, σ′] ≤ h_{φ_{n}}[z, z′] and h_{φ}[σ, σ′] = h_{φ}[z, z′]. Therefore

lim sup_{n→∞} h_{φ_{n}}[σ, σ′] ≤ lim_{n→∞} h_{φ_{n}}[z, z′] = h_{φ}[z, z′] = h_{φ}[σ, σ′].

On the other hand, with the same setting as above,

h_{φ_{n}}[σ, σ′] ≥ ∑_{j} h_{φ_{n}}[σ_{j}, σ′_{j}]

and hence

lim inf_{n→∞} h_{φ_{n}}[σ, σ′] ≥ lim_{n→∞} ∑_{j} h_{φ_{n}}[σ_{j}, σ′_{j}] = ∑_{j} h_{φ}[σ_{j}, σ′_{j}]

= ∑_{j} |β_{j}| > h_{φ}[z, z′] − ε = h_{φ}[σ, σ′] − ε.

This is true for every positive ε and thus proves the assertion.

Using Theorem 4.3, we can derive an inequality for pairs of boundary points.

Corollary 4.3. Let r and s be boundary points of D. Then, under the assumptions of Lemma 4.3, we have

lim inf_{n→∞} h_{φ_{n}}[r, s] ≥ h_{φ}[r, s].

Proof. If h_{φ}[r, s] = 0, there is nothing to prove. So let h_{φ}[r, s] > 0 and let σ and σ′ be two separating totally regular trajectories. Then, by Theorem 4.3

h_{φ_{n}}[r, s] ≥ h_{φ_{n}}[σ, σ′] → h_{φ}[σ, σ′],

and by Theorem 3.4 the last expression is arbitrarily close to h_{φ}[r, s].

5. Existence of the mapping

5.1. We use approximation of φ by polygon differentials to prove the existence of a differential ψ that has the same heights as φ.

Thoerem 5.1. Let φ ≠ 0 be a holomorphic quadratic differential of finite norm in the unit disk D : |z| < 1. Then, φ can be approximated in norm by a sequence of polygon differentials φ_{n} in D for which the maximal length of the sides of the polygons tends to zero.

Proof. Choose a sequence of radii r_{n} → 1. The differential φ_{n} : φ_{n}(z) = φ(r_{n}z) is holomorphic in the closed disk \overline{D}. Choose a number K > 1 and a sequence of polygons P_{j} on \overline{D} with side lenghts tending uniformly to zero. By the well known frame mapping criterion, applied to φ_{n}, the dilatation K and the polygons P_{j}, there exists a sequence of polygon differentials φ_{nj} such that

∥φ_{n} − φ_{nj}∥ → 0, j → ∞.

On the other hand, it is easy to see that

∥φ_{n} − φ∥ → 0, n → ∞.

It follows that there exists a subsequence (j_{n}) → ∞ with the property that

∥φ − φ_{nj_{n}}∥ → 0, n → ∞.

This is the desired approximating sequence; we denote it by (φ_{n}) again.

5.2. Let w = f(z) be a K -qc selfmapping of the unit disk D, and let φ ≠ 0 be a holomorphic quadratic differential of finite norm in D. Let (φ_{n}) be a sequence of polygon differentials with uniformly bounded norm which converges locally uniformly to φ. The vertices of the polygons P_{n} are denoted by ζ_{in} and it is assumed that the maximum of their side lengths tends to zero.

Let P′_{n} be the polygon with the vertices ζ′_{in} = f(ζ_{in}) and denote by ψ_{n} the image by heights of φ_{n} in the polygon P′_{n}. The totally regular trajectories of the polygon differentials are the interior trajectories of the horizontal strips, which is the same as the subintervals, in the disk, of the closed trajectories in Ĉ. In corresponding strips of the two differentials φ_{n} and ψ_{n} we define a mapping of the interior trajectories by equality of heights: if α of φ_{n} subdivides a strip S in a certain ratio, α′ of ψ_{n} subdivides S′ in the same ratio. This establishes a mapping by heights for the trajectories: corresponding pairs of trajectories have the same vertical distance.

It follows from a minimum property of quadratic differentials with closed trajectories (and hence also for polygon differentials) that

\frac{1}{K} ∥φ_{n}∥ ≤ ∥ψ_{n}∥ ≤ K ∥φ_{n}∥

(for a proof see e.g. [3]). We can now show

Theorem 5.2. The sequence (ψ_{n}) converges locally uniformly to a differential ψ ≠ 0. Two boundary points p and q of D are connected by a totally regular trajectory α of φ if and only if the image points p′ = f(p) and q′ = f(q) are connected by a totally regular trajectory α′ of ψ. Corresponding pairs of totally regular trajectories α, α̃ and α′, α̃′ have the same vertical distance (measured in terms of φ and ψ respectively).

