Search results for "quasiregular mapping"

Sharpness of Rickman’s Picard theorem in all dimensions

2015

We show that given \({n \geqslant 3}\), \({q \geqslant 1}\), and a finite set \({\{y_1, \ldots, y_q \}}\) in \({\mathbb{R}^n}\) there exists a quasiregular mapping \({\mathbb{R}^n\to \mathbb{R}^n}\) omitting exactly points \({y_1, \ldots, y_q}\).

Integration by parts on generalized manifolds and applications on quasiregular maps

2016

Quasiregular ellipticity of open and generalized manifolds

2014

We study the existence of geometrically controlled branched covering maps from \(\mathbb R^3\) to open \(3\)-manifolds or to decomposition spaces \(\mathbb {S}^3/G\), and from \(\mathbb {S}^3/G\) to \(\mathbb {S}^3\).

Surface families and boundary behavior of quasiregular mappings

2005

We study the boundary behavior of bounded quasiregular mappings f : Bn(0, 1) → Rn, n ≥ 3. We show that there exists a large family of cusps, with vertices on the boundary sphere S n−1 (0, 1), so that the images of these cusps under f have finite (n − 1)-measure. peerReviewed

Heisenberg quasiregular ellipticity

2016

Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $S^3\backslash L$. The main result is obtained by translating a growth condition on $\pi_1(M)$ into the existence of a supersolution to the $4$-harmonic…

On BLD-mappings with small distortion

2021

We show that every $$L$$ -BLD-mapping in a domain of $$\mathbb {R}^{n}$$ is a local homeomorphism if $$L < \sqrt{2}$$ or $$K_I(f) < 2$$ . These bounds are sharp as shown by a winding map.

Rescaling principle for isolated essential singularities of quasiregular mappings

2012

We establish a rescaling theorem for isolated essential singularities of quasiregular mappings. As a consequence we show that the class of closed manifolds receiving a quasiregular mapping from a punctured unit ball with an essential singularity at the origin is exactly the class of closed quasiregularly elliptic manifolds, that is, closed manifolds receiving a non-constant quasiregular mapping from a Euclidean space.

Mappings of finite distortion : boundary extensions in uniform domains

2015

In this paper, we consider mappings on uniform domains with exponentially integrable distortion whose Jacobian determinants are integrable. We show that such mappings can be extended to the boundary and moreover these extensions are exponentially integrable with quantitative bounds. This extends previous results of Chang and Marshall on analytic functions, Poggi-Corradini and Rajala and Akkinen and Rajala on mappings of bounded and finite distortion.