Existence of optimal flat ribbons

TL;DR

This paper proves that any nonplanar Frenet curve in three-dimensional space can be extended into a flat ribbon with minimal bending energy, revealing properties of the minimizers and their planar points.

Contribution

It demonstrates the existence of minimal bending energy flat ribbons extending nonplanar Frenet curves and analyzes the nature of their planar points.

Findings

Minimizers are generally not free of planar points.

Planar points in minimizers are isolated if torsion does not vanish.

Any nonplanar Frenet curve can be extended to a minimal energy flat ribbon.

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.