Notice that in the proof we only use the fact that the sequence (φ_{n}) tends to φ locally uniformly and has uniformly bounded norm.

Proof. (1) Let α be a totally regular trajectory of φ in D, with end points p and q. Let p′ = f(p), q′ = f(q). Choose a totally regular trajectory α_{n} of φ_{n}, for each n, such that α_{n} → α for n → ∞. (It is enough to have a sequence of points z_{n} ∈ α_{n} tending to a point z ∈ α.) The end points p_{n},q_{n} of α_{n} tend to the end points p and q of α. Denote by α′_{n} the totally regular trajectory of ψ_{n} which has been assigned to α_{n}. By assumption, the distances of neighboring vertices of the polygons P_{n} tend to zero, and by the continuity of f on ∂D the same is true for the polygons P′_{n}. Therefore the end points p′_{n}, q′_{n} of the trajectories α′_{n} tend to p′ and q′ respectively. (Note that in general p′_{n} ≠ f(p_{n}), q′_{n} ≠ f(q_{n}); for polygon differentials this is only true at the vertices.)

We claim that every boundary point ζ′ of D which is different from p′ and q′ has an ε-neighborhood U_{ε}(ζ′) which is free from α_{n} for all n.

Figure 12.

Assume the contrary (Figure 12). Then, there is a point ζ′ and a subsequence of curves α′_{n} (which we call (α′_{n}) again, to avoid double indices) with some z′_{n} ∈ α′_{n}, z′_{n} → ζ′. We choose a second totally regular trajectory α̃, separating ζ = f^{−1}(ζ′) from α, with end points p̃ and q̃. Let α̃_{n} with end points p̃_{n} and q̃_{n} be totally regular trajectories of φ_{n}, with α̃_{n} → α̃ for n → ∞. By Theorem 4.3 (1) the vertical distance h_{φ_{n}}[α_{n}, α̃_{n}] tends to h_{φ}[α, α̃] > 0 for n → ∞. Since by definition

h_{φ_{n}}[α_{n}, α̃_{n}] = h_{ψ_{n}}[α′_{n}, α̃′_{n}],

the vertical distances of the pairs α′_{n}, α̃′_{n} are bounded away from zero. On the other hand, an application of Lemma 1.3 to the point ζ′ shows that the ψ_{n} - distance of the trajectories becomes arbitrarily small (see Figure 12, right side). This is a contradiction and thus proves the assertion.

Figure 13.

(2) The differentials ψ_{n} have uniformly bounded norm. Therefore there exists a subsequence (which we denote by (ψ_{n}) again) which converges locally uniformly to a holomorphic quadratic differential ψ. We claim that ψ ≠ 0, in other words the sequence (ψ_{n}) does not degenerate. To this end we draw a cross cut τ′ of the disk D which separates the two pairs of points p′, p̃′ and q′, q̃′ (Figure 13). We conclude from (1) that the trajectories α′_{n}, α̃′_{n} cut τ′ in a compact subinterval. If ψ_{n} → 0 locally uniformly in D, the ψ_{n} -distance

d_{ψ_{n}}[α′_{n}, α̃′_{n}] ≥ h_{ψ_{n}}[α′_{n}, α̃′_{n}]

tends to zero, a contradiction.

5.3. The next step is to show, that the points p′ and q′ are connected by a horizontal geodesic of ψ.

Lemma 5.3. Let (ψ_{n}) be a sequence of holomorphic quadratic differentials in D which tends locally uniformly to a differential ψ ≠ 0. Assume that the points z′_{n}, z′′_{n} are connected by a ψ_{n} -geodesic γ_{n} which is contained in a disk D_{r} : |z| ≤ r < 1 for all n. If z′_{n} → z′, z′′_{n} → z′′, then, z′ and z′′ are connected by a ψ -geodesic γ in D_{r} and γ_{n} → γ uniformly in the Euclidean metric (Figure 14).

