In this talk I will discuss the evolution of planar polygonal curves under crystalline elastic flow, a (geometric) gradient flow associated with crystalline perimeter. This flow can be seen as a natural perturbation of crystalline curvature flow, where polygonal sides evolve by parallel translation. I will first present a long-time existence and uniqueness result for immersed polygonal curves, including possibly unbounded ones. For closed polygons, I will explain how the flow can be restarted beyond singularities, yielding a global evolution that preserves the topological index of the curve. I will then turn to the long-time behavior: via a Lojasiewicz–Simon type inequality, we prove convergence to stationary configurations. Finally, I will discuss the classification of stationary and translating solutions in the case of square anisotropy.
This is a joint work with Giovanni Bellettini (U. Siena & ICTP Trieste, Italy) and Matteo Novaga (U. Pisa)