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

TL;DR

This paper studies higher order curvature flows of plane curves with generalized Neumann boundary conditions, proving exponential convergence to straight lines under small initial curvature or energy conditions.

Contribution

It establishes convergence results for polyharmonic and gradient flows of curvature derivatives with new boundary conditions, extending previous understanding of curve evolution.

Findings

Curves with small initial curvature converge exponentially to straight lines.

The convergence holds for both polyharmonic and gradient flows under small energy conditions.

Smallness conditions depend only on the order of the derivative, m.

Abstract

We consider the parabolic polyharmonic diffusion and $L^2$-gradient flows 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 $C^\infty$ 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 $m$.