Stability of nearly K\"ahler and nearly parallel $G_2$-manifolds

TL;DR

This paper studies the stability and deformation properties of nearly K"ahler and nearly parallel $G_2$-manifolds through the analysis of generalized Hitchin functionals, their gradient flows, and spectral properties.

Contribution

It introduces the Hitchin index as a Morse-like invariant and explores its relation to Einstein co-index and manifold deformation theory.

Findings

Hitchin index provides a lower bound for the Einstein co-index.

Spectral decomposition of Hessians reveals stability properties.

Connections established between the index and $G_2$- and $ ext{Spin}(7)$-conifold deformations.

Abstract

We investigate generalisations of Hitchin's functionals, whose critical points correspond to nearly K\"ahler and nearly parallel $G_2$-structures. Our focus is on the gradient flow of these functionals and the spectral decomposition of their Hessians with respect to natural indefinite inner products. We introduce a Morse-like index for these functionals, termed the Hitchin index. We prove this index provides a lower bound for the Einstein co-index and explore its relationship with the deformation theory of $G_2$- and $\operatorname{Spin}(7)$-conifolds.