Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 475–494

POINT SHIFT DIFFERENTIALS AND EXTREMAL QUASICONFORMAL MAPPINGS

Kurt Strebel

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

Abstract. We consider quasiconformal selfmappings f of the disk with given boundary values. For fixed z_{0} the set of image points {f_{0}(z_{0})} for all extremal mappings f_{0} is called the variability set V [z_{0}] of z_{0} . Extremal mappings f_{m} which take z_{0} into points w_{m} ∉ V [z_{0}] are called point shift mappings. They are Teichmüller mappings associated with quadratic differentials φ_{m} of norm ∥φ_{m}∥ = 1, called point shift differentials. It is shown that the φ_{m} form Hamilton sequences for the extremal mappings which take z_{0} into boundary points of V [z_{0}]. From that it follows, using the frame mapping criterion, that the φ_{m} depend continuously in norm of the point w_{m}. Their constant dilatation K[w_{m}] is called the dilatation function. Based on a variational method for point shift mappings, it is shown that the level lines of K[w_{m}] are Jordan curves separating V [z_{0}] from ∂D_{w}. Thus, V [z_{0}] is a compact, connected set without holes.

1. Introduction

1. A holomorphic quadratic differential φ(z) dz^{2} induces a conformally invariant metric |φ(z)|^{1/2} |dz|. It is Euclidean in the neighborhood of the non critical points of φ (the critical points are the zeroes and poles): if we apply the conformal mapping

(1)  ζ = Φ(z) = ∫ ^{z} \sqrt{φ(z)} dz

we find dζ^{2} = φ(z) dz^{2} , which is the Euclidean line element in the ζ -plane. This metric and its geodesics (for details see [12]) lead to certain inequalities which constitute a fundamental tool in the theory of extremal quasiconformal mappings.

2. The first of these inequalities is the so-called Main Inequality (see [6] for the disk, and [11] for arbitrary Riemann surfaces).

Let w = f(z) and w̃ = f̃(z) be quasiconformal mappings of the unit disk D_{z} onto D_{w} which agree on the boundary of D_{z}. Let dw = p(z) dz + q(z) dz̄ be the differential of f, κ = q/p its complex dilatation and K its maximal dilatation. The inverse of f̃ is denoted by f_{1} = f̃^{−1} , its complex dilatation by κ_{1} . Of course, its maximal dilatation K_{1} is equal to the maximal dilatation K̃ of f̃. The combination f_{1} ∘ f = f̃^{−1} ∘ f is a qc selfmapping of D_{z} which is equal to the identity on ∂D_{z}.

1991 Mathematics Subject Classification: Primary 30C70; Secondary 30C75.

Let now φ ≢ 0 be any holomorphic quadratic differential in D_{z} of norm ∥φ∥ = ∫∫ |φ(z)| dx dy = 1. It serves as a test quantity. Let α be any non critical trajectory of φ. It has two different end points on ∂D_{z}, and the curve α̃ = f_{1}∘f(α) has the same end points as α. Since α is a geodesic, we have the length inequality, in terms of the φ-metric, |α̃|_{φ} ≥ |α|_{φ}. Using the conformal parameter Φ along the trajectories of φ this length inequality can be integrated over the whole disk D_{z}, which leads to the “Main Inequality”

(2)  1 ≤ ∫∫ |φ(z)|\frac{|1 − κ(φ/|φ|)|^{2}}{1 − |κ|^{2}} \{⋯\} dx dy,

with

(3)  \{⋯\} = \frac{|1 − κ_{1} \frac{\overline{p}}{p} \frac{φ}{|φ|} \frac{1 − \overline{κ}(\overline{φ}/|φ|)}{1 − κ(φ/|φ|)}|^{2}}{1 − |κ_{1}|^{2}}

(see [6], ineq. 1.5.1, and [11] for Riemann surfaces; the method first appears in [8].)

The term (3) in brackets has the upper bound K_{1} = K̃. If we set f̃ = f_{0} , where f_{0} is an extremal quasiconformal mapping for the given boundary values, i.e. one with smallest maximal dilatation K_{0} , we have {⋯} ≤ K_{0} . This turns (2) into the simpler form

(4)  \frac{1}{K_{0}} ≤ ∫∫ |φ(z)|\frac{|1 − κ(φ/|φ|)|^{2}}{1 − |κ|^{2}} dx dy,

which we call the reduced form of the Main Inequality. Its contents is the following: Let f be a qc selfmapping of D and let K_{0} be the smallest maximal dilatation in the class of all qc selfmappings of D with the boundary values of f. Then, for every holomorphic quadratic differential φ of norm one in D we have (4). (See [6], and [11] for Riemann surfaces.)

3. The holomorphic quadratic differentials of finite norm give rise to a very interesting necessary and sufficient condition for the complex dilatation κ of a qc mapping to be extremal. It reads as follows: Let κ be the complex dilatation of a qc mapping f. Then, f is extremal for its boundary values (and in its homotopy class) if and only if

(5)  sup_{∥φ∥=1} Re ∫∫ κφ dx dy = ∥κ∥_{∞}.

The necessity was shown by R.S. Hamilton [1] and simultaneously (for complex dilatations of constant absolute value) by S. Krushkal [2]. The sufficiency is a consequence of the reduced form (4) of the main inequality and was shown, for selfmappings of the disk, by E. Reich and K. Strebel [6] and, for the general case, by K. Strebel [11].

A sequence of holomorphic quadratic differentials φ_{n} of norm one such that

(6)  Re ∫∫ κφ_{n} dx dy → ∥κ∥_{∞}

is called a Hamilton sequence. If there exists such a sequence which converges to a quadratic differential φ in the sense that ∥φ_{n} − φ∥ → 0, it is easily seen that

(7)  κ = k \frac{\overline{φ}}{|φ|}.

This is the complex dilatation of a Teichmüller mapping. The crucial situation is when there is no such sequence, i.e. every Hamilton sequence for κ converges locally uniformly to zero (what we call a “degenerating Hamilton sequence”).

The Main Inequality provides a criterion to the effect that there is no degenerating Hamilton sequence for κ. It is called the Frame Mapping Criterion ([9], [10]) and reads as follows (we only formulate it for the disk here, but it is true in full generality): Let K_{0} be the smallest maximal dilatation for all qc selfmappings of D with a given quasisymmetric boundary homeomorphism h. Assume that h can be continued quasiconformally into a neighborhood of ∂D with a maximal dilatation H < K_{0} . Then, κ does not admit a degenerating Hamilton sequence and thus has the Teichmüller form κ = k\overline{φ}/|φ|, ∥φ∥ = 1. Moreover, every Hamilton sequence (φ_{n}) for κ tends in norm to φ ([10], Theorem 1). According to an earlier theorem (originally in [7], with a new proof in [6]) the mapping f is then uniquely extremal.

4. The existence of a Hamilton sequence (φ_{n}) for every extremal κ was proved by R. Hamilton in an abstract way. It is however possible, for the disk, to give Hamilton sequences in a more concrete manner, namely as differentials of certain Teichmüller mappings.

