Quasiregular maps on Carnot groups

  • Juha Heinonen
  • Ilkka Holopainen
Article

DOI: 10.1007/BF02921707

Cite this article as:
Heinonen, J. & Holopainen, I. J Geom Anal (1997) 7: 109. doi:10.1007/BF02921707

Abstract

In this paper we initiate the study of quasiregular maps in a sub-Riemannian geometry of general Carnot groups. We suggest an analytic definition for quasiregularity and then show that nonconstant quasiregular maps are open and discrete maps on Carnot groups which are two-step nilpotent and of Heisenberg type; we further establish, under the same assumption, that the branch set of a nonconstant quasiregular map has Haar measure zero and, consequently, that quasiregular maps are almost everywhere differentiable in the sense of Pansu. Our method is that of nonlinear potential theory. We have aimed at an exposition accessible to readers of varied background.

Math Subject Classification

30C65 31C45 58G03 

Key Words and Phrases

Quasiregular map Carnot group Heisenberg group nonlinear potential theory 

Copyright information

© Mathematica Josephina, Inc. 1997

Authors and Affiliations

  • Juha Heinonen
    • 1
    • 2
  • Ilkka Holopainen
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsUniversity of HelsinkiFinland

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