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) $S^2$?