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

Abstract

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 \[ \pi:(\mathbb{S}^{2n+1},g_\epsilon)\longrightarrow(\mathbb{C}P^n,h), \] where $g_\epsilon$ is a Berger metric and $h$ is the Fubini--Study metric. For $n\geq3$, 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, $\mathrm{Ind}(\pi)=2n+2$ and $\mathrm{Null}(\pi)=n(3n+5)$. We then extend Rivi\`ere'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 $\mathrm{Ind}(\phi)=2n+2$, 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.