Quasiregular Mappings on Sub-Riemannian Manifolds

Katrin Fässler*, Anton Lukyanenko, Kirsi Peltonen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)

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).

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

Keywords

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

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