Least energy harmonic map onto $\mathbb S^2$

Question

I am reading an article where the author are considering a harmonic map onto the sphere $u: \mathbb R^2 \to \mathbb S^2 \subseteq \mathbb R^3$, i.e. a map satisfying $$\Delta u + |\nabla u|^2 u = 0 \quad \text{in } \mathbb R^2,$$ which is the harmonic map equation for functions valued into spheres (i.e. minimizing the Dirichlet energy). It is a known fact that the energy of harmonic maps are quantized in two dimension in such a way that $$E(u) = \int_{\mathbb R^2} |\nabla u|^2 = 4\pi n \in 4\pi \mathbb N.$$ Now the authors consider the (non-trivial) harmonic map with the least energy, $E(u) = 4\pi$. They also give an explicite form of this function, given by $$u(x) = \frac{1}{1 + |x|^2} \begin{pmatrix} 2x\\ 1 - |x|^2 \end{pmatrix}$$ for $x \in \mathbb R^2$. Now, we directly see that this is the stereographic projection map from $\mathbb R^2 \to \mathbb S^3$. By just computing its energy we indeed see that it's the least energy harmonic map, but my question is why ? I intuitively don't get what's behind this result. Could someone give me an interpretation of this ?