For a given extremal mapping f with complex dilatation κ pick a finite set of boundary points ζ_{i}, i = 1, …, n. The disk D together with these points is called a polygon Π_{n} with vertices ζ_{i}. According to Teichmüller’s theorem there is a unique extremal selfmapping of D which takes the vertices ζ_{i} into their images by f, ζ′_{i} = f(ζ_{i}). It is a Teichmüller mapping associated with a holomorphic quadratic differential φ, ∥φ∥ = 1, which is real along the sides of the polygon, i.e. φ(z) dz^{2} real for tangential dz between the vertices. (The vertices themselves are at worst first order poles.)

Call the φ “polygon differentials”. One can show that, if the vertices of the polygons Π_{n} become more and more dense, the associated polygon differentials form a Hamilton sequence for κ ([6], Theorem 6).

5. The basis of this article is a new kind of Hamilton sequences, namely point shift differentials. They were introduced in ([10], Theorem 7). Pick a point z_{0} ∈ D. Consider again the qc selfmappings f of D with boundary values h. The set of image points of z_{0} by the extremal mappings f_{0} is called the set of variability of z_{0} and denoted by V [z_{0}] (see [10], p. 479). Of course, if h has a unique extremal solution, V [z_{0}] consists of a single point for every z_{0} ∈ D. In general it is a compact subset of D. Choose w_{n} ∉ V [z_{0}]. Then, the extremal qc mapping f_{n} with boundary values h on ∂D and taking z_{0} into w_{n} is a Teichmüller mapping associated with a quadratic differential φ_{n} of norm one which has a first order pole at z_{0} . (This is an immediate consequence of the frame mapping criterion.) φ_{n} is called a point shift differential, f_{n} a point shift mapping.

Let now w_{0} be a boundary point of V [z_{0}], and let f_{0}(z_{0}) = w_{0} . It will be shown that for w_{n} → w_{0} the point shift differentials φ_{n} form a Hamilton sequence for the complex dilatation κ_{0} of f_{0} .

It should be noticed that the polygon differentials provide Hamilton sequences for every extremal mapping, i.e. also for extremal mappings which take z_{0} into an interior point of V [z_{0}]. This is not so for point shift differentials, because they only exist if the shifted point w_{n} is outside V [z_{0}]. In the case of unique extremality this is of course no restriction. The obvious advantage of the point shift differentials is the fact that they only depend on one complex parameter, namely the shifted point w_{n}. On the other hand the polygon differentials depend on an increasing number of parameters, namely the vertices of the polygons. This makes them geometrically extremely hard to control.

Clearly, point shift differentials exist in the most general case, in the same generality as the frame mapping criterion.

6. The simplest but quite formal definition of a Teichmüller mapping is by its complex dilatation, which is κ = k\overline{φ}/|φ|, 0 < k < 1, where φ is a meromorphic quadratic differential. For our purposes it is however necessary to express this in geometric terms. It is done here for the disk (see [8]), but most generally true for arbitrary Riemann surfaces.

Figure 1.

Let f: D_{z} → D_{w} be a qc mapping with complex dilatation (7). Pick a neighborhood U ⊂ D_{z} such that φ is holomorphic in U without zeroes (Figure 1). Let

(8)  ζ = ξ + iη = Φ(z) = ∫ ^{z} \sqrt{φ(z)} dz

in U. By restriction of U, if necessary, we may assume that Φ is schlicht in U. Set U^{∗} = Φ(U) and let F_{K} be the horizontal stretching of U^{∗} by K = (1+k)/(1−k). We have

(9) ζ^{∗} = F_{K}(ζ) = Kξ + iη = \frac{1}{2}(K + 1)ζ + \frac{1}{2}(K − 1)ζ̄ = \frac{1}{2}(K + 1)(Φ(z) + k\overline{Φ}(z)).

From this expression follows that the complex dilatation of the composition F_{K} ∘Φ is given by (7). Therefore the mappings f and F_{K} ∘ Φ are related by a conformal mapping Ψ: V = f(U) → V^{∗} = F_{K}(U^{∗}) such that

(10)  Ψ ∘ f = F_{K} ∘ Φ.

It is easy to see that ψ = Ψ^{′2} is single valued in D_{w}. It is the well defined quadratic differential which is associated to φ. We realize that the horizontal and the vertical trajectories of φ through a point z ∈ U go into the horizontal and vertical trajectories of ψ through w = f(z) respectively. Horizontal lengths are stretched by K, vertical lengths are preserved.

We will call φ and ψ the quadratic differentials associated with the Teichmüller mapping f. Clearly, if ∥φ∥ = 1, ∥ψ∥ = K.

II Point shift differentials

7. Let T [w_{1}, w_{2}] be the extremal qc selfmapping of the unit disk D which takes w_{1} into w_{2} and is equal to the identity on ∂D. The Frame Mapping Criterion assures that it is a Teichmüller mapping associated with a pair of quadratic differentials φ and ψ of finite norm. φ has a first order pole at w_{1} , ψ has one at w_{2} , and if we set ∥φ∥ = 1 and use the horizontal stretching version for Teichmüller mappings (as we always do) we have ∥ψ∥ = K_{T} where K_{T} is the dilatation of T. The mapping T is conjugate, by a Möbius transformation of the disk, to the mapping in Teichmüller’s “Verschiebungssatz”, where w_{1} = 0 and w_{2} = −ϱ. It is not hard to see that K_{T} → 1 for w_{2} → w_{1} . Because of this connection we call T [w_{1}, w_{2}] the Teichmüller shift.

8. Denote the unit disks of the z - and the w-plane by D_{z} and D_{w} respectively. Let h be a fixed quasisymmetric mapping of ∂D_{z} onto ∂D_{w}. Choose a fixed point z_{0} ∈ D_{z} and, for w ∈ D_{w}, let f be an extremal qc mapping of D_{z} onto D_{w} which is equal to h on ∂D_{z} and takes z_{0} into w. Let K be its maximal dilatation.

Theorem 1. The maximal dilatation K is, for fixed z_{0} , a continuous function of w which tends to infinity for |w| → 1. It is called the dilatation function and denoted by K[w].

Proof. To prove the first statement, choose w̃ ≠ w and let K̃ be the maximal dilatation of an extremal mapping f̃, f̃(z_{0}) = w̃, f̃ | ∂D_{z} = h. The composition T [w, w̃]∘f takes z_{0} into w̃ and has therefore a maximal dilatation which is at least equal to K̃. On the other hand it is smaller or equal to K_{T} ·K, thus K_{T} ·K ≥ K̃, which gives

\overline{lim}_{_{w}→w} K̃ ≤ K.

On the other hand, since f is extremal,

\underline{lim}_{_{w}→w} K̃ ≥ K,

which gives

lim_{_{w}→w} K̃ = K,

as claimed.

