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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Mappings of finite distortion: Formation of cusps II
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by Juhani Takkinen PDF
Conform. Geom. Dyn. 11 (2007), 207-218 Request permission

Abstract:

For $s>0$ given, we consider a planar domain $\Omega _s$ with a rectifiable boundary but containing a cusp of degree $s$, and show that there is no homeomorphism $f\colon \mathbb {R}^2\to \mathbb {R}^2$ of finite distortion with $\exp (\lambda K)\in L^1_{\mathrm {loc}}(\mathbb {R}^2)$ so that $f(B)=\Omega _s$ when $\lambda >4/s$ and $B$ is the unit disc. On the other hand, for $\lambda <2/s$ such an $f$ exists. The critical value for $\lambda$ remains open.
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Additional Information
  • Juhani Takkinen
  • Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Finland
  • Email: juhani@maths.jyu.fi
  • Received by editor(s): May 21, 2007
  • Published electronically: October 18, 2007
  • Additional Notes: The author was partially supported by the foundation 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), 207-218
  • MSC (2000): Primary 30C62, 30C65
  • DOI: https://doi.org/10.1090/S1088-4173-07-00170-1
  • MathSciNet review: 2354095