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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2024 MCQ for Conformal Geometry and Dynamics is 0.49.

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On distortion of Hausdorff measures under quasiconformal mappings
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by István Prause
Conform. Geom. Dyn. 11 (2007), 219-223
DOI: https://doi.org/10.1090/S1088-4173-07-00171-3
Published electronically: October 31, 2007
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Abstract:

Astala (Acta Math. 173 (1994), no. 1, 37–60) gave optimal bounds for the distortion of Hausdorff dimension under planar quasiconformal maps. The corresponding estimates on the level of Hausdorff measures remain open. We show that these techniques allow for establishing absolute continuity for some weaker Hausdorff measures.
References
  • Kari Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), no. 1, 37–60. MR 1294669, DOI 10.1007/BF02392568
  • Kari Astala and Vincenzo Nesi, Composites and quasiconformal mappings: new optimal bounds in two dimensions, Calc. Var. Partial Differential Equations 18 (2003), no. 4, 335–355. MR 2020365, DOI 10.1007/s00526-003-0145-9
  • K. Astala, A. Clop, J. Mateu, J. Orobitg and I. Uriarte-Tuero, Distortion of Hausdorff measures and improved Painlevé removability for bounded quasiregular mappings, Duke Math. J., to appear, preprint version: arXiv:math.CV/0609327.
  • C.J. Bishop, Distortion of disks by conformal maps, preprint: \verb+http://www.math.sunysb.edu/ bishop/papers/papers.html+
  • I. Prause, Distortion of dimension under quasiconformal mappings, Preprint 462, Reports in Mathematics, University of Helsinki, 2007.
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Bibliographic Information
  • István Prause
  • Affiliation: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland
  • Address at time of publication: Institute of Mathematics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
  • Email: istvan.prause@helsinki.fi
  • Received by editor(s): August 10, 2007
  • Published electronically: October 31, 2007
  • Additional Notes: The author was supported by the Finnish Academy of Science and Letters through Vilho, Yrjö ja Kalle Väisälän rahasto.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 11 (2007), 219-223
  • MSC (2000): Primary 30C62
  • DOI: https://doi.org/10.1090/S1088-4173-07-00171-3
  • MathSciNet review: 2354096