We propose a method by modulus of curve families to identify extremal quasiconformal mappings in the Heisenberg group.This approach allows study minimizers not only for maximal distortion but also mean functional, where candidate map is required have constant distortion.As counterpart classical Euclidean problem, we consider class between two spherical annuli group.Using logarithmic-type coordinates can define an analog radial stretch and discuss its properties both with respect...

Let π be the fundamental group of a closed surface Σ genus g > 1. One problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations into SU(2, 1), (the triple cover of) holomorphic isometries H2C. In particular, given representation ρ0 π1, can we an open neighbourhood comprising with these properties. We show that this indeed case when preserves totally real Lagrangian plane.

Abstract Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic 2-space are completely determined, up to conjugation in PU(2, 1), by three cross-ratios satisfying two real equations. We give global geometrical coordinates resulting variety.

Abstract Complex hyperbolic packs are hypersurfaces of complex plane H 2 ℂ which may be considered as dual to the well known bisectors. In this paper we study geometric aspects associated packs.

The modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary components onto each other. P. Belinskiĭ studied these inequalities in plane and identified family all minimisers. Beyond Euclidean framework, Grötzsch–Belinskiĭ-type inequality has been previously considered Heisenberg group whose boundaries are Korányi spheres. In this note we show that—in contrast to planar situation—the...

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a surface of revolution embedded in the Heisenberg group alttext="German H"> mathvariant="fraktur">H</mml:mi>...

Let $\partial{\bf H}^n_{\mathbb K}$ denote the boundary of a symmetric space rank-one and non-compact type let $d_{\mathfrak{H}}$ be Korányi metric defined in K}$. We prove that if $d$ is on such all Heisenberg similarities are $d$-Möbius maps, then under topological condition constant multiple power $d_{\mathfrak{H}}$.

In this paper we describe the geodesics on K\"ahler cone of Heisenberg group. Furthermore also prove that is not a complete manifold.

We define linear and radial stretch maps in the affine-additive group, prove that they are minimizers of mean quasiconformal distortion functional. For proofs we use a method based on notion modulus curve family minimal stretching property (MSP) afore-mentioned maps. MSP relies certain given families compatible with respective geometric settings strech

We consider the affine-additive group as a metric measure space with canonical left-invariant and sub-Riemannian metric. prove that this is locally 4-Ahlfors regular it hyperbolic, meaning has non-vanishing 4-capacity at infinity. This implies not quasiconformally equivalent to Heisenberg or roto-translation in contrast fact both of these groups are globally contactomorphic group. Moreover, each quasiregular map, from must be constant.

We study quakebend deformations in complex hyperbolic quasi-Fuchsian space Q C .†/ of a closed surface † genus g > 1, that is the discrete, faithful, totally loxodromic and geometrically finite representations fundamental group into isometries space.Emanating from an R-Fuchsian point 2 .†/, we construct curves associated to quakebending prove may always find open neighborhood U. / containing pieces such curves.Moreover, present generalisations well known Wolpert-Kerckhoff formulae for...