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

Abstract

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 $\phi_{n-2k,n}:S^2 \to \mathbb{C}P^n$ of the Veronese sequence satisfies \[ \operatorname{Ind}(\phi_{n-2k,n})=2k(n-k)(n+1),\qquad \operatorname{Nul}(\phi_{n-2k,n})=2(n-1)(n+3). \] 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.