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

Abstract

We study a flow of G2-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 Θ for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi [CM12] on the mean curvature flow, we define an entropy functional and after proving an ε-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 G2-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth G2-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 admits a subsequence that converges to a shrinking soliton for the flow.

Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere

“…This framework will lead to Theorem 2, which situates some distinguished types of G 2 -structures on S 7 , within a same family of (isometric classes of) G 2 -structures. The main inspirations mobilised here come from [Bry06,DGK19,IKS97].…”

“…The existence of critical points of (48), and specifically of minimisers, has been studied using the associated gradient flow [DGK19, Gri19, LSE19] (49) ∂ϕ t ∂t = (div T t ) ψ t and ϕ(0) = ϕ r , known as the isometric flow, since it preserves isometric classes of G 2 -structures. In particular, Dwivedi et al [DGK19] proved that ( 49) is equivalent to…”

“…Within a given isometric class B r , the G 2 -structures with divergence-free torsion T were first highlighted by Grigorian [Gri17], in relation to a Coulomb gauge condition for octonionic connections. Their interpretation as stationary points for isometric flows of G 2 -structures was thoroughly explored in [DGK19,Gri19], and subsequently their characterisation in terms of harmonicity was systematised in [LSE19]. In §3, we describe the Sp(2)-invariant G 2 -structures of B r with div T = 0 (Proposition 3.4), i.e.…”

New G2-conifolds in M-theory and their field theory interpretation

“…Advances have been made both from the physics [9][10][11][12][13][14] and the mathematics [1,[15][16][17][18][19][20][21], and involves aspects such as a string-worldsheet interpretation [22][23][24], a better understanding of exceptional gauge bundles and instantons [25][26][27][28][29][30][31], in addition to non-perturbative aspects such as M 2 instantons and their relation to exceptional enumerative geometry [32][33][34].…”

Harmonic flow of $\mathrm{Spin}(7)$-structures

“…The analytic core of the paper is developed in § §4,5, following closely the study of the harmonic G 2 -flow in [DGK21] and the methods therein. We prove a general Cheeger-Gromov-type compactness for Spin(7)-structures in Theorem 4.18 and then use it to prove a Hamilton-type compactness theorem for solutions of (HF) in Theorem 4.19.…”