Poincar\'e inequality for one forms on four manifolds with bounded Ricci curvature

TL;DR

This paper establishes a quantitative Poincaré inequality for one forms on closed four-dimensional Riemannian manifolds with bounded Ricci curvature, using geometric bounds and orbifold techniques.

Contribution

It provides the first non-trivial global Poincaré inequality for one forms under only Ricci curvature bounds without higher curvature assumptions.

Findings

Derived a global Poincaré inequality for one forms on four-manifolds.

Utilized orbifold Hodge theory and Gromov-Hausdorff convergence.

Connected geometric bounds to spectral properties of the manifold.

Abstract

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 giving such an inequality without any higher curvature assumptions. The proof is based on a Hodge theoretic result on orbifolds, a comparison for fundamental groups, and a spectral convergence with respect to Gromov-Hausdorff convergence, via a degeneration result to orbifolds by Anderson.