Abstract
We study a geometric flow on curves, immersed in \({\mathbb {R}}^3\), that have strictly positive torsion. The evolution equation is given by
where \(\tau \) is the torsion and \(\mathbf{B} \) is the unit binormal vector. In the case of constant curvature, we find all of the stationary solutions and linearize the PDE for the torsion around stationary solutions admitting an explicit formula. Afterwards, we prove the \(L^2({\mathbb {R}})\) linear stability of the stationary solutions corresponding to helices with constant curvature and constant torsion.
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Acknowledgements
We would like to thank Richard Schwartz and Benoit Pausader for helpful discussions about this topic. We would also like to thank Wolfgang Schief for pointing out his joint paper [11] with C. Roger. Lastly, we would like to acknowledge the anonymous referee and their very helpful comments, especially about Corollary 3.2.
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Coiculescu, M.P. Stationary solutions of the curvature preserving flow on space curves. Arch. Math. 116, 457–467 (2021). https://doi.org/10.1007/s00013-020-01563-z
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DOI: https://doi.org/10.1007/s00013-020-01563-z