Consider $(S^1 \times \Sigma^2, g)$, where $g$ is any Riemannian metric on the compact and closed $3$-manifold $S^1 \times \Sigma^2$.
Question: Does there always exist a nowhere vanishing harmonic $1$-form on $S^1 \times \Sigma^2$? If the answer to this question is No, how about the generalisation to $k$-parameter families of metrics?
I am particularly interested in the case of $\Sigma=S^1 \times S^1$. This is a crosspost from https://math.stackexchange.com/questions/4262844/nowhere-vanishing-harmonic-1-forms-on-3-manifolds.