Four-manifolds of pinched sectional curvature

In this paper, we study closed four-dimensional manifolds. particular, show that under various new pinching curvature conditions (for example, the sectional is no more than 5/6 of smallest Ricci eigenvalue) then manifold definite. If restricting to a metric with harmonic Weyl tensor, it must be self-dual or anti-self-dual same conditions. Similarly, if an Einstein metric, either complex projective space its Fubini-Study round sphere their quotients. Furthermore, also classify manifolds positive intersection form and upper bound on curvature.