Proof. The lengths of the ψ_{n} -geodesics γ_{n} are bounded, |γ_{n}|_{ψ_{n}} ≤ M, say. This is so, because the ψ_{n} are bounded in D_{r} and therefore the ψ_{n} -distance of any two points in D_{r} is bounded. Choose d > 0 such that any two points z_{1} ,

Figure 14.

z_{2} in D_{r} with ψ -distance ≤ d can be joined by a ψ -geodesic, not necessarily in D_{r} (see [1, Theorem 8.1]). Fix N and subdivide each γ_{n} into N pieces of equal ψ_{n} -length less than \frac{1}{2}d. Let z_{n0} = z′_{n}, z_{n1}, …, z_{nN} = z′′_{n} be the subdividing points. By passing to a subsequence we can assume that z_{nk} → z_{k} ∈ D_{r} for every k = 0, 1, …, N, z_{0} = z′, z_{N} = z′′. Clearly, the ψ_{n} -geodesic connection ofz_{nk} and z_{n,k+1} , which is the subinterval of γ_{n} connecting the two points, tends to the ψ -geodesic between z_{k} and z_{k+1} which is therefore in D_{r}. Moreover, the arc z_{k−1}, z_{k}, z_{k+1} is the shortest ψ -connection between z_{k−1} and z_{k+1} . Therefore, altogether, the points z′ and z′′ are connected by a ψ -geodesic γ in D_{r} and γ_{n} → γ uniformly. Because of the uniqueness of γ, the original sequence (γ_{n}) converges to γ.

Figure 15.

We are now able to show that p′ and q′ are connected by a horizontal geodesic of ψ. Fix a double sequence of circles σ_{k}, −∞ < k < ∞, centered at p′ and q′ and tending to these points for k → ±∞ respectively (Figure 15). The trajectory α′_{n} of ψ_{n}, connecting p′_{n} → p′ with q′_{n} → q′ has (for large enough n) a last intersection with σ_{k} (k < 0) and a first one with σ_{l} (l > 0). The subinterval of α′_{n} between these two points is denoted by \overline{α}_{n}. Because of (1) there is a subsequence of the sequence (\overline{α}_{n}) which tends uniformly to a ψ -geodesic between two points of σ_{k} and σ_{l} in D. Passing on to σ_{k−1} , σ_{l+1} etc. we end up with a diagonal sequence which tends uniformly in D, to a geodesic α′ of ψ. It contains sequences of points (on the σ_{k}, σ_{l}) tending to p′ and q′ respectively. Therefore it connects the two points. Since all the α′_{n} are horizontal with respect to ψ_{n}, and the sequence (ψ_{n}) tends locally uniformly to ψ, α′ itself must be a horizontal geodesic of ψ. Let α and α̃ be totally regular trajectories of φ. It follows from Theorem 4.3 (1) that the corresponding horizontal geodesics α′ and α̃′ have the same vertical distance.

Figure 16.

5.4. It is now easy to see that α′ is in fact a totally regular trajectory of ψ. Assume, first, that α′ is not regular. Then, it passes through a zero w of ψ, where at least one other trajectory γ′ starts, which can be continued as a horizontal geodesic to a boundary point r′ of D (Figure 16, right side). Of course, r′ is different from p′ and q′, and hence r = f^{−1}(r′) is different from p and q. Since α is totally regular, it is approximated by totally regular trajectories of φ. Choose such a trajectory α̃, with end points p̃ and q̃ separating r from p and q (Figure 16 left side). The points p̃′, q̃′ are connected by a horizontal geodesic α̃′. By the invariance of heights we have

h_{ψ}[α′, α̃′] = h_{φ}[α, α̃] > 0.

This is impossible, because α′ and α̃′ necessarily belong to the same component of the horizontal graph of ψ and hence have vertical distance zero. This proves that α′ is regular.

Now α is assumed to be totally regular. It can therefore be approximated, from either side, by a sequence of totally regular trajectories α_{n} of φ. Their end points p_{n} and q_{n} tend to p and q respectively. Therefore, the trajectories α′_{n} of ψ have end points p′_{n} → p′, q′_{n} → q′. But then, it is easy to see that the sequence (α′_{n}) tends itself to α′. Otherwise, there would exist a vertical interval β′ with initital point on α′, pointing to α′_{n} but disjoint from all α′_{n}. We would then have a regular trajectory α′′ ≠ α′ with end points p′ and q′, contradicting the uniqueness of geodesic connections of boundary points. Since the approximation can be performed from either side, α′ is totally regular.

Conversely: Assume that p′ and q′ are connected by a totally regular trajectory α′ of ψ. Then, p = f^{−1}(p′) and q = f^{−1}(q′) are connected by a totally regular trajectory α of φ.

