Poincaré inequality for one forms on four manifolds with bounded Ricci curvature

In this short note, we provide a quantitative global Poincar´e inequality for one forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on Ricci curvature. This seems to be the first non-trivial result givi...