The Weyl functional on 4-manifolds of positive Yamabe invariant

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 ? ? ? ( ? ) 3 ? - 8 1 , where $$W^+_g$$ $$\chi (M)$$ and $$\tau respectively, denote the self-dual Weyl tensor of g, Euler characteristic signature M. This generalizes Gursky’s inequality [15] for case $$b_1(M)>0$$ b > 0 in a much simpler way. We also extend all such lower bounds functional to 4-orbifolds including inequalities $$b_2^+(M)>0$$ or $$\delta _gW^+_g=0$$ ? = obtain topological obstructions existence orbifold metrics curvature.