Rendiconti Lincei - Matematica e Applicazioni


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

Published online: 2006-09-30

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) Università degli Studi di Torino, Italy
(2) Università degli Studi di 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