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Quasiregular mappings between equiregular SubRiemannian manifolds

Chang-Yu Guo, Sebastiano Nicolussi Golo, Marshall Williams, Yi Xuan

Abstract

In this paper, we provide an alternative appraoch to an expectaion of Fässler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis-Mostow, Karmanova and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam and Tyson to establish the necessary analytic foundations.

Quasiregular mappings between equiregular SubRiemannian manifolds

Abstract

In this paper, we provide an alternative appraoch to an expectaion of Fässler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular subRiemannian manifolds of homogeneous dimension has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis-Mostow, Karmanova and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam and Tyson to establish the necessary analytic foundations.

Paper Structure

This paper contains 29 sections, 46 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev spaces
  4. Carnot groups
  5. Equiregular subRiemannian manifolds
  6. Tangent cone of an equiregular subRiemannian manifold
  7. Definition of a quasiregular mapping
  8. Analytic foundations of quasiregular mappings
  9. Analytic properties of quasiregular mappings
  10. Differentiability of quasiregular mappings
  11. The minimal upper gradient and Jacobian
  12. Equivalence of the two definitions of metric quasiregularity
  13. Size of the branch set
  14. Open questions
  15. Differentiability of Lipschitz mappings
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $f\colon M\to N$ be a weakly metrically quasiregular mapping between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$. Then

Theorems & Definitions (118)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 2.6: subRiemannian manifold
  • ...and 108 more