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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Contracting convex immersed closed plane curves with slow speed of curvature


Authors: Yu-Chu Lin, Chi-Cheung Poon and Dong-Ho Tsai
Journal: Trans. Amer. Math. Soc. 364 (2012), 5735-5763
MSC (2010): Primary 53C44, 35K15, 35K55
Published electronically: June 20, 2012
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Abstract: The authors study the contraction of a convex immersed plane curve with speed $ \frac {1}{\alpha }k^{\alpha }$, where $ \alpha \in (0,1]$ is a constant, and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. They also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when $ \alpha =1$), this translational self-similar solution is the familiar ``Grim Reaper'' (a terminology due to M. Grayson).


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Additional Information

Yu-Chu Lin
Affiliation: Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
Email: yclin@math.ncku.edu.tw

Chi-Cheung Poon
Affiliation: Department of Mathematics, National Chung Cheng University, Chiayi 621, Taiwan
Email: ccpoon@math.ccu.edu.tw

Dong-Ho Tsai
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan
Email: dhtsai@math.nthu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05611-X
PII: S 0002-9947(2012)05611-X
Received by editor(s): October 12, 2010
Published electronically: June 20, 2012
Additional Notes: The third author’s research was supported by the NCTS and the NSC of Taiwan under grant number 96-2115-M-007-010-MY3.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.