Contracting Convex Immersed Closed Plane Curves with Slow Speed of Curvature

TL;DR

This paper investigates the contraction behavior of convex immersed plane curves under a curvature-dependent flow with slow speed, revealing convergence to self-similar solutions and extending understanding of curve evolution dynamics.

Contribution

It introduces a new class of curvature flows with slow speed and characterizes their long-term self-similar solutions, including symmetric cases and connections to known solutions.

Findings

Curves with type one blow-up rate converge to homothetic solutions.

Type two blow-up symmetric cases converge to translational solutions.

Special case with =1 recovers the Grim Reaper solution.

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]).