Cohomogeneity one solitons for the isometric flow of \(\mathrm{G}_2\)-structures (Q6623100)

One approach to the study of \(G_2\)-holonomy manifolds is to see them as special holonomy Riemannian metrics. Another approach is to see them as \(G_2\)-structures with vanishing torsion \(2\)-tensor. At once we encounter a subtlety: many \(G_2\)-structures induce the same Riemannian metric. A \(G_2\)-structure is precisely a nondegenerate \(3\)-form on a \(7\)-manifold. The torsion of a \(G_2\)-structure is the obstruction to the \(G_2\)-structure arising from a \(G_2\)-holonomy metric, and is a first order invariant of the \(3\)-form. Within the class of \(G_2\)-structures with a given underlying Riemannian metric, consider the negative gradient flow of the \(L^2\)-norm of the torsion. This gradient flow is given by an explicit partial differential equation, the isometric flow. The associated steady state, i.e., the Euler-Lagrange equation, is the equation that the torsion be divergence free.\N\NMore generally, one considers isometric flow solitons, for which the \(3\)-form is merely rescaled under the flow by a scale factor depending only on time. We can then rescale to make the metric vary by a scale factor depending only on time, fixing the \(3\)-form, so that the divergence is instead expressed in terms of a vector field whose flow preserves the metric up to a constant factor. A soliton is expanding, steady or contracting if the metric is. The authors study isometric flow solitons on various \(7\)-manifolds equipped with fixed Riemannian metrics of \(G_2\)-holonomy. Besides the torsion-free \(G_2\)-structure that induces that \(G_2\)-holonomy metric, there are many \(G_2\)-structures inducing the same Riemannian metrics, and the authors search for isometric flow solitons among these \(G_2\)-structures. The carry out this work on flat Euclidean space in \(7\)-dimensions, on metric cylinders over Calabi-Yau \(3\)-folds, on metric cylinders on nearly Kähler \(6\)-manifolds, and on the Bryant-Salamon manifolds. The authors prove that there are global isometric flow solitons, and they study the asymptotic growth or decay of the torsion. Their proof is by analysis of symmetry solutions, giving rise to ordinary differential equations with regular singular points.