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Conformal Geometry and Dynamics

ISSN 1088-4173



Boundary behavior of quasi-regular maps and the isodiametric profile

Authors: Bruce Hanson, Pekka Koskela and Marc Troyanov
Journal: Conform. Geom. Dyn. 5 (2001), 81-99
MSC (2000): Primary 30C65
Published electronically: September 6, 2001
MathSciNet review: 1872158
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Abstract | References | Similar Articles | Additional Information

Abstract: We study obstructions for a quasi-regular mapping $f:M\rightarrow N$of finite degree between Riemannian manifolds to blow up on or collapse on a non-trivial part of the boundary of $M$.

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  • 1. L.V. Ahlfors, On quasi-conformal mappings. J. Analyse Math. 3 (1954), 1-58.
  • 2. M. Bonk, P. Koskela and S. Rohde, Conformal metrics on the unit ball in Euclidean space. Proc. London Math. Soc. 77 (1998), 635-664. MR 99f:30033
  • 3. I. Chavel, Riemannian Geometry - A Modern Introduction. Cambridge University Press (1993). MR 95j:53001
  • 4. L.C. Evans and R.F. Gariepy, Measure Theory and Fine properties of Functions. Studies in Advanced Mathematics (1992). MR 93f:28001
  • 5. B. Fuglede, Extremal length and functional completion. Acta. Math. 98 (1957), 171-219. MR 20:4187
  • 6. F.W. Gehring, Rings and quasi-conformal mappings in space. Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 25:3166
  • 7. J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs (1993). MR 94e:31003
  • 8. J. Heinonen and S. Rohde, The Gehring-Hayman inequality for quasihyperbolic geodesics. Math. Proc. Cambridge Philos. Soc. 114 (1993), 393-405. MR 94j:30019
  • 9. O. Martio and W.P. Ziemer, Lusin's condition (N) and mappings with nonnegative jacobian. Michigan Math. J. 39 (1992), 495-508. MR 93h:26021
  • 10. G.D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms. Publ. Math. IHES 34 (1969), 53-104. MR 38:4679
  • 11. Yu.G. Reshetnyak, Space Mappings with Bounded Distortion. Translations of Mathematical Monographs 73, 1989. MR 90d:30067
  • 12. Yu.G. Reshetnyak, Two-dimensional manifolds of bounded curvature. Encyclopedia of Math. Sciences, vol. 70 Geometry IV, Springer 1993. CMP 94:08
  • 13. S. Rickman, Quasi-regular Mappings. Springer Ergebnisse der Mathematik 26, 1993. MR 95g:30026
  • 14. M. Troyanov, Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 (1991), 793-821. MR 91h:53059
  • 15. M. Troyanov, Parabolicity of Manifolds. Siberian Adv. Math. 9 (1999), 125-150. MR 2001e:31013
  • 16. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings. Springer Lect. Notes. in Math. 229, 1971. MR 56:12260
  • 17. V.A. Zorich and V.M. Kesel'man, On the Conformal Type of a Riemannian Manifold. Funct. Anal. Appl. 30 (1996), 106-117. MR 97f:31022

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Additional Information

Bruce Hanson
Affiliation: Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057

Pekka Koskela
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

Marc Troyanov
Affiliation: Department of Mathematics, Ecole Polytechnique Federale de Lausanne (EPFL), 1015 Lausanne, Switzerland

Received by editor(s): June 4, 2001
Published electronically: September 6, 2001
Additional Notes: The second author was supported in part by the Academy of Finland grants 39788 and 41933
Article copyright: © Copyright 2001 American Mathematical Society

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