Nearly $G_2$-manifolds and $G_2$-Laplacian co-flows
Jason D. Lotay, Jakob Stein
TL;DR
The paper investigates the stability of nearly G2-structures (nG2) as critical points for normalized and modified G2-Laplacian co-flows on compact 7-manifolds. A key contribution is a spectral stability criterion expressed in terms of the spectrum of the operator $d^*$ acting on exact $4$-forms in the $27$-component, $\Omega^4_{27,\mathrm{exact}}$, which is then used to prove instability of the standard round $S^7$ nG2-structure (very large index) and the canonical nG2-structures arising from 3-Sasakian geometry (index at least 1). In contrast, within the normalized co-flow (without the modified term) the round and squashed Einstein nG2-structures on $S^7$ can be stable within the 3-Sasakian family, highlighting a sharp contrast with Ricci flow. The results suggest the modified co-flow has destabilizing tendencies for many nG2-structures, though the normalized flow can exhibit stability in certain subfamilies, and the instability insights may help in understanding singularity formation and perturbations in $G_2$-geometric flows.
Abstract
Nearly $G_2$-structures define positive Einstein metrics in $7$ dimensions and are critical points, up to scale, for a geometric flow of co-closed $G_2$-structures with good analytic properties called the modified $G_2$-Laplacian co-flow. We introduce a suitable normalization of this flow so that nearly $G_2$-structures are stable under rescaling. However, we show that many nearly $G_2$-structures are unstable for this flow: specifically, all those naturally arising from 3-Sasakian geometry. In particular, we demonstrate that the standard nearly $G_2$-structure on the round 7-sphere is an unstable critical point with high index.