Recent developments on stability for the elastic flow of curves

The free elastic flow is the \(L^2(ds)\) steepest descent gradient flow for Euler’s elastic energy defined on curves. Among closed curves, circles and the lemniscate of Bernoulli expand self-similarly under the elastic flow, and there are no stationary solutions. This means that any stabilising effect must be modulo a rescaling that fixes, for instance, length. This additional difficulty has led authors (following Euler) to introduce a constraint or a penalisation in the energy, which changes the dynamics substantially. In particular, there are a plethora of stability and convergence results in a variety of settings, both planar and space, and with a number of boundary conditions. The free elastic flow itself remained untouched, until recently: In 2024, joint with Miura, we were able to establish convergence of the asymptotic profile, through the use of a new quantity depending on the derivative of the curvature. Then, in 2025, joint with Andrews, this has been improved and a sharp convergence rate has been obtained. In this talk, I describe these new results, and also explain what more may be on the horizon for the free elastic flow. If time permits this will include evidence contradicting a conjecture for the free boundary free elastic flow, and our initial work on the space curve case, where an unexpected phenomenon related to turning number appears.