In mathematics, the Douady–Earle extension, named after Adrien Douady and Clifford Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed unit disk, such that the extension is a diffeomorphism of the open disk. The extension is analytic on the open disk. The extension has an important equivariance property: if the homeomorphism is composed on either side with a Möbius transformation preserving the unit circle the extension is also obtained by composition with the same Möbius transformation. If the homeomorphism is quasisymmetric, the diffeomorphism is quasiconformal. An extension for quasisymmetric homeomorphisms had previously been given by Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia. Equivariant extensions have important applications in Teichmüller theory, for example they lead to a quick proof of the contractibility of the Teichmüller space of a Fuchsian group.
Definition
By the Radó–Kneser–Choquet theorem, the Poisson integral of a homeomorphism f of the circle defines a harmonic diffeomorphism of the unit disk extending f. If f is quasisymmetric, the extension is not necessarily quasiconformal, i.e. the complex dilatation does not necessarily satisfy However F can be used to define another analytic extensionHf of f−1 which does satisfy this condition. It follows that is the required extension. For |a| < 1 define the Möbius transformation It preserves the unit circle and unit disk sending a to 0. If g is any Möbius transformation preserving the unit circle and disk, then For |a| < 1 define to be the unique w with |w| < 1 and For |a| =1 set
Smoothness and non-vanishing Jacobian on open disk.Hf is smooth with nowhere vanishing Jacobian on |z| < 1. In fact, because of the compatibility with Möbius transformations, it suffices to check that Hf is smooth near 0 and has non-vanishing derivative at 0.
Homeomorphism on closed disk and diffeomorphism on open disk. It is enough to show that Hf is a homeomorphism. By continuity its image is compact so closed. The non-vanishing of the Jacobian, implies that Hf is an open mapping on the unit disk, so that the image of the open disk is open. Hence the image of the closed disk is an open and closed subset of the closed disk. By connectivity, it must be the whole disk. For |w| < 1, the inverse image of w is closed, so compact, and entirely contained in the open disk. Since Hf is locally a homeomorphism, it must be a finite set. The set of pointsw in the open disk with exactly n preimages is open. By connectivity every point has the same number N of preimages. Since the open disk is simply connected, N = 1.
Extension of quasi-Möbius homeomorphisms
In this section it is established that the extension of a quasisymmetric homeomorphism is quasiconformal. Fundamental use is made of the notion of quasi-Möbius homeomorphism. A homeomorphism f of the circle is quasisymmetric if there are constants a, b > 0 such that It is quasi-Möbius is there are constants c, d > 0 such that where denotes the cross-ratio. If f is quasisymmetric then it is also quasi-Möbius, with c = a2 and d = b: this follows by multiplying the first inequality for and. It is immediate that the quasi-Möbius homeomorphisms are closed under the operations of inversion and composition. The complex dilatation μ of a diffeomorphism F of the unit disk is defined by If F and G are diffeomorphisms of the disk, then In particular if G is holomorphic, then When F = Hf, where To prove that F = Hf is quasiconformal amounts to showing that Since f ia a quasi-Möbius homeomorphism the compositions g1 ∘ f ∘ g2 with gi Möbius transformations satisfy exactly the same estimates, since Möbius transformations preserve the cross ratio. So to prove that Hf is quasiconformal it suffices to show that if f is any quasi-Möbius homeomorphism fixing 1, i and −i, with fixed c and d, then the quantities have an upper bound strictly less than one. On the other hand if f is quasi-Möbius and fixes 1, i and −i, then f satisfies a Hölder continuity condition: for another positive constant C independent of f. The same is true for the f−1's. But then the Arzelà–Ascoli theorem implies these homeomorphisms form a compact subset in C. The non-linear functional Λ is continuous on this subset and therefore attains its upper bound at some f0. On the other hand Λ < 1, so the upper bound is strictly less than 1. The uniform Hölder estimate for f is established in as follows. Take z, w in T.
If |z − 1| ≤ 1/4 and |z - w| ≤ 1/8, then |z ± i| ≥ 1/4 and |w ± i| ≥ 1/8. But then
If |z - w| ≥ 1/8, the Hölder estimate is trivial since |f - f| ≤ 2.
If |z - 1| ≥ 1/4, then |w - ζ| ≥ 1/4 for ζ = i or −i. But then
Comment. In fact every quasi-Möbius homeomorphism f is also quasisymmetric. This follows using the Douady–Earle extension, since every quasiconformal homeomorphism of the unit disk induces a quasisymmetric homeomorphism of the unit circle. It can also be proved directly, following
