Elastic plane curves

Finite type curves in 2-dimensional spaceforms
These curve minimize the total squared curvature. These curves were investigated by Euler in 1744Â [2].
The first figure depicts the elastic plane curves in R2 of spectral genus 1 are shown in the first figure. The 1-looped 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.
The second figure depicts a sampling of closed finte type curves in S2 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 toriÂ [1,3].
The third figure depicts a sampling of elastic curves in H2 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 revolutionÂ [3,5].

References
	1.	C. Bohle, G. Peters, and U. Pinkall, Constrained Willmore surfaces, Calc. Var. Partial Differential Equations 32 (2008), no. 2, 263—277 [2389993].
	2.	L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, E065, eulerarchive.org (174).
	3.	L. Heller, Constrained Willmore tori and elastic curves in 2-dimensional space forms, Comm. Anal. Geom. 22 (2014), no. 2, 343—369 [3210758].
	4.	L. Heller, Constrained Willmore tori and elastic curves in 2-dimensional space forms, Comm. Anal. Geom. 22 (2014), no. 2, 343—369 [3210758].
	5.	M. Kilian, M. Schmidt, and N. Schmitt, Flows of constant mean curvature tori in the 3-sphere: the equivariant case, arXiv:1011.2875, to appear in J. reine angew. Math., DOI 10.1515/crelle-2013-0079 (2010).
	6.	

Circletons

Finite type curves in Euclidean space
The first image depicts closed planar simple circletons (simple factor dressings of the circle) in R2. These curves are parametrized by pairs of positive integers (ℓ, w) satisfying ℓ < w, where ℓ is the lobe count and w is the wrapping number of the underlying circle. The image depicts circletons for small ℓ and w: the rows are indexed by w=2,…,7 and the columns by ℓ1,…,w-1
The second image depects the border line elastic curve, the unique (up to homothety) elastic planecurve in $\bbR^2$ with singular spectral curve. With arclength parametrization $(t-tanh 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.
The third image depicts an isospectral flow through closed simple circletons (simple factor dressings of the circle) in R3. From right to left, the planar circleton (also appearing in the top row of the first figure) deforms to a 2-wrapped circle in the limit. The spectral curve for this family is singular with algebraic genus 2 and geometric genus 0.
The fourth image depicts The Euler lemniscate (first curve), the unique closed elastic planecurve in R2 up to homothety; it is finite type with spectral genus 1. The image shows two views of the spacecurve in R3 constructed by dressing the Euler lemniscate by a simple factor, preserving closing Its spectral curve is singular.

Flows through finite type spacecurves

Finite type curves in Euclidean 3-space
The first figure depicts an isospectral flow through closed finite type spacecurves in R3 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.
The second image depicts an isoperiodic flow through elastic torus knots in R3of spectral genus 1, from a 2-wrapped circle to a 3-wrapped circle. In contrast to isospectral flows, the spectral curve changes during isoperiodic flow. Both flows preserve the closing of the spacecurves.