Quasiregular Mappings on Sub-Riemannian Manifolds

Research output: Contribution to journalArticleScientificpeer-review

Researchers

Research units

  • University of Bern
  • University of Michigan, Ann Arbor

Abstract

We study mappings on sub-Riemannian manifolds which are quasiregular with respect to the Carnot–Carathéodory distances and discuss several related notions. On H-type Carnot groups, quasiregular mappings have been introduced earlier using an analytic definition, but so far, a good working definition in the same spirit is not available in the setting of general sub-Riemannian manifolds. In the present paper we adopt therefore a metric rather than analytic viewpoint. As a first main result, we prove that the sub-Riemannian lens space admits nontrivial uniformly quasiregular (UQR) mappings, that is, quasiregular mappings with a uniform bound on the distortion of all the iterates. In doing so, we also obtain new examples of UQR maps on the standard sub-Riemannian spheres. The proof is based on a method for building conformal traps on sub-Riemannian spheres using quasiconformal flows, and an adaptation of this approach to quotients of spheres. One may then study the quasiregular semigroup generated by a UQR mapping. In the second part of the paper we follow Tukia to prove the existence of a measurable conformal structure which is invariant under such a semigroup. Here, the conformal structure is specified only on the horizontal distribution, and the pullback is defined using the Margulis–Mostow derivative (which generalizes the classical and Pansu derivatives).

Details

Original languageEnglish
Pages (from-to)1754-1794
Number of pages41
JournalJOURNAL OF GEOMETRIC ANALYSIS
Volume26
Issue number3
Publication statusPublished - 1 Jul 2016
MoE publication typeA1 Journal article-refereed

    Research areas

  • Conformal structures, Lens spaces, Metric spaces, Quasiregular mappings, Sub-Riemannian manifolds, Uniformly quasiregular mappings

ID: 6632419