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The free elastic flow for closed planar curves

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arxiv 2404.12619 v3 pith:73XZSE5R submitted 2024-04-19 math.AP math.DG

Tatsuya Miura , Glen Wheeler This is my paper · ORCID
classification math.APmath.DG
keywords flowelasticcurvesfreeasymptoticclosedinitialappropriate
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abstract

The free elastic flow is the $L^2$-gradient flow for Euler's elastic energy, or equivalently the Willmore flow with translation invariant initial data. In contrast to elastic flows under length penalisation or preservation, it is more challenging to study the free elastic flow's asymptotic behavior, and convergence for closed curves is lost. In this paper, we nevertheless determine the asymptotic shape of the flow for initial curves that are geometrically close to circles, possibly multiply-covered, proving that an appropriate rescaling smoothly converges to a unique round circle.

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  1. Length-constrained, length-penalised and free elastic flows of planar curves inside cones

    math.AP 2024-11 conditional novelty 5.0 of 10

    For small curvature oscillation, length-penalized and length-constrained elastic flows in a cone converge exponentially to circular arcs, while the free elastic flow converges smoothly to an expanding circular arc.

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