Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 405–412

CONFORMAL WELDING OF JORDAN CURVES USING WEIGHTED DIRICHLET SPACES

Gavin L. Jones

University of Liverpool, Department of Mathematical Sciences, Liverpool, UK Current address: 67 Manor Way, London SE3 9XG, UK

Abstract. This paper gives an alternative description of Lehto’s results on homeomorphisms which admit conformal welding [6]. A normal families argument is controlled by the action of the homeomorphism as a composition operator on weighted Dirichlet spaces. This point of view admits a discussion of uniqueness.

1. Introduction

Let J be a Jordan curve on the Riemann sphere S^{2} . Then S^{2} \ J = D_{1} ∪ D_{2} , where D_{1} and D_{2} are disjoint simply connected subdomains of the sphere. The conformal maps f: U → D_{1} , g: U^{∗} → D_{2} , where U is the unit disc, and U^{∗} the complement of the closed unit disc, extend as homeomorphisms from S^{1} , the unit circle, and the welding h = g^{−1} ∘ f is an element of Homeo_{+}(S^{1}). It is an open question to establish the relationship between J and h. The case that is well understood is where J is a quasi-circle which corresponds to h being quasisymmetric [2] . One may rephrase this as

Theorem 1.1 [2]. The homeomorphism h: S^{1} ⟶ S^{1} is a welding of a quasicircle if and only if V_{h}: f ⟶ f ∘h is a bounded operator on 𝒟 the Dirichlet space.

The Dirichlet space 𝒟 consists of f ∈ L^{2}(S^{1}) with Poisson extensions to the unit disc f with ∫ |∇f|^{2} dA < ∞. The contribution of this paper is to weaken the hypothesis to obtain for certain spaces D_{w} given by ∫ |∇f(z)|^{2}w(z) dA(z) < ∞ and certain weights w (defined in Section 3).

Theorem 1.2. The homeomorphism h: S^{1} ⟶ S^{1} is a welding of a Jordan curve if h is a uniform limit of quasi-symmetric mappings, h_{n}, so that V_{h_{n}}: f ⟶ f ∘ h_{n} are uniformly bounded operators from D_{w} to 𝒟, where D_{w} is a weighted Dirichlet space.

1991 Mathematics Subject Classification: Primary 30C55; Secondary 30C60.

Supported by EPSRC grant GR/K/75453.

The aim is to obtain some information on the welding problem without being too dependent on the Beltrami equation cf. [6], [3]. The paper falls into two parts, in the first we give a simple argument for welding to exist cf. [4] and in the second we give an example where we can explicitly calculate with the weight for Theorem 1.2. Sadly the elementary calculations only give the same conclusions as [6], and are weaker than [4] but they do offer a different framework to discuss uniqueness.

2. Outline

The argument is simple. Consider a homeomorphism h, fixing three points, 1, i, −1, say. Then we may approximate h by h_{n} uniformly, where the h_{n} are nice normalized homeomorphisms for which we can solve the welding. These give Jordan curves J_{n} with Riemann maps f_{n} to the interior, and we may assume the f_{n} fix 1, i, −1. So the {f_{n}} form a normal family and on a subsequence converge to f a univalent map of the unit disc, U. If f(S^{1}) is a Jordan curve J then all of the convergence is uniform on S^{1} and h is the welding of J. Further f takes the unit disc to the interior of a Jordan curve if and only if f extends continuously and injectively to the boundary. This holds if the family {f_{n}} is uniformly continuous and uniformly injective on S^{1} , i.e. every distinct pair of points on S^{1} have images uniformly separated under the f_{n}. We verify the uniform conditions in terms of module calculations for a set of annuli. Recall that any doubly connected region of the Riemann sphere, A, is conformally equivalent to one of {z : 1 < |z| < r}, or the punctured plane. The modulus of A is then Mod A = log r/2π, or ∞ for the punctured plane.

