## Contracting convex immersed closed plane curves with slow speed of curvature

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- by Yu-Chu Lin, Chi-Cheung Poon and Dong-Ho Tsai PDF
- Trans. Amer. Math. Soc.
**364**(2012), 5735-5763 Request permission

## 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
- MR Author ID: 843221
- 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
- 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.
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 5735-5763 - MSC (2010): Primary 53C44, 35K15, 35K55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05611-X
- MathSciNet review: 2946930