Stability of nearly Kähler and nearly parallel $G_2$-manifolds

We investigate generalisations of Hitchin’s functionals, whose critical points correspond to nearly Kähler 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 $\mathrm{Spin}(7)$-conifolds.