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

Abstract

We prove that torsion-free $\mathrm{G}_2$ structures are (weakly) dynamically stable along the Laplacian flow for closed $\mathrm{G}_2$ structures. More precisely, given a torsion-free $\mathrm{G}_2$ structure $\overline{\varphi}$ on a compact $7$-manifold $M$, the Laplacian flow with initial value in $[\overline{\varphi}]$, sufficiently close to $\overline{\varphi}$, will converge to a point in the $\mathrm{Diff}^0 (M)$-orbit of $\overline{\varphi}$. We deduce, from fundamental work of Joyce, that the Laplacian flow starting at any closed $\mathrm{G}_2$ structure with sufficiently small torsion will exist for all time and converge to a torsion-free $\mathrm{G}_2$ structure.