To prove the second statement, let w_{n} → ω ∈ ∂D_{w}. Let ζ = h^{−1}(ω) and choose two disjoint closed intervals δ_{1} and δ_{2} on ∂D_{z} none of which contains the point ζ. The set of cross cuts γ joining δ_{1} to δ_{2} in D_{z} and separating z_{0} from ζ goes over by f_{n} into a set of curves γ′ connecting δ′_{1} = h(δ_{1}) and δ′_{2} = h(δ_{2}) and separating w_{n} from ω. Here, f_{n} is an extremal mapping, with maximal dilatation K_{n}, taking z_{0} into w_{n}. We have the extremal length inequality λ{γ′} ≤ K_{n}λ{γ}, and since the left hand side evidently tends to infinity, we find K_{n} → ∞, as claimed.

Corollary. The set of variability of z_{0} ,

V [z_{0}] = {w = f(z_{0}) ; f extremal for the boundary values h}

is compact.

This is so because all the extremal mappings f_{0} for all points w_{0} ∈ V [z_{0}] have the same maximal dilatation K_{0} . Therefore V [z_{0}] is bounded away from ∂D_{w}, and because of the continuity of the maximal dilatation it contains all its accumulation points.

9. Let w_{n} be a point of D_{w} outside the variablity set V [z_{0}]. Then, any extremal qc mapping f_{n} of D_{z} onto D_{w} with the boundary values h and f_{n}(z_{0}) = w_{n} has a maximal dilatation K_{n} > K_{0} , K_{0} being the smallest maximal dilatation without the additional condition that z_{0} → w_{n}. We can therefore use any extremal mapping f_{0} near ∂D_{z} as a frame mapping (near z_{0} we just use the translation z_{0} → w_{n}, which is of course conformal). The frame mapping condition applied to the punctured disks tells us that f_{n} is a Teichmüller mapping associated with a quadratic differential φ_{n} of finite norm (∥φ_{n}∥ = 1, say). φ_{n} has a first order pole at z_{0} and is otherwise holomorphic (without the pole, f_{n} would be uniquely extremal for its boundary values f_{n} | ∂D_{z} = h, see [8] and [6]). We call f_{n} a point shift mapping and φ_{n} the associated point shift differential (see [10], p. 479).

The next result is basic for the whole paper.

10.

Theorem 2. Let w_{0} be a boundary point of the variablity set V [z_{0}], and let f_{0} , with complex dilatation κ_{0} and maximal dilatation K_{0} be an extremal qc mapping which takes z_{0} into w_{0} (for brevity, we call such a mapping a “boundary element”, see also [5], p. 290). Let w_{n} ∈ D_{w} \ V [z_{0}], and let f_{n}, φ_{n} and K_{n} be as in Section 9. Then, for w_{n} → w_{0} the sequence (φ_{n}) is a Hamilton sequence for the complex dilatation κ_{0} of f_{0} , i.e. Re ∫∫ κ_{0}φ_{n} dx dy → k_{0} = ∥κ_{0}∥_{∞}.

Proof. Let T_{n} ≡ T [w_{0}, w_{n}] be the Teichmüller shift which takes w_{0} into w_{n}. Let f̃_{n} = T_{n} ∘ f_{0} . This is a qc mapping of D_{z} onto D_{w}, with the same boundary values as f_{n}, sending z_{0} into w_{n}, and with a maximal dilatation K̃_{n} which tends to K_{0} for w_{n} → w_{0} .

Consider a regular trajectory α_{n} of φ_{n}, i.e. one which does not go to a zero or the simple pole of φ_{n}. It connects two different boundary points of the disk D_{z}. The image α′_{n} of α_{n} by f_{n} is a well defined regular trajectory of the image differential ψ_{n} connecting the images of the end points of α_{n}. Let α̃_{n} be the image of α_{n} by f̃_{n}; it has the same end points as α′_{n}. Since f_{n}(z_{0}) = f̃_{n}(z_{0}) = w_{n}, the two mappings f_{n} and f̃_{n} are homotopic in D_{w} \ {w_{n}} modulo the boundary. Therefore α_{n} and α̃_{n} are in the same homotopy class in D_{w} \ {w_{n}} and thus satisfy the length inequality with respect to the ψ_{n} -metric, namely

(11)  |α̃_{n}|_{ψ_{n}} ≥ |α′_{n}|_{ψ_{n}} = K_{n}|α_{n}|_{φ_{n}}.

This inequality can now be integrated over the disk D_{z} (see [8], or [6] p. 384–385 for polygon differentials).

The image of α_{n} by ζ = Φ_{n}(z) is an open horizontal interval α_{n}^{∗} in the ζ = ξ + iη -plane. We have, with w̃_{n} = f̃_{n}(z),

(12)  K_{n}|α_{n}|_{φ_{n}} = K_{n} ∫_{α_{n}^{∗}} dξ ≤ ∫_{α_{n}} |dΨ_{n}(w̃_{n})| = ∫_{α_{n}} |\frac{dΨ_{n}(w̃_{n})}{dw̃_{n}} | |dw̃_{n}|.

With the notations dw̃_{n} = p̃_{n}(z) dz + q̃_{n}(z) dz̄, dζ = Φ′_{n}(z) dz, we find along α_{n}^{∗}

(13) dw̃_{n} = \{p̃_{n}(z)\frac{1}{Φ′_{n}(z)} +q̃_{n}(z)\frac{1}{\overline{Φ′_{n}}(z)}\} dξ = \frac{1}{Φ′_{n}(z)}\{p̃_{n}(z)+q̃_{n}(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}\} dξ.

We put this expression into the equation (12) and integrate it over a short vertical interval in the ζ = ξ + iη -plane. The domain of integration is a (Euclidean) horizontal strip Σ in the ζ -plane. Going back to the z -plane with dξ dη = |Φ′_{n}(z)|^{2} dx dy = |φ_{n}(z)| dx dy the domain of integration is a strip S swept out by horizontal trajectories of φ_{n}. Finally, summing up over a denumerable set of such strips forming an exhaustion of D_{z} (up to a set of measure zero) we get

(14) K_{n}∥φ_{n}∥ ≤ ∫∫_{|z|<1} |ψ_{n}(w̃_{n})|^{1/2}|φ_{n}(z)|^{1/2}|p̃_{n}(z) + q̃_{n}(z)\frac{φ_{n}(z)}{|φ_{n}(z)|}| dx dy.

Before we apply the Schwarz inequality to this expression we insert the square root of the functional determinant

(15)  J(w̃_{n} | z) = |p̃_{n}(z)|^{2} − |q̃_{n}(z)|^{2}

and thus have

(16) K_{n}∥φ_{n}∥ ≤ ∫∫_{|z|<1} |ψ_{n}(w̃_{n})|^{1/2}J(w̃_{n} | z)^{1/2} \frac{|φ_{n}|^{1/2}|p̃_{n} + q̃_{n}(φ_{n}/|φ_{n}|)|}{J(w̃_{n} | z)^{1/2}} dx dy.

Finally, the Schwarz inequality yields (17)

K_{n}^{2}∥φ_{n}∥^{2} ≤ ∫∫_{|z|<1} |ψ_{n}(w̃_{n})|J(w̃_{n} | z) dx dy·∫∫_{|z|<1} |φ_{n}|\frac{|1 + κ̃_{n}(φ_{n}/|φ_{n}|)|^{2}}{1 − |κ̃_{n}|^{2}} dx dy,

with κ̃_{n} = q̃_{n}(z)/p̃_{n}(z) the complex dilatation of f̃_{n}.

