Contracting convex immersed closed plane curves with slow speed of curvature

Lin, Yu-Chu; Poon, Chi-Cheung; Tsai, Dong-Ho

doi:10.1090/S0002-9947-2012-05611-X

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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