Building Foliations from Heegaard Diagrams

Every closed orientable 3-manifold admits both a Heegaard splitting and a coorientable codimension-one foliation. We address the foliation realization problem: constructing a foliation directly from a Heegaard diagram of arbitrary genus. Using Gabai's sutured manifold theory, we introduce the notion of a meridional sutured handlebody and prove that every such handlebody decomposes, via basic sutured decomposition operations, into a Reeb component together with product disks. We then present a three-step construction: (i) building two meridional sutured handlebodies from a given Heegaard diagram, (ii) gluing them along compatible disk regions by reversing disk decompositions, and (iii) filling the remaining cavities with a meridional sutured handlebody and some Reeb components. The construction applies to Heegaard diagrams of any genus.