To see that, we use the same approximating sequences of polygon differentials φ_{n} → φ, φ_{n} ↔ ψ_{n} ψ_{n} → ψ. We now just reverse the argument. Let (α′_{n}) be a sequence of totally regular trajectories of the differentials ψ_{n} which tends to α′. (It suffices to choose a sequence of points w_{n} → w ∈ α′ such that w_{n} lies on a totally regular trajectory α′_{n} of ψ_{n}.) Let α_{n} be the trajectory of φ_{n} which corresponds to α′_{n}. Then, the end points p_{n} and q_{n} of α_{n} tend to p and q respectively, because the end points p′_{n} and q′_{n} tend to p′ and q′ respectively. The argument is a repetition of the last part of the earlier one, showing that p and q are connected by a totally regular trajectory α of φ. Because the two points p′ and q′ can only be connected by one trajectory, α′ is the one corresponding to α. We therefore have a 1-1- correspondence of the totally regular trajectories of φ and those of ψ.

5.5. It follows readily that φ and ψ generate the same vertical distance for all corresponding pairs of boundary points. For, let r′ = f(r), s′ = f(s). If h_{φ}[r, s] = 0, there are no totally regular trajectories of φ separating r and s. Since the totally regular trajectories of φ and ψ correspond to each other, there are no totally regular trajectories of ψ separating r′ and s′. Thus, h_{φ}[r′, s′] = 0. The same argument goes in the reversed direction.

Let h_{φ}[r, s] > 0. Let σ be a totally regular trajectory separating r and s. Then, the totally regular trajectory σ′ separates r′ and s′, and conversely. Moreover, the vertical distances of corresponding pairs of totally regular trajectories are the same. Therefore, by Theorem 3.4,

h_{φ}[r, s] = sup h_{φ}[σ, τ] = sup h_{ψ}[σ′, τ′] = h_{ψ}[r′, s′],

where σ and τ are running over all totally regular trajectories of φ which separate r and s.

Theorem 5.5. Let f be a quasisymmetric mapping of ∂D onto itself. Then, to every holomorphic quadratic differential φ of finite norm corresponds a differential ψ which satisfies

h_{φ}[r, s] = h_{ψ}[r′, s′] for all r, s ∈ ∂D,

with r′ = f(r), s′ = f(s).

5.6. The following uniqueness theorem is based on the vertical distance of pairs of boundary points.

Theorem 5.6 (Uniqueness). Let φ and φ̃ be holomorphic quadratic differentials of finite norm in the disk D. Assume that the vertical distance of any pair of boundary points p, q is the same with respect to φ as with respect to φ̃. Then, φ = φ̃.

Clearly, φ = 0 if and only if all its heights are zero. For, if φ ≠ 0, it has a regular vertical trajectory β, connecting two points r and s. The vertical distance of r and s is equal to the length of β, which is

|β|_{φ} = ∫_{β} |φ(z)|^{1/2}|dz| > 0.

Let φ ≠ 0. Choose a denumerable dense set of regular horizontal trajectories α_{ν} of φ. By the vertical strip S_{ν} based on α_{ν} we mean the domain swept out by the set of vertical trajectories which intersect α_{ν}. Progressive cancelling of intersections leads to a system {S_{ν}} of non overlapping strips which cover D up to the critical points of φ (for details see [1, Theorem 19.2]).

Let β be a regular vertical trajectory of φ connecting the boundary points r and s. Then, with

w̃ = ũ + iṽ = Φ̃(z) = ∫ ^{z} \sqrt{φ̃(z)} dz

we find

∫_{β} |dṽ| ≥ hφ̃[r, s] = h_{φ}[r, s] = ∫_{β} dv,

where w = u + iv = Φ(z) = ∫^{z} \sqrt{φ(z)} dz. The strips S_{ν} are oriented in the increasing direction of v, which is well determined on each individual S_{ν}.

We now introduce the parameter w in the individual strips S_{ν}. We then get, by first integrating the above inequality over u in each S_{ν} and then summing up,

∫∫_{ΣS_{ν}} |\frac{∂ṽ}{∂v}| du dv ≥ ∫∫_{ΣS_{ν}} du dv = ∥φ∥ .

