A New Family of Buckled Rings on the Two-Sphere

Buckled Rings, also known as Pressurized Elastic Circles, were first studied by Maurice Lévy , and then by Halphen and Greenhill . These planar curves can be described as critical points for a variational problem, namely the integral of a quadratic polynomial in the geodesic curvature of a curve. Thus they are a generalization of elastic curves, and they are solitary wave solutions to a flow in the (three-dimensional) filament hierarchy. An example of such a curve is the Kiepert Trefoil, which has three leaves meeting at a central singular point. Such a variational problem can be considered for curves in other surfaces. In particular, a recent paper of I. Castro et al found many examples of such curves in the unit sphere. In this article, which is in part an extension of that paper, we consider a new family of such curves, having a discrete dihedral symmetry about a central singular point. That is, these are spherical analogues of the Kiepert curve. We determine such curves explicitly using the notion of a Killing field, which is a vector field along a curve which is the restriction of an isometry of the sphere. The curvature k of each such curve is given explicitly by an elliptic function. If the curve is centered at the south pole of the sphere and has minimum value ρ, then k−ρ is linear in the height above the pole.