Taking into account that

(18)  ∫∫_{|z|<1} |ψ_{n}(w̃_{n})|J(w̃_{n} | z) dx dy = ∥ψ_{n}∥ = K_{n}∥φ_{n}∥

and normalizing ∥φ_{n}∥ = 1 we get

(19)  K_{n} ≤ ∫∫_{|z|<1} |φ_{n}(z)|\frac{|1 + κ̃_{n}(z)(φ_{n}(z)/|φ_{n}(z)|)|^{2}}{1 − |κ̃_{n}(z)|^{2}} dx dy.

We call this the “point shift inequality”. It resembles very much the “polygon inequality” (see [6], p. 385).

11. To evaluate the point shift inequality, we write

(20)  |1 + κ̃_{n} \frac{φ_{n}}{|φ_{n}|}|^{2} = (1 + |κ̃_{n}|)^{2} − 2\{|κ̃_{n}| − Re κ̃_{n} \frac{φ_{n}}{|φ_{n}|}\}.

Since

(21)  \frac{1 + |κ̃_{n}|}{1 − |κ̃_{n}|} ≤ \frac{1 + k̃_{n}}{1 − k̃_{n}} = K̃_{n}  a.e.

the point shift inequality (19) gives

(22) \begin{aligned}K_{n} & ≤ ∫∫_{|z|<1} |φ_{n}(z)|\frac{1 + |κ̃_{n}|}{1 − |κ̃_{n}|} dx dy − 2 ∫∫_{|z|<1} \frac{|κ̃_{n}| |φ_{n}| − Re κ̃_{n}φ_{n}}{1 − |κ̃_{n}|^{2}} dx dy \\ & ≤ ∫∫_{|z|<1} |φ_{n}|\frac{1 + |κ̃_{n}|}{1 − |κ̃_{n}|} dx dy ≤ K̃_{n}. \end{aligned}

Let now w_{n} → w_{0} ∈ ∂V [z_{0}]. Then, because of the continuity of K[w_{n}] and since K[w_{0}] = K_{0} we have K_{n} → K_{0} . On the other hand, we have K̃_{n} ≤ K_{T_{n}} · K_{0} and thus

\overline{lim}_{w_{n}→w_{0}} K̃_{n} ≤ K_{0}.

From (22) we conclude that K̃_{n} → K_{0} for w_{n} → w_{0} .

Another use of (22) shows that for w_{n} → w_{0}

(23)  ∫∫_{|z|<1} |φ_{n}(z)|\frac{1 + |κ̃_{n}|}{1 − |κ̃_{n}|} dx dy → K_{0}

and

(24)  ∫∫_{|z|<1} \frac{|κ̃_{n}| |φ_{n}| − Re κ̃_{n}φ_{n}}{1 − |κ̃_{n}|^{2}} dx dy → 0.

Therefore necessarily

(25)  ∫∫_{|z|<1}{|κ̃_{n}| |φ_{n}| − Re κ̃_{n}φ_{n}} dx dy → 0.

The quantity κ̃_{n} is the complex dilatation of the composition f̃_{n} = T_{n} ∘ f_{0} , with T_{n} = T [w_{0}, w_{n}]. Denote the complex dilatation of T_{n} by κ_{T_{n}}; since it is a Teichmüller mapping, its absolute value is constant, |κ_{T_{n}}| = k_{T_{n}}. The formula for the complex dilatation of the composition gives

(26)  κ̃_{n} = \frac{κ_{0} + κ_{T_{n}}τ_{0}}{1 + κ_{T_{n}} \overline{κ}_{0}τ_{0}} ,  τ_{0} = \frac{\overline{p}_{0}}{p_{0}} ,

(the differential of the extremal mapping w = f_{0}(z) is dw = p_{0}(z) dz + q_{0}(z) dz̄). From this is follows that

(27)  κ̃_{n} − κ_{0} = κ_{T_{n}} \frac{τ_{0}\{1 − |κ_{0}|^{2}\}}{1 + κ_{T_{n}} \overline{κ}_{0}τ_{0}}.

Since |κ_{T_{n}}(z)| ≡ k_{T_{n}} → 0, |κ_{0}(z)| ≤ k_{0} < 1 and |τ_{0}| = 1 we conclude that

(28)  | |κ̃_{n}| − |κ_{0}|| ≤ |κ̃_{n} − κ_{0}| ≤ k_{T_{n}} \frac{1}{1 − k_{T_{n}}k_{0}} → 0

uniformly in z with n → ∞. We can therefore replace κ̃_{n} by κ_{0} in relations (23) and (25) and get

(23′)  ∫∫_{|z|<1} |φ_{n}(z)|\frac{1 + |κ_{0}(z)|}{1 − |κ_{0}(z)|} dx dy → K_{0}

and

(25′)  ∫∫_{|z|<1}{|κ_{0}(z)| |φ_{n}(z)| − Re κ_{0}(z)φ_{n}(z)} dx dy → 0

for w_{n} → w_{0} .

If |κ_{0}(z)| = k_{0} a.e., relation (25′) gives

(29)  Re ∫∫ κ_{0}(z)φ_{n}(z) dx dy → k_{0},

which is the Hamilton relation (6) for κ_{0} .

If, however, |κ_{0}(z)| is not equal to k_{0} a.e., it follows from relation (23′) that ∫∫_{E} |φ_{n}(z)| dx dy → 0 on every measurable set E on which |κ_{0}(z)| ≤ k_{0} − ε, for every ε > 0. Since the φ_{n} are analytic and the sequence (φ_{n}) is locally bounded in D_{z}, it must be degenerating. Writing

(30)  ∫∫_{|z|<1} |κ_{0}(z)| |φ_{n}(z)| dx dy = ∫∫_{E} + ∫∫_{D_{z}\E}

and considering that on D_{z} \ E we have k_{0} − ε < |κ_{0}(z)| ≤ k_{0} we get

∫∫_{|z|<1} |κ_{0}(z)| |φ_{n}(z)| dx dy → k_{0}

as before. (25′) thus leads to (29) again, which proves the Theorem.

12. Summing up, we have considered the following situation. The qc mapping f_{0} has the boundary values h on ∂D_{z} and takes z_{0} into w_{0} . Its maximal dilatation is K_{0} , and it is extremal for the above data. Moreover there is a sequence of points w_{n} ∈ D_{z}, w_{n} → w_{0} , such that the corresponding extremal mappings f_{n}, f_{n}(z_{0}) = w_{n}, are Teichmüller mappings associated with quadratic differentials φ_{n} of norm one. The φ_{n} are defined and holomorphic in D_{z} \ {z_{0}}. They have a first order pole at z_{0} , because otherwise the mapping f_{n} would be uniquely extremal for its boundary values on ∂D_{z}, without any further condition. Then, we have shown that these φ_{n} form a Hamilton sequence for the complex dilatation κ_{0} of f_{0} .

