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arXiv · 2405.19168

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

Shouhei Honda·Andrea Mondino
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Abstract

In this short note, we provide a quantitative global Poincaré 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.

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Shouhei Honda, Andrea Mondino. 2024-12-18. Poincaré inequality for one forms on four manifolds with bounded Ricci curvature. https://arxiv.org/abs/2405.19168

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