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

Area of Science:

• Differential Geometry
• Geometric Analysis
• Topology

Background:

• Poincaré inequalities are fundamental in analysis and geometry, relating function norms to their derivatives.
• Global inequalities on Riemannian manifolds are crucial for understanding geometric properties.
• Previous results often required strong curvature conditions.

Purpose of the Study:

• To establish a quantitative global Poincaré inequality for one-forms on closed Riemannian four manifolds.
• To provide such an inequality using only bounds on diameter, volume, and Ricci curvature.
• To achieve this without imposing higher-order curvature assumptions.

Main Methods:

• Utilizing a Hodge theoretic result on orbifolds.
• Employing a comparison for fundamental groups.
• Leveraging spectral convergence with respect to Gromov-Hausdorff convergence.
• Applying Anderson's degeneration result to orbifolds.

Main Results:

• A quantitative global Poincaré inequality for one-forms on closed Riemannian four manifolds was derived.
• The inequality is expressed in terms of upper bounds on the diameter, a positive lower bound on the volume, and a two-sided bound on the Ricci curvature.
• This represents a novel result as it does not require higher curvature assumptions.

Conclusions:

• The established Poincaré inequality offers a new tool for studying geometric properties of Riemannian four manifolds.
• The methods used demonstrate the power of orbifold techniques and spectral convergence in geometric analysis.
• The result advances the understanding of global analytic inequalities in the context of lower curvature bounds.