Rigidity of Einstein 4-manifolds with positive curvature

An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε₀, ε₀≡(-23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S ⁴, RP ⁴ with constant sectional curvature K=1/3, or CP ² with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S ⁴ and CP ² contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S ⁴, RP ⁴, or CP ² but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions. Oblatum 4-II-1999 & 4-V-2000?Published online: 16 August 2000