Elastic curves with constant curvature at rest in the hyperbolic plane (Q719389)

\(\lambda\)-elastica are critical curves of the Willmore energy functional \[ \gamma \mapsto \int_\gamma(\kappa+\lambda)^2, \] where \(\kappa\) is the geodesic curvature of \(\gamma\), and \(-\lambda \in \mathbb {R}\) is the constant mean curvature in the undeformed state (the curve is a circle at rest). The purpose of this article is to study closed \(\lambda\)-elastica (for \(\lambda \neq 0\)) in the hyperbolic plane \(\mathbf{H}^2(-1)\) of constant mean curvature \(\lambda=-1\). By computing the Euler-Lagrange equation, the authors classify these critical curves into three categories, according to their symmetries: there are curves of rotational, of translational, and of horocyclical type. They show the existence of a \(2\)-parameter family of curves of rotational type, that there is an ``eight''-shaped example of a translational one, while there are no examples of curves of horocyclical type. Then an analysis of the second variation is performed to study the stability of these closed \(\lambda\)-elastica. Eventually, numerical simulations are performed, revealing possible further properties, some of them being already proved by the authors in some cases.