Quasiconformal Homeomorphisms Research Articles

Quasiconformal homeomorphisms between Riemann surfaces

Theorem. Let f : Ĉ → Ĉ be a K-qc (K-quasiconformal) homeomorphism with f(∞) = ∞. If zj , j = 0, 1, 2, are three distinct points in C and z′ j = f(zj), then |z1 − z0| ≤ |z2 − z0| implies |z′ 1 − z′ 0| ≤ e |z′ 2 − z′ 0| . Our initial aim was to extend this theorem to K-qc homeomorphisms between Riemann surfaces. It was an immediate observation that Gehring’s theorem may be rewritten by means of Evans–Selberg potential or the Sario capacity function and we shall use the latter to cover both the parabolic and the hyperbolic cases. Indeed, the Sario capacity function of C with respect to z0 is pC(z, z0) = log |z−z0|. On the other hand, f may be interpreted by restriction as a K-qc homeomorphism f |C : C → C, so that if we denote pC(z, z0) by τ and pC(z, z′ 0) by τ ′, Gehring’s theorem takes the form:

Uniform and Sobolev Extension Domains

We prove that if a domain $D \subset {{\mathbf {R}}^n}$ is quasiconformally equivalent to a uniform domain, then $D$ is an extension domain for the Sobolev class $W_n^1$ if and only if $D$ is locally uniform. We provide examples which suggest that this result is best possible. We exhibit a list of equivalent conditions for domains quasiconformally equivalent to uniform domains, one of which characterizes the quasiconformal homeomorphisms between uniform and locally uniform domains.

Uniform and Sobolev extension domains

We prove that if a domain D ⊂ R n D \subset {{\mathbf {R}}^n} is quasiconformally equivalent to a uniform domain, then D D is an extension domain for the Sobolev class W n 1 W_n^1 if and only if D D is locally uniform. We provide examples which suggest that this result is best possible. We exhibit a list of equivalent conditions for domains quasiconformally equivalent to uniform domains, one of which characterizes the quasiconformal homeomorphisms between uniform and locally uniform domains.

The Limiting Behavior of Sequences of Quasiconformal Mappings

AbstractThe limiting behavior of sequences of quasiconformal homeomorphisms of the n-sphere Sn is studied using a substitute to the Poincaré extension of Möbius transformations introduced by Tukia. Adapted versions of the limit set and the conical limit set known in the theory of Kleinian groups are utilized. Most of the results also hold for families of homeomorphisms of Sn with the convergence property introduced by Gehring and Martin.

Quasiconformal Mappings Onto John Domains

In this paper we study quasiconformal homeomorphisms of the unit ball B = Bn = {x I Rn: |x| < 1} of Rn onto John domains. We recall that John domains were introduced by F. John in his study of rigidity of local quasi-isometries [J]; the term John domain was coined by O. Martio and J. Sarvas seventeen years later [MS]. From the various equivalent characterizations we shall adapt the following definition based on diameter carrots, cf. [V4], [V5], [NV].

Quasiconformal 4-manifolds

For any pseudo-group of homeomorphism of Euclidean space one can define the corresponding category of manifolds. The most familiar examples in Topology are the full pseudo-group of homeomorphisms, giving rise to the theory of topological manifolds, and the subgroup of smooth differomorphisms giving rise to the theory of C ~ manifolds. In this paper, we discuss an intermediate category--quasiconformal homeomorphisms and manifolds. Recall that a homeomorphism 9 : D ~ R n from a domain D in R n to its image 9(D) is K quasiconformal if for all x in D

The argument of an extremal dilatation

Suppose κ \kappa is the complex dilatation of an extremal quasiconformal homeomorphism of the unit disk onto itself. Then, except in special cases, the width of the set of arguments of κ \kappa must strictly exceed π \pi .

The rectifiability of certain quasicircles

Suppose that the quasiconformal homeomorphism fof the complex plane fixes the point at infinity, is conformal in the exterior of the unit disk, and that the complex dilatation μ of f satisfies for some positive δ. It is shown that then fmaps the unit circle onto a rectifiable quasicircle.

Stability property of Möbius mappings

Let F F be an arbitrary class of continuous mappings acting and ranging on domains in R n {{\mathbf {R}}^n} which is invariant under similarity transformations of R n {{\mathbf {R}}^n} and the restriction of a map to any subdomain. The class Möb of Möbius mappings acting in R n {{\mathbf {R}}^n} is of particular interest. Assume that the class F F is " c c -uniformly close" to Möb. Then we show that any map in F F is either constant or a local quasiconformal homeomorphism. As a corollary we obtain a distinctly elementary proof of the Local Injectivity Theorem for quasiregular mappings.

