A Gradient Flow of Isometric \(\mathrm {G}_2\)-Structures

We study a flow of \(\mathrm {G}_2\)-structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist 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 underlying connection yields a nice reaction–diffusion equation for the torsion along the flow. We define a scale-invariant quantity \(\Theta \) for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding–Minicozzi (Ann Math (2) 175(2):755–833, 2012) on the mean curvature flow, we define an entropy functional and after proving an \(\varepsilon \)-regularity theorem, we show that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a \(\mathrm {G}_2\)-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth \(\mathrm {G}_2\)-structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-\(\text {I}\) then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.