The free elastic flow for closed planar curves

TL;DR

This paper studies the free elastic flow of closed planar curves, showing that curves initially close to circles evolve towards a round circle under rescaling, despite the lack of convergence in general.

Contribution

It establishes the asymptotic behavior of the free elastic flow for curves near circles, including multiply-covered ones, and proves convergence to a round circle after rescaling.

Findings

Rescaled flow converges smoothly to a round circle.

Convergence holds for initial curves close to circles, including multiply-covered cases.

The asymptotic shape is uniquely determined as a circle.

Abstract

The free elastic flow is the $L^2$-gradient flow for Euler's elastic energy, or equivalently the Willmore flow with translation invariant initial data. In contrast to elastic flows under length penalisation or preservation, it is more challenging to study the free elastic flow's asymptotic behavior, and convergence for closed curves is lost. In this paper, we nevertheless determine the asymptotic shape of the flow for initial curves that are geometrically close to circles, possibly multiply-covered, proving that an appropriate rescaling smoothly converges to a unique round circle.