Let us now consider a point w_{m} in D_{w} outside the variability set V [z_{0}]. The mapping f_{m}: z_{0} → w_{m}, with boundary values h on ∂D_{z} is a uniquely extremal Teichmüller mapping with a (constant) dilatation K_{m} > K_{0} . Let w_{n} → w_{m}. Then, the mappings f_{n}, f_{n}(z_{0}) = w_{n}, tend to f_{m} uniformly in D_{z} (because f_{m} is uniquely extremal) and the associated normalized quadratic differentials φ_{n}, all defined and holomorphic in D_{z} \ {z_{0}}, form a Hamilton sequence for the complex dilatation κ_{m} = k_{m}(\overline{φ}_{m}/|φ_{m}|) of f_{m}. But now the frame mapping criterion says: there is no degenerating Hamilton sequence for κ_{m} (as a frame mapping we can use any extremal mapping f_{0} near ∂D_{z} and the translation z → w_{m} + z − z_{0} near z_{0} , since K_{m} > K_{0} ). Therefore the φ_{n} tend to φ_{m} in norm. We thus have proved the

Theorem 3. Let w_{m} ∈ D_{z} \ V [z_{0}], and let w_{n} → w_{m}. Then, the normalized point shift differentials φ_{n} tend in norm to the Teichmüller differential φ_{m} of f_{m}, i.e. ∥φ_{n} − φ_{m}∥ → 0.

13. Actually, every Teichmüller mapping is associated with a pair of quadratic differentials, φ on the original domain and ψ on the image domain, and the mapping sends the trajectories and orthogonal trajectories of φ into the trajectories and orthogonal trajectories respectively of ψ. We have seen, that, for w_{n} → w_{m}, φ_{n} → φ_{m} in norm, and hence, because of analyticity, locally uniformly; we will need the analogue for the ψ_{n}.

To fix the ideas, take, for a given w_{m}, a closed, regular φ_{m} -rectangle S_{m} and the corresponding ψ_{m} -rectangle S′_{m}. S_{m} is bounded by two opposite horizontal and two opposite vertical intervals, and there are no zeroes of φ_{m} on S_{m}. The same holds for S′_{m} with respect to ψ_{m}. ζ = Φ_{m}(z), ζ^{∗} = Ψ_{m}(w) take S_{m} and S′_{m} into horizontal Euclidean rectangles R_{m}, R′_{m} respectively. By a proper choice of the additive constants we can achieve that ζ = ζ^{∗} = 0 in the lower left corners of the rectangles. We express everything in terms of these parameters ζ and ζ^{∗} respectively. Since dζ^{2} = φ_{m}(z) dz^{2} , we have φ_{m}(ζ) ≡ 1, and analogously ψ_{m}(ζ^{∗}) ≡ 1. Moreover, Φ_{m}(ζ) ≡ ζ, Ψ_{m}(ζ^{∗}) ≡ ζ^{∗}, and the Teichmüller mapping f_{m} is

ζ^{∗} = f_{m}(ζ) = K_{m}ξ + iη ≡ F_{K_{m}}(ζ), ζ = ξ + iη.

F_{K_{m}} denotes the horizontal stretching of R_{m} onto R′_{m} by K_{m}.

We have seen, in the preceding section, that, with w_{n} → w_{m} and the normalization ∥φ_{n}∥ = ∥φ_{m}∥ = 1, we have ∥φ_{n} − φ_{m}∥ → 0. In terms of the parameter ζ we therefore have φ_{n}(ζ) → 1 uniformly in R_{m}. Therefore Φ_{n}(ζ) → ζ, also uniformly in R_{m}, and clearly f_{n}(ζ) → f_{m}(ζ), also uniformly.

The mapping f_{n} has the representation, locally

f_{n} = Ψ_{n}^{−1} ∘ F_{K_{n}} ∘ Φ_{n},

and thus

Ψ_{n}(ζ^{∗}) = F_{K_{n}}(Φ_{n}(f_{n}^{−1}(ζ^{∗}))).

Taking into account that, with w_{n} → w_{m}, f_{n}^{−1}(ζ^{∗}) → ζ, Φ_{n}(ζ) → ζ, K_{n} → K_{m}, hence F_{K_{n}}(ζ) → ζ^{∗} uniformly, we find

Ψ_{n}(ζ^{∗}) → ζ^{∗}

uniformly in R′_{m}. This gives, because of the analyticity, ψ_{n}(ζ^{∗}) → 1 uniformly in R′_{m}.

The natural normalization of the ψ -differentials is, according to the Euclidean geometry of the Ψ-planes and the horizontal stretching by the dilatation K, ∥ψ_{m}∥ = K_{m}, ∥ψ_{n}∥ = K_{n}. With this we have, in any finite collection B = ⋃_{j} S_{m}^{j} of ψ_{m} -rectangles,

∥ψ_{n} − ψ_{m}∥_{B} → 0.

On the other hand we can clearly approximate D_{w}, for any positive ε, by a finite system B of non-overlapping ψ_{m} rectangles, such that ∥ψ_{m}∥_{D_{w}\B} < ε and ∥ψ_{n}∥_{D_{w}\B} < ε for all w_{n} which are lying sufficiently close to w_{m}. This gives the result ∥ψ_{n} − ψ_{m}∥ → 0. We have proved the following

Theorem 4. Let w_{m} ∉ V [z_{0}], w_{n} → w_{m}, and let f_{n}, f_{m} be a corresponding point shift mappings. Let ψ_{n}, ψ_{m} be the image differentials of the φ_{n}, φ_{m} in the normalization ∥ψ_{n}∥ = K_{n}, ∥ψ_{m}∥ = K_{m}. Then, ∥ψ_{n} − ψ_{m}∥ → 0.

We can interpret that as continuity of the ψ_{n} with respect to the norm topology. By the mean value theorem we get ψ_{n} → ψ_{m} locally uniformly in D_{w} \{w_{m}}.

III A method of variation for point shift mappings

14. The quadratic differential

(31)  φ(z) dz^{2} = \frac{1}{z} dz^{2}

is basic in the following considerations. Its regular (horizontal) trajectories are the parabolas with focus z_{0} = 0 and with vertex on the negative real axis: this is geometrically evident, since the field of line element (1/z) dz^{2} > 0, i.e. arg dz = \frac{1}{2} arg z (mod π), is equal to the field of tangential elements of the parabolas. The only critical trajectory is the positive real half line. The vertical trajectories are the orthogonal parabolas and the negative real half line. Clearly, the field of line elements as well as the trajectories and orthogonal trajectories are invariant with respect to homotheties with center z_{0} .

Figure 2.

Let α be one of the parabolas. The interior of it is denoted by Π_{α} = Π. It is mapped by ζ = Φ(z) = ∫ \sqrt{φ(z)} dz = 2z^{1/2} onto a horizontal parallel half strip (Figure 2). The interval between the vertex z_{1} and the focus z_{0} is opened up into a vertical interval with center ζ_{0} = 0 = Φ(0), whereas the two branches of α go over into the two horizontal half lines which bound the strip. This again is geometrically evident because the tangential elements of the parabolas are turned into horizontal position by Φ. A Teichmüller mapping f of Π_{α} associated with the quadratic differential φ and any constant dilatation K > 1 is represented by the horizontal stretching ζ = ξ + iη → Kξ + iη of the half strip. It is therefore a self-mapping of Π_{α} with fixed point z_{0} . Actually all the points of the horizontal interval [z_{1}, z_{0}] are fixed points of f. Each trajectory of φ is mapped onto itself, whereas every vertical trajectory is transformed into another one, further away from z_{0} .

