I think the following two statements are true, and most likely are in the literature. If so, could someone point me to some references? If not, counterexamples?

1. Let $Y$ be a closed oriented connected 3-manifold, and $[\theta]\in H^1(Y;\mathbb{Z})$ be a primitive class. Then a minimal genus embedded surface representing the Poincar'e dual of $[\theta]$ may be chosen to be connected.
2. In the preceding context, suppose $\theta=df$ where $f$ is a circle-valued Morse function with no extrema. Then one of the regular level surfaces of $f$ is a minimal genus embedded surface representing the Poincar'e dual of $[\theta]$.

Many thanks in advance!

I think the following two statements are true, and most likely are in the literature. If so, could someone point me to some references? If not, counterexamples?

1. Let $Y$ be a closed oriented connected 3-manifold, and $[\theta]\in H^1(Y;\mathbb{Z})$ be a primitive class. Then a minimal genus embedded surface representing the Poincar'e dual of $[\theta]$ may be chosen to be connected. (Here, the embedded surface has "minimal genus" if $\chi_-(\Sigma):=\max (-\chi(\Sigma),0)$ is minimal among embedded surfaces representing the same homology class.)
2. In the preceding context, suppose $\theta=df$ where $f$ is a circle-valued Morse function with no local extrema. Then one of the regular level surfaces of $f$ is a minimal genus embedded surface representing the Poincar'e dual of $[\theta]$.

Many thanks in advance!