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

Abstract

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.