A new energy method for shortening and straightening complete curves

Abstract

We introduce a novel energy method that reinterprets ``curve shortening'' as ``tangent aligning''. This conceptual shift enables the variational study of infinite-length curves evolving by the curve shortening flow, as well as higher order flows such as the elastic flow, which involves not only the curve shortening but also the curve straightening effect. For the curve shortening flow, we prove convergence to a straight line under mild assumptions on the ends of the initial curve. For the elastic flow, we establish a global well-posedness theory, and investigate the precise long-time behavior of solutions. In fact, our method applies to a more general class of geometric evolution equations including the surface diffusion flow, Chen's flow, and the free elastic flow.