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

TL;DR

This paper investigates the Morse-like indices of nearly K"ahler and Einstein structures with cohomogeneity one symmetry, reducing the problem to an ODE eigenvalue analysis and applying it to specific examples, including $S^3\times S^3$.

Contribution

It introduces a method to analyze the Hitchin index in cohomogeneity one nearly K"ahler structures, providing bounds and insights into their stability properties.

Findings

Derived lower bounds on Hitchin index for specific structures

Reduced index analysis to an ODE eigenvalue problem

Answered a question regarding inhomogeneous nearly K"ahler structures on $S^3\times S^3$

Abstract

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.