Morse functions and contact convex surfaces

Let $f$ be a Morse function on closed surface $\Sigma$ such that zero is regular value and admits neither positive minima nor negative maxima. In this expository note, we show $\Sigma\times \mathbb{R}$ an $\mathbb{R}$-invariant contact form $\alpha=fdt+\beta$ whose characteristic foliation along the section (negative) weakly gradient-like with respect to $f$. The proof self-contained gives explicit constructions of any structure in $\Sigma \times \mathbb{R}$, up isotopy. As application, give alternative geometric homotopy classification structures terms their dividing set.