Waist inequality for 3-manifolds with positive scalar curvature

We construct singular foliations of compact three-manifolds $(M^3,h)$ with scalar curvature $R_h\ge {\mathrm{\Lambda }}_0>0$ by surfaces of controlled area, diameter and genus. This extends Urysohn and waist inequalities of Gromov-Lawson and Marques-Neves.