Nearly $G_2$-manifolds and $G_2$-Laplacian co-flows

TL;DR

This paper investigates the stability of nearly $G_2$-structures under a modified $G_2$-Laplacian co-flow, revealing instability in many natural examples including the standard structure on the 7-sphere.

Contribution

It introduces a normalized flow for nearly $G_2$-structures and analyzes their stability, showing many are unstable, especially those from 3-Sasakian geometry.

Findings

Nearly $G_2$-structures are unstable under the flow.

Standard nearly $G_2$-structure on the 7-sphere is an unstable critical point.

The flow normalization stabilizes nearly $G_2$-structures under rescaling.

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.