Morse functions and contact convex surfaces

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.

15 pages, 5 figures. Expository note

References in corpus (1)

• Dividing sets as nodal sets of an eigenfunction of the Laplacian