Hi there!

This gallery features images and animations from my research on elastic curves and related geometric problems.

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.

From left to right, the figures show the teardrop--heart, cage, and catapult. The cage and teardrop--herat are previously obtained as stable configurations of closed wires in a figure-eight knot class. It was long time conjectured that the teardrop--heart is an energy-minimal elastic knot in the figure-eight knot class. 

In my paper, I proved that elastic knots in every non-torus twist knot class are non-planar, thus disproving the above conjecture for the figure-eight knot class. However their precise shapes remain unknown. In the same paper I conjectured that the catapult is an elastic figure-eight knot; indeed, the catapult has lower numerically computed bending energy than the teardrop–heart and cage configurations.

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}.

The figures show super-lemniscates, a family of closed stationary curves for scale-critical curve diffusion flows. The flow is introduced in our paper with Glen Wheeler. From left to right, the curves correspond to j = 1, 2, 3, 4, 10, and 100, with parameter c = 2/(4j − 1)² in the flow. The first is nothing but Bernoulli’s lemniscate. We have classified all closed stationary solutions as circles and super-lemniscates.

The figure shows a three-dimensional flat-core p-elastica for p = 4, consisting of two planar loops connected by a straight segment. A p-elastica is a critical point of the integral of the pth power of curvature under a length constraint.

In our paper with Florian Gruen, we showed that these solutions form the exceptional class among non-planar p-elasticae for p > 2: they can lose regularity and extend beyond three dimensions.