Abstract

Evolutions by self-similarity constitute an important paradigm in geometric flows. They often convey information about the dynamical properties of a given flow. This principle is particularly evident in the well-known curve-shortening flow (CSF), where homothetic shrinkers and translators naturally arise in singularity analysis. Furthermore, stability analysis shows that closed CSF-shrinkers act as saddle points for the asymptotic behavior of the flow. Such shrinkers are fully classified as either circular or belonging to the family of Abresch-Langer curves.

In this talk, we present an analog classification scheme for a non-local, area-preserving version of CSF. As a corollary, we obtain a rigidity result for closed $\lambda$-curves, confirming a conjecture by J.-E. Chang. Finally, we establish a saddle-point property for this constrained setting.