Space Curves
Nicholas SchmittDescription
Elastic plane curves are finite type curves in 2dimensional spaceforms. These curve minimize the total squared curvature. These curves were investigated by Euler in 1744 [1].
Figure 1depicts the elastic plane curves in $\mathbb{R}^2$ of spectral genus 1 are shown in the first figure. The 1looped example (fourth from left) with singular spectral curve demarcates the orbitlike (left) from the wavelike (right). The orbitlike starting at the singular example flow left, converging to a circle. The wavelike starting at the singular example flow right, passing through the closed Euler lemniscate, converging to a straight line.
Figure 2 depicts a sampling of closed finte type curves in $S^2$ with low spectral genus. The elastic curve (first curve) with spectral genus 1 and six lobes, is symmetric about the equator. It can flow isoperiodically through constrained elastic curves with spectral genus 2 (second curve). The lemniscate (third curve) has spectral genus 1. These curves are profile curves of constrained Willmore Hopf tori[2][3].
Figure 3 depicts a sampling of elastic curves in $H^2$ of spectral genus 1. The first three curve are wavelike. The last two closed orbitlike curves are related to profile curves of constant mean curvature tori of revolution[3][4].
Circletons are finite type curves in Euclidean space. Figure 4 depicts closed planar simple circletons (simple factor dressings of the circle) in $\mathbb{R}^2$. These curves are parametrized by pairs of positive integers $(\ell, w)$ satisfying $\ell < w$, where $\ell$ is the lobe count and $w$ is the wrapping number of the underlying circle. The image depicts circletons for small $\ell$ and $w$: the rows are indexed by $w=2,…,7$ and the columns by $\ell=1,…,w1$.
Figure 5 depicts the border line elastic curve, the unique (up to homothety) elastic planecurve in $\mathbb{R}^2$ with singular spectral curve. With arclength parametrization $(ttanh 2t,sech 2t)$, it is a simple factor dressing of the vacuum $(t,\,0)$. As indicated by the struts, the pointwise distance between the straight line and the Euler loop as arclength parametrized curves is constant 1.
Figure 6 depicts an isospectral flow through closed simple circletons (simple factor dressings of the circle) in $\mathbb{R}^3$. From right to left, the planar circleton (also appearing in the top row of the first figure) deforms to a 2wrapped circle in the limit. The spectral curve for this family is singular with algebraic genus 2 and geometric genus 0.
Figure 7 depicts The Euler lemniscate (first curve), the unique closed elastic planecurve in $\mathbb{R}^2$ up to homothety; it is finite type with spectral genus 1. The image shows two views of the spacecurve in $\mathbb{R}^3$ constructed by dressing the Euler lemniscate by a simple factor, preserving closing Its spectral curve is singular.
At last we see flows through finite type spacecurves in Euclidean 3space. Figure 8 depicts an isospectral flow through closed finite type spacecurves in $\mathbb{R}^3$ of spectral genus 2. The curve returns to a translation of itself. This is an example of a vortex filament flow. The spacecurves are spherical by with changing sphere radius.
Figure 9 depicts an isoperiodic flow through elastic torus knots in $\mathbb{R}^3$ of spectral genus 1, from a 2wrapped circle to a 3wrapped circle. In contrast to isospectral flows, the spectral curve changes during isoperiodic flow. Both flows preserve the closing of the spacecurves.
References

Leonhard Euler.
Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti.
1744.
URL: eulerarchive.org. 
Christoph Bohle, Paul Peters, and Ulrich Pinkall.
Constrained Willmore surfaces.
Calc. Var. Partial Differential Equations, 2008.
URL: https://mathscinet.ams.org/mathscinetgetitem?mr=2389993. 
Lynn Heller.
Constrained Willmore tori and elastic curves in 2dimensional space forms.
Comm. Anal. Geom., 2014.
URL: https://mathscinet.ams.org/mathscinetgetitem?mr=3210758. 
Martin Kilian, Martin Schmidt, and Nicholas Schmitt.
Flows of constant mean curvature tori in the 3sphere: The equivariant case.
preprint, 2010.
arXiv:1011.2875.