The Schwarz inequality leads to

\begin{aligned}∥φ∥^{2} & ≤ ∥φ∥ ∫∫_{ΣS_{ν}} (\frac{∂ṽ}{∂v})^{2} du dv ≤ ∥φ∥ ∫∫_{ΣS_{ν}} \{(\frac{∂ṽ}{∂u})^{2} + (\frac{∂ṽ}{∂v})^{2}\} du dv \\ & = ∥φ∥ · ∥Φ̃′∥^{2} = ∥φ∥ · \left∥{ \atop φ̃}\right∥ . \end{aligned}

This gives ∥φ∥ ≤ ∥φ̃∥, and by reversing the argument, ∥φ∥ = \left∥{ \atop φ̃}\right∥. But then, ∂ṽ/du ≡ 0, and hence ṽ = ṽ(v). From the Schwarz inequality, applied to the union of the strips S_{ν} in terms of the parameter w we find that ṽ = a · v + b. Now the equality of the norms gives a = ±1, hence Φ̃ = ±Φ+const, and finally φ̃ = φ, as claimed.

Corollary 5.6. Let f be a K -quasiconformal selfmapping of the disk D and let ψ be the induced image by heights of φ. Then,

\frac{1}{K} ∥φ∥ ≤ ∥ψ∥ ≤ K ∥φ∥ .

Proof. Let (φ_{n}) be a sequence of polygon differentials approximating φ in norm, ∥φ_{n} − φ∥ → 0. The image by heights ψ_{n} of φ_{n} satisfies

\frac{1}{K} ∥φ_{n}∥ ≤ ∥ψ_{n}∥ ≤ K ∥φ_{n}∥ .

Since ψ_{n} → ψ locally uniformly, we have

∥ψ∥ ≤ lim inf_{n→∞} ∥ψ_{n}∥ ≤ K ∥φ∥ .

Let the sequence of polygon differentials (ψ̃_{n}) approximate ψ in norm, and let φ̃_{n} be the image by heights of ψ̃_{n}. Let φ̃ be the locally uniform limit of the sequence (φ̃_{n}). It produces the same heights as ψ, and thus because of the uniqueness theorem φ = φ̃. We find, as above,

∥φ∥ = \left∥{ \atop φ̃}\right∥ ≤ K ∥ψ∥ ,

which completes the double inequality.

5.7. The definition and the uniqueness of the mapping by heights is based on the vertical distance of pairs of boundary points, whereas for the existence we use totally regular trajectories and their vertical distance. The two properties are in fact equivalent.

Theorem 5.7. Let f be a qc selfmapping of the unit disk D. Let φ and ψ be holomorphic quadratic differentials of finite norm in D. Then, the following two properties are equivalent:

(1)  h_{ψ}[r′, s′] = h_{φ}[r, s],  r′ = f(r), s′ = f(s),

for all r, s ∈ D.

(2) a) p and q ∈ ∂D are connected by a totally regular trajectory α of φ if and only if p′ = f(p) and q′ = f(q) are connected by a totally regular trajectory α′ of ψ.

b) If α, α̃ and α′, α̃′ are corresponding pairs of totally regular trajectories, then

h_{φ}[α, α′] = h_{ψ}[α′, α̃′].

We call ψ the image by heights of φ and set ψ = H_{f}(φ). H_{f} is the mapping by heights induced by the quasisymmetric mapping f | ∂D.

Proof. It has been shown in Section 5.5 that (2) implies (1).

Let (1) hold. By Theorem 5.2 we construct a quadratic differential ψ̃ which has the properties (2). Again by Section 5.5 ψ̃ satisfies (1). The Uniqueness Theorem 5.6 shows that ψ̃ = ψ. Therefore ψ has the properties (2).

5.8. Every quasisymmetric mapping of ∂D onto itself induces a mapping by heights of the space of holomorphic quadratic differentials of finite norm. We now show that conversely, if a homeomorphism f of ∂D onto itself induces a mapping by heights with a Lipschitz condition for the norm, then it is quasisymmetric.

Theorem 5.8. Let f: ∂D → D′ be a homeomorphism. Assume that there is a bijection H_{f}: φ → ψ = H_{f}(φ) of the space of holomorphic quadratic differentials of finite norm onto itself satisfying

\frac{1}{K} ∥φ∥ ≤ ∥ψ∥ ≤ K ∥φ∥

for some constant K ≥ 1. Then, f is quasisymmetric.

Figure 17.

Proof. Choose four points ζ_{1} , ζ_{2} , ζ_{3} , ζ_{4} in this order on ∂D and let Φ: z → ζ^{∗} = Φ(z) be the conformal mapping of the quadrilateral Q = (D; ζ_{1}, …, ζ_{4}) onto a rectangle R with side lengths a and b (Figure 17).

