Stability of torsion-free $\mathrm{G}_2$ structures along the Laplacian flow

Abstract

We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structureφ on a compact 7-manifold M , the Laplacian flow with initial value in [φ], sufficiently close toφ, will converge to a point in the Diff 0 (M )-orbit ofφ. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.2010 Mathematics Subject Classification. 53C44, 53C25, 53C10.

Laplacian flow of homogeneous G2-structures and its solitons

“…provide a continuous family of G 2 -structures (G At , ϕ), or equivalently a family (G A1 , ϕ t ), where the Lie algebra of G A1 is n 6 , such that there is no any pair which is equivalent up to scaling. Indeed, by (29), the Ricci operator of (G At , ·, · ϕ ) is given by…”

Laplacian coflow for warped G2-structures

“…Remark 3.8. For the Laplacian coflow we chose the parameters pα, βq to be p0, 1q in order to obtain equations depending on the torsion forms σ 0 , σ 2 and ν 3 (see (2)) which are the ones that appear in the canonical definitions of the SUp3q-structures, nearly Kähler, symplectic half-flat and balanced, respectively (see equations (18), (21) and (27) in the next sections).…”

“…The nearly Kähler case. Recall that a nearly Kähler SU(3)-structure satisfies (18) dω "´3 2 σ 0 ψ`, dψ`" 0, dψ´" σ 0 ω 2 .…”

“…Recent papers by Lotay and Wei [17,18,19] derived important properties of the Laplacian flow as long time existence or convergence results. Even more recently Fino and Raffero on [11] obtained sufficient conditions for the existence of solution of this flow on warped products of the form M 6ˆf S 1 with M 6 a 6-dimensional manifold endowed with an SUp3q-structure.…”

Cohomogeneity‐one G2‐Laplacian flow on the 7‐torus

“…Since , the solution to the hypersymplectic flow on gives a solution to that is, the ‐ Laplacian flow on . There are several important results regarding the ‐Laplacian flow: the short‐time existence was proved by , a Shi‐type estimate and compactness result were proved by , the dynamical stability was proved in . The problem of long‐time existence has seen a lot of advances: the case of left‐invariant closed structures on nilpotent Lie groups was obtained by ; the case of homogeneous ‐Laplacian flows on solvable Lie groups with a codimension‐one Abelian normal subgroup was obtained by , and there are lots of studies of the corresponding homogeneous Laplacian solitons in .…”