The $\infty$-elastica problem on a Riemannian manifold

TL;DR

This paper studies the problem of minimizing the maximum curvature of curves with fixed length and boundary conditions on Riemannian manifolds, deriving a second order ODE system that characterizes solutions and reveals geometric properties.

Contribution

It introduces a new variational problem on Riemannian manifolds and derives the governing differential equations for the extremal curves, extending the understanding of curvature optimization.

Findings

Solutions satisfy a second order ODE system.

Derived geometric insights about the behavior of optimal curves.

Extended the theory of curvature minimization to Riemannian manifolds.

Abstract

We consider the following problem: on any given complete Riemannian manifold $(M,g)$, among all curves which have fixed length as well as fixed end-points and tangents at the end-points, minimise the $L^\infty$ 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.