Existence of optimal flat ribbons

Abstract

We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in ${\mathbb{R}}^3$ can be extended to an infinitely narrow flat ribbon having minimal bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.

11 pages, no figures. Minor revision

References in corpus (7)

• A non-linear rod model for folded elastic strips
• Statistical Mechanics of Developable Ribbons
• Translation of Michael Sadowsky's paper "An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem"
• Variational analysis of inextensible elastic curves
• Forceless Sadowsky strips are spherical
• Stability of boundary conditions for the Sadowsky functional
• Nonrigidity of flat ribbons