We consider two families of annuli 𝒜 and ℬ defined as follows. Given any pair of points x, y on the unit circle with 0 < d(x, y) < \frac{1}{5} , not both in \{1, i, −1\}, form the annulus A_{x,y}. This is given by deleting the shorter arc (x, y) ⊂ S^{1} and one of the arcs of the unit circle I ∈ \{(1, i), (i, −1), (−1, 1)\} so that d(I, (x, y)) > \frac{1}{5} from the Riemann sphere. The annuli A_{x,y} form the family 𝒜 and we will use them to test the uniform continuity. To test injectivity consider the family ℬ consisting of the B_{x,y} defined next. Form arcs I_{1}, I_{2} where I_{1} has endpoint x and one of 1, i, −1 and I_{2} with endpoints y and one of 1, i, −1 so that 0 < d(I_{1}, I_{2}) ≤ d(x, y), then B_{x,y} is the annulus formed by deleting I_{1}, I_{2} from the Riemann sphere. There are corresponding versions of these annuli under the weldings h_{n}. If E, F are arcs in the unit circle then we can consider the annulus A_{E,F} = S^{2} \ (E ∪ F), we denote the annulus in the sphere formed by welding h_{n} as A_{E,F}^{n} = S^{2} \ (f_{n}(E) ∪ f_{n}(F)). Then following Lehto [6], [7] we have the following lemma.

Lemma 2.1. Let h be an orientation preserving homeomorphism of the unit circle, fixing {1, i, −1}, and let h_{n} be a normalized family of quasi-symmetric homeomorphisms converging uniformly to h. If for all annuli in 𝒜 Mod A^{n} ⟶ ∞ as Mod A ⟶ ∞ uniformly in Mod A, and if for all annuli in ℬ, Mod B^{n} ⟶ 0 implies Mod B ⟶ 0 uniformly then h welds a Jordan curve.

Proof. The proof is [7] where euclidean bounds on separation of boundary components of annuli are obtained from modulus inequalities and normalization of annuli. Then as outlined h welds a Jordan curve. To obtain estimates on the moduli of annuli we use an alternative definition. From [1] Mod A_{E,F} = L_{E,F}^{−1/2} where L_{E,F} is the square of the Dirichlet norm of the harmonic function u on A_{E,F} = S^{2} \ (E ∪ F) with boundary values 0 on E, 1 on F. In particular we have

L_{E,F} ≤ inf{|u|_{D}^{2} : u|_{E} = 0, u|_{F} = 1, u ∈ C^{1}(A_{E,F})}.

Therefore we may rewrite Lemma 2.1 in terms of the extremal length L^{1/2} .

Corollary 2.1. Let h be an orientation preserving homeomorphism of the unit circle, fixing {1, i, −1}, and let h_{n} be a normalized family of quasi-symmetric homeomorphisms converging uniformly to h. If for all annuli in 𝒜 , L^{n} ⟶ 0 as L ⟶ 0 uniformly, and if for all annuli in ℬ, L^{n} ⟶ ∞ implies L ⟶ ∞ uniformly, then h welds a Jordan curve J.

It remains to determine inequalities for this to hold for a given homeomorphism.

3. The welding as operator

Given arcs E, F on the unit circle we have formed the annulus A_{E,F} and the welded annulus A_{E,F}^{n} = S^{2} \ (f_{n}(E) ∪ f_{n}(F)). We wish to compute the extremal length of A^{n} by considering functions which take values 0 on f_{n}(E) and 1 on f_{n}(F) and evaluating the Dirichlet norm. So given a function u ∈ C^{0}(J_{n}) say with u|_{f_{n}(E)} = 0, u|_{f_{n}(F)} = 1 we may form the harmonic extensions u_{+} , u_{−} to Ω_{n}, Ω_{n}^{∗} the interior and exterior of J_{n}. Since J_{n} is a quasi-circle the Dirichlet norm may be calculated by integrating off J_{n} and we have

|u|_{D}^{2} = ∫_{Ω_{n}} |∇u|^{2} dA + ∫_{Ω_{n}^{∗}} |∇u|^{2} dA.

Denoting the family of such u as U^{n}(E, F) we have

L_{E,F}^{n} = inf{|u|_{D}^{2} : u ∈ U^{n}(E, F)}.

