Minimal two-spheres of low index in manifolds of positive complex sectional curvature

Suppose that $S^n$ is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real sectional curvatures $K_r(\sigma )$ satisfy $$1\mathrm{/}2<K_r(\sigma )\le 1.$$ Then the number of minimal two spheres of Morse index $\lambda$, for $n-2\le \lambda \le 2n-5$, is at least $p_3(\lambda -n+2)$, where $p_3(k)$ is the number of $k$-cells in the Schubert cell decomposition for $G_3({\mathbb{R}}^{n+1})$.