11 problems

Global asymptotic-shape conjecture. The rescaled flow $\hat\gamma(\cdot,t)$ converges as $t\to\infty$ exponentially fast in the smooth topology to a limit $\gamma_\infty$. Furtherm…

Let $n\geq 2$ and let $\gamma\in W^{2,2}_{loc}(\mathbf{R};\mathbf{R}^n)$ satisfy … A curve has a self-intersection if there exist distinct parameters with the same image point. Hig…

Optimal-threshold conjecture. These criteria yield the optimal thresholds in dimension $n\geq3$ for many concrete $p$-elastic flows studied in the literature. The conjecture concer…

For an integer $r\geq4$, a closed $r$-leafed elastica is a closed elastic curve consisting of $r$ leaves. A stable elastic knot for the unknot class is a curve that locally minimiz…

For a tame knot class $K$, let $\mathcal{K}$ be the $W^{2,2}$-closure of the class of unit-length, arclength-parametrized closed curves in $K$: … A stable elastic knot for $K$ is a…

Let the figure-eight knot class be the knot class represented by the figure-eight knot, and let an elastic knot be understood in the sense of Gerlach–Reiter–von der Mosel. Elastic…

Let $N$ be a positive integer, and let a knot class have braid index and bridge index both equal to $N$. An $N$-fold circle is the corresponding circle traversed $N$ times, and an…

Lemniscate stability conjecture. As in the circle case, an analogous stability result should hold for the lemniscate under the free elastic flow.

Boundary-attachment and curvature-selection conjecture. Attaching to the boundary remains optimal in convex domains $C^2$-close to a disk. In domains with non-constant boundary cur…

Boundary-attachment and monotonicity conjecture. Any minimiser in this regime touches the boundary of the unit disk, and the energy is increasing as the length $L$ increases.

Consider stationary solutions of the elastic flow, namely curves whose curvature satisfies … An open solution is properly immersed if its immersion is proper. Open-elastica conject…