The length-constrained ideal curve flow

TL;DR

This paper studies a length-constrained sixth order curvature flow that deforms closed planar curves to multiply-covered circles, proving long-term existence and exponential convergence under small initial energy conditions.

Contribution

It introduces a length-constrained version of the ideal curve flow, simplifying analysis and establishing convergence results for small initial energy.

Findings

Flow exists for all time under certain conditions

Flow converges exponentially to a multiply-covered circle

Initial energy bounds ensure smooth long-term behavior

Abstract

A recent article by the first two authors together with B Andrews and V-M Wheeler 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 $L^2$ 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.