The Weyl functional on 4-manifolds of positive Yamabe invariant

Abstract

It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $${\int }_M|W_g^{+}{|}^{2}d{\mu }_g\ge 2{\pi }^2(2\chi (M)+3\tau (M))-\frac{8{\pi }^2}{|{\pi }_1(M)|},$$ 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 ${\delta }_g{W}_g^{+}=0$, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.