Morse functions and contact convex surfaces

TL;DR

This paper constructs explicit $ $-invariant contact forms on $ $-product surfaces using Morse functions, providing a new geometric proof of the classification of such contact structures via dividing sets.

Contribution

It offers explicit constructions of $ $-invariant contact structures on surface products and a new geometric proof of their classification based on dividing sets.

Findings

Explicit construction of $ $-invariant contact forms on $ $-product surfaces.

Proof that the characteristic foliation is weakly gradient-like.

Alternative geometric proof of the homotopy classification of contact structures.

Abstract

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