A classical result ofT. Takahashi [8] is generalized to the case of hypersurfaces in the Euclidean space E ~' . More concretely, we classify Euclidean hypersurfaces whose coordinate functions in E" are eigenfunctions of their Laplacian.

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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.