Quasiconformal Homeomorphisms Research Articles

Noded Teichmüller spaces

LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmuller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmuller space ofG supported in Δ. Some concrete examples are discussed.

Quasiconformal extension of biholomorphic mappings in several complex variables

Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC n. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of\(\bar B\). We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of\(\overline {R^{2n} } \) onto itself and give some applications.

Dirichlet solutions on bordered Riemann surfaces and quasiconformal mappings

In this paper, we consider quasiconformal homeomorphisms ϕn ;S 0→S n (n = 1, 2, ...) of a bordered Riemann surfaceS 0 and discuss how the Dirichlet solutions $$H_{fo\varphi n^{ - 1} }^{S_n } $$ for a continuous functionf on ϖS 0 vary when the maximal dilatations of ϕn converge to one. Furthermore, we consider the smoothness of Dirichlet solutions for parameters of the quasiconformal deformation.

Quasiconformal homeomorphisms and the convex hull boundary

We investigate the relationship between an open simply-connected region Ω ⊂ S^2 and the boundary Y of the hyperbolic convex hull in H^3 of S^2∖Ω. A counterexample is given to Thurston’s conjecture that these spaces are related by a 2-quasiconformal homeomorphism which extends to the identity map on their common boundary, in the case when the homeomorphism is required to respect any group of Mobius transformations which preserves Ω. We show that the best possible universal lipschitz constant for the nearest point retraction r:Ω → Y is 2. We find explicit universal constants 0 < c_2 < c_1, such that no pleating map which bends more than c_1 in some interval of unit length is an embedding, and such that any pleating map which bends less than c_2 in each interval of unit length is embedded. We show that every K-quasiconformal homeomorphism D^2 → D^2 is a (K,ɑ(K))-quasi-isometry, where ɑ(K) is an explicitly computed function. The multiplicative constant is best possible and the additive constant ɑ(K) is best possible for some values of K.

The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces

We consider the action of the Teichmuller modular group on the Teichmuller space of a topologically infinite Riemann surface. For a Riemann surface S, the Teichmuller space TðSÞ is the set of all equivalence classes of the pair ð f ; sÞ, where f : S ! Ss is a quasiconformal homeomorphism of S onto another Riemann surface Ss of a complex structure s. Two pairs ð f1; s1Þ and ð f2; s2Þ are considered to be equivalent if s1 1⁄4 s2 and f2 f 1 1 is isotopic to a conformal map. Here the isotopy is considered to be relative to the boundary at infinity. A distance between p1 1⁄4 1⁄2 f1; s1 and p2 1⁄4 1⁄2 f2; s2 in TðSÞ is defined by dðp1; p2Þ 1⁄4 log KðhÞ for an extremal quasiconformal homeomorphism h whose maximal dilatation KðhÞ is minimal in the isotopy class of f2 f 1 1 . Then d becomes a complete metric on TðSÞ, which is called the Teichmuller distance. The Teichmuller modular group ModðSÞ of S is a group of the isotopy classes of quasiconformal automorphisms of S. An element g of ModðSÞ acts on TðSÞ in such a way that 1⁄2 f ; s 7! 1⁄2 f g ; s , where g also denotes a representative of the isotopy class. It is evident from definition that ModðSÞ acts on TðSÞ isometrically with respect to the Teichmuller distance. In the case that TðSÞ is finite dimensional (equivalently S is of analytically finite type), ModðSÞ acts on TðSÞ properly discontinuously and the orbit of any point p A TðSÞ is discrete. However, when TðSÞ is infinite dimensional, these are not always true. See recent works [4], [5] and a monograph [6, Chap. 10]. For finite dimensional Teichmuller spaces, Bers [2] classified the elements of ModðSÞ by certain analytic criteria in comparison with Thurston’s topological classification. This can be extended to infinite dimensional Teichmuller spaces in the same way. For example, an element g A ModðSÞ is elliptic if g has a fixed point in TðSÞ, and parabolic if inf dðp; gðpÞÞ 1⁄4 0 where the infimum is taken over all points p in TðSÞ. An elliptic element is realized as a conformal automorphism of the Riemann surface corresponding to the fixed point of g. A

Compact families ofBMO-quasiconformal mappings I

Families of quasiconformal homeomorphisms of bounded mean oscillation (BMO-qc) are studied from the point of view of compacity (normality + closedness). Using Tukia's convergence theorems and David's existence and factorization theorem, a normality criterion for families ofBMO-qc homeomorphisms and a compacity criterion for families transforming a given compact subsetM1 in a compact subsetM2 are established.

