#### Volume 24, issue 2 (2020)

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Closed ideal planar curves

### Ben Andrews, James McCoy, Glen Wheeler and Valentina-Mira Wheeler

Geometry & Topology 24 (2020) 1019–1049
##### Abstract

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 ${L}^{3}\parallel {k}_{s}{\parallel }_{2}^{2}$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.

##### Keywords
constant mean curvature, curvature flow, geometric evolution equation
##### Mathematical Subject Classification 2010
Primary: 35K25, 53C44
Secondary: 58J35