15. Let S be a segment of the parabola domain Π_{α}, bounded by a symmetric interval of α and a symmetric interval of an orthogonal parabola β. For short, we call these intervals α and β again. Let us denote the restriction of f to S by f_{0} , its dilatation by K_{0} . S′ = f_{0}(S) is a segment of the same parabola domain Π_{α}. In the ζ -plane the corresponding quantities are rectangles R and R′ and a horizontal stretching of R onto R′ by K_{0} (Figure 2). The mapping f_{0}: S → S′ is uniquely extremal for its boundary values on ∂S and with z_{0} as fixed point. This follows from the fact that it is a Teichmüller mapping associated with a holomorphic quadratic differential of finite norm, namely the restriction of φ to S \ {z_{0}}.

We now look at the extremal mapping f̃_{0}: S → S′ with the boundary values of f_{0} but without the assumption that z_{0} be fixed. The boundary dilatation of f_{0} on ∂S is determined by the local boundary dilatation at the two points z_{2} and z_{3} on ∂S, because f_{0} is analytic on the boundary intervals in-between these two points. However, at these two points the local boundary dilatation is smaller than K_{0} (see [7]). Therefore, H < K_{0} (see [10], p. 475). If the maximal dilatation K̃_{0} of f̃_{0} were known to be greater than H, an application of the frame mapping condition would tell us that f̃_{0} is a uniquely extremal Teichmüller mapping associated with a holomorphic quadratic differential φ̃_{0} of finite norm. However, the only thing which is known is K̃_{0} ≤ K_{0} . This is the reason for an additional step in the argument.

Move the vertical interval β to the right. Then, the segment S tends to Π_{α}. In the ζ -plane again, this corresponds to a shift of the right hand vertical boundary interval of R to the right. f̃_{0} goes over into a K̃_{0} -qc mapping of a half strip, equal to the stretching by K_{0} on its horizontal boundaries. It is then well known (see [7]) that the maximal dilatation K̃_{0} tends to K_{0} . Therefore there is a segment S such that K̃_{0} > H, and for this S the above conclusion holds. Also, because of the unique extremality of f̃_{0} and the symmetry of the boundary values, the point z̃_{0} = f̃_{0}(z_{0}) lies on the real axis. Again because of the unique extremality and since φ̃ ≠ φ, f̃_{0}(z_{0}) ≠ z_{0} and K̃_{0} < K_{0} .

An extremal length argument shows that z̃_{0} cannot lie between z_{0} and the vertex z_{1} of α. To this end, look at the upper half of the parabola segment S. It is mapped by f̃_{0} onto the upper half of S′. The extremal distance of the interval (z_{1}, z_{0}) from the upper half of the orthogonal parabola segment β is 2a/b, a being the length of the rectangle R and b its height. The analogous quantity in S′ is K_{0} ·2a/b. Assume that z̃_{0} ∈ (z_{1}, z_{0}). Then, the extremal distance of (z_{1}, z̃_{0}) from the upper half of β′ is > K_{0}2a/b. On the other hand, since f̃_{0} is K̃_{0} -qc, it is ≤ K̃_{0}2a/b which is smaller than K_{0}2a/b, a contradiction.

16. Next, we claim that for all points z̃ ∈ (z_{0}, z̃_{0}) on the real axis the extremal mapping f̃: z_{0} → z̃, f̃ | ∂S = f_{0} | ∂S is a Teichmüller mapping with a dilatation K̃, K̃_{0} < K̃ < K_{0} . Its quadratic differential has finite norm and a first order pole at z_{0} .

To this end we use a homothety with center z_{0} , retracting the segments S, S′ into inner parabola segments S̃, S̃′. The extremal mapping f̃_{0}: S → S′ goes over into the extremal mapping of S̃ onto S̃′, with the boundary values of f_{0} on ∂S̃. It takes z_{0} into the point z̃ ∈ (z_{0}, z̃_{0}) which is the image of z̃_{0} by the homothety. Of course, its dilatation is again K̃_{0} . This retracted mapping can be continued, by the orginal mapping f_{0} , to a qc mapping of S onto S′. Its dilatation is K̃_{0} in S̃ and K_{0} in S \ S̃, and it takes z_{0} into z̃_{0} . The extremal mapping f̃ with these conditions must have a dilatation K̃ > K̃_{0} , because it is subject to the additional condition z_{0} → z̃, whereas f̃_{0} was free on z_{0} . It is therefore a uniquely extremal Teichmüller mapping. Because of that, clearly K̃ < K_{0} , because the latter is the maximal dilatation of a competing mapping.

The construction with the homothety is possible for all z̃ ∈ (z_{0}, z̃_{0}), just by changing the homothety factor. We have thus shown the claim.

The final step is the following. The parabola segment S we were forced to take in the first step and in which we have constructed the variations of f_{0} could be quite long. However, by means of a homothety of S we can make it as small as we want. We have thus proved the following

Parabola Lemma. Consider the quadratic differential φ(z) dz^{2} = (1/z) dz^{2} and a number K_{0} > 1. Let f_{0} be the corresponding Teichmüller mapping of the plane. Then, in any neighborhood of z_{0} = 0 there are parabola segments S, S′, associated with φ, and mappings f̃: S → S′ with maximal dilatations K̃ < K_{0} , f̃(z_{0}) = z̃, f̃ | ∂S = f_{0} | ∂S. The points z̃ fill out an interval (z_{0}, z̃_{0}] of positive length δ. The extremal mappings satisfying the above boundary conditions are actually Teichmüller mappings, the quadratic differentials of which have finite norm and, if z̃ ≠ z̃_{0} , a first order pole at z_{0} .

17. Application to the point shift mappings.

Theorem 5. Let f_{m}: D_{z} → D_{w} be extremal for its boundary values, with f_{m}(z_{0}) = w_{m} ∉ V [z_{0}]. Let the associated quadratic differentials be φ_{m} in D_{z} and ψ_{m} in D_{w}, and let K_{m} be the dilatation of f_{m}. Denote the critical trajectory of ψ_{m} emanating from w_{m} by γ_{m}. Then, there exists a closed subinterval Δγ_{m} of γ_{m} with w_{m} one of its endpoints such that for every w_{n} ∈ Δγ_{m}, w_{n} ≠ w_{m}, the point shift mapping f_{n}, f_{n}(z_{0}) = w_{n}, has a dilatation K_{n} < K_{m} (Figure 3).

If the point w_{m} varies on a compact set C ⊂ D_{w}, C ∩ V [z_{0}] = ∅, the Euclidean length of the intervals Δγ_{m} has a positive lower bound.

Figure 3.

Proof. We introduce the normal parameter for φ_{m} near z_{0} and for ψ_{m} near w_{m}. That means that we apply local conformal mappings z ↔ ζ and w ↔ ζ such that φ_{m} and ψ_{m} both have the representation, in terms of ζ, φ_{m}(z) dz^{2} = (1/ζ) dζ^{2} and ψ_{m}(w) dw^{2} = (1/ζ) dζ^{2} (see [12]).