Quasiconformal and Bi-Lipschitz Homeomorphisms, Uniform Domains and the Quasihyperbolic Metric

Let $D$ be a proper subdomain of ${R^n}$ and ${k_D}$ the quasihyperbolic metric defined by the conformal metric tensor $d{\overline s ^2} = \operatorname {dist} {(x,\partial D)^{ - 2}}d{s^2}$. The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for ${k_D}$; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in ${R^n}$, in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.

Quasiconformal Homeomorphisms and Dynamics I. Solution of the Fatou-Julia Problem on Wandering Domains

On demontre un resultat dynamique sur l'iteration d'une application analytique complexe sur la sphere de Riemann

Deformations of hyperbolic structures

In this paper I shall define a deformation of hyperbolic structures on n-manifolds, called ‘bending’, by constructing a family of quasiconformal homeomorphisms of hyperbolic (n+ l)-space. The deformation can be applied to manifolds which contain totally geodesic submanifolds of codimension 1.

Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups

Sous groupe de PSl(2,C). Familles holomorphes d'homeomorphismes de C•λ-lemme. Champs de ligne invariants. Familles holomorphes de groupes abstraitement isomorphes, car finiment genere. La stabilite structurelle implique geometriquement fini ou rigide. Theoreme A avec paraboles. Stabilite structurelle au sens de la dynamique differentiable

Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric

Let D D be a proper subdomain of R n {R^n} and k D {k_D} the quasihyperbolic metric defined by the conformal metric tensor d s ¯ 2 = dist ⁡ ( x , ∂ D ) − 2 d s 2 d{\overline s ^2} = \operatorname {dist} {(x,\partial D)^{ - 2}}d{s^2} . The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for k D {k_D} ; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in R n {R^n} , in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.

Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms

Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms

Remarks on the isomorphisms of certain spaces of harmonic differentials induced from quasiconformal homeomorphisms

Remarks on the isomorphisms of certain spaces of harmonic differentials induced from quasiconformal homeomorphisms

EXTREMAL PROBLEMS ON CLASSES OF QUASI-CONFORMAL HOMEOMORPHISMS

This work is devoted to obtaining necessary conditions for the extremum of an arbitrary smooth functional on sets of quasi-conformal homeomorphisms of one finite Riemann surface onto another, in terms of its differential. Bibliography: 10 titles.

One-parameter groups of quasi-conformal homeomorphisms in Euclidean space

One-parameter groups of quasi-conformal homeomorphisms in Euclidean space

Square integrable differentials on Riemann surfaces and quasiconformal mappings

If $f:R \to R’$ is a quasiconformal homeomorphism of Riemann surfaces, then Marden showed that $f$ naturally induces an isomorphism of the corresponding Hilbert spaces of square integrable first-order differential forms. It is demonstrated that this isomorphism preserves many important subspaces. Preliminary to this, various facts about the subspace of semiexact differentials are derived; especially, the orthogonal complement is identified. The norm of the isomorphism is $K(f)^{1/2}$ where $K(f)$ is the maximal dilatation of $f$. In addition, $f$ defines an isomorphism of the square integrable harmonic differentials and some important subspaces are preserved. It is shown that not all important subspaces are preserved. The relationship of this to other work is investigated; in particular, the connection with the work of Nakai on the isomorphism of Royden algebras induced by a quasiconformal mapping is explored. Finally, the induced isomorphisms are applied to the classification theory of Riemann surfaces to show that various types of degeneracy are quasiconformally invariant.

Quasiconformal Mappings and Royden Algebras in Space

On every open connected set $G$ in Euclidean $n$-space ${R^n}$ and for every index $p > 1$, we define the Royden $p$-algebra ${M_p}(G)$. We use results by F. W. Gehring and W. P. Ziemer to prove that two such sets $G$ and $G’$ are quasiconformally equivalent if and only if their Royden $n$-algebras are isomorphic as Banach algebras. Moreover, every such algebra isomorphism is given by composition with a quasiconformal homeomorphism between $G$ and $G’$. This generalizes a theorem by M. Nakai concerning Riemann surfaces. In case $p \ne n$, the only homeomorphisms which induce an isomorphism of the $p$-algebras are the locally bi-Lipschitz mappings, and for $1 < p < n$, every such isomorphism arises this way. Under certain restrictions on the domains, these results extend to the Sobolev space $H_p^1(G)$ and characterize those homeomorphisms which preserve the $H_p^1$ classes.