On the curve diffusion flow of closed plane curves

TL;DR

This paper studies the curve diffusion flow of closed plane curves, proving long-term convergence to circles for curves close to a circle, and providing estimates on waiting time and embeddedness preservation.

Contribution

It establishes conditions under which the flow exists globally and converges exponentially to a circle, including optimal waiting time estimates and embeddedness control.

Findings

Curves close to a circle remain smooth and converge exponentially to a circle.

A quantified waiting time ensures the flow becomes strictly convex.

Embeddedness is preserved under certain initial conditions.

Abstract

In this paper we consider the steepest descent $H^{-1}$-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves which develop at least one singularity in finite time and initially embedded curves which self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised $L^2$ oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large `waiting time' the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.