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Closed ideal planar curves

Journal Article


Abstract


  • 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.

Publication Date


  • 2020

Citation


  • Andrews, B., McCoy, J., Wheeler, G., & Wheeler, V. M. (2020). Closed ideal planar curves. Geometry and Topology, 24(2), 1019-1049. doi:10.2140/gt.2020.24.1019

Scopus Eid


  • 2-s2.0-85092247476

Start Page


  • 1019

End Page


  • 1049

Volume


  • 24

Issue


  • 2

Abstract


  • 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.

Publication Date


  • 2020

Citation


  • Andrews, B., McCoy, J., Wheeler, G., & Wheeler, V. M. (2020). Closed ideal planar curves. Geometry and Topology, 24(2), 1019-1049. doi:10.2140/gt.2020.24.1019

Scopus Eid


  • 2-s2.0-85092247476

Start Page


  • 1019

End Page


  • 1049

Volume


  • 24

Issue


  • 2