A gradient flow of isometric ${\mathrm{G}}_2$ structures

We study a flow of $G_2$ structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the connection gives a nice diffusion-reaction equation for the torsion along the flow. We define a quantity for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi on the mean curvature flow, we define an entropy functional and after proving an $\epsilon$-regularity theorem, we show that low entropy initial data lead to solutions of the flow which exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We also study the finite-time singularities and show that at the singular time the flow converges to a smooth $G_2$ structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton for the flow.