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Conformal Geometry and Dynamics

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Mappings of finite distortion: Formation of cusps II


Author: Juhani Takkinen
Journal: Conform. Geom. Dyn. 11 (2007), 207-218
MSC (2000): Primary 30C62, 30C65
DOI: https://doi.org/10.1090/S1088-4173-07-00170-1
Published electronically: October 18, 2007
MathSciNet review: 2354095
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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

Keywords: Cusp, homeomorphism, mapping of finite distortion
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.