The Hitchin index in cohomogeneity one nearly K\"ahler structures

Nearly K\"ahler and Einstein structures admit a variational characterization, where the second variation is associated with a strongly elliptic operator. This allows us to associate a Morse-like index to each structure. Our study focuses on how these indices behave under the assumption that the nearly K\"ahler structure admits a cohomogeneity one action. Specifically, we investigate elements of the index that also exhibit cohomogeneity one symmetry, reducing the analysis to an ODE eigenvalue problem. We apply our discussion to the two inhomogeneous examples constructed by Foscolo and Haskins. We obtain non-trivial lower bounds on the Hitchin index and Einstein co-index for the inhomogeneous nearly K\"ahler structure on $S^3\times S^3$, answering a question of Karigiannis and Lotay.