Cohomogeneity one solitons for the isometric flow of $$\textrm{G}_2$$-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>G</mml:mtext> <mml:mn>2</mml:mn></mml:msub> </mml:math> -structures on the following classes of torsion-free <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>G</mml:mtext> <mml:mn>2</mml:mn></mml:msub> </mml:math> -manifolds: the Euclidean <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mn>7</mml:mn></mml:msup> </mml:math> with its standard <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>G</mml:mtext> <mml:mn>2</mml:mn></mml:msub> </mml:math> -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mtext>G</mml:mtext> <mml:mn>2</mml:mn></mml:msub> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mn>7</mml:mn></mml:msup> </mml:math> 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.