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?
So far I tried to find an example of a harmonic $1$-form on $T^3=S^1 \times S^1 \times S^1$ that does have a zero but did not succeed.
I have cross-posted this question to: https://mathoverflow.net/questions/407340/nowhere-vanishing-harmonic-1-forms-on-3-manifolds.