The $$\infty $$-Elastica Problem on a Riemannian Manifold

We consider the following problem: on any given complete Riemannian manifold ( M , g ), among all curves which have fixed length as well as fixed endpoints and tangents at the endpoints, minimise the $$L^\infty $$ L ∞ norm of the curvature. We show that the solutions of this problem, as well as a wider class of curves, must satisfy a second-order ODE system. From this system,we obtain some geometric information about the behaviour of the curves.