Abstract
We consider the existence of cohomogeneity one solitons for the isometric flow of G2 -structures on the following classes of torsion-free G2 -manifolds: the Euclidean R7 with its standard G2 -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon G2 -manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7 is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.