Infimum and Morse index of nonzero critical values of the Cartan-form functional

Determine whether 1/(2(\rho^{\vee},\rho^{\vee})h^{\vee}) is the infimum of the nonzero critical values of the function W associated with the Cartan 3-form, and determine whether a critical point realizing this infimum must have index D.

Background

The paper studies the critical points of the functions F, K, and W, where W is the normalized squared absolute value of the Cartan 3-form on three copies of a complex simple Lie algebra. Critical points arising from subalgebras isomorphic to sl(2) have values determined by their Dynkin indices, and the principal subalgebra yields the smallest value among that family, namely 1/(2(\rho^{\vee},\rho^{\vee})h^{\vee}). The paper also establishes a lower bound D for the Morse index of a nonzero critical point and shows that the principal critical point has index D. It remains unresolved whether these two properties extend to a globally minimizing nonzero critical point, including critical points not arising from the sl(2)-subalgebra construction.