The square of the derivative of Φ, i.e., φ = (dΦ/dz)^{2} is a quadratic differential associated with the given quadrilateral Q. All its trajectories in D are totally regular, and the vertical distance of any two boundary points is the Euclidean vertical distance of the corresponding boundary points of the rectangle R. Let ψ = H_{f}(φ). All its trajectories in D′ are again totally regular. Therefore it has no zeroes and the function Ψ = ∫ \sqrt{ψ} is a conformal mapping onto a domain S shaped in Figure 17. The trajectories are the horizontal crosscuts, and the Euclidean vertical distances in R and in S are the same. They connect boundary points corresponding by f. This serves as an illustration for the mapping by heights.

However, for our present purposes, we map the quadrilateral Q′ = (D′; ζ′_{1}, …, ζ′_{4}), ζ′_{i} = f(ζ_{i}), conformally onto a rectangle R′. We double R′ by reflection on one of its vertical sides and identify the two free vertical sides of the new rectangle R′′ to form a cylinder. The quadratic differential ψ has a representation in terms of the parameter ζ of the R′ -plane. By the Dirichlet principle, applied to the cylinder, we find that

a′b′ ≤ ∥ψ∥ ≤ K ∥φ∥ = Kab.

We end up, because of b′ = b, with

\frac{a′}{b′} ≤ K \frac{a}{b},

which is the module inequality for inscribed quadrilaterals. It is well known that the inequality proves the quasisymmetry of f.

5.9. The same considerations which served to show the existence of an image by heights can be used to prove the weak convergence of these images.

Theorem 5.9. Let f be a quasiconformal selfmapping of D and let ψ = H_{f}(φ) be the image by heights of φ. Assume that the sequence (φ_{n}) tends to φ ≠ 0 locally uniformly and has uniformly bounded norm. Then, the images by heights ψ_{n} = H_{f}(φ_{n}) converge to ψ in the same sense.

Proof. (This is a repetition of the proof of Theorem 5.2) (1) From the norm inequality ∥ψ_{n}∥ ≤ K ∥φ_{n}∥ and the boundedness of the sequence (∥φ_{n}∥) we conclude that the sequence (∥ψ_{n}∥) is bounded. Therefore there exists a subsequence (ψ_{n_{i}}) which converges locally uniformly to a differential ψ̃ of finite norm.

(2) Let p and q be the end points of a totally regular trajectory α of φ. Choose, for each n, a totally regular trajectory α_{n} of φ_{n}, such that α_{n} → α in the Euclidean metric (it suffices to have z_{n} ∈ α_{n}, z_{n} → z ∈ α). Denote the end points of α_{n} by p_{n} and q_{n} respectively. We have p_{n} → p, q_{n} → q, and hence p′_{n} = f(p_{n}) → p′ = f(p), q′_{n} = f(q_{n}) → q′ = f(q). The points p′_{n}, q′_{n} are connected by a totally regular trajectory α′_{n} of ψ_{n}. Using these α′_{n} it follows as before that ψ̃ ≠ 0.

(3) In the next step we show, as in the earlier proof, that the sequence (α′_{n}) actually converges pointwise to a horizontal geodesic γ of ψ̃ with end points p′ and q′.

(4) Let α and α̃ be two totally regular trajectories of φ, with end points p, q and p̃, q̃ respectively. The points p̃′ = f(p̃), q̃′ = f(q̃) are connected by a horizontal geodesic γ̃ of ψ̃. It follows from Theorem 4.3 that

h_{φ}[α, α̃] = hψ̃[γ, γ̃].

From this we easily conclude that γ is actually a regular trajectory α̃′ of ψ̃.

(5) We then show, again as before, that α̃′ is totally regular. Theorem 3.4 then shows that the heights (vertical distances of pairs of boundary points) with respect to ψ and ψ̃ are the same, and by Theorem 5.6 ψ̃ = ψ. A standard argument then gives ψ_{n} → ψ locally uniformly for the original sequence. It seems reasonable to expect that if the convergence φ_{n} → φ is in norm, so is the convergence ψ_{n} → ψ.

6. Extremal Teichmüller mappings

6.1. A Teichmüller mapping f: D → D′ is a quasiconformal mapping with a complex dilatation of the form κ = k\overline{φ}/|φ|, where φ is a holomorphic quadratic differential and k a real constant 0 < k < 1 (we do not admit conformal mappings, where k = 0). It is well known that the mapping generates a holomorphic quadratic differential ψ in the image domain and that in the Φ- and Ψ-planes the mapping is represented by a horizontal stretching by the factor K = (1+k)/(1−k) (Figure 18).

