Abstract:
The curve shortening flow classically arises as the -gradient flow of the length functional. In this talk, we will discuss a reinterpretation of "curve shortening" as “tangent aligning”. This allows us to apply energy-based methods to study the long-time behavior of curves of infinite length. Our method also applies to higher order flows such as the curve diffusion flow, Chen’s flow, and the elastic flow, which involves not only the curve shortening but also the curve straightening effect. This is joint work with T. Miura (Kyoto).