It is shown that on every closed oriented Riemannian 4-manifold (M, g) with positive scalar curvature,

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 [15] 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.

结果表明，在每个具有正标量曲率的封闭定向黎曼四流形 ( M ,  g ) 上，

其中\(W^+_g\)、\(\chi (M)\)和\(\tau (M)\)分别表示g的自对偶 Weyl 张量、欧拉特征和M的签名. 这以更简单的方式概括了 Gursky 不等式 [15] 对于\(b_1(M)>0\) 的情况。我们还将 Weyl 泛函的所有此类下界扩展到 4-orbifolds，包括在\(b_2^+(M)>0\)或\(\delta _gW^+_g=0\)的情况下的 Gursky 不等式，并获得拓扑阻碍正标量曲率的自对偶轨道度量的存在。