Rigidity of Einstein 4-manifolds with positive curvature
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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 S4, RP4 with constant sectional curvature K=1/3, or CP2 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 S4 and CP2 contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S4, RP4, or CP2 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.
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