Heisenberg quasiregular ellipticity

Katrin Fässler; Anton Lukyanenko; Jeremy T. Tyson

doi:10.4171/rmi/1060

Heisenberg quasiregular ellipticity

Katrin Fässler, Anton Lukyanenko, Jeremy T. Tyson

Mathematics

Research output: Contribution to journal › Article › peer-review

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)
Pages (from-to)	471-520
Number of pages	50
Journal	Revista Matematica Iberoamericana
Volume	35
Issue number	2
DOIs	https://doi.org/10.4171/rmi/1060
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)

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title = "Heisenberg quasiregular ellipticity",

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.",

keywords = "3-sphere, Capacity, Contact manifold, Hopf link, Isoperimetric inequality, Link complement, Morphism property, Nonlinear potential theory, Quasiregular mapping, Sobolev–Poincar{\'e} inequality, Sub-Riemannian manifold",

author = "Katrin F{\"a}ssler and Anton Lukyanenko and Tyson, {Jeremy T.}",

note = "Funding Information: K. F. was supported by the Academy of Finland through the grant 285159 {\textquoteleft}Sub-Riemannian manifolds from a quasiconformal viewpoint{\textquoteright}. A. L. was supported by NSF RTG grant DMS-1045119. J.T.T. was supported by Simons Collaboration Grant 353627 {\textquoteleft}Geometric analysis in sub-Riemannian and metric spaces{\textquoteright} and NSF grants DMS-1201875 {\textquoteleft}Geometric mapping theory in sub-Riemannian and metric spaces{\textquoteright}. and DMS-1600650 {\textquoteleft}Mappings and measures in sub-Riemannian and metric spaces{\textquoteright}. Publisher Copyright: c European Mathematical Society",

year = "2019",

doi = "10.4171/rmi/1060",

language = "English (US)",

volume = "35",

pages = "471--520",

journal = "Revista Matematica Iberoamericana",

issn = "0213-2230",

publisher = "Universidad Autonoma de Madrid",

number = "2",

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T1 - Heisenberg quasiregular ellipticity

AU - Fässler, Katrin

AU - Lukyanenko, Anton

AU - Tyson, Jeremy T.

N1 - Funding Information: K. F. was supported by the Academy of Finland through the grant 285159 ‘Sub-Riemannian manifolds from a quasiconformal viewpoint’. A. L. was supported by NSF RTG grant DMS-1045119. J.T.T. was supported by Simons Collaboration Grant 353627 ‘Geometric analysis in sub-Riemannian and metric spaces’ and NSF grants DMS-1201875 ‘Geometric mapping theory in sub-Riemannian and metric spaces’. and DMS-1600650 ‘Mappings and measures in sub-Riemannian and metric spaces’. Publisher Copyright: c European Mathematical Society

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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KW - Capacity

KW - Contact manifold

KW - Hopf link

KW - Isoperimetric inequality

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KW - Morphism property

KW - Nonlinear potential theory

KW - Quasiregular mapping

KW - Sobolev–Poincaré inequality

KW - Sub-Riemannian manifold

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Heisenberg quasiregular ellipticity

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