A survey of the elastic flow of curves and networks

TL;DR

This survey consolidates recent research on the elastic flow of curves and networks, providing a comprehensive overview, proofs of key results, and discussing open problems in the field.

Contribution

It offers a unified presentation of recent results, including a complete proof of global existence and convergence for elastic flow of closed curves in the plane.

Findings

Complete proof of global existence for elastic flow of closed curves.

Proof of smooth convergence to critical points.

Discussion of open problems in elastic flow research.

Abstract

We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in $\mathbb{R}^2$. In the last section of the paper we also discuss a list of open problems.