Setting z = x + iy, w = u + iv for the variables in the Φ- and Ψ-plane respectively the mapping f has locally and away from the zeroes the respresentation

f = Ψ^{−1} ∘ F ∘ Φ, w = F (z) = Kx + iy.

The two differentials φ and ψ are called the Teichmüller differentials associated with f. One can read off several properties of f from the figure, in particular:

(1) The trajectories of φ (which are the horizontals in the Φ-plane) are taken into the trajectories of ψ. Vertical distances of pairs of trajectories stay the same.

(2) The height of a curve γ with respect to φ, which is the Euclidean height in the Φ-plane, is the same as the height of its image with respect to ψ. Therefore the vertical distances of the pairs of boundary points in the z -plane and their images in the w-plane are the same (in terms of φ and ψ respectively). This means that ψ is the image by heights of φ.

(3) The vertical trajectories of φ (which are the verticals in the Φ-plane) are taken into the vertical trajectories of ψ. The horizontal distance of corresponding pairs of vertical trajectories is multiplied by K.

(4) The norm of φ, which is the Euclidean area in the Φ-plane, is multiplied by K: ∥ψ∥ = K ∥φ∥.

It is also known that a Teichmüller mapping associated with a quadratic differential of finite norm is uniquely extremal for its boundary values.

6.2. The mapping by heights permits to formulate the existence problem for Teichmüller mappings associated with quadratic differentials of finite norm as an extremum problem for norms.

Theorem 6.2. Let f: ∂D → ∂D′ be a quasisymmetric mapping, and let H_{f} be the associated mapping by heights. Then, f allows for an extension to a Teichmüller mapping associated with the holomorphic quadratic differentials φ and ψ of finite norm if and only if the quotient ∥ψ∥ / ∥φ∥, ψ = H_{f}(φ), assumes a maximum greater than one. The differentials φ_{0} , ψ_{0} = H_{f}(φ_{0}) of the maximum are the Teichmüller differentials of the mapping and the maximum is its dilatation K_{0} .

Proof. Let

L = max \{\frac{∥ψ∥}{∥φ∥}, \frac{∥φ∥}{∥ψ∥}\} > 1

and suppose that it is assumed by φ_{0} , ∥ψ_{0}∥ = L·∥φ_{0}∥. Let β be an open, regular vertical interval with end points on totally regular trajectories α_{1} , α_{2} of φ. They connect p_{1} , q_{1} and p_{2} , q_{2} respectively on ∂D. The domain swept out by the

Figure 19.

trajectories through β is called the horizontal strip S. Let α be a variable totally regular trajectory through β.

The images α′_{1} , α′_{2} of α_{1} and α_{2} , connecting p′_{1} = f(p_{1}), q′_{1} = f(q_{1}) etc., form the upper and lower boundaries of a horizontal strip S′ in D′. We now apply the conformal mappings Φ, Ψ to the strips S, S′ respectively. Using the same notations downstairs, we arrive at Figure 19. Note that the widths of the strips S and S′ as well as the heights of α and its image α′ are the same in the two planes.

We now apply the mapping by heights to −ψ in the reversed direction, i.e. we look at φ̃ = (H_{f})^{−1}(−ψ) = H_{f^{−1}}(−ψ). We partition D into non overlapping horizontal strips S_{i} and represent φ̃ in each strip in terms of the z -parameter (for details see [1, Theorem 19.2]). We have w̃ = ũ + iṽ = Φ̃ = ∫ \sqrt{φ̃}, and the computation of ∥φ̃∥ looks as follows. We first integrate |dṽ| along α in the z - plane and remember that its heights are the same as the heights of −ψ, which are the horizontal lenghts of ψ. We get

∫_{α} |dṽ| = ∫_{α} |\frac{∂ṽ}{dx}| dx ≥ hφ̃[p, q] = h_{−ψ}[p′, q′] = ∫_{α′} du.

Since φ and ψ have the same vertical distances, we get, in S and S′, dy = dv.

Therefore integration over y in S and over v in S′ gives

∫∫_{S} |\frac{∂ṽ}{dx}| dx dy ≥ ∫∫_{S′} du dv.

