Let $M$ be a closed Riemannian manifold. A circle-valued function $u : M \to \mathbb{S}^1$ is harmonic if the associated one form $h_u = u^*(d \theta)$ is harmonic in the Hodge sense: $dh_u = 0$ and $d^* h_u = 0$. I would like to know non-constant examples of such maps on some $3$-manifolds like the $3$-sphere, the $3$-torus, the projective $3$-space, etc.
More generally, one could consider manifolds with boundary. If $M$ is such a manifold, then a natural condition to impose to a map $u : M \to \mathbb{S}^1$ is that $\frac{\partial u}{\partial \nu} = 0$, where $\nu$ is the unit outer normal to $\partial M$ (Neumann condition). Are there non-constant examples of harmonic maps as defined above in this setting?
Let $h : M \to S^1$ be harmonic, and consider the associated harmonic $1$-form $h_u$, as in the question. It is a standard fact in Hodge theory that the vector space of harmonic $k$-forms on $M$ is naturally isomorphic to the de Rham cohomology vector space $H^k(M;\mathbb{R})$. Thus, if $H^1(M;\mathbb{R}) = 0$ (which includes the $3$-sphere and the projective space), $h_u$ must be identically $0$, which implies that $h$ is constant.
For a non-constant example, consider the 3-torus $T= S^1 \times S^1 \times S^1$ and the projection $T \to S^1$ of, e.g., the first $S^1$ factor. For an example with a manifold with boundary, take the projection of the first $S^1$ factor from $S^1 \times S^1 \times [0,1]$.