The Morse Index, Nullity and Jacobi Fields of Constant-Curvature 2-Spheres in Complex Projective Spaces

We determine the Morse index and normal nullity of minimal immersions $S^2\to \mathbb{C}P^N$ with constant Gauss curvature. For integers $n\ge1$ and $k\in\{0,\ldots,n\}$, the member $ϕ_{n-2k,n}:S^2 \to \mathbb{C}P^n$ of the Veronese sequence satisfies \[ \operatorname{Ind}(ϕ_{n-2k,n})=2k(n-k)(n+1)