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A quasisymmetric surface
with no rectifiable curves

Author: Christopher J. Bishop
Journal: Proc. Amer. Math. Soc. 127 (1999), 2035-2040
MSC (1991): Primary 30C65
Published electronically: February 18, 1999
MathSciNet review: 1610908
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Abstract | References | Similar Articles | Additional Information

Abstract: There is a quasiconformal mapping $f$ of ${\Bbb R}^3$ to itself such that the image of ${\Bbb R}^2 \times \{0\}$ contains no rectifiable curves.

References [Enhancements On Off] (What's this?)

  • 1. Assouad, P. (1983) Prolongements Lipschitziens dans ${\Bbb R}^n$, Bull. Soc. Math. France 111, 429-448. MR 86f:54050
  • 2. Heinonen, J. and Semmes, S. (1997) Thirty-three yes or no questions about mappings, measures and metrics, Conf. Geometry and Dynamics 1, 1-12. CMP 97:13
  • 3. Heinonen, J. and Koskela, P. (1994) The boundary distortion of a quasiconformal mapping, Pacific J. Math 165, 93-114. MR 95f:30031
  • 4. P. Tukia and J. Väisälä (1980) Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5, 97-114. MR 82g:30038

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Additional Information

Christopher J. Bishop
Affiliation: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651

Keywords: Quasisymmetric maps, quasiconformal mappings, rectifiable curves, Jacobian
Received by editor(s): September 22, 1997
Published electronically: February 18, 1999
Additional Notes: The author was supported in part by NSF grant # DMS 95-00577.
Communicated by: Frederick W. Gehring
Article copyright: © Copyright 1999 American Mathematical Society

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