Curve diffusion flows in cones

We study families of smooth, embedded, regular planar curves with generalised Neumann boundary conditions inside cones, satisfying three variants of the fourth-order nonlinear curve diffusion flow: curve diffusion flow with length penalisation and two forms of constrained curve diffusion flow with fixed length. Assuming neither end of the evolving curve reaches the cone tip, existence of smooth solutions for all time given quite general initial data is well known for the constrained flows is well known, but classification of limiting shapes is generally not known. We prove for the constrained flows smooth exponential convergence of solutions in the \(C^\infty \)-topology to a circular arc centred at the cone tip with the same length as the initial curve. In the length penalised case, we show smooth exponential convergence under suitable rescaling to a circular arc centred at the cone tip.

This is joint work with Mashniah Gazwani.