Scale-critical curve diffusion flows

TL;DR

This paper introduces a scale-critical curve diffusion flow with a cubic curvature term, classifies stationary solutions, and analyzes stability, revealing complex stability patterns depending on the circle's winding number.

Contribution

It provides a classification of stationary solutions and a stability analysis of circles under a new scale-critical flow using variational methods.

Findings

Stationary solutions are circles or super-lemniscates.

Small perturbations of circles tend to rescaled circles under stability conditions.

Stability depends non-monotonically on the circle's winding number.

Abstract

We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of ``super-lemniscates''. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an $\omega$-fold circle monotonically approaches the unit $\omega$-circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all $\omega$-circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of $\omega$-circles depends non-monotonically on $\omega$.