Quasiconformal mappings in the hyperbolic Heisenberg group and a lifting theorem (original) (raw)

Quasiconformal mappings on the Heisenberg group:An overview

We present a brief overview of the Korányi-Reimann theory of quasiconformal mappings on the Heisenberg group stressing on the analogies as well as on the differences between the Heisenberg group case and the classical two-dimensional case. We examine the extensions of the theory to more general spaces and we state some known results and open problems.

Hyperbolic Geometry and Quasiconformal Mappings

2005

The interaction between hyperbolic geometry and conformal analysis is a beautiful and fruitful aspect of the fields of analysis and low-dimensional geometry-topology. In particular, the study of hyperbolic geometry intertwines complex analysis, geometric function theory (especially in the guise of the study of quasiconformal mappings), and topology in a way that allows one to study a fixed object from diverse perspectives.

K.: Weak contact equations for mappings into Heisenberg groups

2016

Let k > n be positive integers. We consider mappings from a subset of R k to the Heisenberg group H n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H n is purely k-unrectifiable. We also prove that for an open set Ω ⊂ R k , the rank of the weak derivative of a weakly contact mapping in the Sobolev space W 1,1 loc (Ω; R 2n+1) is bounded by n almost everywhere, answering a question of Magnani. Finally we prove that if f : Ω → H n is α-Hölder continuous, α > 1/2, and locally Lipschitz when considered as a mapping into R 2n+1 , then f cannot be injective. This result is related to a conjecture of Gromov.

Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group

Journal d'Analyse Mathématique, 2001

We construct quasiconformal mappings on the Heisenberg group which change the Hansdorffdimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortion.

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

Journal of Geometric Analysis, 2010

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactificationH 1 = H 1 ∪ {∞} of the Heisenberg group H 1 equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere S n was proven by Martin (Conform. Geom. Dyn. 1:24-27, 1997). Moreover, we construct uniformly quasiregular mappings on H 1 with large-dimensional branch sets. We prove that for any uniformly quasiregular map g onH 1 there exists a measurable CR structure μ which is equivariant under the semigroup generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.

Weak contact equations for mappings into Heisenberg groups

Let k > n be positive integers. We consider mappings from a subset of R k to the Heisenberg group H n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H n is purely k-unrectifiable. We also prove that for an open set Ω ⊂ R k , the rank of the weak derivative of a weakly contact mapping in the Sobolev space W 1,1 loc (Ω; R 2n+1) is bounded by n almost everywhere, answering a question of Magnani. Finally we prove that if f : Ω → H n is α-Hölder continuous, α > 1/2, and locally Lipschitz when considered as a mapping into R 2n+1 , then f cannot be injective. This result is related to a conjecture of Gromov.

Analytic properties of quasiconformal mappings on Carnot groups

Siberian Mathematical Journal, 1995

In a series of recent articles, the properties of nilpotent Lie groups and related objects have undergone intensive study in connection with various problems of sub-Riemannian geometry, analysis, and subelliptic differential equations. The analytic questions in such problems are primarily connected with the presence of nontrivial commutation relations which, as a rule, prohibit straightforward translation of the technique developed for similar problems in Euclidean space. Such difficulties appear, for instance, in the problem concerning the differential properties of quasiconformal mappings on Carnot groups which is studied in the present article. A metric definition of a quasiconformal mapping can be given in an arbitrary metric space (see, for instance, [1]). However, for developing the theory of quasiconformal mappings, in particular for establishing their analytic properties, the domain of definition must possess some extra structure.

Complex hyperbolic quasi-Fuchsian groups

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current state of the art of the theory of complex hyperbolic quasi-Fuchsian groups.

Hyperbolic structures on surfaces

Cambridge University Press eBooks, 1988

We give a brief introduction to hyperbolic structures on surfaces. Using the concepts of developing map and holonomy, we sketch a proof that every surface equipped with a complete hyperbolic metric is isometric to a quotient of H by a Fuchsian group. We then define Teichmüller spaces and explain Fenchel-Nielsen coordinates. Finally, we introduce mapping class groups and show that they act properly discontinuously on Teichmüller space.