A new conformal invariant on 3-dimensional manifolds

By improving the analysis developed in the study of σkσk-Yamabe problem, we prove in this paper the following new inequality for any 3-dimensional Riemannian manifold (M3,g)(M3,g) of non-negative scalar curvature(∫Mσ1(g)dv(g))2⩾3vol(g)∫Mσ2(g)dv(g). Equality holds if and only if (M3,g)(M3,g) is a space form.