Calculus of Variations and Geometric Measure Theory

S. Honda - A. Mondino

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

created by mondino on 29 May 2024



Inserted: 29 may 2024
Last Updated: 29 may 2024

Year: 2024


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.