The mapping f_{m} is locally represented in the ζ -plane, by a parabola mapping f_{m}^{∗}, taking a parabola segment S^{∗} into a parabola segment S^{∗′} and with fixed point ζ_{0} = 0. The segments are bounded by subintervals of the same parabola α^{∗} and two different vertical parabola segments β^{∗} and β^{∗′} respectively.

The corresponding domains in the z - and w-planes are denoted by S_{m} and S′_{m} respectively. The points z_{0} and w_{m} go over, by the parameter transformations, into ζ_{0} = 0.

The Parabola Lemma tells us that there exists a closed interval [ζ_{0}, ζ̃_{0}] on the positive real axis of the ζ -plane such that for every ζ̃ ∈ (ζ_{0}, ζ̃_{0}] the extremal mapping f̃: S^{∗} → S^{∗′}, with the induced boundary values and with f̃(ζ̃_{0}) = ζ̃ has a dilatation K̃ < K_{m}. The mapping f_{m}: S_{m} → S′_{m} can now be replaced by the mapping f̃ composed with the parameter transformations z → ζ and ζ → w. This local mapping can be continued to the whole disks D_{z}, D_{w} outside S_{m}, S′_{m} by f_{m}. Its maximal dilatation is K_{m} but it takes z_{0} into a point w_{n} ∈ Δγ_{m} which corresponds to ζ̃. The extremal mapping with this property is f_{n}. Because of the unique extremality of f_{n} we have K_{n} < K_{m}. Taking into account that the interval [ζ_{0}, ζ̃_{0}] corresponds to a subinterval Δγ_{m} of γ_{m} of positive length, we have proved the theorem. The last statement follows because, due to the locally uniform convergence φ_{n} → φ_{m}, ψ_{n} → ψ_{m} for w_{n} → w_{m} we can use the same parabola segments S^{∗}, S^{∗′} in the natural parameter plane for all n and m.

18. Whereas in the previous section we have found points w_{n} with K[w_{n}] < K[w_{m}] we are now going to show the existence of points w_{n}, in every neighborhood of w_{m}, with K[w_{n}] > K[w_{m}].

Theorem 6. Let w_{m} ∉ V [z_{0}] and let K[w_{m}] be the dilatation of the point shift mapping f_{m}, f_{m}(z_{0}) = w_{m}. Then, in every neighborhood of w_{m} there is a point w_{n} such that the point shift mapping f_{n}, f_{n}(z_{0}) = w_{n}, has a dilatation K[w_{n}] > K[w_{m}].

Proof. The proof consists in finding points w_{n} ≠ w_{m} such that w_{m} ∈ Δγ_{n}. We can then apply the previous result to get the desired inequality.

Figure 4.

We start with a fixed parabola segment S in the ζ -plane (Figure 4). Let g_{m} be the inverse of the parameter mapping for ψ_{m}, with g_{m}(S) = S′_{m}, and g_{m}([ζ_{0}, ζ̃_{0}]) = Δγ_{m} the distinguished critical trajectory interval of ψ_{m}, with end points w_{m}, w̃_{m}. Likewise, let g_{n} be the inverse of the parameter mapping for ψ_{n}, g_{n}(S) = S′_{n}, g_{n}(ζ_{0}) = w_{n}, g_{n}([ζ_{0}, ζ̃_{0}]) = Δγ_{n} the distinguished critical parameter interval of ψ_{n}. Because of the convergence ∥ψ_{n} − ψ_{m}∥ → 0 for w_{n} → w_{m} the mappings g_{n} tend to g_{m} uniformly in S. In particular, the images g_{n}(ζ) and g_{m}(ζ), ζ ∈ [ζ_{0}, ζ̃_{0}], have a distance |g_{n}(ζ) − g_{m}(ζ)| which is arbitrarily small, and the directions, in which the critical trajectories γ_{n} and γ_{m} leave the points w_{n} and w_{m} respectively are arbitrarily close. This is so because this direction is the same as minus the arg of the residue, and the residues are arbitrarily close by the locally uniform convergence ψ_{n} → ψ_{m} away from the pole w_{m} and the Cauchy representation of the residue.

Choose now ϱ > 0 small enough and let w_{n} turn around w_{m} on the circle |w_{n} − w_{m}| = ϱ. Then, it is evident that the interval Δγ_{n} must pass through the point w_{m} at least once, and its origin is the point w_{n} we are looking for.

IV The level sets of K[w]

19. Given a quasisymmetric boundary homeomorphism h: ∂D_{z} → ∂D_{w} and a fixed point z_{0} ∈ D_{z}. Recall that for every w ∈ D_{w} we define K[w] to be the smallest maximal dilatation of all quasiconformal mappings f: D_{z} → D_{w} which agree with h on ∂D_{z} and satisfy f(z_{0}) = w. The smallest maximal dilatation of all qc mappings f with f | ∂D_{z} = h but without a given value at a fixed point is called K_{0} . Every extremal qc mapping f_{0}: D_{z} → D_{w}, f_{0} | ∂D_{z} = h, has maximal dilatation K_{0} , and the set of images of z_{0} by all extremal qc mappings f_{0} is called the variability set of z_{0} and denoted by V [z_{0}]. Clearly, if the boundary homeomorphism h admits a unique extremal solution, V [z_{0}] consists of a single point for every z_{0} . In general, this is however not the case, but for every w_{0} ∈ V [z_{0}] there exists at least one extremal mapping f_{0} with f_{0}(z_{0}) = w_{0} , by definition.

If w_{m} ∉ V [z_{0}] the value K[w_{m}] is realized as the (constant) dilatation of a unique extremal Teichmüller mapping f_{m} associated with a pair of quadratic differentials φ_{m}, ψ_{m} of norm ∥φ_{m}∥ = 1, ∥ψ_{m}∥ = K[w_{m}]. φ_{m} has a first order pole at z_{0} , while ψ_{m} has one at w_{m}; both differentials are holomorphic otherwise.

We already know that V [z_{0}] is compact. But in addition to that we can now prove the following

Theorem 7. The variability set V [z_{0}] is simply connected, i.e. connected and without holes.

The proof of the second statement is immediate. For, let G be a component of D_{w}\V [z_{0}] which is bounded away from ∂D_{w}. Then, the boundary ∂G lies in V [z_{0}] and the continuous function K[w] must attain its maximum in a point w_{m} ∈ G, K[w_{m}] > K_{0} . But by Theorem 6 there is a point w in every neighborhood of w_{m} with K[w] > K[w_{m}], contradicting that K[w_{m}] is the maximum value of K[w] in G.

The first statement will follow from the theorem in the next section.

