I have been recently reading the following paper about harmonic maps, where the authors prove the non-degeneracy of harmonic maps from $\mathbb{R}^2\to \mathbb{S}^2$. In their analysis they make use of the following classification result from harmonic maps:
Corollary 2.1. A map $u$ from $\mathbb{R}^2$ to $\mathbb{S}^2$ is harmonic with $\operatorname{deg}(u) \geq 0$ (resp. $\operatorname{deg}(u)<0)$ if and only if $u=\text{S}(p / q)$ where $p$ and $q$ are complex polynomials of $z$ (resp. $\bar{z}$) and $\text{S}$ denotes the stereographic projection. That is, $u$ is a harmonic map if and only if $u$ is a stereographic projection of a rational function of $z$ (resp. $\bar{z}$ ). We call (irreducible) rational function $p / q$ the generate function of the harmonic map $u$. Moreover, all harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$ are stable.
Question: I was interested in knowing whether some version of this result is known in for higher dimensional targets in particular consider harmonic maps from $\mathbb{R}^2\to \mathbb{S}^n$ or $\mathbb{S}^2\to \mathbb{S}^n$ for $n>2$
- Is the energy of the harmonic map in this case quantized? When $n=2$ then we have that $E[u]=4\pi \operatorname{deg}(u).$ Can we expect some higher dimensional analogue of this fact?
- Are there some characterizations of harmonic maps like the one in Cor. 2.1 known when $n>2$?
