Every homotopy class contains at least a harmonic representative

Question

Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is harmonic in the Hodge sense: $d \omega_v = 0$ and $\delta \omega_v = 0$. It can be seen that this happens precisely when $v$ minimizes the Dirichlet energy in its homotopy class $[v] \in [M:S^1]$. Thus, by Hodge theory, each homotopy class of a circle-valued map contains a harmonic representative.

My question is whether every homotopy class of $S^2$-valued maps contains a harmonic representative. More precisely: given $u : M \to S^2$ a smooth map, does there exist a harmonic map $u_0 : M \to S^2$ such that $u$ is smooth and homotopic to $u$?

A parallel question: if $u_0 : M \to S^2$ is any harmonic map, can we say that $u_0^* \sigma \in \Omega_2(M)$ is a harmonic $2$-form, where $\sigma$ is the area form of (the round) $S^2$?

Answer 1

As Andy says, the answer is 'no': It is known that there is no harmonic map of degree $1$ from the torus to the $2$-sphere. I forget who first observed this. (Amended after Andy's comment: It's originally due to J. C. Wood in the early 1970s, see Andy's comment for the exact reference.)

If I have time, I can put in the argument, but the essential outline of the argument is this:

There are two kinds of harmonic maps from the torus to the $2$-sphere. Those that are conformal and those that are not.

If it is conformal, then, up to reversing the orientation on the torus, it is a holomorphic map, and it is well-known that a non-constant holomorphic map from the torus to the $2$-sphere has degree at least 2. (In fact, there is such a holomorphic map of any degree $d\ge 2$.)

If it is not conformal, then a simple calculation shows that the degree of the mapping is zero. (Essentially, one produces an explicit $1$-form on the torus whose differential is the pullback of the area form on the $2$-sphere.)

Thus, there is no harmonic map of degree 1 from the torus to the $2$-sphere.