Cohomogeneity one solitons for the isometric flow of G 2 -structures.

Abstract

We consider the existence of cohomogeneity one solitons for the isometric flow of ${\text{G}}_2$ -structures on the following classes of torsion-free ${\text{G}}_2$ -manifolds: the Euclidean $R^7$ with its standard ${\text{G}}_2$ -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon ${\text{G}}_2$ -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 $R^7$ 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.