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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
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\bR^2\to\mathbb{R}^2$ of finite distortion with $ \exp(\lambda K)\in L^1_{\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

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.

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