Equivalence of Harmonic Maps and Conformal Maps on Genus-0 closed surfaces 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?
 A: MR0164306 (29 #1603) Reviewed 
Eells, James, Jr.; Sampson, J. H.
Harmonic mappings of Riemannian manifolds. 
Amer. J. Math. 86 1964 109–160. 
Example on p. 118
A: I think it is a very good exercise to show that a conformal map between Riemann surfaces (equipped with metrics in the conformal classes) are harmonic. 
The converse direction is more interesting, and does depend on the assumption that  the domain is  a compact surface of genus $0.$ In general, one can consider for  maps $f\colon M\to S^2$ the so called Hopf differential which is given in a local holomorphic coordinate $z$ on $M$ as $$Q=<\frac{\partial f}{\partial z},\frac{\partial f}{\partial z}>(dz)^2.$$
It turns out that $f$ is harmonic if and only if $Q$ is holomorphic. On a surface of genus $0$ there do not exist any non-zero holomorphic quadratic differentials, so under your assumption, $Q=0.$ By looking at the definition of $Q$ this implies conformality.
Besides the book of Helein mentioned by Paul, the book "Riemannian Geometry and Geometric Analysis" by Jost contains the details of these computations.
A: There is also the excellent book of Frederic Helein which will provide you many good references.
