2016/05/03 by Chang-Yu Guo, Chang‐Yu Guo, Guo, Chang-Yu +2
Mathematics · #30C65 #53C17 #58C06 #58C25 #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.CV #math.DG #msc:30C65 #msc:53C17 #msc:58C06 #msc:58C25
paper · pdf · doi:10.48550/arxiv.1605.00916
subRiemannian manifolds, quasiregular mappings, Popp extension, P-differential, conformality
arxiv created 2016/05/03 · openalex publication_date 2016/05/03 · arxiv updated 2016/05/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We show that all the common definitions of quasiregular mappings f\colon M→ N between two equiregular subRiemannian manifolds of homogeneous dimension Q≥ 2 are quantitatively equivalent with precise dependences of the quasiregularity constants. As an immediate consequence, we obtain that if f is 1-quasiregular according to one of the definitions, then it is also 1-quasiregular according to any other definition. In particular, this recovers a recent theorem of Capogna et al. on the equivalence of 1-quasiconformal mappings. Our main results answer affirmatively a few open questions from the recent research. The main new ingredient in our proofs is the distortion estimates for particular local extensions of the horizontal metrics. These extensions are named "Popp extensions", and based on these extensions, we introduce a new natural and invariant definition of quasiregularity in the equiregular subRiemannian setting. The analysis on Popp extensions and on the implied distortion is also of independent interest.