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