The Weyl functional on 4-manifolds of positive Yamabe invariant

TL;DR

This paper establishes lower bounds for the Weyl functional on 4-manifolds with positive scalar curvature, extending Gursky's inequalities to orbifolds and providing topological obstructions for certain self-dual metrics.

Contribution

It generalizes Gursky's inequality to include 4-manifolds with positive scalar curvature and extends these bounds to orbifolds, offering new topological obstructions.

Findings

Derived lower bounds for the Weyl functional on 4-manifolds.

Extended inequalities to 4-orbifolds including Gursky's cases.

Identified topological obstructions to self-dual orbifold metrics.

Abstract

It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$\int_M|W^+_g|^2d\mu_{g}\geq 2\pi^2(2\chi(M)+3\tau(M))-\frac{8\pi^2}{|\pi_1(M)|},$$ where $W^+_g$, $\chi(M)$ and $\tau(M)$ respectively denote the self-dual Weyl tensor of $g$, the Euler characteristic and the signature of $M$. This generalizes Gursky's inequality \cite{gur} for the case of $b_1(M)>0$ in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky's inequalities for the case of $b_2^+(M)>0$ or $\delta_gW^+_g=0$, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.