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

Abstract

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