Rendiconti Lincei - Matematica e Applicazioni


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Volume 17, Issue 3, 2006, pp. 227–242
DOI: 10.4171/RLM/466

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

Paolo Caldiroli[1] and Michela Guida[2]

(1) Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto,10, 10123, TORINO, ITALY
(2) Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto,10, 10123, TORINO, ITALY

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 $\tau\equiv 0$.

Keywords: Prescribed curvature and torsion, perturbative methods, Fredholm operators

Caldiroli Paolo, Guida Michela: Closed curves in $\mathbb{R}^{3}$ with prescribed curvature and torsion in perturbative cases. Part 1: Necessary condition and study of the unperturbed problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), 227-242. doi: 10.4171/RLM/466