The length-constrained ideal curve flow

Abstract. A recent article [AMWW] considered the so-called ‘ideal curve flow’, a sixth order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the L sense. Critical in the analysis there was a length bound on the evolving curves. It is natural to suspect therefore that the length-constrained ideal curve flow should permit a more straightforward analysis, at least in the case of small initial ‘energy’. In this article we show this is indeed the case, with suitable initial data providing a flow that exists for all time and converges smoothly and exponentially to a multiply-covered round circle of the same length and winding number as the initial curve.