Closed curves in R3 with prescribed curvature and torsion in perturbative cases - Part 2: Sufficient conditions

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 $\kappa=\kappa_{\varepsilon}(p)$ and $\tau=\tau_{\varepsilon}(p)$ with $\kappa_{\varepsilon}$ and $\tau_{\varepsilon}$ converging to the constants 1 and 0, respectively, as $\varepsilon\to 0$. We prove existence of branches of $(\kappa_{\varepsilon},\tau_{\varepsilon})$-loops (for small $|\varepsilon|$) emanating from circles which correspond to stable zeroes of a suitable vector field $M\colon\mathbb{T}^{2}\times\mathbb{R}^{3}\to\mathbb{R}^{5}$.