Curvature in the Balance: The Weyl Functional and Scalar Curvature of 4-Manifolds

TL;DR

This paper investigates the minimal values of the Weyl functional on 4-manifolds with positive scalar curvature and compares scalar and self-dual Weyl curvatures in almost-Kaehler 4-manifolds, revealing new geometric insights.

Contribution

It demonstrates that the Weyl functional can be surprisingly small on certain 4-manifolds and systematically compares scalar and self-dual Weyl curvatures in almost-Kaehler cases.

Findings

Weyl functional infimum is very small on many 4-manifolds with positive scalar curvature

Systematic comparison of scalar and self-dual Weyl curvatures in almost-Kaehler 4-manifolds

New geometric relationships between curvature quantities in 4-manifolds

Abstract

The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kaehler 4-manifolds.