Harmonic 2-forms and positively curved 4-manifolds

TL;DR

This paper investigates the properties of compact Riemannian 4-manifolds with positive sectional curvature, proving a key definiteness result under a Kato inequality and offering new insights into their structure.

Contribution

It establishes that such manifolds are definite if they satisfy a Kato type inequality, providing a novel criterion for understanding their geometric structure.

Findings

Proves that manifolds satisfying the Kato inequality are definite.

Provides new insights into the structure of positively curved 4-manifolds.

Highlights conditions under which positive curvature implies definiteness.

Abstract

We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a Kato type inequality, then it is definite. We also discuss some new insights for compact Riemannian 4-manifolds of positive sectional curvature.