Stability of torsion-free G_2 structures along the Laplacian flow

TL;DR

This paper proves that torsion-free G_2 structures on compact 7-manifolds are dynamically stable under the Laplacian flow, ensuring convergence to a torsion-free structure when starting close enough.

Contribution

It establishes the weak dynamical stability of torsion-free G_2 structures along the Laplacian flow for closed G_2 structures.

Findings

Laplacian flow converges to torsion-free G_2 structures from nearby initial data

Stability holds for initial structures cohomologous and sufficiently close

Flow preserves the cohomology class of the initial G_2 structure

Abstract

We prove that torsion-free G_2 structures are (weakly) dynamically stable along the Laplacian flow for closed G_2 structures. More precisely, given a torsion-free G_2 structure $\varphi$ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to $\varphi$ will converge to a torsion-free G_2 structure which is in the orbit of $\varphi$ under diffeomorphisms isotopic to the identity.