Closed ideal planar curves

We use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the L-2 sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply covered circle. Moreover, we show that curves in any homotopy class with initially small L-3 parallel to k(s)parallel to(2)(2) enjoy a uniform length bound under the flow, yielding the convergence result in these cases.