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The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

Authors: Gaven Martin, Volker Mayer and Kirsi Peltonen
Journal: Proc. Amer. Math. Soc. 134 (2006), 2091-2097
MSC (2000): Primary 30D05; Secondary 32H02
Published electronically: January 6, 2006
MathSciNet review: 2215779
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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.

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  • [F] J. Ferrand, Transformations conformes et quasi-conformes des variétés riemanniennes compactes (démonstration de la conjecture de A. Lichnerowicz), Acad. Roy. Belg. Cl. Sci. Mém. Coll. in-8$ \deg $(2), 39, 44 pp. (1971). MR 0322739 (48:1100)
  • [GM] F. Gehring and G. Martin, Discrete quasiconformal groups I, Proc. London Math. Soc. (3), 55, (1987), 331-358. MR 0896224 (88m:30057)
  • [HMM] A. Hinkkanen, G. Martin and V. Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand., 95, (2004), 80-100. MR 2091483 (2005f:37094)
  • [HR] I. Holopainen and S. Rickman, Ricci curvature, Harnack functions and Picard type theorems for quasiregular maps, Analysis and Topology, World Sci. Publ., (1998), 315-326. MR 1667818 (99j:30026)
  • [IM1] T. Iwaniec and G. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Ser. AI Math., 21, (1996), 241-254. MR 1404085 (97i:30032)
  • [IM2] T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, 2001. MR 1859913 (2003c:30001)
  • [L] A. Lichnerowicz, Sur les transformations conformes d'une variété riemannienne compacte. (Italian) C. R. Acad. Sci. Paris, 259, (1964), 697-700. MR 0166734 (29:4007)
  • [MM] G. Martin and V. Mayer, Rigidity in holomorphic and quasiregular dynamics, Trans. Amer. Marth. Soc. 355, (2003), 4349-4363. MR 1990755 (2004i:37095)
  • [M1] V. Mayer, Cyclic parabolic quasiconformal groups that are not the quasicoformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. AI Math., 18, (1993), 147-154. MR 1207901 (95f:30032)
  • [M2] V. Mayer, Uniformly quasiregular mappings of Lattès type, Conformal Geometry and Dynamics, 1, (1997), 104-111. MR 1482944 (98j:30017)
  • [P] K. Peltonen, Examples of UQR mappings, Conformal Geometry and Dynamics, 3, (1999), 158-163. MR 1718708 (2001i:30017)
  • [Ri] S. Rickman, Simply connected quasiregularly elliptic $ 4$-manifolds, preprint.
  • [R] W. Rinow, Die innere Geometrie der Metrischen Räume, Springer, 1961. MR 0123969 (23:A1290)
  • [TV] P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension., Ann. Acad. Sci. Fenn. Ser. A I Math., 6, (1981), 303-342 (1982) MR 0658932 (84a:57016)
  • [VSC] N.Th. Varopoulos, L. Saloff-Coste and T.Coulhon, Analysis and geometry on groups, Cambridge Univ. Press, Cambridge, 1992. MR 1218884 (95f:43008)
  • [W] J.A. Wolf, Spaces of constant curvature, McGraw-Hill Series in Higher Mathematics (1967). MR 0217740 (36:829)

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Additional Information

Gaven Martin
Affiliation: Institute of Information and Mathematical Sciences, Massey University, Auckland, New Zealand

Volker Mayer
Affiliation: UFR de Mathématiques, UMR 8524 du CNRS, Université de Lille I, 59655 Villeneuve d’Ascq Cedex, France

Kirsi Peltonen
Affiliation: Helsinki University of Technology, P.O. Box 1100, FIN-02015 Espoo, Finland

Keywords: Uniformly quasiregular mappings, space forms
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
Article copyright: © Copyright 2006 American Mathematical Society

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