Published Paper

Inserted: 29 may 2024
Last Updated: 24 feb 2025

Journal: Archiv Der Mathematik
Year: 2024
Doi: https://doi.org/10.1007/s00013-024-02091-w

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.

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