But as Dirchlet norms are conformally invariant we pull back by the conformal maps f_{n} and g_{n} to the round sphere. Then if we pick u ∈ U^{n}(E, F) we consider v = u ∘ f_{n}, r = u ∘ g_{n} defined on U, U^{∗} respectively. In which case

L_{E,F}^{n} = inf{|v|_{D}^{2} + |r|_{D}^{2}}

where the infimum is over v harmonic on U, continuous on S^{1} , v|_{E} = 0, v|_{F} = 1 and over r harmonic on U^{∗} continuous on S^{1} , r|_{h_{n}(E)} = 0, r|_{h_{n}(F)} = 1. That is

L_{E,F}^{n} ≤ inf{|v|_{D}^{2} + |v ∘ h_{n}|_{D}^{2} : v ∈ U(E, F)}.

We require that if L tends to 0, L^{n} tends to 0 and as L^{n} tends to infinity L does, uniformly. So it is clear if V_{h_{n}} are uniformly bounded on 𝒟 the above argument produces a Jordan curve i.e. welding for quasi-symmetric maps. We give a slight weakening.

Theorem 3.1. Let D_{w} be a weighted Dirichlet space, so that the weight has properties (P) outlined below. If h is a homeomorphism of the circle which is the uniform limit of quasi-symmetric homeomorphisms h_{n}, for which the operator norm V_{h_{n}} : D_{w} ⟶ 𝒟 is uniformly bounded, then h welds a Jordan curve.

Given an annulus A_{E,F} determined by a pair of arcs E, F on the unit circle, we have

L_{E,F} = 2 inf{|u|_{D}^{2} : u ∈ U(E, F)}.

We define

L_{E,F}^{w} = 2 inf{|u|_{w}^{2} : u ∈ U(E, F)}

where

|u|_{w}^{2} = ∫_{U} |∇u(z)|^{2}w(z) dA(z).

Property (P) of the weight w is now the three conditions:

(1) w(z) > c > 0.

(2) For A_{E,F} ∈ 𝒜 , L_{E,F} ⟶ 0 implies L_{E,F}^{w} ⟶ 0 uniformly.

(3) For B_{E,F} ∈ ℬ L_{E,F}^{w} ⟶ ∞ implies L_{E,F} ⟶ ∞, uniformly.

Proof of 3.1. We have

L_{E,F}^{n} ≤ inf{|v|_{D}^{2} + |v ∘ h_{n}|_{D}^{2} : v ∈ U(E, F)}
≤ inf{|v|_{w}^{2} + |v ∘ h_{n}|_{D}^{2} : v ∈ U(E, F)} ≤ inf{C|v|_{w}^{2} : v ∈ U(E, F)}

where C is the bound on the operator norms of V_{h_{n}}: D_{w} ⟶ 𝒟 . But parts (2) and (3) of (P) now imply the appropriate modulus estimates for the annuli A^{n}(E, F), and h welds by Corollary 2.1.

4. Examples

We look at examples of radially symmetric weights for which the computation is simple. Let α < 1 and let w(z) = O(| log(1 − |z|)^{−1}|^{α}) as |z| ⟶ 1. We show for some h, homeomorphisms with V_{h}: D_{w} ⟶ 𝒟 bounded, that h welds.

Step 1. First we check that annuli are behaved appropriately in D_{w}. For this step by conformal invariance we may as well work on the upper-half plane. Let f_{0,a,1,∞} denote the unique conformal map of the upper-half plane with distinguished points {0, a, 1, ∞} to the rectangle with corners at 0, 1 and ib, 1 + ib. Then Re f is a candidate for U(E, F) where F = (0, a) and E = (1, ∞). By radial symmetry to check properties (2) and (3) of (P) we need that |f|_{w} tends to zero as a tends to zero and |f|_{w} is uniformly bounded as a stays away from 1. But the derivative is given (up to a constant O(1)) as

f′(z) = \frac{1}{log |a|\sqrt{z(z − a)(z − 1)}}.

So with w(z) = O(| log Im z|^{α}) in a neighbourhood of the origin an integration verifies D_{w} has properties (P).