20. For every w_{m} ∈ D_{w} \ V [z_{0}] we consider the following three disjoint subsets of D_{w}. We set K[w_{m}] = K_{m}.

\begin{aligned}ΓK_{m}^{−} & := \{w; K[w] < K_{m}\} \\ Γ_{K_{m}} & := \{w; K[w] = K_{m}\} \\ ΓK_{m}^{+} & := \{w; K[w] > K_{m}\} \end{aligned}

Theorem 8. The level set Γ_{K_{m}} is a Jordan curve, ΓK_{m}^{−} its interior and ΓK_{m}^{+} its extrerior (in D_{w}). The sets ΓK_{m}^{−} are decreasing with K_{m} ↘ K_{0} and

V [z_{0}] = ⋂_{K_{m}>K_{0}} ΓK_{m}^{−}.

Proof. (1) Since K[w] → ∞ for |w| → 1 the set ΓK_{m}^{+} contains an annulus r < |w| < 1 for sufficiently large r. As an open set, ΓK_{m}^{+} decomposes into domains, i.e. maximal connected open subsets. Let G be such a domain and assume that it is bounded away from ∂D_{w}. Then, its boundary consists of points w with K[w] = K_{m}, whereas for w ∈ G K[w] > K_{m}. Therefore the function K[w] assumes its maximum at a point ŵ ∈ G. But this is impossible, since in every neighborhood of ŵ there are points w with K[w] > K[ŵ]. We conclude that every component of ΓK_{m}^{+} contains points arbitrarily close to ∂D_{w}. This means that all components contain the annulus r < |w| < 1 and are thus connected themselves, hence ΓK_{m}^{+} is a domain.

(2) Let C be a component of its complement. It consists of points w with K[w] ≤ K_{m}. Every boundary point of C is a point w_{m} ∈ Γ_{K_{m}}. The trajectory interval Δγ_{m} lies in ΓK_{m}^{−}, except for its initial point w_{m}. On the other hand, as Δγ_{m} is connected, it is contained in C. We conclude that C has interior points, and in fact that every boundary point of C is a boundary point of its interior.

Let G^{ν} be a component of its interior, hence a subdomain of C. For every point w ∈ G^{ν}K[w] < K_{m}, whereas on ∂G^{ν}K[w] = K_{m}. Therefore K[w] must assume its minimum at a point ŵ ∈ G^{ν}, and by the same reason as above K[ŵ] = K_{0} . We conclude that every component G^{ν} of int C contains points of V [z_{0}]. From this it follows, because of the uniform continuity of K[w] in C, that int C consists of finitely many components G^{ν} only. Otherwise, we could pick a point w_{0}^{ν} ∈ V [z_{0}] out of every component G^{ν}. An accumulation point w_{0} of these points would lie in V [z_{0}] and in ∂C, an impossibility. Every G^{ν} is clearly simply connected.

(3) The next step is to show that there can indeed be only one component of int C. For, let G^{0} be a maximal subdomain of int C and assume there is more than one. The boundary ∂G^{0} is a subset of the connected set ∂C. It is now easy to see that there exists a point w_{m} ∈ ∂G^{0} which is also a boundary point of at least one other subdomain of int C, because otherwise we could find a point in ∂C which is not a boundary point of int C. Assume that the points w ∈ Δγ_{m} \ {w_{m}} lie in G^{0} , and let G^{1} be another subdomain of int C which has w_{m} as boundary point. For w^{1} ∈ G^{1} , sufficiently close to w_{m}, the interval Δγ^{1} with initial point w^{1} would be arbitrarily close to Δγ_{m}, and it would thus have to cut ∂G^{0} . But since K[w^{1}] < K_{m} and hence K[w] < K_{m} for every w ∈ Δγ^{1} this would give a contradiction. We conclude that int C consists of a single simply connected domain G.

(4) We now proceed to show that G is a Jordan domain. Consider a boundary point w_{m} of G. The interval Δγ_{m} with initial point w_{m} lies in G, except for w_{m}. This means that every boundary point of G is the end point of an end-cut (the same as “erreichbarer Randpunkt”, see [3], pp. 357–363). Consider two end-cuts with the same end point w_{m}. Their initial points, which lie in G, can be joined by an arc in G. We thus have a Jordan curve which lies in G, except for the point w_{m} itself. If the two end-cuts were not equivalent, the domain interior to this Jordan curve would contain boundary points of G, i.e. points of Γ_{K_{m}}. But in every neighborhood of such a point there is a point of ΓK_{m}^{+} , which is impossible.

Therefore the equivalence classes of end-cuts are in one-to-one correspondence with the boundary points of G. By standard arguments the Riemann mapping of G onto the disk has a homeomorphic extension to ∂G, which shows that ∂G is a Jordan curve.

(5) The last step is to show that ΓK_{m}^{−} consists of only one Jordan domain and hence the level set Γ_{K_{m}} is a single Jordan curve for every K_{m} > K_{0} . We have seen that the set ΓK_{m}^{−} consists of finitely many Jordan domains each of which contains points of V [z_{0}], and the boundary curves of these Jordan domains are the components of Γ_{K_{m}}. For K_{n} > K_{m} the set ΓK_{n}^{−} contains ΓK_{m}^{−} and therefore Γ_{K_{m}} has at least as many components as Γ_{K_{n}}. Choose r < 1 such that the set V [z_{0}] is contained in the disk |w| < r. For K_{m} > min{K[w] ; |w| = r} the circle |w| = r lies in ΓK_{m}^{−} and hence in one of its components. This component contains all of V [z_{0}] and is hence the only one. We conclude that Γ_{K_{m}} consists of a single Jordan curve for all sufficiently large K_{m}.

Let K > K_{m}, and denote the finitely many components of Γ_{K_{m}} by Γ_{K_{m}}^{ν}, hence Γ_{K_{m}} = ∪Γ_{K_{m}}^{ν}, ν = 1, …, N. The level set Γ_{K} is composed of finitely many Jordan curves Γ_{K}^{μ}, μ = 1, …, M, M ≤ N. They contain the curves Γ_{K_{m}}^{ν} in their interior, and it is easy to see, by contradiction, that for K ↘ K_{m} the level set Γ_{K} tends to Γ_{K_{m}}. Therefore there cannot be fewer curves, one Γ_{K}^{μ} for each Γ_{K_{m}}^{ν}, thus M = N, as soon as K is sufficiently close to K_{m}.

Assume now that K < K_{m}. Again, for K ↗ K_{m} the limit set of Γ_{K} is Γ_{K_{m}} = ∪Γ_{K_{m}}. The Jordan curves Γ_{K}^{μ} tend to the Jordan curves Γ_{K_{m}}^{μ}, and eventually in a neighborhood of each Γ_{K_{m}}^{ν} there can be only one Γ_{K}^{μ}, μ = 1, …, M. We conclude that M = N for all K < K_{m} which are sufficiently close to K_{m}.

The number of components of Γ_{K_{m}} is thus a continuous function of K_{m}, and being entire, it is constant, q.e.d.

494 Kurt Strebel

The set V [z_{0}] is evidently equal to the intersection V [z_{0}] = ⋂_{K_{m}>K_{0}} ΓK_{m}^{−}. This proves the theorem.

Added in proof. In a forthcoming paper, Clifford Earle and Nikola Lakic are going to generalize most of the theorems of this article to the case of Riemann surfaces, using Teichmüller space methods.

References

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Received 4 February 1997