Harmonic maps from $\mathbb{S}^{2n+1}$ into $\mathbb{CP}^n$ with least Morse index

Abstract

Let $u:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if \begin{equation*} u = h \circ \pi \circ \Xi, \end{equation*} where $\Xi:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ is an isometric transformation, $\pi:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ is the Hopf map and $h:\mathbb{CP}^n\to\mathbb{CP}^n$ is a holomorphic map with degree one. When $n=1$, it refines a classical result of Urakawa and a recent result of Rivi\`{e}re. Moreover, we prove that $h\circ\pi\circ \Xi$ has nullity $3n^2+5n$.