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.

Rectifying separated nets

In our earlier paper [BuK] (see also McMullen’s paper [M]), we constructed examples of separated nets in the plane R2 which are not biLipschitz equivalent to the integer lattice Z2. These examples gave a negative answer to a question raised by H. Furstenberg and M. Gromov. Furstenberg asked this question in connection with Kakutani equivalence for R2-actions, [F]. Return times for a section of an R2-action form a separated net, and to represent the returns of an R2-action by a Z2-action, one has to have a biLipschitz identification of the return times for each point with Z2 (depending measurably on the point). As was pointed out to us by A. Katok, one can use a standard construction of R2-actions to represent our example as the set of return times for points from a set of positive measure, thus showing that not every section can be used (it is worth mentioning here that an old result of Katok [K] asserts that every R 2-action admits a section whose return times are biLipschitz equivalent to Z2). Gromov’s motivation for the question came from large scale geometry, and the definition of quasi-isometries. Two metric spaces are quasiisometric if they contain biLipschitz equivalent separated nets; hence one would like to know if the choice of separated net matters, and if a given space can contain nets which are not biLipschitz equivalent. This question is particularly interesting for spaces with cocompact isometry groups. The counterexample in [BuK] was based on a counterexample to another question which had been posed by J. Moser and M. Reimann in the 60’s, namely whether every positive continuous function on the plane is locally the Jacobian of a biLipschitz homeomorphism. Using well-known properties of quasi-conformal homeomorphisms, one can actually show that any

Direct products of quasiregular mappings

We prove that direct products of two quasiregular mappings are quasiregular under certain conditions of compatibility. Such a condition has been introduced by A.P. Karmazin for direct products of two quasiconformal homeomorphisms. Instead of the Markushevich–Pesin definition he used, we start with Reshetnyak's one, which allows us to consider normal neighborhoods, to establish properties of their direct products and to define the compatibility.

The Hilbert-Smith conjecture for quasiconformal actions

This note announces a proof of the Hilbert–Smith conjecture in the quasiconformal case: A locally compact group G G of quasiconformal homeomorphisms acting effectively on a Riemannian manifold is a Lie group. The result established is true in somewhat more generality.

The angular distribution of mass by Bergman functions

Let \mathbb D = {z : |z| &lt; 1} be the unit disk in the complex plane and denote by d\mathcal A two-dimensional Lebesgue measure on \mathbb D . For \epsilon &gt; 0 we define \sum_\epsilon = z:| arg z | &lt; \epsilon . We prove that for every \epsilon &gt; 0 there exists a \delta &gt; 0 such that if f is analytic, univalent and area-integrable on \mathbb D , and f(0) = 0 , then \int _{f^–1(\sum_\epsilon)} | f | d\mathcal A &gt; \delta \int_\mathbb D | f | d\mathcal A . This problem arose in connection with a characterization by Hamilton, Reich and Strebel of extremal dilatations for quasiconformal homeomorphisms of \mathbb D .

The groups of quasiconformal homeomorphisms on Riemann surfaces

Suppose M M is a connected Riemann surface. Let H ( M ) {\mathcal H}(M) denote the homeomorphism group of M M with the compact-open topology, and H Q C ( M ) {\mathcal H}^{\mathrm {QC}}(M) denote the subgroup of quasiconformal mappings of M M onto itself, and let H ( M ) 0 {\mathcal H}(M)_0 and H Q C ( M ) 0 {\mathcal H}^{\mathrm {QC}}(M)_0 denote the identity components of H ( M ) {\mathcal H}(M) and H Q C ( M ) {\mathcal H}^{\mathrm {QC}}(M) respectively. In this paper we show that the pair ( H ( M ) 0 , H Q C ( M ) 0 ) ({\mathcal H}(M)_0, {\mathcal H}^{\mathrm {QC}}(M)_0) is an ( s , Σ ) (s, \Sigma ) -manifold, and determine their topological types.

Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings

In this paper we shall show that the boundary 9Ip,q of the hyperbolic building Ip,q considered by M. Bourdon admits Poincare type inequalities. Then by using Heinonen-Koskela's work, we shall prove Loewner capacity estimates for some families of curves of WIp,q and the fact that every quasiconformal homeomorphism f : 0Ip,q d9Ip,q is quasisymmetric. Therefore by these results, the answer to questions 19 and 20 of Heinonen and Semmes (Thirty-three YES or NO questions about mappings, measures and metrics, Conform Geom. Dyn. 1 (1997), 1-12) is NO.

Bending deformations of complex hyperbolic surfaces.

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, our main goal is to show that they enjoy nevertheless the flexibility of low-dimensional real hyperbolic manifolds. Namely we define a class of ``bending deformations of a given (Stein) complex surface $M$ associated with its closed geodesics provided that $M$ is homotopy equivalent to a Riemann surface whose embedding in $M$ has a non-trivial totally real geodesic part. Such bending deformations bend $M$ along its closed geodesics and are induced by equivariant quasiconformal homeomorphisms of the complex hyperbolic space and its Cauchy-Riemannian structure at infinity.

Quasiconformally Equivalent Domains and Quasiconformal Extensions

This paper concerns the extendability of boundary quasiconformal homeomorphisms of domains in and the problem of determining whether or not a domain is quasiconformally equivalent to a given domain. In particular, we prove that any boundary QC homeomorphism of domains bounded by finitely many QC-flat spheres has a QC extension to the entire domain, generalizing some well known results. We also show that if a portion of a domain D near its boundary can be mapped quasiconformally into a portion of a topological n-ball (n ≠ 4) D′ near its boundary, then D is QC equivalent to D′ generalizing a result of Gehring.

Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Let X X and Y Y be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of X X onto the Teichmüller space of Y Y is induced by a quasiconformal homeomorphism of X X onto Y Y . These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.

Uniform domains and quasiconformal mappings on the Heisenberg group

We prove that in the Heisenberg group the image of a uniform domain under a global quasiconformal homeomorphism is still a uniform domain. As a consequence, the class of NTA (non-tangentially accessible) domains in the Heisenberg group is also quasiconformally invariant. A large class of non-differentiable Lipschitz quasiconformal homeomorphisms is constructed. The images of smooth domains under these rough mappings give a class of non-smooth NTA domains in the Heisenberg group.

Homeomorphisms that induce monomorphisms of Sobolev spaces

LetG, G′ be domains in ℝn. We obtain a geometrical description of the class of all homeomorphisms ϕ:G→ G′ that induce bounded operators ϕ* from the seminormed Sobolev spaceL p 1 (G′) toL p 1 (G) by the rule ϕ* u =u o ϕ. Forp-Poincare domains the same classes of homeomorphisms induce bounded operators for classical Sobolev spacesW p 1 . These classes of homeomorphisms are natural generalizations of the class of quasiconformal homeomorphisms that correspond to the casep=n. We demonstrate some applications of our results for embedding theorems in domains with Holder singularities.

Reconstruction of domains from their groups of quasiconformal autohomeomorphisms

For an open set U ⊆ R n , let QC( U) denote the group of all quasiconformal homeo morphism of U. The following is our first main result. Let U ⊆ R m and V ⊆ R n be open, and suppose that τ is a group isomorphism between QC( U) and QC( V). Then there is a quasiconformal homeomorphism ϑ from U onto V such that ϕ induce τ. That is, for every - f ϵ QC( U): τ(-) = ϑ - to - to ϑ −1.

Quasiconformal homeomorphisms on CR 3-manifolds with symmetries

An extremal quasiconformal homeomorphisms in a class of homeomorphisms between two CR 3-manifolds is an one which has the least conformal distortion among this class. This paper studies extremal quasiconformal homeomorphisms between CR 3-manifolds which admit transversal CR circle actions. Equivariant $K$-quasiconformal homeomorphisms are characterized by an area-preserving property and the $K$-quasiconformality of their quotient maps on the spaces of $S^1$-orbits. A large family of invariant CR structures on $S^3$ is constructed so that the extremal quasiconformal homeomorphisms among the equivariant mappings between them and the standard structure are completely determined. These homeomorphisms also serve as examples showing that the extremal quasiconformal homeomorphisms between two invariant CR manifolds are not necessarily equivariant.