Step 2. To construct the approximations we need to control the homeomorphisms, we work on the real line (by Möbius conjugation). First consider h a quasi-symmetric homeomorphism, we estimate |V_{h}|_{D_{w},𝒟} the norm of the composition operator V_{h}: D_{w} ⟶ 𝒟. Given H an extension of h to the upper half-plane W, we apply the observation [1]

(∗)  |f ∘ h^{−1}|_{D}^{2} ≤ 2 ∫_{W} |∂f(z)|^{2} \frac{1}{1 − |μ_{H}(z)|} dA(z).

Thus we obtain |V_{h^{−1}}|_{D_{w},𝒟} ≤ [(1 − μ_{H}(z))w(z)]^{−1} . Writing

sup_{x} \frac{h(x + t) − h(x)}{h(x) − h(x − t)} = ρ_{t}

the Beurling–Ahlfors extension H of h satisfies (1 − |μ_{H}(z)|)^{−1} ≤ cρ_{Im z}. So for h a homeomorphism of the reals we have

|V_{h^{−1}}|_{D_{w},𝒟} ≤ c sup ρ_{t}/w(t).

Therefore given any homeomorphism h satisfying

O(| log t|^{−α}) ≤ \frac{h(x + t) − h(x)}{h(x) − h(x − t)} ≤ O(| log t|^{α})

we may approximate h by piecewise linear quasi-symmetric maps h_{n} with

sup_{n} |V_{h_{n}^{−1}}|_{D_{w},𝒟} ≤ C sup ρ_{t}(h)/w(t) < ∞

and applying Theorem 3.1 h^{−1} welds. This implies that h welds.

To prove Lehto’s result [6] with weight w(z) = O(| log(1 − |z|)|) we apply the above argument modifying Step 1. Consider f_{0,a,1,2} mapping the upper half-plane to a rectangle with vertices 0, 1, 1 + ib, ib. Let ψ be a conformal map of the upper half-plane to a smooth region W in the upper half-plane, where \overline{W} meets the real line in [0, 2], so that ψ(0) = 0, ψ(2) = 2. Then we consider up to a constant O(1)

F_{a}(z) = \frac{1}{log | log a| log(ψ(z)/4)\sqrt{z(z − a)(z − 1)(z − 2)}}.

The additional logarithmic singularity at the origin is sufficient to force |F_{a}|_{w} to zero as a tends to zero, and h welds.

5. Uniqueness

The operator formulation allows a different approach to the uniqueness of the welding following [5]. A welding h is said to be unique if the Jordan curve J with welding h forces any homeomorphism of the sphere, conformal off J to be Möbius. It is well known [5] that if functions continuous on the sphere with ∫_{S^{2}\J} |∇f|^{2} dA < ∞ automatically satisfy ∫_{S^{2}} |∇f|^{2} dA < ∞ (‘J Dirichlet removable’) then J is unique (‘conformally removable’). We outline this for quasi-circles and then point out the extension to curves with weldings of the previous section (for which uniqueness follows from [3]).

Let J be a quasi-circle. Pick F continuous, ∫_{S^{2}\J} |∇F |^{2} dA < ∞, and harmonic off J. We need to show that ∫_{S^{2}} |∇F |^{2} dA < ∞. As [5] we show F is the uniform limit of smooth F_{n} with ∫_{S^{2}} |∇F_{n}|^{2} < C. Think of F as F_{+} = F |_{intJ} ∘ f defined on the unit disc and the harmonic extension to the exterior of F_{−} = F_{+} ∘ h|_{S^{1}} . Then ∫_{S^{2}\J} |∇F |^{2} dA is ∫_{U} |∇F_{+}|^{2} dA + ∫_{U^{∗}} |∇F_{−}|^{2} dA by conformal invariance. Approximate F_{+} by smooth harmonic G_{n} on U and h by smooth h_{n}. The functions G_{n} on the unit disc and the harmonic extensions of G_{n} ∘ h_{n} on the exterior represent, after welding, smooth functions F_{n} on the sphere which converge uniformly to F. Further the Dirichlet norms of F_{n} are controlled by |F_{+}|_{D} < ∞ and |F_{n} ∘ h_{n}|_{D} ≤ K|F_{n}|_{D} with K the operator norm on the composition V_{h_{n}}. So F is uniformly approximable by smooth Dirichlet bounded functions and J is Dirichlet removable, hence unique. We repeat this argument with a different norm to obtain

Theorem 5.1. Let α < 1 and h be a homeomorphism with

\frac{h(x + t) − h(x)}{h(x) − h(x − t)} ≤ O(| log t|^{α}).

