Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature

Details Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature

2010-09-24

arxiv.org/abs/1009.4777v1

Mathematics

Differential Geometry

24 pages, 3 figures

Scientific paper

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]).

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