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, $$\begin{aligned} \int _M|W^+_g|^2d\mu _{g}\ge 2\pi ^2(2\chi (M)+3\tau (M))-\frac{8\pi ^2}{|\pi _1(M)|}, \end{aligned}$$ ∫ M | W g + | 2 d μ g ≥ 2 π 2 ( 2 χ ( M ) + 3 τ ( M ) ) - 8 π 2 | π 1 ( M ) | , where $$W^+_g$$ W g + , $$\chi (M)$$ χ ( M ) and $$\tau (M)$$ τ ( M ) , respectively, denote the self-dual Weyl tensor of g, the Euler characteristic and the signature of M. This generalizes Gursky’s inequality [15] for the case of $$b_1(M)>0$$ 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 $$b_2^+(M)>0$$ b 2 + ( M ) > 0 or $$\delta _gW^+_g=0$$ δ g W g + = 0 and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.