Morse functions and contact convex surfaces

Abstract

Let f be a Morse function on a closed surface Σ such that zero is a regular value and such that f admits neither positive minima nor negative maxima. In this note, we show that Σ×R admits an R-invariant contact form α=fdt+β 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 R-invariant contact structure in Σ×R, up to isotopy. It provides evidence for these functions to be eigenfunctions of the Laplacian for some metric on the surface. As an application, we give an alternative geometric proof of the homotopy classification of R-invariant contact structures in terms of their dividing set.