# 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?

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.

• Thank you, Sebastian, for your excellent answer. The book of Jost indeed goes through all the details nicely. – Skrodde Sep 29 '15 at 12:55
• Dear Skrodde, you are welcome. – Sebastian Sep 30 '15 at 6:52

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

• Thank you, Igor, for your reply to my question. However, I fail to see how any of the three examples given on page 118 fits my problem. In the first example, the authors suppose $M'$ to be flat, where I need $M'=\mathbb{S}^2$. In the second example, they only establish $f$ homomorphism $\Rightarrow$ $f$ harmonic. But not the converse and conformal maps don't even enter the picture here. Finally, in the third example it is $f$ holomorphic $\Rightarrow$ $f$ harmonic, but again not the converse, which I would need. Could you maybe clarify a little? Thanks in advance. – Skrodde Sep 16 '15 at 8:08

There is also the excellent book of Frederic Helein which will provide you many good references.