The Shrinking Figure Eight and Other Solitons for the Curve Diffusion Flow

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (3) an explicit parametrisation of a shrinking figure eight curve.