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.