Douady-Earle Extension Research Articles

Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space

In this survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or C^{1+\alpha} ) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension BA(h) of a C^{1+\mathrm{Zygmund}} orientation-preserving diffeomorphism h of the real line, and show that the Beltrami coefficient \mu (BA(h))(x+iy) vanishes as O(y) uniformly on x near the boundary of the upper half plane if and only if h is C^{1+\mathrm{Lipschitz}} . Finally, we show this criterion is indeed true when h is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.

Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

BMO-Teichmüller spaces revisited

In [6] the equivalence among three definitions of BMO-Teichmüller spaces associated with a Fuchsian group was proven using the Douady–Earle extension operator. In this paper, we show that these equivalences are actually biholomorphisms. In [6] it was further shown that the Douady–Earle extension operator is continuous at the origin. We improve this result by showing Gâteaux-differentiability at this point.

Computing the Douady–Earle extension using Kuramoto oscillators

We demonstrate that the Douady–Earle extension can be computed by solving the specific set of ODE’s. This system of ODE’s has several interpretations in Mathematical Physics, such as Kuramoto model of coupled oscillators or Josephson junction arrays. This method emphasizes the key role of conformal barycenter in some theories of Mathematical Physics. On the other hand, it indicates that variations of Kuramoto model might be used in different problems of computational quasiconformal geometry. The idea can be extended to compute the D–E extension in higher dimensions as well.

Douady-Earle extension of the strongly symmetric homeomorphism

It is shown that the complex dilatation of the Douady-Earle extension of a strongly symmetric homeomorphism induces a vanishing Carleson measure on the unit disk D. As application, it is proved that the VMO-Teichmuller space is a subgroup of the universal Teichmuller space.

On Douady–Earle extension and the contractibility of the VMO-Teichmüller space

Douady–Earle extension provides a quasiconformal extension of a quasisymmetric homeomorphism to the unit disk and induces a continuous self-map σ of Beltrami coefficients. In this note, we show that σ is also continuous (under a stronger topology) on those Beltrami coefficients which induce vanishing Carleson measures and thus project to the VMO-Teichmüller space. An immediate consequence is that the VMO-Teichmüller space is contractible.

Some characterizations of the integrable Teichmüller space

In this paper, we prove that the Bers projection of the integrable Teichmuller space is holomorphic. By using the Douady-Earle extension, we obtain some characterizations of the integrable Teichmuller space as well as the p-integrable asymptotic affine homeomorphism.

Cross-ratio distortion and Douady-Earle extension: II. Quasiconformality and asymptotic conformality are local

Let f be an orientation-preserving circle homeomorphism and Φ the Douady-Earle extension of f. In this paper, we show that the quasiconformality and asymptotic conformality of Φ are local properties; i.e., if f is quasisymmetric or symmetric on an arc of the unit circle, then Φ is quasiconformal or asymptotically conformal nearby. Furthermore, our methods enable us to conclude the global quasiconformality and asymptotic conformality from local properties. In the quasiconformal case, our methods also enable us to provide an upper bound for the maximal dilatation of Φ on a neighborhood of the arc in the open unit disk in terms of the cross-ratio distortion norm of f on the arc.

Cross-ratio distortion and Douady-Earle extension: I. A new upper bound on quasiconformality

Let f be an orientation-preserving circle homeomorphism and Φ (f) the Douady–Earle extension of f, and let ||f||cr be the cross-ratio distortion norm of f and K(Φ (f)) be the maximal dilatation of Φ (f). As a consequence of results in [A. Douady and C. J. Earle, ‘Conformally natural extension of homeomorphisms of circle’, Acta Math. 157 (1986) 23–48 and M. Lehtinen, ‘The dilatation of Beurling–Ahlfors extensions of quasisymmetric functions’, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983) 187–191], ln K(Φ (f)) has an upper bound depending on ||f||cr exponentially. In this paper, we first show that ln K(Φ (f)) has an upper bound depending on ||f||cr linearly. Then we extensively study the Douady–Earle extension of a very simple map fλ which depends on a real non-negative parameter λ. For this example, we show that, as λ→∞, (1) our new upper bound on ln K(Φ (fλ)) is substantially smaller than the one given by Douady and Earle in terms of the maximal dilatation of the extremal extension of fλ, and (2) the Douady–Earle extension Φ (fλ) stays exponentially far away from being extremal. Finally, we show that, in general, our upper bound on ln K(Φ (f)) implies the one of Douady and Earle when ||f||cr is large enough.

Möbius transformations and extended diffusion above the homeomorphisms of the disk

With the Douady–Earle extension of vector fields, see Douady and Earle (Acta Math 157(1–2):23–48, 1986) and Earle (Proc Am Soc 102(1):145–149, 1988), we construct the extension inside and outside the unit disk for the stochastic differential equation (SDE) defining the canonic Brownian motion above the homeomorphisms of the circle. The inside extension had been explicited in Airault et al. (J Funct Anal 259(12):3037–3079, 2010). We prove that the same extended SDE is valid inside and outside the unit disk. For this extended SDE, we discuss the behaviour of the solutions, in particular we prove that a solution with initial point inside (or outside) the unit disk, almost surely never reaches the unit circle in finite time. We obtain stochastic log-Lipschitz flows of homeomorphisms defined inside the unit disk as in Airault et al. (J Math Pures Appl (9) 83(8):955–1018, 2004), and also outside the unit disk.

On Conformal Mappings onto Quasidisks

In this paper, we prove a distortion theorem for conformal mappings whose images are quasidisks. As an application, we construct a quasicon-formal reflection with bounded partial derivatives near the reflection boundary based on the Douady-Earle extension.

Some estimates for normalized quasiconformal self-mappings of the unit disk

In this article, we provide an integral estimate for the boundary values of a normalized quasiconformal self-mapping of the unit disk. Estimates are also given for the partial derivative of the Douady–Earle extension.

Harmonic extensions of symmetric maps

We examine the behavior of the Douady-Earle extension and we prove that a symmetric self-map of the circle can be harmonically extended to a conformal self-map of the hyperbolic disc.

BMO-Teichmüller spaces

We show that the complex dilatation of the Douady-Earle extension of a strongly quasisymmetric homeomorphism produces a Carleson measure. As an application, we study the BMO-Teichmuller theory compatible with a Fuchsian group.

Conformal barycenters and the Douady-Earle extension—a discrete dynamical approach

The Douady-Earle extension produces a homeomorphism of a disk from a homeomorphism of its bounding circle. It is based on a center of mass computation at the center and is extended to the disk by naturality. Quasiconformal homeomorphisms are the extensions of quasisymmetric ones. There are several approaches to the numerical computation of the extension defined by a redistribution of mass. The algorithms of Abikoff-Ye and Milnor turn out to be the same —even in the more general situation of nontrivial probability measures on the unit circle. The numerical computation uses measures with finite support; in that case, the iterator has rational square. We obtain an easy approach to the proof of the validity of the algorithm and to its calculation. New proofs and additional properties of the computation of the barycenter and the extension are also presented.