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A quasisymmetric surface with no rectifiable curves

Author(s): Christopher J. Bishop
Journal: Proc. Amer. Math. Soc. 127 (1999), 2035-2040.
MSC (1991): Primary 30C65
Posted: February 18, 1999
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Abstract: There is a quasiconformal mapping of ${\Bbb R}^3$ to itself such that the image of ${\Bbb R}^2 \times \{0\}$ contains no rectifiable curves.

References:

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
Email: bishop@math.sunysb.edu

DOI: 10.1090/S0002-9939-99-04900-X
PII: S 0002-9939(99)04900-X
Keywords: Quasisymmetric maps, quasiconformal mappings, rectifiable curves, Jacobian
Received by editor(s): September 22, 1997
Posted: February 18, 1999
Additional Notes: The author was supported in part by NSF grant # DMS 95-00577.
Communicated by: Frederick W. Gehring
Copyright of article: Copyright 1999, American Mathematical Society


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