The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms
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- by Gaven Martin, Volker Mayer and Kirsi Peltonen PDF
- Proc. Amer. Math. Soc. 134 (2006), 2091-2097 Request permission
Abstract:
A uqr mapping of an $n$-manifold $M$ is a mapping which is rational with respect to a bounded measurable conformal structure on $M$. Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.References
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Additional Information
- Gaven Martin
- Affiliation: Institute of Information and Mathematical Sciences, Massey University, Auckland, New Zealand
- MR Author ID: 120465
- Email: g.j.martin@massey.ac.nz
- Volker Mayer
- Affiliation: UFR de Mathématiques, UMR 8524 du CNRS, Université de Lille I, 59655 Villeneuve d’Ascq Cedex, France
- MR Author ID: 333982
- Email: volker.mayer@univ-lille1.fr
- Kirsi Peltonen
- Affiliation: Helsinki University of Technology, P.O. Box 1100, FIN-02015 Espoo, Finland
- Email: kirsi.peltonen@helsinki.fi
- Received by editor(s): August 28, 2003
- Received by editor(s) in revised form: February 17, 2005
- Published electronically: January 6, 2006
- Additional Notes: This research was supported in part by the Marsden Fund (NZ) and The Royal Society of NZ (James Cook Fellowship)
- Communicated by: Juha M. Heinonen
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2091-2097
- MSC (2000): Primary 30D05; Secondary 32H02
- DOI: https://doi.org/10.1090/S0002-9939-06-08201-3
- MathSciNet review: 2215779