High order curvature flows of plane curves with generalised Neumann boundary conditions

We consider the parabolic polyharmonic diffusion and the L-2-gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L-2, then the evolving curve converges exponentially in the C-infinity topology to a straight horizontal line segment. The same behaviour is shown for the L-2-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

Automated summary

The study focuses on the evolution of regular closed curves, showing that when the curvature or energy of the initial curve is small enough, the evolving curve converges to a straight horizontal line segment. The smallness conditions depend only on m.