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

MathSciNet review:
2946930

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Abstract | References | Similar Articles | Additional Information

Abstract: The authors study the contraction of a convex immersed plane curve with speed , where 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 ), 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

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.