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 $n\ge 1$ and $k\in \{0,\dots ,n\}$, the member ${\phi }_{n-2k,n}:S^2\to \mathbb{C}{P}^n$ of the Veronese sequence satisfies $$\mathrm{Ind}({\phi }_{n-2k,n})=2k(n-k)(n+1),\mathrm{Nul}({\phi }_{n-2k,n})=2(n-1)(n+3)\mathrm{.}$$ We also obtain the corresponding formulas for its totally geodesic extensions to $\mathbb{C}{P}^N$, $N\ge n$, and identify the normal Jacobi kernel with infinitesimal deformations obtained by post-composing the rational normal directrix with projective linear embeddings into $\mathbb{C}{P}^N$. In particular, every normal Jacobi field is integrable through a family of minimal $2$-spheres.