Harmonic maps from ${\mathbb{S}}^{2n+1}$ to $\mathbb{C}{P}^n$ with low Morse index

We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to $$\pi :({\mathbb{S}}^{2n+1},g_{\epsilon })⟶(\mathbb{C}{P}^n,h),$$ where $g_{\epsilon }$ is a Berger metric and $h$ is the Fubini–Study metric. For $n\ge 3$, we determine all negative and zero eigenvalues and obtain explicit formulas for the Morse index and nullity for every $\epsilon >0$. In particular, for the unit round metric, $\mathrm{I}\mathrm{n}\mathrm{d}(\pi )=2n+2$ and $\mathrm{N}\mathrm{u}\mathrm{l}\mathrm{l}(\pi )=n(3n+5)$. We then extend Rivière’s rigidity argument from ${\mathbb{S}}^3\to {\mathbb{S}}^2$ to higher dimensions. If $\phi :{\mathbb{S}}^{2n+1}\to \mathbb{C}{P}^n$ is harmonic and $\mathrm{I}\mathrm{n}\mathrm{d}(\phi )=2n+2$, then $d{\phi }_x\circ d{\phi }_x^{\ast }$ commutes with the complex structure of $\mathbb{C}{P}^n$ at every point. Consequently, up to an isometry of the domain, $\phi$ factors through the Hopf fibration as $\phi =P\circ \pi \circ f$, where $P:\mathbb{C}{P}^n\to \mathbb{C}{P}^n$ is holomorphic.