JournalsrlmVol. 17 , No. 4DOI 10.4171/rlm/470

Closed curves in <strong>R</strong><sup>3</sup> with prescribed curvature and torsion in perturbative cases — Part 2: Sufficient conditions

  • Paolo Caldiroli

    Università degli Studi di Torino, Italy
  • Michela Guida

    Università degli Studi di Torino, Italy
Closed curves in <strong>R</strong><sup>3</sup> with prescribed curvature and torsion in perturbative cases — Part 2: Sufficient conditions cover

Abstract

We investigate the problem of (κ,τ)(\kappa,\tau)-loops, namely closed curves in the three-dimensional Euclidean space, with prescribed curvature κ\kappa and torsion τ\tau. In particular we focus on some perturbative cases, taking κ=κ\eps(p)\kappa=\kappa_{\eps}(p) and τ=τ\eps(p)\tau=\tau_{\eps}(p) with κ\eps\kappa_{\eps} and τ\eps\tau_{\eps} converging to the constants 1 and 0, respectively, as \eps0\eps\to 0. We prove existence of branches of (κ\eps,τ\eps)(\kappa_{\eps},\tau_{\eps})-loops (for small \eps|\eps|) emanating from circles which correspond to stable zeroes of a suitable vector field M ⁣:\T2×R3R5M\colon\T^{2}\times\R^{3} \to\R^{5}.