Elastic curves with constant curvature at rest in the hyperbolic plane

We extend the classical variational model for elastic curves that are circular at rest to the hyperbolic plane H2(−1)H2(−1). For simplicity, we call them λλ-elastica. We show that there are three types of critical curves according to their symmetries: rotational, translational and horocyclical type curves. By explicitly solving the Euler–Lagrange equation and giving a closedness criterion in each case, we can show that there exists a 2-parameter family of closed rotational λλ-elastica and that there exists an “eight”-shaped example of closed translational λλ-elastica in H2(−1)H2(−1). However, we prove that there are no examples of closed λλ-elastica of horocyclical type. The second variation formula is applied to study the stability of the constant curvature solution multiple covers. As an application, we combine these results with a Lorentzian version of the Hopf map to construct examples of closed elastic membranes in the anti de Sitter 3-space. A numerical approach is used to gain insight into the space of closed λλ-elastica. One plausible consequence of this numerics is that the “eight”-shaped critical curve mentioned before appears to be the only closed translational λλ-elastica and that it is also a candidate for a local minimum of elastic energy.