Complex vs. convex Morse functions and geodesic open books

Abstract

Suppose that Sigma is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of Sigma, having complex, contact and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on Sigma. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on Sigma. Moreover, we observe that if Sigma has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.