Contracting convex immersed closed plane curves with slow speed of curvature (Q2844728)

The authors study the evolution of convex immersed plane curves with rotation index \(m\geq 2\) in \(\mathbb R^2\) and speed \(\frac{1}{\alpha}k^{\alpha}\), \(0<\alpha\leq 1\), in the normal direction. They prove that if the blow-up rate of the curvature is of type one, then it will converge to a homothetic self-similar solution. In the symmetric case of type-two blow-up, they prove that the curve converges to a translational self-similar solution. When \(\alpha=1\), this translational self-similar solution is the familiar ``Grim Reaper'' of the curve shortening flow.