By the uniformization theorem, for every genus-0 closed surface $\mathcal{M}\subset\mathbb{R}^3$, there is a conformal map $f:\mathcal{M}\rightarrow \mathbb{S}^2$. Furthermore consider the Dirichlet Energy $$E(f)=\int_{\mathcal{M}}\left|\nabla f\right|^2\:d\lambda_{\mathcal{M}}.$$ A critical point of this energy functional is called harmonic map. Now the intersting statement is:

For a genus-0 closed surface $\mathcal{M}$, the conformal maps $f:\mathcal{M}\rightarrow\mathbb{S}^2$ are equivalent to the harmonic maps.

Can you please provide a source with a rigorous prove of this statement, or a short explanation as to why it holds?