The ideal flow for planar closed curves with local length constraint

Abstract

We construct the L ² -gradient flow of the ideal functional, which is defined as the squared integral of the derivative of curvature, under the local length constraint. We prove that (i) the Cauchy problem of the gradient flow has a unique global-in-time solution, and (ii) the global-in-time solution converges to a critical point of the ideal functional under the total length constraint. As a corollary, we show the existence of a critical point with a rotation number of zero.