Closed ideal planar curves

Authors

Ben Andrews
James McCoy
Glen Wheeler, University of WollongongFollow
V.-M Wheeler, University of WollongongFollow

RIS ID

145959

Publication Details

Andrews, B., McCoy, J., Wheeler, G. & Wheeler, V. (2020). Closed ideal planar curves. Geometry and Topology, 24 (2), 1019-1049.

Abstract

2020, Mathematical Sciences Publishers. All rights reserved. We use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the L2 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 L3kks k22 enjoy a uniform length bound under the flow, yielding the convergence result in these cases.