FOUR-MANIFOLDS OF PINCHED SECTIONAL CURVATURE

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