A summation over all S_{i} and subsequent application of the Schwarz inequality yields

∥ψ∥ = ∫∫_{ΣS′_{i}} du dv ≤ ∫∫_{ΣS_{i}} |\frac{∂ṽ}{∂x}| dx dy,

\begin{aligned}∥ψ∥^{2} & ≤ \{ ∫∫_{ΣS_{i}} |\frac{∂ṽ}{∂x}| dx dy\}^{2} ≤ ∫∫_{ΣS_{i}} dx dy · ∫∫_{ΣS_{i}} (\frac{∂ṽ}{∂x})^{2} dx dy \\ & ≤ ∥φ∥ ∫∫_{ΣS_{i}} \{(\frac{∂ṽ}{∂x})^{2} + (\frac{∂ṽ}{∂y})^{2}\} dx dy = ∥φ∥ · ∥φ̃∥. \end{aligned}

Remember that ∥ψ∥ = L ∥φ∥ with maximal L. Therefore

L ∥ψ∥ ≤ ∥φ̃∥ ≤ L ∥−ψ∥ = L ∥ψ∥ ,

which gives ∥φ̃∥ = L ∥ψ∥. We thus must have equality all over. This implies ∂ṽ/∂y ≡ 0. Therefore the orthogonal trajectories of φ (locally the curves x =const) are the trajectories of φ̃ (locally the curves ṽ =const). Therefore φ̃ = c(−φ), with a positive constant c. From ∥φ̃∥ = L ∥ψ∥ = L^{2} ∥φ∥ we finally get φ̃ = L^{2}(−φ).

Since φ̃ = H_{f^{−1}}(−ψ), H_{f}(φ̃) = H_{f}(L^{2}(−φ)) = −ψ. The interpretation of this functional equation is as follows.

Let β be a totally regular vertical trajectory of φ, connecting the boundary points r, s. Then, ψ has a totally regular vertical trajectory β′ connecting r′ = f(r) with s′ = f(s). The horizontal distances of β_{1} and β_{2} , connecting r_{1} , s_{1} and r_{2} , s_{2} respectively, measured in terms of the differential L^{2}φ is the same as the horizontal distance of β′_{1} , β′_{2} connecting r′_{1} , s′_{1} and r′_{2} , s′_{2} respectively, measured in terms of ψ. In other words, measuring in terms of φ and ψ, the horizontal distance of corresponding pairs of totally regular vertical trajectories is multiplied by L.

So far we only have a mapping of trajectories and orthogonal trajectories. We now define a mapping of points. Let z ∈ D be the intersection of a totally regular trajectory α of φ connecting p and q with a totally regular vertical trajectory β of φ connecting r and s. Then, their images α′ and β′ cut at a point w, because their pairs of end points separate each other. This is a bijection of a dense set of points of D onto a dense set of points on D′. We are actually right back at Figure 18: the completion of the mapping can be carried out in the Φand Ψ-planes, as horizontal stretching by the factor L of small rectangles with sides parallel to the axes. We end up with an L-quasiconformal mapping of the disk D punctured at the zeroes of φ onto D′ punctured at the zeroes of ψ. It can therefore be extended to these zeroes and to the boundary. Because the end points of the totally regular horizontal and vertical trajectories are everywhere dense on ∂D, the boundary values are the given ones, and we have found an L-qc extension of the quasisymmetric boundary mapping which is a Teichmüller mapping associated with the differentials φ and ψ = H_{f}(φ). Such a mapping is known to be uniquely extremal, and so is φ, if we normalize it by ∥φ∥ = 1. In correspondence with the usual notation we set L = K_{0} .

We have the horizontal stretching version of the Teichmüller mapping from left to right. If the maximum were taken by the quotient ∥φ∥ / ∥ψ∥ = L, we would have a horizontal stretching from the right to the left. Replacing φ by φ̃ = (−φ)/L^{2} , ψ by ψ̃ = −ψ, we have again ∥ψ̃∥/∥φ̃∥ = L, so we are back at the former case.

The converse is immediate. Let f be a Teichmüller mapping associated with the differentials φ in D and ψ in D′, of finite norm, in the horizontal stretching version. Then, ψ = H_{f}(φ), ∥ψ∥ = K_{0} ∥φ∥, with K_{0} the extremal dilatation. Since the mapping by heights satisfies the double inequality

\frac{1}{K_{0}} ∥φ∥ ≤ ∥ψ∥ ≤ K_{0} ∥φ∥ ,

φ gives the maximal value of ∥ψ∥ / ∥φ∥.

References

[1] Strebel, K.: Quadratic differentials. - Ergeb. Math. Grenzgeb. (3) 5, Springer-Verlag, 1984, 1–184.

[2] Strebel, K.: On the geometry of quadratic differentials in the disk. - Results in Mathematics (to appear).

[3] Marden, A., and K. Strebel: A characterization of Teichmüller differentials. - J. Differential Geom. (to appear).

Received 28 April 1992