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 π\circ Ξ, \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 Rivière. Moreover, we prove that $h\circ \pi \circ \Xi$ has nullity $3n^2+5n$.

33 pages