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We study the following problem: existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in CP 2 (4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional.

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We study curve motion by the binormal flow with curvature and torsion depending velocity and sweeping out immersed surfaces. Using the Gauss-Codazzi equations, we obtain filaments evolving with constant torsion which arise from extremal curves of curvature energy functionals. They are “soliton” solutions in the sense that they evolve without changing shape.

No abstract