Abstract
Following the Euclidean results of Varopoulos and Pankka–Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold M to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group H. As an application, we show that a link complement S3\L has a sub-Riemannian metric admitting such a mapping only if L is empty, an unknot or Hopf link. In the converse direction, if L is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from H to S3\L. The main result is obtained by translating a growth condition on π1(M) into the existence of a supersolution to the 4-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.
Original language | English (US) |
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Pages (from-to) | 471-520 |
Number of pages | 50 |
Journal | Revista Matematica Iberoamericana |
Volume | 35 |
Issue number | 2 |
DOIs | |
State | Published - 2019 |
Keywords
- 3-sphere
- Capacity
- Contact manifold
- Hopf link
- Isoperimetric inequality
- Link complement
- Morphism property
- Nonlinear potential theory
- Quasiregular mapping
- Sobolev–Poincaré inequality
- Sub-Riemannian manifold
ASJC Scopus subject areas
- Mathematics(all)