Motion of an elastic wire with thickness in a Riemannian manifold

TL;DR

This paper extends the existence results for elastic wire equations, accounting for thickness, from Euclidean spaces to general Riemannian manifolds, broadening the mathematical understanding of elastic wire dynamics.

Contribution

It proves the existence of solutions for elastic wire equations with thickness on any Riemannian manifold, generalizing previous Euclidean space results.

Findings

Existence of solutions established on Riemannian manifolds.

Generalization from Euclidean spaces to curved manifolds.

Extension of previous work by Caflisch, Maddocks, Koiso, and Sugimoto.

Abstract

There are several types of equation of motion of elastic wires. In this paper, we treat an equation taking account of the thickness of wire. The equation was introduced by Caflisch and Maddocks on plane curves, and they proved the existence of solutions. Koiso and Sugimoto generalized the result to any dimensional Euclidean space. In this paper, we will prove the existence of solutions on any riemannian manifold.