Kirchhoff elastic rods in a Riemannian manifold

Abstract

Imagine a thin elastic rod like a piano wire. We consider the situation that the elastic rod is bent and twisted and both ends are welded together to form a smooth loop. Then, does there exist a stable equilibrium? In this paper, we generalize the energy of uniform symmetric Kirchhoff elastic rods in the $3$-dimensional Euclidean space to consider such a variational problem in a Riemannian manifold. We give the existence and regularity of minimizers of the energy in a compact or homogeneous Riemannian manifold.