Closed ideal planar curves

TL;DR

This paper introduces a gradient flow method to deform closed planar curves towards circles with minimal geodesic curvature variation, proving convergence under certain conditions.

Contribution

It establishes the existence, long-term behavior, and convergence of the flow to multiply-covered circles for smooth initial curves.

Findings

Flow exists for all time for smooth initial curves.

Curves with small initial $L^3\|k_s\|_2^2$ have bounded length and converge to circles.

Convergence to multiply-covered circles is proven under bounded length assumption.

Abstract

In this paper 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 $L^3\lVert k_s\rVert_2^2$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.