In my paper I proved a Li-Yau inequality for curves in Euclidean space of any codimension. Equality is attained by a new family of curves, which I named leafed elasticae. 

An m-leafed elastica consists of m leaves of equal length, and each leaf is given by half of the figure-eight elastica discovered by Euler in 1744. In particular, the 2-leafed elastica is uniquely given by the (planar) figure-eight elastica itself, and the 3-leafed elastica is uniquely given by a new (spatial) shape. I call it the elastic propeller.

The elastic propeller can be reproduced by a stable equilibrium of a closed wire.

The following movie shows that the figure-eight elastica is (spatially) unstable but the elastic propeller is indeed stable.

In our paper jointly with Tomoya Kemmochi, we obtained numerical examples of migrating and non-migrating elastic flows.

Here "migration" means that an initial curve is contained in the upper half-plane but later contained in the lower half-plane at a positive time. Such a migration phemomenon is a typical example of "positivity breaking" phenomena in higher order flows, and never occurs in the (2nd order) curve shortening flow.

The first three movies are about length-preserving elastic flows. All are migrating examples.

The last four are about length-penalized elastic flows. The first and third examples migrate, but the others do not.