The Jordan curve J with welding h is unique.

Proof. Consider functions F continuous on the sphere harmonic off J. Let f, g be the conformal maps from U, U^{∗} to the interior and exterior of J, respectively. We consider the norm

|F |_{α,1} = ∫_{U} |∇(F ∘ f)|^{2}w_{α}(z) dA + ∫_{U^{∗}} |∇(F ∘ g)|^{2} dA.

We show if |F |_{α,1} < ∞ then

|F |_{W}^{2} = ∫_{S^{2}} |∇F |^{2}W (z) dA < ∞

where W (z) = 1 for z ∈ \overline{ext J} and W (z) = w_{α}(f^{−1}(z)) for z ∈ int J. This follows as above by writing F as two distinct functions F_{+} = F |_{intJ} ∘ f on the unit disc and F_{−} = F |_{ext J} ∘ g on the exterior—i.e. F_{+}, F_{+} ∘ h. Now approximate F_{+} by smooth harmonic functions Φ_{n}, and approximate h by smooth h_{n} with uniformly bounded operator norm D_{w} ⟶ 𝒟 . We regard the harmonic functions Φ_{n} on the unit disc and the harmonic extension of Φ_{n} ∘ h_{n} as representing, after welding, a smooth function on the sphere F_{n}, harmonic off J_{n}. Now |F_{n}|_{w_{α}} ⟶ |F |_{w_{α}} and the smoothness forces |F_{n}|_{W} = |F_{n}|_{α,1} which are uniformly bounded, hence |F |_{W} is finite.

To show that J is conformally removable we wish to show that for Ψ a homeomorphism of the sphere conformal off J, |Ψ(z) − z|_{W} < ∞, hence finite Dirichlet norm and by Weyl’s lemma is thus Möbius. To do this we just need to check that f ∈ D_{w_{α}}, and that J has zero area. For this we have recourse to the powerful estimates of [3], which give |J| = 0 immediately. We require

∫_{U} |f′(z)|^{2}| log(1 − |z|)|^{α} dA(z) < ∞

that is

∑_{n>0} n^{α} ∫_{A_{n}} |f′(z)|^{2} dA(z) < ∞,

where A_{n} = {z : 1 − 2^{−n} > |z| ≥ 1 − 2^{−n−1}}. But the bounds from [3] show

∫_{A_{n}} |f′(z)|^{2} dA(z) < O(n^{−γ})

for some γ > 2 so f ∈ D_{w} and the welding is unique.

412 Gavin L. Jones

References

[1] Ahlfors, L.V.: Lectures on Quasiconformal Mappings. - Wadsworth, 1966.

[2] Beurling, A., and L.V. Ahlfors: The boundary correspondence under a quasi-confor-mal mapping. - Acta Math. 96, 1956, 125–142.

[3] David, G.: Solutions de l’equation de Beltrami avec |μ|_{∞} = 1. - Ann. Acad. Sci. Fenn. Math. 13, 1988, 25–70.

[4] Hamilton, D.H.: Generalized conformal welding. - Ann. Acad. Sci. Fenn. Math. 16, 1991, 333–345.

[5] Jones, P.W.: On removable sets for Sobolev spaces in the plane. - In: Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton Univ. Press, 1995, pp. 250–267.

[6] Lehto, O.: Homeomorphisms with a given dilatation. - In: Proceedings of the 15^{th} Scandinavian Congress, Oslo, 1968. Lecture Notes in Math., Springer-Verlag, 1970, pp. 58– 73.

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

[8] Reed, T.J.: Quasiconformal mappings with given boundary values. - Duke Math. J. 33, 1966, 459–464.

Received 12 October 1998