Closed curves in ${\mathbb{R}}^3$ with prescribed curvature and torsion in perturbative cases. Part 1: Necessary condition and study of the unperturbed problem

Abstract

We study the problem of $(\kappa ,\tau )$-loops, namely closed curves in the three-dimensional Euclidean space, with prescribed curvature $\kappa$ and torsion $\tau$. We state a necessary condition for the existence of a bounded sequence of $({\kappa }_n,{\tau }_n)$-loops when the functions ${\kappa }_n$ and ${\tau }_n$ converge to the constants 1 and 0, respectively. Moreover we prove some Fredholm-type properties for the “unperturbed” problem, with $\kappa \equiv 1$ and $\tau \equiv 0$.