Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature

Abstract

We study the contraction of a convex immersed plane curve with speed (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. We 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 [GR]).