The figures show the elastic propeller, which is first mathematically found and later experimentally reproduced by a stable equilibrium of a closed wire.

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, which I named the elastic propeller.

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 possibility of migration is theoretically shown here in the length-preserving case, and here in the length-penalized case.

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.

These are newly discovered non-classical shapes that appear as critical configurations giving optimal energy thresholds for "positivity-preserving" properties of the elastic flow, a typical fourth-order geometric flow obtained as a gradient flow of the elastic energy.

The left figure shows the elastic two-teardrop, found here together with Marius Müller and Fabian Rupp, which gives the optimal energy threshold for preserving embeddedness of the elastic flow for closed planar curves. In the same work we also showed that the classical figure-eight elastica gives the threshold in codimension two or higher.

The right figures show the elastic pendant (top) and the elastic serpent (bottom), obtained here together with Fabian Rupp. These provide nontrivial optimal thresholds for preserving planar embeddedness and graphicality, respectively, of the elastic flow for non-closed complete curves of infinite length. A key difficulty here is that the standard elastic energy is always infinite, but we overcame this by using our previously developed direction energy method, which yields a finite energy even for infinite-length curves.

Those shapes are non-classical in that they lose regularity: the elastic two-teardrop and the elastic pendant are of class C^{2,1}, while the elastic serpent is of class C^{1,1}.